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+ANALYSIS OF THE SMOOTHLY AMNESIA-REINFORCED
+MULTIDIMENSIONAL ELEPHANT RANDOM WALK
+JIAMING CHEN AND LUCILE LAULIN
+Abstract. In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant
+random walk (MARW). The scaling limit of the MARW is shown to exist in the diffusive, critical
+and superdiffusive regimes. We also establish the almost sure convergence in all of the three
+regimes. The quadratic strong law is displayed in the diffusive regime as well as in the critical
+regime. The mean square convergence towards a non-Gaussian random variable is established
+in the superdiffusive regime. Similar results for the barycenter process are also derived. Finally,
+the last two Sections are devoted to a discussion of the convergence velocity of the mean square
+displacement and the Cram´er moderate deviations.
+Contents
+1.
+Introduction
+1
+2.
+The amnesia-reinforced elephant random walk
+4
+3.
+A correlated martingale approach
+6
+4.
+Scaling limit and convergence
+7
+5.
+Scaling limit of the barycenter process
+13
+6.
+Velocity of quadratic mean displacement
+21
+7.
+Cram´er moderate deviations
+23
+Appendix A.
+Technical Lemmas
+24
+References
+35
+1. Introduction
+The study of reinforced processes and reinforced random walks has known a growing interest
+over the last decades. In particular, random walks on graphs, or more precisely edge [37] or vertex
+[39] reinforced random walks, have been the subject of a great number of contributions, see also
+[1, 12, 27] and the references therein. The insight of introducing reinforcement mechanisms to
+stochastic processes has also shed light on more applied models. In [30], the adaptive strategy of
+an agent who plays a two-armed bandit machine was described as a self-reinforced random walk.
+The philosophy of stochastic reinforcement has also been discussed in the topics of evolutionary
+ecology [4] and machine learning theory [17]. Another manifestation of reinforced P´olya urn models
+on financial economics can be found in [35]. We also refer the readers to [38] for a comprehensive
+and extensive survey on the subject.
+The Elephant Random Walk (ERW) is a discrete-time random walk, introduced by Sch¨utz and
+Trimper [40] in 2004. It was referred to as the ERW in allusion to the traditional saying that
+elephants can always remember anywhere they have been. As it was pointed out [12] by Bertoin
+2010 Mathematics Subject Classification. 60G50, 60G42, 62M09.
+Key words and phrases. Reinforced random walk, scaling limit, Cram´er moderate deviation, martingale.
+1
+arXiv:2301.08644v1  [math.PR]  20 Jan 2023
+
+2
+JIAMING CHEN AND LUCILE LAULIN
+(a) Diffusive regime
+(b) Critical regime
+(c) Superdiffusive regime
+Figure 1. The two-dimensional ERW with amnesia (in blue) and its barycenter
+(in red).
+who relied on K¨ursten’s work [29], the ERW is a special case of step-reinforced random walk. In
+fact, the ERW is reinforced because its behavior is influenced by its past : the ERW may have
+a tendency to do the same thing over and over, or on the contrary, it may try to compensate its
+previous steps. This different types of behavior, here-called regimes, are ruled by the memory
+parameter p and it is well-known that the ERW shows three regimes of behavior and that the
+critical value is p = 3/4.
+The ERW in dimension d = 1 has received a lot of attention from mathematicians and physicists
+over the last two decades.
+The almost sure convergence and the asymptotic normality of the
+position of the ERW were established in the diffusive regime p < 3/4 and the critical regime
+p = 3/4, see [3, 9, 16] and the references therein. In the superdiffusive regime p > 3/4, Bercu
+[5] proved that the limit of the position of the ERW is not Gaussian and Kubota and Takei [28]
+showed that the fluctuation of the ERW around this limit is Gaussian. To obtain those asymptotics,
+various approaches have been followed : Baur and Bertoin [3] went with the connection to P´olya-
+type urns while martingales were used by Bercu [5] and Coletti et al. [16] and the construction of
+random trees with Bernoulli percolation have been explicited by K¨ursten [29] and Businger [13].
+Other quantities of interest regarding the ERW have been studied. For example, Fan et al.
+[20] provided the Cramer moderate deviations associated with the ERW in dimension 1 and, more
+recently, Hayashi et al. [26] studied the rate of quadratic mean displacement.
+Bercu and Laulin [9] introduced the multidimensional ERW (MERW), where d ≥ 1, and estab-
+lished the natural extensions of the results [5] in dimension d = 1. Then, they investigated the
+center of mass of the MERW [8]. In both papers, they extensively used a martingale approach.
+Bertenghi [10] made use of the connection to P´olya-type urns in order to establish functional
+results for the MERW.
+Finally, the ERW with changing memory has also been introduced. The ERW with linearly
+reinforced memory has been studied by Baur [1] via the urn approach, and Laulin [31] using
+martingales. Gut and Stadm¨uller [25] proposed an amnesic ERW where the elephant could stop
+and only remember the first (and second) step it tooks. They also investigated the case where
+the elephant only remembered a fixed or time-evolving portion of its past (recent or distant)
+[24]. In the recent work [32], Laulin introduced smooth amnesia to the memory of the ERW and
+established the asymptotic behavior of this new process.
+The idea of our paper is to generalise the work [32] in dimension 1 to the dimension d ≥ 1.
+In other words, we introduce smooth amnesia to the memory of the multidimensional elephant
+random walk.
+
+40
+19
+30
+20
+10
+0
+-10
+-20
+0
+10
+20
+30
+40
+50
+60400
+300
+200
+100
+0
+0
+50
+100
+150
+200
+250
+35060
+50
+40
+30
+20
+10
+0
+0
+200
+400
+600
+800MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+3
+Our paper is organized as follows. In Section 2, we introduce the basic setting of the elephant
+random walk (Sn)n∈N placed under an amnesia reinforcement mechanism, which is controlled
+by the memory sequence (βn)n∈N. This type of multidimensional reinforced random walked is
+named as the multidimensional amnesia-reinforced elephant random walk (MARW). Similar to
+the ERW with the amnesia reinforcement, the MARW also admits a martingale structure, which
+is discussed in Section 2. Unlike the usual ERW, the additional amnesia-reinforcement induces
+two discrete-time martingales, instead of a single martingale, which are strongly correlated in a
+nontrivial fashion. Such strong correlation of martingales will eventually pose some computational
+difficulties when we analyze the limiting behavior of the MARW in Section 4. For instance, when
+we compute the pointwise limit and the scaling limit of (Sn)n∈N in the diffusive regime, the two
+strongly correlated martingales have to be dealt with separately, see [8, 31, 32] for the same
+methodology.
+As a courtesy to our readers, we give a preview of some of our main results, whose proofs will
+be deferred to Theorem 4.1, Theorem 4.2, and Theorem 4.3. In the diffusive regime, we have the
+almost sure convergence,
+1
+nSn → 0
+as
+n → ∞
+P − a.s.
+Another logarithmic scaling to the MARW yields the quadratic strong law,
+1
+log n
+n
+�
+k=1
+SkST
+k
+k2
+→ C(p, (βn)n∈N) · 1
+dId
+as
+n → ∞
+P-a.s.
+where the constant C(p, (βn)n∈N) > 0 depends only on the parameter p and the control sequence
+(βn)n∈N of the amnesia-reinforcement.
+Using square-root scaling factor, we observe that the
+MARW also admits a scaling limit in the diffusive regime, or convergence in distribution, in the
+Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+� 1
+√nS⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+Wt, t ≥ 0
+�
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with
+covariance structure given in (4.6).
+It is also of interest to look at the barycenter process (Gn)n∈N of the MARW. Its definition as
+well as its limiting behavior are discussed in Section 5. Similar to the discussion of the MARW,
+we obtain its pointwise convergence, quadratic strong law, and its scaling limit. In particular,
+Theorem 5.5 states that the barycenter process admits a scaling limit at the diffusive regime, or
+convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+� 1
+√nG⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+Wtr dr, t ≥ 0
+�
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process defined in Theorem 4.3 with
+its covariance structure defined in (4.6).
+A natural question to ask is how fast the limiting Theorems in Section 4 are carried on. Section
+6 provides a quantitative estimate on the mean square convergence velocity of the pointwise limit,
+quadratic strong law, and the scaling limit of the MARW. It should be possible to derive similar
+convergence velocity to the barycenter process, which is not computed in this work. In Section
+7, we end this work with a discussion on the Cram´er moderate deviations of the MARW in the
+diffusive and critical regimes. As a preview of our result in this Section, let (ϑn)n∈N ⊆ R be a
+non-decreasing sequence so that ϑn/√n → 0 as n → ∞, and wn the sequence with asymptotic
+
+4
+JIAMING CHEN AND LUCILE LAULIN
+behavior described in Lemma A.1. Take any non-empty Borel set B ⊆ Rd, then we have
+−
+inf
+x∈int B
+1
+2∥x∥2 ≤ lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ lim sup
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ − inf
+x∈cl B
+1
+2∥x∥2,
+(1.1)
+where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively. This is the
+Cram´er moderate deviations for the MARW in the diffusive and critical regimes.
+Moreover, we chose to postpone some technicalities regarding the analysis of the random walk
+to the Appendix A. That way, the reader can focus on the main Theorems and the ideas of their
+proofs. However, some analogous technicalities are displayed in the proof of the Theorems on the
+barycenter such that the reader can also have a complete overview of the work needed.
+Other probabilistic aspects of interest to the MARW include the statistical inference and an
+analysis on the Fisher information, see [7], as well as the Wasserstein distance of the reinforced
+random walk, see [21]. Perturbations of the amnesia intensity and its stability for the MARW is
+also of independent interest. A similar topic for another type of stochastic process, the Schramn-
+Loewner evolution, has been considered in [2, 15]. The transience and recurrence property of the
+MARW remains unknown, to the best of our knowledge. Readers are referred to [11, 20] for an
+exposition on the ERW without the amnesia reinforcement mechanism.
+2. The amnesia-reinforced elephant random walk
+To begin with, let us properly introduce the MARW. It is the natural extension to higher
+dimensions of the one-dimensional MARW, defined in [31]. For an arbitrarily given dimension
+d ≥ 1, let (Sn)n∈N be a (reinforced) random walk on Zd starting from the origin at time n = 0,
+i.e. S0 = 0. At time n = 1, the reinforced random walk moves to one of the 2d nearest-neighbors
+with equal probability 1/2d. After that, at time n ≥ 1, the reinforced random walk chooses at
+random an integer 1 ≤ k ≤ n among the past times and performs the same step with probabily p,
+or goes in any of the 2d − 1 other directions with probability (1 − p)/(2d − 1). This random walk
+possesses the amnesia property, in the sense that it remembers its most recent past steps better
+than its remote past steps. Colloquially, this random walk has higher probability to choose its
+recent steps than its earlier steps.
+From a mathematical perspective, the position of this reinforced random walk at time n+1 ≥ 1
+is given by
+Sn+1 = Sn + Xn+1
+with Xn+1 being defined as the step of this random walk at time n + 1, satisfying
+Xn+1 = An+1Xβn+1.
+Here An+1 is a random d × d matrix given by
+P(An = +Id) = p,
+and, for all 1 ≤ k ≤ d − 1,
+P(An = −Id) = P(An = +Jk
+d ) = P(An = −Jk
+d ) = 1 − p
+2d − 1
+where Id is the identity matrix of order d, Id = (δi,j)d and Jd = C(0, 1, 0, . . . , 0) is the circulant
+matrix of order d such that J = (δi+1,j)d. It is easy to observe that the fixed permutation matrix
+Jd satisfied Jd
+d = Id. The distribution of the memory βn of the reinforced random walk is such
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+5
+that the probability of choosing a fixed past time k ∈ N decays approximately with rate kβ/nβ+1,
+where β ≥ 0 is the amnesia parameter.
+(a) n = 10
+(b) n = 100
+Figure 2. Evolution of the distribution of the memory β depending on the value
+of β and the time.
+To be precise, this random walk chooses βn+1 according to
+P
+�
+βn+1 = k
+�
+= (β + 1)Γ(β + k)Γ(n)
+Γ(k)Γ(β + n + 1)
+= β + 1
+n
+·
+µk
+µn+1
+for all
+1 ≤ k ≤ n,
+where
+µn =
+n−1
+�
+k=1
+�
+1 + β
+k
+�
+=
+Γ(β + n)
+Γ(n)Γ(β + 1).
+(2.1)
+(a) d = 1
+(b) d = 2
+(c) d = 3
+(d) d = 10
+Figure 3. Competition between the dimension and the amnesia.
+Figure 3 aims to give a better understanding on how amnesia affects the MARW in various
+dimensions. The horizontal axis corresponds to p (from 0 to 1) and the vertical axis corresponds to
+β (from 0 to 10, arbitrary chosen). The diffusive regime, ie. when p < 4dβ+2d+1
+4d(β+1)
+or a < 1−
+1
+2(β+1),
+is in blue while the superdiffusive regime is in red, see Lemma A.1 for the definition of the regimes.
+One can observe that when the amnesia parameter β grows, the superdiffusive regime tends to be
+less represented. It should also be noted that when the dimension grows the superdiffusive regime
+is more important. Hence, the amnesia is somehow leading the MARW to a behavior closer to
+the one in dimension 1. When β vanishes, i.e. β = 0, the MARW reduces to the multidimensional
+elephant random walk (MERW) introduced in [9].
+The two random variables An and βn are constructed to be conditionally independent. At each
+time n, define the σ-algebra Fn = σ(X1, . . . , Xn). Then (Fn)n∈N is a discrete-time filtration to
+which the MARW is clearly adapted.
+
+β= 0
+0.5
+β= 1
+β= 2
+0.4 
+β= 5
+β= 10
+0.3
+0.2
+0.1
+0.0
+2
+4
+9
+00
+100.10
+β= 0
+β= 1
+0.08
+β= 2
+β= 5
+0.06 
+β= 10
+0.04-
+0.02
+0.00
+0
+20
+40
+60
+80
+10010
+8 -
+6
+4 
+2 -
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p10
+8 -
+6
+B
+4 
++0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p10
+8 -
+6
+B
+4 
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p10
+8 -
+6
+B
+4 
++0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+p6
+JIAMING CHEN AND LUCILE LAULIN
+Since An and βn are conditionally independent, we clearly have
+E
+�
+Xn+1|Fn
+�
+= E
+�
+An
+�
+E
+�
+Xβn+1|Fn
+�
+= 2dp − 1
+2d − 1 E
+�
+n
+�
+k=1
+Xk1{βn+1=k}|Fn
+�
+= 2dp − 1
+2d − 1 · β + 1
+nµn+1
+n
+�
+k=1
+µkXk.
+(2.2)
+We further denote
+a = 2dp − 1
+2d − 1
+and
+Yn =
+n
+�
+k=1
+µkXk
+(2.3)
+such that
+E
+�
+Yn+1|Fn
+�
+=
+�
+1 + a(β + 1)
+n
+�
+Yn = γnYn
+with γn = 1 + a(β + 1)/n. Hereafter, for each n ≥ 1, let
+an =
+n−1
+�
+k=1
+γ−1
+k
+= Γ(n)Γ(a(β + 1) + 1)
+Γ(a(β + 1) + n)
+and
+wn =
+n
+�
+k=1
+(akµk)2.
+(2.4)
+From a Gamma function estimate, also see in [31], we have that
+na(β+1)an → Γ(a(β + 1) + 1)
+as
+n → ∞
+(2.5)
+and
+n−βµn → Γ(β + 1)−1
+as
+n → ∞.
+(2.6)
+3. A correlated martingale approach
+Define the following two Rd-valued processes by
+Mn = anYn
+and
+Nn = Sn +
+a(β + 1)
+β − a(β + 1)µ−1
+n Yn.
+(3.1)
+Proposition 3.1. The Rd-valued processes (Mn)n∈N and (Nn)n∈N defined in (3.1) are locally
+square-integrable martingales adapted to (Fn)n∈N.
+Proof. Since, both Mn and Nn are finite sums for each n ≥ 1, the square-integrability and adapt-
+ness are immediate. By (2.3) and (2.4), we have
+E
+�
+Mn+1|Fn
+�
+= anγ−1
+n Yn + anµnγ−1
+n E
+�
+Xn+1|Fn
+�
+= anYn.
+And by (2.2), we have
+E
+�
+Nn+1|Fn
+�
+= E
+�
+Sn+1 +
+a(β + 1)
+β − a(β + 1)µ−1
+n+1Yn+1|Fn
+�
+= Sn +
+a(β + 1)
+β − a(β + 1)µ−1
+n Yn.
+Hence the assertion is verified.
+□
+Notice that via introducing the martingales (Mn)n∈N and (Nn)n∈N, we can write Sn as
+Sn = Nn −
+a(β + 1)
+β − a(β + 1)(anµn)−1Mn.
+(3.2)
+This writing is the key on which rely all of our analysis and our martingale approach.
+Moreover, the asymptotic behavior of (Mn)n∈N is closely related to wn defined in (2.4). In fact,
+we have the following asymptotic result, which states the three regimes of the MARW.
+Lemma 3.1. In the diffusive regime when p < 4dβ+2d+1
+4d(β+1)
+or a < 1 −
+1
+2(β+1), we have
+wn
+n1−2(a(β+1)−β) → l(β)
+as
+n → ∞
+(3.3)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+7
+with
+l(β) =
+1
+1 + 2(β − a(β + 1))
+�Γ(a(β + 1) + 1)
+Γ(β + 1)
+�2
+.
+In the critical regime when p = 4dβ+2d+1
+4d(β+1)
+or a = 1 −
+1
+2(β+1), we have
+wn
+log n →
+�Γ(β + 1 + 1
+2)
+Γ(β + 1)
+�2
+as
+n → ∞.
+(3.4)
+In the superdiffusive regime when p > 4dβ+2d+1
+4d(β+1)
+or a > 1 −
+1
+2(β+1), we have
+wn →
+∞
+�
+k=1
+�Γ(a(β + 1) + 1)Γ(β + k)
+Γ(a(β + 1) + k)Γ(β + 1)
+�2
+< ∞
+as
+n → ∞.
+(3.5)
+In order to investigate the asymptotic behavior of (Sn)n∈N, we first introduce an arbitrarily
+fixed test non-zero vector u ∈ Rd and we define
+Mn(u) = uT Mn
+and
+Nn(u) = uT Nn
+for each
+n ∈ N.
+It is then clear that (Mn(u))n∈N (Nn(u))n∈N are real-valued locally square-integrable martingales
+for each fixed u ∈ Rd.
+We further infer that (Sn(u))n∈N satisfies an equation analogous to (3.2). In
+this setting, we have reduced the multidimensional martingales to real-valued martingales without
+loss of generality. This technique greatly simplifies our martingale analysis. From now on, we fix
+the test vector u ∈ Rd and we introduce the two-dimensional martingale (Ln(u))n∈N defined as
+Ln(u) =
+�
+Nn(u)
+Mn(u)
+�
+for each
+n ∈ N.
+(3.6)
+Denote the martingale increment ϵn+1 = Yn+1 − γnYn for each n.
+Then (ϵn)n∈N satisfies the
+martingale difference relation E[ϵn+1|Fn] = 0. We obtain that
+∆Ln+1(u) = Ln+1(u) − Ln(u) =
+�
+Sn+1(u) − Sn(u) +
+a(β+1)
+β−a(β+1)
+�
+µ−1
+n+1Yn+1(u) − µ−1
+n Yn(u)
+�
+an+1Yn+1(u) − anYn(u)
+�
+=
+�
+βµ−1
+n+1
+β−a(β+1)
+�
+µn+1Xn+1(u) − (γn − 1)Yn(u)
+�
+an+1ϵn+1(u)
+�
+=
+�
+βµ−1
+n+1
+β−a(β+1)
+an+1
+�
+ϵn+1(u).
+(3.7)
+4. Scaling limit and convergence
+In this section, we discuss the scaling limit as well as the almost sure convergence in the
+diffusive, critical and the superdiffusive regimes, depending on the value of p with respect to
+(4dβ +2d+1)/(4d(β +1)). We also give the quadratic strong law in the diffusive regime as well as
+in the critical regime. Afterwards, the mean square convergence is established in the superdiffusive
+regime.
+4.1. The diffusive regime.
+Theorem 4.1. We have the almost sure convergence
+1
+nSn → 0
+as
+n → ∞
+P-a.s.
+
+8
+JIAMING CHEN AND LUCILE LAULIN
+Proof. We have from [18, Theorem 4.3.15] again that, for all γ > 0,
+∥Mn∥2
+λmax⟨M⟩n
+= o
+��
+log Tr⟨M⟩n
+�1+γ�
+P-a.s.
+(4.1)
+From equation (A.9) and the fact that λmax⟨M⟩n ≤ Tr⟨M⟩n ≤ wn, we get
+∥Mn∥2 = o
+�
+wn
+�
+log wn
+�1+γ�
+P-a.s.
+(4.2)
+By (3.3), we observe
+∥Mn∥2 = o
+�
+n1−2(a(β+1)−β)�
+log n
+�1+γ�
+P-a.s.
+Since Mn = anYn, we have from equations (2.5) and (2.6)
+∥Yn∥2
+(nµn+1)2 = o
+�
+n−1�
+log n
+�1+γ�
+P-a.s.
+which implies
+Yn
+nµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+By (A.10) and [18, Theorem 4.3.15] again, we find that
+∥Nn∥2 = o
+�
+n
+�
+log n
+�1+γ�
+P-a.s.
+(4.3)
+Moreover, we obtain from equation (3.2)
+1
+n2
+����Sn +
+a(β + 1)
+(β − a(β + 1))µn+1
+Yn
+����
+2
+= o
+�
+n−1�
+log n
+�1+γ�
+P-a.s.
+Hence, we conclude that
+Sn
+n +
+a(β + 1)
+β − a(β + 1) ·
+Yn
+nµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+and the proof is complete.
+□
+Theorem 4.2. We have the quadratic strong law
+1
+log n
+n
+�
+k=1
+SkST
+k
+k2
+→
+2β + 1 − a
+(1 − a)(1 − 2(a(β + 1) − β)) · 1
+dId
+as
+n → ∞
+P-a.s.
+Proof. We will check that all the conditions of [32, Theorem A.3] are satisfied, see also [14, 41].
+The condition (H.1) is satisfied thanks to Lemma A.4 while the condition (H.2) directly follows
+from Lemma A.5 and the condition (H.4) is exactly the statement of Lemma A.7. Therefore,
+1
+log
+�
+det V −1
+n
+�2
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkLk(u)Lk(u)T V T
+k → 1
+duT uVt=1
+as n → ∞ P-a.s. On the one hand, we have from (A.24) that
+1
+log n
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkLk(u)Lk(u)T V T
+k → 2(1 − a)(β + 1)
+d
+uT uVt=1
+(4.4)
+as n → ∞ P-a.s. On the other hand, by (2.5), (2.6) and (A.24), we have
+n
+�(det Vn)2 − (det Vn+1)2
+(det Vn)2
+�
+→ 2(1 − a)(β + 1)
+as
+n → ∞
+P-a.s.
+Finally, we obtain from (A.17) and (4.4) that
+1
+log n
+n
+�
+k=1
+uT SkST
+k u
+k2
+=
+1
+log n
+n
+�
+k=1
+vT VkLk(u)Lk(u)T V T
+k v
+k
+→ vT Vt=1v · 1
+duT u
+(4.5)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+9
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (4.5).
+□
+Theorem 4.3. The MARW admits a scaling limit at the diffusive regime, or convergence in
+distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+� 1
+√nS⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+Wt, t ≥ 0
+�
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with
+covariance
+E
+�
+WsW T
+t
+�
+=
+a(β + 1)(1 − a) + aβ
+(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)s
+� t
+s
+�a−β(1−a)
+· 1
+dId
++
+β
+(β(1 − a) − a)(1 − a)s · 1
+dId
+for all
+0 ≤ s ≤ t < ∞.
+(4.6)
+Proof. We will check that all the conditions of [32, Theorem A.2] are satisfied, see also [14, 41].
+The condition (H.1) is satisfied thanks to Lemma A.4 while the condition (H.2) directly follows
+from Lemma A.5 and the condition (H.3) is exactly the statement of Lemma A.6. Consequently,
+we have the convergence in distribution in the Skorokhod space D(R+) such that
+�
+VnL⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+Wt(u), t ≥ 0
+�
+where (Wt(u))t≥0 is a continuous R2-valued centered Gaussian process such that W0 = 0 and with
+covariance
+E
+�
+Ws(u)Wt(u)T �
+= 1
+duT uVs
+for all
+0 ≤ s ≤ t < ∞.
+From (2.5), (2.6), and (3.2), we see that S⌊nt⌋(u) is asymptotically equivalent to
+N⌊nt⌋(u) + tβ−a(β+1)
+a(β + 1)
+β − a(β + 1)(anµn)−1M⌊nt⌋(u)
+P-a.s.
+Multiplying on the left side by vt = (1, ta(β+1)−β)T , we obtain
+� 1
+√nS⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+Wt(u), t ≥ 0
+�
+with Wt(u) = vT
+t Wt(u). Hereafter, when 0 ≤ s ≤ t < ∞, we have the covariance
+E
+�
+Ws(u)Wt(u)T �
+= vT
+s E
+�
+Ws(u)Wt(u)T �
+vt = 1
+d(uT u)vT
+s Vsvt.
+(4.7)
+Solving (4.7), we have
+E
+�
+WsW T
+t
+�
+= 1
+dvT
+s Vsvt
+for all
+0 ≤ s ≤ t < ∞
+and the assertion (4.6) is verified.
+□
+4.2. The critical regime.
+Theorem 4.4. We have the almost sure convergence
+1
+√n log nSn → 0
+as
+n → ∞
+P-a.s.
+Proof. We still have (4.1) and (4.2) such that
+∥Mn∥2 = o
+�
+wn
+�
+log wn
+�1+γ�
+for all
+γ > 0
+P-a.s.
+However, in the critical regime, we have (3.4) rather than (3.3), and
+wn
+log n →
+�Γ(β + 1 + 1
+2)
+Γ(β + 1)
+�2
+as
+n → ∞.
+
+10
+JIAMING CHEN AND LUCILE LAULIN
+Since (2.5), (2.6), and since Mn = anYn, we observe for all γ > 0 that
+∥Yn∥2
+n(log n)2µ2n
+= o
+�
+(log n)−1�
+log log n
+�1+γ�
+P-a.s.
+In this regard
+Yn
+√n log nµn
+→ 0
+as
+n → ∞
+P-a.s.
+(4.8)
+Similarly, we still have (A.10) and
+∥Nn∥2 = o
+�
+n
+�
+log n
+�1+γ�
+for all
+γ > 0
+P-a.s.
+Then
+∥Nn∥2
+n(log n)2 = o
+�
+(log n)γ−1�
+for all
+γ ∈ (0, 1)
+P-a.s.
+and therefore
+Nn
+√n log n → 0
+as
+n → ∞
+P-a.s.
+By (3.2), we can hereafter conclude that
+Sn
+√n log n +
+a(β + 1)
+β − a(β + 1) ·
+Yn
+√n log nµn
+→ 0
+as
+n → ∞
+P-a.s.
+Combining with (4.8), the assertion is verified.
+□
+Theorem 4.5. We have the quadratic strong law
+1
+log log n
+n
+�
+k=1
+SkST
+k
+(k log k)2 → (2β + 1)2 · 1
+dId
+as
+n → ∞
+P-a.s.
+Proof. We will check that all the conditions of [32, Theorem A.3] are satisfied. The condition
+(H.1) is satisfied thanks to Lemma A.8 while the condition (H.2) directly follows from Lemma
+A.9 and the condition (H.4) is exactly the statement of Lemma A.10. Therefore,
+1
+log
+�
+det W −1
+n
+�2
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkLk(u)Lk(u)T W T
+k → 1
+duT uW
+(4.9)
+as n → ∞ P-a.s. On the one hand, we have from (A.34)
+1
+log log n
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkLk(u)Lk(u)T W T
+k → 1
+duT uW
+as n → ∞ P-a.s. On the other hand, by (2.5), (2.6), and (A.33), we have
+n log n
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+→ (2β + 1)2
+as
+n → ∞
+P-a.s.
+Then, we obtain from (A.17) and (4.9) that
+1
+log log n
+n
+�
+k=1
+uT SkST
+k u
+(k log k)2 =
+1
+log log n
+n
+�
+k=1
+wT WkLk(u)Lk(u)T W T
+k w
+k log k
+→ (2β + 1)2
+d
+uT u
+(4.10)
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (4.10).
+□
+Theorem 4.6. The MARW admits a scaling limit at the critical regime, or convergence in dis-
+tribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+�
+1
+�
+nt log n
+S⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+(2β + 1)Bt, t ≥ 0
+�
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+11
+where (Bt)t≥0 is a continuous d-dimensional canonical Brownian motion with covariance
+E
+�
+BsBT
+t
+�
+= s · 1
+dId
+for all
+0 ≤ s ≤ t < ∞.
+Proof. We will check that all the three conditions of [32, Theorem A.2] are satisfied, see also [42,
+Theorem 1]. First of all, by (3.4) and (A.7) we know that
+w−1/2
+n
+⟨M(u)⟩⌊nt⌋w−1/2
+n
+→ t
+d · uT u
+as
+n → ∞
+P-a.s.
+(4.11)
+Hence the condition (H.1) is satisfied. Notice that
+⌊nt⌋
+�
+k=1
+1
+wn
+E
+�
+∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1
+�
+≤
+⌊nt⌋
+�
+k=1
+�w⌊nt⌋
+wn
+�2
+1
+ϵ2w2
+⌊nt⌋
+E
+�
+∆Mk(u)4|Fk−1
+�
+,
+(4.12)
+since (2.5), (2.6), and (A.21), we observe that
+⌊nt⌋
+�
+k=1
+��∆Mk(u)4�� ≤ C1(β)∥u∥4
+⌊nt⌋
+�
+k=1
+(akµk)4 ≤ C2(β)∥u∥4
+⌊nt⌋
+�
+k=1
+1
+k2
+P-a.s.
+(4.13)
+with constants C1(β), C2(β) > 0. Therefore, by (4.12) and (4.13), we have
+⌊nt⌋
+�
+k=1
+1
+wn
+E
+�
+∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1
+�
+≤ C3(β)∥u∥4 · t2
+ϵ2 ·
+1
+nt(log nt)2
+P-a.s.
+Simplifying the above expression, we obtain
+⌊nt⌋
+�
+k=1
+1
+wn
+E
+�
+∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+(4.14)
+Then the condition (H.2), or the Lindeberg condition, is satisfied by (4.14). In this particular case
+at critical regime, (4.11) implies that the condition (H.3) is satisfied. Hence
+�
+1
+√wn
+M⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+Bt(u), t ≥ 0
+�
+where (Bt(u))t≥0 is a continuous real-valued centered Gaussian process such that B0(u) = 0 and
+with covariance
+E
+�
+Bs(u)Bt(u)
+�
+= s
+d · uT u
+for all
+0 ≤ s ≤ t < ∞.
+In the critical regime, from (3.2) we can write
+S⌊nt⌋(u) = N⌊nt⌋(u) + (2β + 1) M⌊nt⌋(u)
+a⌊nt⌋µ⌊nt⌋
+.
+(4.15)
+From (A.8) we know that
+⟨N(u)⟩⌊nt⌋
+nt log n
+→ 0
+and
+N⌊nt⌋(u)
+�
+nt log n
+→ 0
+as
+n → ∞
+P-a.s.
+(4.16)
+Using (2.5), (2.6), and (3.4) again, we conclude that
+�
+1
+�
+nt log n
+S⌊nt⌋(u), t ≥ 0
+�
+=⇒
+�
+(2β + 1)Bt(u), t ≥ 0
+�
+with
+E
+�
+Bs(u)Bt(u)
+�
+= s · uT u
+d
+for all
+0 ≤ s ≤ t.
+(4.17)
+Solving (4.17), we get
+E
+�
+BsBT
+t
+�
+= s · 1
+dId
+for all
+0 ≤ s ≤ t.
+
+12
+JIAMING CHEN AND LUCILE LAULIN
+which completes the proof.
+□
+4.3. The superdiffusive regime.
+Theorem 4.7. We have the almost sure convergence
+1
+na(β+1)−β Sn → Lβ
+as
+n → ∞
+P-a.s.
+where the limiting Lβ is an Rd-valued random variable.
+Remark 4.1. In fact, from Theorem 4.8 below, we will see the random vector Lβ is non-degenerate.
+Proof. From (3.5) and (A.7), in the superdiffusive regime, we have
+Tr⟨M⟩n ≤ wn ≤
+∞
+�
+k=1
+�Γ(a(β + 1) + 1)Γ(β + k)
+Γ(a(β + 1) + k)Γ(β + 1)
+�2
+< ∞
+for all
+n ∈ N.
+By [18, Theorem 4.3.15], this leads to
+Mn → M
+as
+n → ∞
+P-a.s.
+with
+M =
+∞
+�
+k=1
+akϵk.
+By (3.1), Mn = anYn, and by (2.5), we observe that
+Yn
+na(β+1) →
+1
+Γ(a(β + 1) + 1)M
+as
+n → ∞
+P-a.s.
+(4.18)
+Moreover, equations (4.3) still holds and, as 2a(β + 1) > 2β + 1 in the superdiffusive regime, we
+find that
+1
+n2(a(β+1)−β)
+����Sn +
+a(β + 1)
+(β − a(β + 1))µn+1
+Yn
+����
+2
+= o
+�
+n−(1−2a(β+1)+2β)�
+log n
+�1+γ�
+P-a.s.
+Thanks to (2.6), we obtain
+Sn
+na(β+1)−β +
+a(β + 1)
+β − a(β + 1) · Γ(β + 1)Yn
+na(β+1)
+→ 0
+as
+n → ∞
+P-a.s.
+(4.19)
+Combining (4.18), it yields
+Sn
+na(β+1)−β → Lβ
+as
+n → ∞
+P-a.s.
+where
+Lβ =
+a(β + 1)
+a(β + 1) − β ·
+Γ(β + 1)
+Γ(a(β + 1) + 1)M
+(4.20)
+and the assertion follows.
+□
+Theorem 4.8. We have the following mean square convergence
+E
+�����
+1
+na(β+1)−β Sn − Lβ
+����
+2�
+→ 0
+as
+n → ∞.
+(4.21)
+Proof. For each test vector u ∈ Rd, we have
+E
+�
+Mn(u)2�
+= E
+�
+⟨M(u)⟩n
+�
+≤ 1
+dwnuT u
+for all
+n ∈ N.
+From (3.5), we obtain
+sup
+n≥1
+E
+�
+Mn(u)2�
+< ∞
+which implies that (Mn(u))n∈N is a martingale bounded in L2. Therefore
+E
+�
+|Mn(u) − M(u)|2�
+→ 0
+as
+n → ∞.
+(4.22)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+13
+Moreover, on the one hand (4.22) together with (4.18) implies that
+E
+�����
+1
+na(β+1) Yn(u) − Y (u)
+����
+2�
+→ 0
+as
+n → ∞.
+(4.23)
+On the other hand, from (A.8) we know that
+E
+�
+Nn(u)2�
+= E
+�
+⟨N(u)⟩n
+�
+≤ 1
+d
+�
+β
+β − a(β + 1)
+�2
+nuT u
+for all
+n ∈ N.
+Since a(β + 1) > β + 1
+2 in the superdiffusive regime, we have
+E
+�����
+1
+na(β+1)−β Nn(u)
+����
+2�
+→ 0
+as
+n → ∞.
+(4.24)
+The proof is complete by combining (4.23) and (4.24).
+□
+Remark 4.2. The expected value of Lβ is
+E
+�
+Lβ
+�
+= 0
+(4.25)
+whereas its quadratic deviation is
+E
+�
+LβLT
+β
+�
+=
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(β + 1)2Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2
+· 1
+dId.
+(4.26)
+Theorem 4.9. The MARW admits a scaling limit at the superdiffusive regime, or convergence in
+distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that
+�
+1
+na(β+1)−β S⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+Qt, t ≥ 0
+�
+(4.27)
+with the limiting Qt = ta(β+1)−βLβ for all t ≥ 0.
+Proof. For all t ≥ 0 and from (4.19), we observe that
+S⌊nt⌋
+⌊nt⌋a(β+1)−β +
+a(β + 1)
+β − a(β + 1) ·
+Y⌊nt⌋
+⌊nt⌋a(β+1) → 0
+as
+n → ∞
+P-a.s.
+which implies
+1
+na(β+1)−β Sn → ta(β+1)−βLβ
+as
+n → ∞
+P-a.s.
+(4.28)
+The P-a.s. convergence in (4.28)holds in all finite-dimensional distributions which characterizes
+the Skorokhod space topology. Hence, we have (4.27) and the assertion is verified.
+□
+5. Scaling limit of the barycenter process
+The study of the scaling limit of the MARW (Sn)n∈N gives us some information on its asymptotic
+behavior.
+Nonetheless, to understand its pathwise geometric features, we need to discuss its
+barycenter, or center of mass process. Such topics have been raised and discussed in [36, 43]. In
+this Section, we turn our attention to the above-mentioned barycenter process (Gn)n∈N defined
+by
+Gn := 1
+n
+n
+�
+k=1
+Sk
+(5.1)
+Our work contains the discussion on the scaling limit and the almost sure convergence in the
+diffusive, critical and superdiffusive regimes. The quadratic strong law in the diffusive and crit-
+ical regimes is also discussed while the mean square convergence in the superdiffusive regime is
+established.
+
+14
+JIAMING CHEN AND LUCILE LAULIN
+5.1. Almost sure convergence. The barycenter process was discussed in [8] for the elephant
+random walk in dimension d, which is a special case of the process we study here when β = 0. We
+first begin with the almost sure convergence.
+Theorem 5.1. We have the almost sure convergence, in the diffusive regime,
+1
+nGn → 0
+as
+n → ∞
+P-a.s.
+(5.2)
+while in the critical regime,
+1
+√n log nGn → 0
+as
+n → ∞
+P-a.s.
+(5.3)
+and, in the superdiffusive regime,
+1
+na(β+1)−β Gn →
+1
+1 + a(β + 1) − β Lβ
+as
+n → ∞
+P-a.s.
+(5.4)
+where Lβ was characterized in Theorems 4.7 and 4.2.
+Proof. In the diffusive regime, from (5.1) we observe that
+1
+nGn =
+n
+�
+k=1
+k
+n2 · 1
+k Sk =
+n
+�
+k=1
+1
+k Ska′
+n,k
+with
+a′
+n,k = k
+n2 .
+Since �n
+k=1 a′
+n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.1, from Lemma
+A.12 we can conclude that
+1
+nGn =
+n
+�
+k=1
+1
+k Ska′
+n,k → 0
+as
+n → ∞
+P-a.s.
+such that (5.2) is verified. In the critical regime, we have from (5.1) that
+1
+√n log nGn =
+1
+n3/2 log n
+n
+�
+k=1
+Sk =
+n
+�
+k=1
+1
+√
+k log k
+Ska′′
+n,k
+with
+a′′
+n,k = k1/2 log k
+n3/2 log n.
+Since �n
+k=1 a′′
+n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.4 holds, we get
+from Lemma A.12 hat
+1
+√n log nGn =
+n
+�
+k=1
+1
+√
+k log k
+Ska′′
+n,k → 0
+as
+n → ∞
+P-a.s.
+and we obtain (5.3). Finally, in the superdiffusive regime, we also get from (5.1) that
+1
+na(β+1)−β Gn =
+1
+n1+a(β+1)−β
+n
+�
+k=1
+Sn =
+n
+�
+k=1
+1
+ka(β+1)−β Ska′′′
+n,k
+with
+a′′′
+n,k =
+ka(β+1)−β
+n1+a(β+1)−β .
+Since
+n
+�
+k=1
+a′′′
+n,k →
+1
+1 + a(β + 1) − β
+as
+n → ∞
+by a simple calculation, and because of the almost sure convergence in Theorem 4.7, we can
+conclude using Lemma A.13
+1
+na(β+1)−β Gn →
+1
+1 + a(β + 1) − β Lβ
+as
+n → ∞
+P-a.s.
+and (5.4) is verified.
+□
+5.2. Quadratic strong law.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+15
+Theorem 5.2. In the diffusive regime, we have the quadratic strong law
+1
+log n
+n
+�
+k=1
+GkGT
+k
+k2
+→ 4I(a, β) · 1
+dId
+as
+n → ∞
+P-a.s.
+where I(a, β) is given explicitly
+I(a, β) =
+1
+Γ(a(β + 1) + 1)2Γ(β + 1)2 ·
+2a2(1 − a)(β + 1)3
+3(β − a(β + 1))2(1 − a(β + 1) + β).
+Proof. We will check that all the three conditions of [32, Theorem A.2] are satisfied. Looking back
+to (5.1), we observe that
+Gn = 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+1
+akµk
+Mk = 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+1
+akµk
+k
+�
+l=1
+alϵl.
+Then, changing the order of summation, we have
+Gn = 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+akϵk
+n
+�
+l=k
+1
+alϵl
+= 1
+n
+n
+�
+k=1
+Nk − 1
+n
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+ak(δn − δk−1)ϵk
+where we define δn = �n
+k=1(akµk)−1 for all n ∈ N. Moreover, we denote
+Zn =
+n
+�
+k=1
+Nk −
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+akδk−1ϵk.
+such that we have
+Gn = 1
+nZn − δn
+n ·
+a(β + 1)
+β − a(β + 1)
+n
+�
+k=1
+akϵk = 1
+n
+�
+Zn −
+a(β + 1)
+β − a(β + 1)δnMn
+�
+.
+For a fixed text vector u ∈ Rd, we define
+Hn(u) =
+�
+Zn(u)
+Mn(u)
+�
+for all
+n ∈ N.
+(5.5)
+which implies
+∆Hn(u) = Hn+1(u) − Hn(u) =
+�
+Nn+1(u)ϵn+1(u)−1 −
+a(β+1)
+β−a(β+1)an+1δn
+an+1
+�
+ϵn+1(u).
+Then, let
+Vn =
+1
+n3/2
+�
+1
+0
+0
+a(β+1)
+β−a(β+1)δn
+�
+and
+v =
+�
+1
+−1
+�
+.
+Then it is immediate that
+vT VnHn(u) =
+1
+√nGn
+for all
+n ∈ N
+(5.6)
+and that
+lim
+n→∞ Vn⟨H(u)⟩nV T
+n = lim
+n→∞
+1
+n3
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+�
+a(β + 1)
+β − a(β + 1)
+�2
+δ2
+ka2
+k+1E
+�
+ϵk+1(u)2|Fk
+�
+= lim
+n→∞
+1
+n3 ·
+a2(1 − a)(β + 1)3uT u
+d(β − a(β + 1))2(1 − a(β + 1) + β)
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+δ2
+ka2
+k+1µ2
+k+1
+P-a.s.
+
+16
+JIAMING CHEN AND LUCILE LAULIN
+By (2.5) and (2.6), we know that
+n−(1+a(β+1)−β)δn →
+1
+1 + a(β + 1) − β ·
+1
+Γ(a(β + 1) + 1)Γ(β + 1)
+as
+n → ∞.
+Hence the above calculation leads us to
+Vn⟨H(u)⟩nV T
+n → I(a, β)uT u · 1
+d
+�
+1
+−1
+−1
+1
+�
+as
+n → ∞
+P-a.s.
+(5.7)
+where
+I(a, β) =
+1
+1 − 2(a(β + 1) − β) ·
+a2(1 − a)(β + 1)3
+(β − a(β + 1))2(1 − a(β + 1) + β).
+(5.8)
+Consequently, (5.7) ensures that the condition (H.1) is satisfied. Notice that by (2.3) and (3.1),
+there exists some constant C1(a, β) > 0 and similarly, by (2.5), (2.6), (A.22), there exists some
+other constant C2(a, β) > 0 such that
+∥Nn∥2 ≤ C1(a, β)n2
+and
+a2
+kϵk(u)2 ≤ C2(a, β)n2δ−2
+n
+for all
+1 ≤ k ≤ n.
+Moreover, notice that for all 1 ≤ k ≤ n,
+Vn∆Hk(u) =
+1
+n3/2
+�
+Nk(u)ϵk(u)−1 −
+a(β+1)
+β−a(β+1)akδk−1
+a(β+1)
+β−a(β+1)akδn
+�
+ϵk(u).
+Hence, for all 1 ≤ k ≤ n, we observe that
+∥Vn∆Hk(u)∥2 ≤ 4a2
+k
+n3
+�
+a(β + 1)
+β − a(β + 1)
+�2��β − a(β + 1)
+aka(β + 1)
+Nk(u)
+ϵk(u)
+�2
++ δ2
+k−1 + δ2
+n
+�
+ϵk(u)2 ≤ C(a, β)
+n
+(5.9)
+for some constant C(a, β) > 0. Consequently, we
+n
+�
+k=1
+E
+�
+∥Vn∆Hk(u)∥4�
+≤ 1
+nC(a, β) → 0
+as
+n → ∞
+P-a.s.
+since, for all ϵ > 0,
+n
+�
+k=1
+E
+�
+∥Vn∆Hk(u)∥21{∥Vn∆Hk(u)∥>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Vn∆Hk(u)∥4�
+→ 0
+as
+n → ∞
+P-a.s.
+(5.10)
+Then the condition (H.2), or the Lindeberg condition, is satisfied by (5.10). Hereafter, by (2.5),
+(2.6), and by the definition of δn, we know there exists some constant C′(a, β) ̸= 0 such that
+log
+�
+det V −1
+n
+�2
+log n
+→ C′(a, β)
+as
+n → ∞.
+This ensures that there exists some other constant C′′(a, β) > 0 such that
+∞
+�
+n=1
+1
+�
+log
+�
+det V −1
+n
+�2�2 E
+�
+∥Vn∆Hn(u)∥4|Fn−1
+�
+≤ C2(a, β)
+∞
+�
+n=1
+1
+(log n)2 E
+�
+∥Vn∆Hn(u)∥4|Fn−1
+�
+.
+Finally, using (5.9) leads to
+∞
+�
+n=1
+1
+(log n)2 ∥Vn∆Hn(u)∥4 ≤ C(a, β)
+∞
+�
+n=1
+1
+(n log n)2 < ∞
+P-a.s.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+17
+for some constant C(a, β) > 0 depending only on a and β. The condition (H.4) is satisfied by
+combining the above with (5.10). On the one hand,
+1
+log
+�
+det V −1
+n
+�2
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkHk(u)Hk(u)T V T
+k → 1
+duT uV
+(5.11)
+as n → ∞ P-a.s. where
+V =
+�
+1
+−1
+−1
+1
+�
+I(a, β)
+and I(a, β) has been specified in (5.8). Then, we have
+1
+log n
+n
+�
+k=1
+�(det Vk)2 − (det Vk+1)2
+(det Vk)2
+�
+VkHk(u)Hk(u)T V T
+k → 4 − 2(a(β + 1) − β)
+d
+uT uV
+as n → ∞ P-a.s. since
+log n
+log
+�
+det V −1
+n
+�2 → 4 − 2(a(β + 1) − β)
+as
+n → ∞.
+On the other hand, by (2.5) and (2.6), we have
+n
+�(det Vn)2 − (det Vn+1)2
+(det Vn)2
+�
+→ 4 − 2
+�
+a(β + 1) − β
+�
+as
+n → ∞
+P-a.s.
+Using (5.6) and (5.11), we observe that
+1
+log n
+n
+�
+k=1
+uT GkGT
+k u
+k2
+=
+1
+log n
+n
+�
+k=1
+vT VkHk(u)Hk(u)T V T
+k v
+k
+→ vT V v · 1
+duT u
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (4.5).
+□
+Theorem 5.3. In the critical regime, we have the quadratic strong law
+1
+log log n
+n
+�
+k=1
+GkGT
+k
+(k log k)2 → 4(2β + 1)2
+9
+· 1
+dId
+as
+n → ∞
+P-a.s.
+Proof. We will check that all the three conditions of [32, Theorem A.2] are satisfied. Denote
+Wn =
+1
+n√n log n
+�
+1
+0
+0
+a(β+1)
+β−a(β+1)δn
+�
+and
+w =
+�
+1
+−1
+�
+.
+Then, for H defined in (5.5), it is clear that
+wT WnHn(u) =
+1
+√n log nGn
+for all
+n ∈ N
+and that
+lim
+n→∞ Wn⟨H(u)⟩nW T
+n = lim
+n→∞
+1
+n3 log n
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+(2β + 1)2δ2
+ka2
+k+1E
+�
+ϵk+1(u)2|Fk
+�
+= lim
+n→∞
+(2β + 1)2
+n3 log n · uT u
+d
+�
+1
+−1
+−1
+1
+� n−1
+�
+k=1
+δ2
+ka2
+k+1µ2
+k+1
+P-a.s.
+By (2.5) and (2.6), we know that
+n−3/2δn → 2
+3 ·
+Γ(β + 1)
+Γ(β + 1 + 1
+2)
+as
+n → ∞.
+
+18
+JIAMING CHEN AND LUCILE LAULIN
+Hence, the above calculation leads us to
+Wn⟨H(u)⟩nW T
+n → I(β)uT u · 1
+d
+�
+1
+−1
+−1
+1
+�
+as
+n → ∞
+P-a.s.
+with
+I(β) = 4(2β + 1)2
+9
+.
+(5.12)
+Consequently, the condition (H.1) is satisfied thanks to (5.12). Notice that by (2.3) and (3.1),
+there exists some constant C1(β) > 0 and similarly, there exists some constant C2(β) > 0 such
+that
+∥Nn∥2 ≤ C1(β)n2
+and
+a2
+kϵk(u)2 ≤ C2(β)n2δ−2
+n log n
+for all
+1 ≤ k ≤ n.
+Then, notice for all 1 ≤ k ≤ n that
+Wn∆Hk(u) =
+1
+n√n log n
+�
+Nk(u)ϵk(u)−1 −
+a(β+1)
+β−a(β+1)akδk−1
+a(β+1)
+β−a(β+1)akδn
+�
+ϵk(u).
+The ensures that, for all 1 ≤ k ≤ n,
+∥Wn∆Hk(u)∥2 ≤
+4a2
+k
+n3 log n(2β + 1)2
+��
+(2β + 1)−2 Nk(u)
+ϵk(u)
+�2 + δ2
+k−1 + δ2
+n
+�
+ϵk(u)2 ≤ C(β)
+n
+(5.13)
+for some constant C(β) > 0. Hence,
+n
+�
+k=1
+E
+�
+∥Wn∆Hk(u)∥4�
+≤ 1
+nC(β) → 0
+as
+n → ∞
+P-a.s.
+since, for all ϵ > 0,
+n
+�
+k=1
+E
+�
+∥Wn∆Hk(u)∥21{∥Wn∆Hk(u)∥>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Wn∆Hk(u)∥4�
+→ 0
+as
+n → ∞.
+(5.14)
+Therefore, the condition (H.2), or the Lindeberg condition, is satisfied using (5.14). Hereafter, we
+know that
+log
+�
+det W −1
+n
+�2
+log log n
+→ 4
+as
+n → ∞.
+This ensures that there exists some constant C2(β) > 0 such that
+∞
+�
+n=1
+1
+�
+log
+�
+det W −1
+n
+�2�2 E
+�
+∥Wn∆Hn(u)∥4|Fn−1
+�
+≤
+∞
+�
+n=1
+C2(β)
+(log log n)2 E
+�
+∥Wn∆Hn(u)∥4|Fn−1
+�
+.
+(5.15)
+We get from (5.13) that
+∞
+�
+n=1
+1
+(log log n)2 ∥Wn∆Hn(u)∥4 ≤ C(β)
+∞
+�
+n=1
+1
+(n log n log log n)2 < ∞
+P-a.s.
+for some constant C(β) > 0 depending only onβ. The condition (H.4) is satisfied using the above
+together with (5.15). Then,
+1
+log
+�
+det W −1
+n
+�2
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkHk(u)Hk(u)T W T
+k → 1
+duT uW
+as n → ∞ P-a.s. where
+W = 4(2β + 1)2
+9
+�
+1
+−1
+−1
+1
+�
+.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+19
+Furthermore, on the one hand we have
+1
+log log n
+n
+�
+k=1
+�(det Wk)2 − (det Wk+1)2
+(det Wk)2
+�
+WkHk(u)Hk(u)T W T
+k → 1
+duT uW
+as n → ∞ P-a.s. since
+log log n
+log
+�
+det W −1
+n
+�2 → 1
+4
+as
+n → ∞.
+On the other hand, we have
+n log n
+�(det Wn)2 − (det Wn+1)2
+(det Wn)2
+�
+→ 1
+as
+n → ∞
+P-a.s.
+By (5.6) and (5.11), we observe that
+1
+log log n
+n
+�
+k=1
+uT GkGT
+k u
+(k log k)2 =
+1
+log log n
+n
+�
+k=1
+wT WkHk(u)Hk(u)T W T
+k w
+4k log k
+→ wT Ww · 1
+4duT u
+(5.16)
+as n → ∞ P-a.s. Since u ∈ Rd is arbitrary, the assertion follows from (5.16).
+□
+Theorem 5.4. In the superdiffusive regime, we have the mean square convergence, given by
+E
+�����
+1
+na(β+1)−β Gn −
+1
+1 + a(β + 1) − β Lβ
+����
+2�
+→ 0
+as
+n → ∞.
+(5.17)
+Proof. For all test vector u ∈ Rd, it is immediate that
+E
+�����
+1
+na(β+1)−β Gn(u) −
+1
+1 + a(β + 1) − β Lβ(u)
+����
+2�
+≤ 2E
+�����
+1
+n1+a(β+1)−β Zn(u)
+����
+2�
++ 2E
+�����
+1
+n1+a(β+1)−β ·
+a(β + 1)
+a(β + 1) − β δnMn −
+1
+1 + a(β + 1) − β Lβ
+����
+2�
+.
+(5.18)
+By (4.20) and (5.7), the second term converges to zero. Looking back to the first term in (5.18),
+we observe
+E
+�����
+1
+n1+a(β+1)−β Zn(u)
+����
+2�
+≤
+4
+n1+2(a(β+1)−β)
+n
+�
+k=1
+E
+�
+Nk(u)2�
++
+4
+n1+2(a(β+1)−β)
+�
+a(β + 1)
+a(β + 1) − β
+�2
+E
+������
+n
+�
+k=1
+akδk−1ϵk(u)
+�����
+2�
+.
+(5.19)
+The first term in (5.19) converges to zero because E[Nk(u)] ≤ (uT u)n for all 1 ≤ k ≤ n, and
+moreover, in the superdiffusive regime we have a(β +1) > β +1/2. The second term in (5.19) also
+converges to zero because
+n−(1+a(β+1)−β)δn →
+1
+1 + a(β + 1) − β ·
+1
+Γ(1 + a(β + 1))Γ(β + 1)
+as
+n → ∞.
+Finally, using the above and that M(u) = �∞
+k=1 akϵk(u) is bounded in L2, the assertion follows.
+□
+5.3. Scaling limit.
+Theorem 5.5. The barycenter process admits a scaling limit at the diffusive regime, or conver-
+gence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+� 1
+√nG⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+Wtr dr, t ≥ 0
+�
+
+20
+JIAMING CHEN AND LUCILE LAULIN
+where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.3 with its
+covariance defined in (4.6). In particular,
+E
+��
+1
+�
+0
+Wsv dv
+��
+1
+�
+0
+Wtu du
+�T �
+=
+β
+3(β(1 − a) − a)(1 − a)s · 1
+dId
++
+2(a(β + 1)(1 − a) + aβ)
+3(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)(1 + (1 − a)(β + 1))ta−β(1−a)s1−a+β(1−a) · 1
+dId
+(5.20)
+for all 0 ≤ s ≤ t < ∞.
+Proof. An easy calculation leads to
+lim
+n→∞
+1
+√nG⌊nt⌋ = lim
+n→∞
+1
+�
+0
+1
+√nS⌊ntr⌋ dr =⇒
+1
+�
+0
+Wtr dr
+which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]). Then, the last convergence
+in law is due to the functional central limit Theorem 4.3, with (Wt)t≥0 defined there. Hence, the
+barycenter process (Gn)n∈N admits a Gaussian scaling limit in the diffusive regime as well, with
+covariance
+E
+��
+1
+�
+0
+Wsv dv
+��
+1
+�
+0
+Wtu du
+�T �
+= 2
+1
+�
+0
+u
+�
+0
+E
+�
+WsvW T
+tu
+�
+dv du.
+Using (4.6), the formula (5.20) and the assertion follows.
+□
+Theorem 5.6. The barycenter process admits a scaling limit at the critical regime, or convergence
+in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+�
+1
+�
+nt log n
+G⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+(2β + 1)Btr dr, t ≥ 0
+�
+where (Bt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.6 with its
+covariance defined in (4.17).
+Proof. For each r ∈ [0, 1], (3.2) and (4.11) implies that
+lim
+n→∞
+1
+�
+nt log n
+· M⌊ntr⌋(u)
+a⌊ntr⌋µ⌊ntr⌋
+= lim
+n→∞
+1
+�
+nt log n
+�
+ntr(log n + r
+t log r)
+�1/2Btr(u)
+P-a.s.
+for all u ∈ Rd. Moreover, (4.16) yields
+lim
+n→∞
+1
+�
+nt log n
+N⌊ntr⌋(u) = lim
+n→∞ r1/2 ·
+1
+�
+ntr log n
+N⌊ntr⌋(u) = 0
+P-a.s.
+for all u ∈ Rd. By (4.15), we have
+�
+1
+�
+nt log n
+S⌊ntr⌋(u), t ≥ 0
+�
+=⇒
+�
+(2β + 1)Btr(u), t ≥ 0
+�
+for all u ∈ Rd and r ∈ [0, 1]. Hence, we use again
+lim
+n→∞
+1
+�
+nt log n
+G⌊nt⌋ = lim
+n→∞
+1
+�
+0
+1
+�
+nt log n
+S⌊ntr⌋ dr =⇒
+1
+�
+0
+(2β + 1)Btr dr
+and the assertion is verified.
+□
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+21
+Theorem 5.7. The barycenter process admits a scaling limit at the superdiffusive regime, or
+convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that
+�
+1
+na(β+1)−β G⌊nt⌋, t ≥ 0
+�
+=⇒
+�
+1
+�
+0
+Qtr dr, t ≥ 0
+�
+with the covariance specified in (5.3) and the limiting Lβ characterized in Theorem 4.8 and Qt =
+ta(β+1)−βLβ characterized in Theorem 4.9 for all t ≥ 0.
+Proof. Again, we find that
+lim
+n→∞
+1
+na(β+1)−β G⌊nt⌋ =
+1
+�
+0
+1
+na(β+1)−β S⌊ntr⌋ dr =⇒
+1
+�
+0
+Qtr dr
+which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]). Then, the last convergence
+in law is due to the functional central limit Theorem 4.9. Hence the barycenter process (Gn)n∈N
+admits a non-degenerate scaling limit in the superdiffusive regime as well, with covariance
+E
+��
+1
+�
+0
+Qsv dv
+��
+1
+�
+0
+Qtu du
+�T �
+= 2
+1
+�
+0
+u
+�
+0
+E
+�
+QsvQT
+tu
+�
+dv du = ta(β+1)−βsa(β+1)−β
+(1 + a(β + 1) − β)2 E
+�
+LβLT
+β
+�
+= ta(β+1)−βsa(β+1)−β
+(1 + a(β + 1) − β)2
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2 · 1
+dId
+for all 0 ≤ s ≤ t < ∞.
+□
+6. Velocity of quadratic mean displacement
+In this Section, we investigate the velocity of the mean square displacement of the MARW.
+This quantitative estimates give us the information on how fast the limit Theorems in Section 4
+are carried on. Similar convergence velocities have been discussed in [20, 26], where the authors
+analyzed the convergence velocity of the moments of a one-dimensional elephant random walk of
+all orders. In the superdiffusive regime, the convergence velocity was discussed in [6]. Here, only
+the rate of quadratic moment convergence for the MARW in all of the three (diffusive, critical,
+and superdiffusive) regimes are discussed.
+Following the limit Theorems in Section 4, we expect the asymptotic behavior of the mean
+square displacement is as follows,
+E
+�
+SnST
+n
+�
+∼
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+n ·
+(a−2β)(1−a)(β+1)+β(a+1)
+(2(β+1)(1−a)−1)(a−β(1−a))(1−a) · 1
+dId
+when
+a < 1 −
+1
+2(β+1)
+n log n · (2β + 1)2 · 1
+dId
+when
+a = 1 −
+1
+2(β+1)
+n2(a(β+1)−β) ·
+�
+a(β+1)
+β−a(β+1)
+�2
+Γ(2(a−1)(β+1)+1)
+Γ((2a−1)(β+1)+1)2 · 1
+dId
+when
+a > 1 −
+1
+2(β+1),
+(6.1)
+where the notation ∼ indicates two sequences an ∼ bn if and only if an/bn → 1 as n → ∞.
+The aim of this Section is not only to show that the above estimates (6.1) are valid, but also
+to investigate the exact velocity of their convergence in the diffusive and critical regime.
+6.1. Diffusive regime.
+Theorem 6.1. For all p < (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,
+1
+nE
+�
+SnST
+n
+�
+−
+(a − 2β)(1 − a)(β + 1) + β(a + 1)
+(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1
+dId
+∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1
+dId.
+
+22
+JIAMING CHEN AND LUCILE LAULIN
+Proof. Take the vector v = (1, −1)T and Vn ∈ R2×2 as in (A.16). Then,
+1
+√nSn(u) = vT VnLn(u),
+where Ln(u) = (Nn(u), Mn(u))T is as in (3.6). In particular,
+1
+nuT E
+�
+SnST
+n
+�
+u = vT VnE
+�
+Ln(u)Ln(u)T �
+V T
+n v
+= vT VnE
+� �
+E
+�
+Nn(u)2�
+E
+�
+Nn(u)Mn(u)
+�
+E
+�
+Mn(u)Nn(u)
+�
+E
+�
+Mn(u)2�
+� �
+V T
+n v
+= vT VnE
+� �
+E
+�
+⟨N(u)⟩n
+�
+E
+�
+⟨N(u), M(u)⟩n
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+E
+�
+⟨M(u)⟩n
+�
+� �
+V T
+n v.
+Therefore,
+1
+nuT E
+�
+SnST
+n
+�
+u = 1
+nE
+�
+⟨N(u)⟩n
+�
++
+1
+na2nµ2n
+�
+a(β + 1)
+β − a(β + 1)
+�2
+E
+�
+⟨M(u)⟩n
+�
+−
+2
+nanµn
+�
+a(β + 1)
+β − a(β + 1)
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+.
+Since the test vector u ∈ Rd is taken arbitrarily, we get from Lemmas A.15 and A.16 that
+1
+nE
+�
+SnST
+n
+�
+−
+(a − 2β)(1 − a)(β + 1) + β(a + 1)
+(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1
+dId
+∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1
+dId
+as
+n → ∞.
+□
+6.2. Critical regime.
+Theorem 6.2. When p = (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,
+1
+n log nE
+�
+SnST
+n
+�
+− (2β + 1)2 · 1
+dId ∼ −(C1(log n)−1 + C2n−1) · 1
+dId.
+Proof. Take w = (1, −1)T and Wn ∈ R2×2 as in (A.28). Then
+1
+√n log nSn(u) = wT WnLn(u) as in
+(A.29) for all u ∈ Rd. In particular,
+1
+n log nuT E
+�
+SnST
+n
+�
+u = wT WnE
+�
+Ln(u)Ln(u)T �
+W T
+n w.
+Hence,
+1
+n log nuT E
+�
+SnST
+n
+�
+u = wT WnE
+� �
+E
+�
+⟨N(u)⟩n
+�
+E
+�
+⟨N(u), M(u)⟩n
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+E
+�
+⟨M(u)⟩n
+�
+� �
+W T
+n w.
+Therefore, we get by (3.4) as n → ∞,
+1
+n log nuT E
+�
+SnST
+n
+�
+u =
+1
+n log n
+�
+E
+�
+⟨N(u)⟩n
+�
++ (2β + 1)2
+a2nµ2n
+E
+�
+⟨M(u)⟩n
+��
+,
+which implies
+1
+n log nE
+�
+SnST
+n
+�
+− (2β + 1)2 · 1
+dId ∼ −(C1(log n)−1 + C2n−1) · 1
+dId
+as
+n → ∞.
+□
+6.3. Superdiffusive regime.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+23
+Theorem 6.3. When p > (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,
+1
+n2(a(β+1)−β) E
+�
+SnST
+n
+�
+−
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2 · 1
+dId
+∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β)).
+Proof. Similar to previous computations for the diffusive regime, we have for all u ∈ Rd,
+1
+n2(a(β+1)−β) uT E
+�
+SnST
+n
+�
+u =
+1
+n2(a(β+1)−β) E
+�
+⟨N(u)⟩n
+�
++
+1
+n2(a(β+1)−β)a2nµ2n
+�
+a(β + 1)
+β − a(β + 1)
+�2
+E
+�
+⟨M(u)⟩n
+�
+−
+2
+n2(a(β+1)−β)anµn
+�
+a(β + 1)
+β − a(β + 1)
+�
+E
+�
+⟨M(u), N(u)⟩n
+�
+.
+Hence, by (2.5), (2.6), (3.5) and since u ∈ Rd is arbitrary,
+1
+n2(a(β+1)−β) E
+�
+SnST
+n
+�
+−
+�
+a(β + 1)
+β − a(β + 1)
+�2 Γ(2(a − 1)(β + 1) + 1)
+Γ((2a − 1)(β + 1) + 1)2 · 1
+dId
+∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β))
+as
+n → ∞.
+□
+7. Cram´er moderate deviations
+In this Section, we discuss the Cram´er moderate deviations for the multidimensional reinforced
+random walk (Sn)n∈N. The similar statistical quantity as well as the Berry-Esseen bound for the
+one-dimensional elephant random walk (ERW) without amnesia-reinforcement has been given in
+[20]. Our derivation of Cram´er moderate deviations for the MARW does not rely on a Berry-
+Esseen bound. The discussion of such statistical quantities is expected to reveal the transience
+property and the central limit Theorems for the MARW. For this direction, readers are refereed
+to [3, 16]. Thanks to Lemma A.21 and Lemma A.22, we can properly state the Cram´er moderate
+deviations principles for the MARW.
+Theorem 7.1. In the diffusive and critical regimes, we have the following Cram´er moderate
+deviations for the MARW. Let (ϑn)n∈N ⊆ R be a non-decreasing sequence so that ϑn/√n → 0 as
+n → ∞. Take any non-empty Borel set B ⊆ Rd, then we have
+−
+inf
+x∈int B
+1
+2∥x∥2 ≤ lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ lim sup
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ − inf
+x∈cl B
+1
+2∥x∥2,
+where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively.
+Proof. Our proof will only present the Cram´er moderate deviations for the MARW in the diffusive
+regime. The same property for the critical regime follows from exactly the same steps. First, take
+xB = infx∈B ∥x∥.
+Then it is obvious that infx∈cl B ∥x∥ ≤ xB and infx∈cl B ∥x∥2/2 ≤ x2
+B/2.
+Henceforth,
+P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤
+d
+�
+j=1
+P
+�����
+anµnSj
+n
+√wn
+���� ≥ ϑnxB
+d
+�
+≤
+�
+1 − Φ(ϑnxB)
+�
+F(B, ϑ, n),
+(7.1)
+
+24
+JIAMING CHEN AND LUCILE LAULIN
+where we write
+F(B, ϑ, n) := 2Cd · exp
+�
+1
+√n
+� ϑnxB
+2d
+�3 + 1
+n
+� ϑnxB
+2d
+�2 +
+1
+√n(1 + 1
+2 log n)(1 + ϑnxB
+2d )
+�
++ 2Cd · exp
+�
+1
+√n
+� ϑnxB
+2d
+�3 +
+1
+n2(1−a)(β+1)
+� ϑnxB
+2d
+�2 +
+1
+√n(1 + 1
+2 log n)(n1/2−(1−a)(β+1) + ϑnxB
+2d )
+�
+.
+Hence,
+lim sup
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≤ −1
+2x2
+B ≤ − inf
+x∈cl B
+1
+2∥x∥2.
+To achieve the asymptotic lower bound, we first notice that this assertion automatically holds if
+int B = ∅, whence − infx∈∅ ∥x∥2/2 = −∞. Consequently, we assume that int B ̸= ∅. Notice that
+int B is open in Rd. Hence, for all ϵ∗ > 0 sufficiently small, we could find x∗ ∈ int B with
+0 < 1
+2∥x∗∥2 <
+inf
+x∈int B
+1
+2∥x∥2 + ϵ∗
+and
+0 < min
+���xj
+∗
+�� : 1 ≤ j ≤ d
+�
+.
+Choose ϵ∗∗ sufficient small such that 0 < ϵ∗∗ <
+���xj
+∗
+��� for each j = 1, . . . , d. Then,
+U(x∗, ϵ∗∗) ⊆ int B ⊆ B,
+where
+U(x∗, ϵ∗∗) :=
+�
+x ∈ Rd :
+��xj − xj
+∗
+�� < ϵ∗∗ for all j
+�
+.
+On the other hand,
+P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≥ P
+�anµnSn
+√wn
+∈ ϑn · U(x∗, ϵ∗∗)
+�
+≥
+d
+�
+j=1
+P
+�
+ϑn(xj
+∗ + ϵ∗∗) ≥ anµnSj
+n
+√wn
+≥ ϑn(xj
+∗ − ϵ∗∗)
+�
+.
+From Lemma A.21 and Lemma A.22, we know that
+lim
+n→∞ P
+�anµnSj
+n
+√wn
+≥ ϑn(xj
+∗ + ϵ∗∗)
+��
+P
+�anµnSj
+n
+√wn
+≥ ϑn(xj
+∗ − ϵ∗∗)
+�
+= 0
+for each
+j.
+Similar to (7.1),
+lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≥ −1
+2∥x∗ − ϵ∗∗∥2.
+Letting ϵ∗∗ → 0, we observe that
+lim inf
+n→∞ ϑ−2
+n log P
+�anµnSn
+ϑn√wn
+∈ B
+�
+≥ −1
+2∥x∗∥2 ≥ −
+inf
+x∈int B
+1
+2∥x∥2 − ϵ∗.
+Since ϵ∗ > 0 was take arbitrarily, letting ϵ∗ → 0, we verify the assertion.
+□
+Appendix A. Technical Lemmas
+A.1. Asymptotics of the processes. We start by introducing the following processes that are
+of great influence on the behavior of the random walk. Let (e1, e2, . . . , ed) denote a canonical
+Euclidean basis of Rd. For each n ∈ N and 1 ≤ j ≤ d, define
+N X
+n (j) =
+n
+�
+k=1
+1{Xj
+k̸=0}µk
+and
+Σn =
+d
+�
+j=1
+N X
+n (j)ejeT
+j ,
+(A.1)
+such that (Σn)n∈N is a matrix-valued process.
+Lemma A.1. We have the following almost sure convergence in the three regimes.
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+(A.2)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+25
+Proof. For each n ∈ N and 1 ≤ j ≤ d, define
+ΛX
+n (j) = N X
+n (j)
+n
+.
+(A.3)
+It follows from (A.1) that
+ΛX
+n+1(j) =
+n
+n + 1ΛX
+n (j) +
+1
+n + 11{Xj
+n+1̸=0}µn+1.
+Moreover, we observe thanks to (A.12) that
+ΛX
+n+1(j) =
+n
+n + 1 · γnΛX
+n (j) +
+1
+n + 11{Xj
+n+1̸=0}µn+1 − a(β + 1)
+n + 1 ΛX
+n (j)
+=
+n
+n + 1 · γnΛX
+n (j) + µn+1
+n
+δX
+n+1(j) + (1 − a)µn+1
+d(n + 1)
+with
+δX
+n+1(j) = 1{Xj
+n+1̸=0} − P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+.
+Then, by (2.4) we know
+ΛX
+n (j) =
+1
+nan
+�
+ΛX
+1 (j) + 1 − a
+d
+n
+�
+k=2
+akµk + HX
+n (j)
+�
+(A.4)
+with
+HX
+n (j) =
+n
+�
+k=2
+akµkδX
+k (j).
+It is clear that for a fixed 1 ≤ j ≤ d, the real-valued process (HX
+n (j))n∈N is locally square-integrable
+since it is a finite sum. Afterwards, this process appears to be a martingale adapted to (Fn)n∈N
+because (δX
+n (j))n∈N satisfied the martingale difference relation E[δX
+n+1(j)|Fn] = 0. It is obvious
+that
+⟨HX(j)⟩n ≤ wn =
+n
+�
+k=1
+(akµk)2
+P-a.s.
+Hence, we get by [18, Theorem 4.3.15] that for all γ > 0
+HX
+n (j)2
+⟨HX(j)⟩n
+= o
+��
+log⟨HX(j)⟩n
+�1+γ�
+P-a.s.
+(A.5)
+Since ⟨HX(j)⟩n ≤ wn and by (A.5), we obtain that
+HX
+n (j)2 = o
+�
+wn
+�
+log wn
+�1+γ�
+P-a.s.
+In the diffusive regime, by Lemma A.1 and (3.3), we have
+HX
+n (j)2 = o
+�
+n1−2(a(β+1)−β)�
+log n
+�1+γ�
+P-a.s.
+By (2.5) and (2.6), we observe that
+� HX
+n (j)
+nanµn+1
+�2
+= o
+�
+n−1�
+log n
+�1+γ�
+P-a.s.
+Hence
+HX
+n (j)
+nanµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+By (2.5) and (2.6) again, we observe further
+1
+nanµn+1
+n
+�
+k=1
+akµk →
+1
+(1 − a)(β + 1)
+as
+n → ∞.
+(A.6)
+
+26
+JIAMING CHEN AND LUCILE LAULIN
+Hence, we have
+µ−1
+n+1ΛX
+n (j) →→
+1
+β + 1
+as
+n → ∞.
+By (A.3) and (A.4), we can then conclude that
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+in the diffusive regime. In the critical regime, where a = 1 −
+1
+2(β+1), we have from (3.4))
+HX
+n (j)2 = o
+�
+log n
+�
+log log n
+�1+γ�
+P-a.s.
+Hence
+� HX
+n (j)
+nanµn+1
+�2
+= o
+�
+n−1 log n
+�
+log log n
+�1+γ�
+P-a.s.
+which implies that
+HX
+n (j)
+nanµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+Similar to the convergence in (A.6), in the critical regime, we observe
+1
+nanµn+1
+n
+�
+k=1
+akµk → 1
+2
+P-a.s.
+Hence, we conclude that
+µ−1
+n+1ΛX
+n (j) →
+1
+d(β + 1)
+and
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+which proves (A.2). In the superdiffusive regime, we have
+HX
+n (j)2 = o
+�
+1
+�
+P-a.s.
+and then
+� HX
+n (j)
+nanµn+1
+�2
+= o
+�
+n−2(1−a)(β+1)�
+P-a.s.
+which implies
+HX
+n (j)
+nanµn+1
+→ 0
+as
+n → ∞
+P-a.s.
+We can similarly show that
+µ−1
+n+1ΛX
+n (j) →
+1
+β + 1
+as
+n → ∞.
+which then ensures that
+1
+nµn+1
+Σn →
+1
+d(β + 1)Id
+as
+n → ∞
+P-a.s.
+Consequently, the assertion is verified.
+□
+The next result follows directly from the definition of Mn and Nn
+Lemma A.2. We have the following formulas for the predictable matrix-valued quadratic varia-
+tions
+⟨M⟩n = (a1µ1)2E
+�
+X1XT
+1
+�
++
+n−1
+�
+k=1
+a(β + 1)
+ka−2
+k+1
+µk+1Σk + 1 − a
+da−2
+k+1
+µ2
+k+1Id −
+�γk − 1
+a−1
+k+1
+�2
+YkY T
+k ,
+(A.7)
+and
+⟨N⟩n =
+�
+β
+β − a(β + 1)
+�2
+E
+�
+X1XT
+1
+�
++
+n−1
+�
+k=1
+a(β + 1)
+kµk+1
+Σk + 1 − a
+d
+Id −
+�γk − 1
+µk+1
+�2
+YkY T
+k .
+(A.8)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+27
+In particular, we have
+Tr⟨M⟩n = wn −
+n
+�
+k=1
+(γk − 1)2a2
+k+1∥Yk∥2,
+(A.9)
+and
+Tr⟨N⟩n =
+�
+β
+β − a(β + 1)
+�2
+n −
+n−1
+�
+k=1
+�a(β + 1)
+kµk+1
+�2
+∥Yk∥2.
+(A.10)
+Lemma A.3. We have the following estimate for the matrix-valued conditional expectation.
+E
+�
+ϵn+1ϵT
+n+1|Fn
+�
+= a(β + 1)
+n
+µn+1Σn + 1 − a
+d
+µ2
+n+1Id − (γn − 1)2YnY T
+n .
+And as a consequence
+E
+�
+∥ϵn+1∥2|Fn
+�
+= µ2
+n+1 − (γn − 1)2∥Yn∥2.
+Proof. Observe that
+E
+�
+ϵn+1ϵT
+n+1|Fn
+�
+= E
+�
+Yn+1Y T
+n+1|Fn
+�
+− γ2
+nYnY T
+n
+with
+E
+�
+Yn+1Y T
+n+1|Fn
+�
+= YnY T
+n + 2µn+1YnE
+�
+XT
+n+1|Fn
+�
++ µ2
+n+1E
+�
+Xn+1XT
+n+1|Fn
+�
+=
+�
+1 + 2a(β + 1)
+n
+�
+YnY T
+n + µ2
+n+1E
+�
+Xn+1XT
+n+1|Fn
+�
+.
+(A.11)
+For all k ≥ 1, we know that XkXT
+k = �d
+j=1 1{Xj
+k̸=0}ejeT
+j . Then
+P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+=
+n
+�
+k=1
+P
+�
+βn+1 = k
+�
+· P
+�
+(AnXk)j ̸= 0|Fn
+�
+=
+n
+�
+k=1
+1{Xj
+k̸=0}P
+�
+An = ±Id
+�
+· (β + 1)µk
+nµn+1
++
+n
+�
+k=1
+�
+1 − 1{Xj
+k̸=0}
+�
+P
+�
+An = ±Jd
+�
+· (β + 1)µk
+nµn+1
+.
+Hence
+P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+= β + 1
+nµn+1
+·
+�
+P
+�
+An = +Id
+�
+− P
+�
+An = +Jd
+��
+N X
+n (j) + 2P
+�
+An = +Jd
+�
+= a(β + 1)
+nµn+1
+N X
+n (j) + 1 − a
+d
+.
+(A.12)
+Therefore
+E
+�
+Xn+1XT
+n+1|Fn
+�
+=
+d
+�
+j=1
+P
+�
+Xj
+n+1 ̸= 0|Fn
+�
+ejeT
+j = a(β + 1)
+nµn+1
+Σn + 1 − a
+d
+Id.
+(A.13)
+And from (A.11) and (A.13) we can conclude that
+E
+�
+ϵn+1ϵT
+n+1|Fn
+�
+= E
+�
+Yn+1Y T
+n+1|Fn
+�
+− γ2
+nYnY T
+n
+=
+�
+1 + 2a(β + 1)
+n
+�
+YnY T
+n + a(β + 1)
+n
+µn+1Σn + 1 − a
+d
+µ2
+n+1Id − γ2
+nYnY T
+n
+= a(β + 1)
+n
+µn+1Σn + 1 − a
+d
+µ2
+n+1Id − (γn − 1)2YnY T
+n .
+(A.14)
+On the other hand
+Tr(Σn) = nµn+1
+β + 1 .
+(A.15)
+Taking traces in (A.14) and by (A.15), we have
+E
+�
+∥ϵn+1∥2|Fn
+�
+= µ2
+n+1 − (γn − 1)2∥Yn∥2
+which ensures that the assertion is verified.
+□
+
+28
+JIAMING CHEN AND LUCILE LAULIN
+A.2. Scaling limits of the random walk and the barycenter.
+A.2.1. The diffusive regime.
+Lemma A.4. For each n ∈ N and test vector u ∈ Rd, let
+Vn =
+1
+√n
+�
+1
+0
+0
+a(β+1)
+β−a(β+1)(anµn)−1
+�
+and
+v =
+�
+1
+−1
+�
+.
+(A.16)
+Then
+vT VnLn(u) =
+1
+√nSn(u).
+(A.17)
+And for all t ≥ 0, we have
+Vn⟨L(u)⟩⌊nt⌋V T
+n → uT u
+d Vt
+as
+n → ∞
+P-a.s.
+(A.18)
+where
+Vt =
+1
+(β − a(β + 1))2
+�
+β2t
+aβ
+1−at1+β−a(β+1)
+aβ
+1−at1+β−a(β+1)
+a2(β+1)2
+1−2a(β+1)+2β t1+2β−2a(β+1)
+�
+.
+(A.19)
+Proof. From Lemma A.3 and the fact that ⟨M(u)⟩n = uT ⟨M⟩nu, we see that
+⟨M(u)⟩⌊nt⌋ = a2
+1µ2
+1uT E
+�
+X1XT
+1
+�
+u
++
+⌊nt⌋−1
+�
+k=1
+a(β + 1)
+k
+a2
+k+1µk+1uT Σku + 1 − a
+d
+a2
+k+1µ2
+k+1uT u − (γk − 1)2a2
+k+1uT YkY T
+k u
+and
+⟨N(u)⟩⌊nt⌋ =
+�
+β
+β − a(β + 1)
+�2
+uT E
+�
+X1XT
+1
+�
+u
++
+�
+β
+β − a(β + 1)
+�2 ⌊nt⌋−1
+�
+k=1
+a(β + 1)
+kµk+1
+uT Σku + 1 − a
+d
+uT u −
+�γk − 1
+µk+1
+�2
+uT YkY T
+k u.
+Using a similar token and Lemma A.1, we can work out the off-diagonal entries in ⟨L(u)⟩⌊nt⌋, and
+we obtain that
+lim
+n→∞ Vn⟨L(u)⟩⌊nt⌋V T
+n
+= lim
+n→∞
+uT u
+nd(β − a(β + 1))2
+�
+�
+�
+β2⌊nt⌋
+a(β+1)β
+anµn
+�⌊nt⌋−1
+k=0
+ak+1µk+1
+a(β+1)β
+anµn
+�⌊nt⌋−1
+k=0
+ak+1µk+1
+�
+a(β+1)
+anµn
+�2 �⌊nt⌋−1
+k=0
+(ak+1µk+1)2
+�
+�
+�
+=
+uT u
+d(β − a(β + 1))2
+�
+β2t
+aβ
+1−at1−(a(β+1)−β)
+aβ
+1−at1−(a(β+1)−β)
+a2(β+1)2
+1−2(a(β+1)−β)t1−2(a(β+1)−β)
+�
+= uT u
+d Vt
+P-a.s.
+where the last equality is due to (2.5) and (2.6). Thus, it implies that
+1
+nanµn
+n
+�
+k=1
+akµk →
+1
+1 − (a(β + 1) − β)
+and
+1
+n(anµn)2
+n
+�
+k=1
+(akµk)2 →
+1
+1 − 2(a(β + 1) − β)
+as n → ∞. Hence, equation (A.18) holds and the assertion is then verified.
+□
+Lemma A.5. The MARW satisfies the Lindeberg condition in the diffusive regime. That is, for
+all t ≥ 0 and all ϵ > 0,
+⌊nt⌋
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+29
+Proof. On the one hand, it is easy to compute from (3.7) and (A.16) that, for all 1 ≤ k ≤ n,
+Vn∆Lk(u) =
+1
+√n(β − a(β + 1))µn
+�
+β µn
+µk
+a ak
+an
+�
+ϵk(u)
+which implies
+∥Vn∆Lk(u)∥2 =
+1
+n(β − a(β + 1))2
+�β2
+µ2
+k
++
+a2a2
+k
+(anµn)2
+�
+ϵk(u)2.
+Hence
+∥Vn∆Lk(u)∥4 ≤
+2
+n2(β − a(β + 1))4
+�β4
+µ4
+k
++
+a4a4
+k
+(anµn)4
+�
+ϵk(u)4.
+(A.20)
+On the other hand, from (2.5) we observe that
+1
+na2n
+n
+�
+k=1
+a2
+k ≤ C1(a, β)−1
+and
+1
+na4n
+n
+�
+k=1
+a4
+k ≤ C2(a, β)−1
+for all
+n ∈ N
+(A.21)
+and where C1(a, β), C2(a, β) > 0 are constants depending only on a and β. Moreover, we get that
+sup
+1≤k≤n
+|ϵk(u)| ≤
+sup
+1≤k≤n
+∥ϵk∥∥u∥ ≤
+sup
+1≤k≤n
+(β + 2)µk∥u∥ ≤ (β + 2)µn∥u∥.
+(A.22)
+Hence, we deduce from (A.21) and (A.22)
+n
+�
+k=1
+∥Vn∆Lk(u)∥4 ≤
+2
+n2(β − a(β + 1))4
+��
+β(β + 2)
+�4∥u∥4 +
+�
+a(β + 2)
+�4∥u∥4
+C2(a, β)
+�
+→ 0
+(A.23)
+as n → ∞ P-a.s. This implies that
+n
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Therefore, for all ϵ > 0, we obtain
+n
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as n → ∞ P-a.s. This yields finally
+⌊nt⌋
+�
+k=1
+E
+�
+∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+⌊nt⌋
+�
+k=1
+E
+����(VnV −1
+⌊nt⌋)V⌊nt⌋∆Lk(u)
+���
+4
+|Fk−1
+�
+→ 0
+as n → ∞ P-a.s. since VnV −1
+⌊nt⌋ converges as n → ∞.
+□
+Lemma A.6. The deterministic matrix Vt defined in (A.19) can be rewritten as
+Vt = tα1K1 + tα2K2 + · · · + tαqKq
+with q ∈ N, αj > 0 and each Kj is a symmetric matrix for all 1 ≤ j ≤ 1.
+Proof. A direct computation analoguous to the one in [32] shows that Vt = tα1K1+tα2K2+tα3K3,
+where
+α1 = 1,
+α2 = 1 − a(β + 1) > 0,
+α3 = 1 − 2a(β + 1) > 0
+since a < 1 −
+1
+2(β+1) is in the diffusive regime. Moreover
+K1 =
+β2
+(a(β + 1) − β)2
+�
+1
+0
+0
+0
+�
+,
+K2 =
+aβ
+(1 − a)(a(β + 1) − β)2
+�
+0
+1
+1
+0
+�
+,
+K3 =
+a2(β + 1)2
+(1 − 2a(β + 1) + 2β)(a(β + 1) − β)2
+�
+0
+0
+0
+1
+�
+.
+
+30
+JIAMING CHEN AND LUCILE LAULIN
+□
+Lemma A.7. Given the matrix-valued process (Vn)n∈N define in (A.16), we have
+∞
+�
+n=1
+1
+�
+log
+�
+det V −1
+n
+�2�2 E
+�
+∥Vn∆Ln(u)∥4|Fn−1
+�
+< ∞
+P-a.s.
+Proof. From (A.16), it is immediate that
+det V −1
+n
+= β − a(β + 1)
+a(β + 1)
+nanµn.
+(A.24)
+By (2.5) and (2.6), we obtain
+log
+�
+det V −1
+n
+�2
+log n
+→ 2(1 − a)(β + 1)
+as
+n → ∞
+P-a.s.
+(A.25)
+Hence there exists a constant C(a, β) > 0 depending only on a and β such that
+∞
+�
+n=1
+1
+�
+log
+�
+det V −1
+n
+�2�2 E
+�
+∥Vn∆Ln(u)∥4|Fn−1
+�
+≤ C(a, β)
+∞
+�
+n=1
+1
+(log n)2 E
+�
+∥Vn∆Ln(u)∥4|Fn−1
+�
+.
+(A.26)
+Hereafter, equations (A.20), (A.22), (A.23) together imply that
+∞
+�
+n=1
+1
+(log n)2 ∥Vn∆Ln(u)∥4 ≤ C′(a, β)
+∞
+�
+n=1
+1
+(n log n)2 < ∞
+P-a.s.
+(A.27)
+for some other constant C′(a, β) > 0 depending only on a and β. Consequently, equation (A.27)
+together (A.26) ensures that the assertion is verified.
+□
+A.2.2. The critical regime.
+Lemma A.8. For each n ∈ N and test vector u ∈ Rd, let
+Wn =
+1
+√n log n
+�
+1
+0
+0
+2β+1
+anµn
+�
+and
+w =
+�
+1
+−1
+�
+.
+(A.28)
+Then for all t ≥ 0, we have
+wT WnLn(u) =
+1
+√n log nSn(u)
+(A.29)
+and
+Wn⟨L(u)⟩nW T
+n → uT u
+d W
+as
+n → ∞
+P-a.s.
+where
+Wt = (2β + 1)2
+�
+0
+0
+0
+1
+�
+.
+(A.30)
+Proof. It is clear that (A.29) follows from (3.2). Using a similar token than for the proof Lemma
+A.4, we have
+lim
+n→∞ Wn⟨L(u)⟩nW T
+n
+= lim
+n→∞
+4uT u
+(n log n)d
+�
+�
+�
+β2n
+β(β+ 1
+2 )
+anµn
+�n−1
+k=0 ak+1µk+1
+β(β+ 1
+2 )
+anµn
+�n−1
+k=0 ak+1µk+1
+�
+β+ 1
+2
+anµn
+�2 �n−1
+k=0(ak+1µk+1)2
+�
+�
+�
+= 4uT u
+d
+�
+0
+0
+0
+�
+β + 1
+2
+�2
+�
+= uT u
+d W
+P-a.s.
+and the proof is complete.
+□
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+31
+Lemma A.9. The MARW satisfies the Lindeberg condition in the critical regime. That is, for all
+t ≥ 0 and all ϵ > 0, given the (Wn)n∈N defined in (A.16), it satisfies
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Proof. We state that equations (A.20) and (A.21) remain true with Vn replaced by Wn. More
+precisely, they can be rewritten as
+∥Wn∆Lk(u)∥4 ≤
+32
+(n log n)2
+�β4
+µ4
+k
++
+a4a4
+k
+(anµn)4
+�
+ϵk(u)4
+(A.31)
+and
+1
+na4n
+n
+�
+k=1
+a4
+k ≤ C(a, β)−1
+for all
+n ∈ N
+where C(a, β) > 0 is a constant depending only on t, a, and β. Since (A.22) is not affected by
+switching regimes, we have that
+n
+�
+k=1
+∥Wn∆Lk(u)∥4 ≤
+32
+(n log n)2
+��
+β(β + 2)
+�4∥u∥4 +
+�
+a(β + 2)
+�4∥u∥4
+C(t, a, β)
+�
+→ 0
+(A.32)
+as n → ∞ P-a.s. This implies
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Therefore, for all ϵ > 0, we obtain
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1
+�
+≤ 1
+ϵ2
+n
+�
+k=1
+E
+�
+∥Wn∆Lk(u)∥4|Fk−1
+�
+→ 0
+as n → ∞ P-a.s. and the assertion is verified.
+□
+Lemma A.10. Given the matrix-valued sequence (Wn)n∈N define in (A.28), we have
+∞
+�
+n=1
+1
+�
+log
+�
+det W −1
+n
+�2�2 E
+�
+∥Wn∆Ln(u)∥4|Fn−1
+�
+< ∞
+P-a.s.
+Proof. From (A.28), it is immediate that
+det W −1
+n
+=
+1
+2β + 1
+�
+n log n · anµn.
+(A.33)
+Then, we obtain by (2.5) and (2.6) that
+log
+�
+det W −1
+n
+�2
+log log n
+→ 1
+as
+n → ∞
+P-a.s.
+(A.34)
+Hence, there exists a constant C(a, β) > 0 depending only on a and β such that
+∞
+�
+n=1
+1
+�
+log
+�
+det W −1
+n
+�2�2 E
+�
+∥Wn∆Ln(u)∥4|Fn−1
+�
+≤
+∞
+�
+n=1
+C(a, β)
+(log log n)2 E
+�
+∥Wn∆Ln(u)∥4|Fn−1
+�
+.
+(A.35)
+Hereafter, (A.31) together with (A.32) imply that
+∞
+�
+n=1
+1
+(log log n)2 ∥Wn∆Ln(u)∥4 ≤ C′(a, β)
+∞
+�
+n=1
+1
+(n log n log log n)2 < ∞
+P-a.s.
+for some other constant C′(a, β) > 0 depending only on a and β. Finally, using the above equation
+together with (A.35) completes the proof.
+□
+
+32
+JIAMING CHEN AND LUCILE LAULIN
+Lemma A.11. Fix the test vector u ∈ Rd. The growth rate of the compensator of the partial sum
+of (Nn(u)2)n∈N is less than cubic growth, in the sense that
+1
+n3
+n−1
+�
+k=1
+E
+�
+Nk+1(u)2|Fn
+�
+→ 0
+as
+n → ∞
+P-a.s.
+Proof. The law of iterated expectations and (A.8) yields
+1
+nE
+�
+E
+�
+Nn+1(u)2|Fn
+��
+= 1
+nE
+�
+⟨N(u)⟩n
+�
+→
+�
+β
+β − a(β + 1)
+�2
+uT u
+as
+n → ∞
+P-a.s.
+The strong law of large numbers then yields
+1
+n
+n−1
+�
+k=1
+1
+k E
+�
+Nk+1(u)2|Fk
+�
+→
+�
+β
+β − a(β + 1)
+�2
+uT u
+as
+n → ∞
+P-a.s.
+Hence
+1
+n3
+n−1
+�
+k=1
+E
+�
+Nk+1(u)2|Fn
+�
+≤ 1
+n2
+n−1
+�
+k=1
+1
+k E
+�
+Nk+1(u)2|Fk
+�
+→ 0
+as
+n → ∞
+P-a.s.
+□
+A.2.3. The barycenter process. For the following Toeplitz Lemmas, see [18] and [33].
+Lemma A.12. [33, Theorem 1.1 Part I] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers
+such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N
+�kn
+k=1 |an,k| < ∞. Let (xn)n∈N
+be a real sequence. If xn → 0 as n → ∞, then �kn
+k=1 an,kxk → 0 as n → ∞.
+Lemma A.13. [33, Theorem 1.1 Part II] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers
+such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N
+�kn
+k=1 |an,k| < ∞. Let (xn)n∈N
+be a real sequence. If xn → x as n → ∞ with x ∈ R and �kn
+k=1 an,k = 1, then �kn
+k=1 an,kxk → x
+as n → ∞.
+A.3. Quadratic rate estimates. Our first result is about the convergence rate of the process
+(Yn)n∈N defined in (2.3).
+Lemma A.14. For all p ∈ (0, 1), then we have, as n → ∞,
+E[YnY T
+n ] ∼
+n2a(β+1)
+Γ(1 + 2a(β + 1)) · 1
+dId +
+n1+2β
+Γ(β + 1)2(1 + 2β − 2a(β + 1))(β + 1) · 1
+dId.
+Proof. From (A.11) and (A.13), we see
+E
+�
+Yn+1Y T
+n+1|Fn
+�
+=
+�
+1 + 2a(β + 1)
+n
+�
+YnY T
+n + µ2
+n+1
+�a(β + 1)
+nµn+1
+Σn + 1 − a
+d
+Id
+�
+.
+Then, remember that
+E
+�
+Σn
+�
+=
+d
+�
+j=1
+E
+�
+N X
+n (j)
+�
+ejeT
+j =
+d
+�
+j=1
+n
+�
+k=1
+P
+�
+Xj
+k ̸= 0
+�
+µk · ejeT
+j .
+Lemma A.1 yields E[(nµn+1)−1Σn] ∼ (β + 1)−1 · 1
+dId. Hence,
+E
+�
+Yn+1Y T
+n+1
+�
+∼
+�
+1 + 2a(β + 1)
+n
+�
+E
+�
+YnY T
+n
+�
++ µ2
+n+1
+β + 1 · 1
+dId.
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+33
+A recursive argument then gives
+E
+�
+YnY T
+n
+�
+∼
+Γ(n + 2a(β + 1))
+Γ(n)Γ(1 + 2a(β + 1))E
+�
+Y1Y T
+1
+�
++
+n−1
+�
+j=1
+µ2
+j
+β + 1 ·
+�n−1
+k=1(1 + k−12a(β + 1))
+�j−1
+k=1(1 + k−12a(β + 1))
+· 1
+dId
+∼
+Γ(n + 2a(β + 1))
+Γ(n)Γ(1 + 2a(β + 1)) · 1
+dId +
+n−1
+�
+j=1
+µ2
+j
+β + 1 · Γ(n + 2a(β + 1))Γ(j)
+Γ(j + 2a(β + 1))Γ(n) · 1
+dId.
+Employing the asymptotics in (2.1) and (2.6), the assertion follows.
+□
+The process Yn = �n
+k=1 µkXk differs from Sn by a multiplicative factor at each step. When
+there is no amnesia, the asymptotics of these two processes coincide. However, when β ≥ 0, we
+have to treat the general case in another way.
+Lemma A.15. For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞,
+E
+�
+⟨M(u)⟩n
+�
+∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u,
+and
+E
+�
+⟨N(u)⟩n
+�
+∼
+�
+β
+β − a(β + 1)
+�2
+nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.
+Proof. By Lemma A.2
+E
+�
+⟨M(u)⟩n
+�
+= E
+�
+Tr⟨M⟩n
+�
+uT u = wnuT u −
+n
+�
+k=1
+(γk − 1)2a2
+k+1uT E
+�
+YkY T
+k
+�
+u.
+By Lemma A.14 and a finite summation,
+E
+�
+⟨M(u)⟩n
+�
+∼ wnuT u −
+n−1
+�
+k=1
+a2(β + 1)2
+k2
+(k + 1)−2a(β+1)(C1k2a(β+1) + C2k1+2β)uT u
+∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u.
+Similarly,
+E
+�
+⟨N(u)⟩n
+�
+= E
+�
+Tr⟨N⟩n
+�
+uT u =
+�
+β
+β − a(β + 1)
+�2
+nuT u −
+n−1
+�
+k=1
+a2(β + 1)2
+k2
+µ−2
+k+1uT E
+�
+YkY T
+k
+�
+u.
+Hence, using Lemma A.14 again, we observe
+E
+�
+⟨N(u)⟩n
+�
+∼
+�
+β
+β − a(β + 1)
+�2
+nuT u −
+n−1
+�
+k=1
+a2(β + 1)2
+k2
+(k + 1)−2β(C1k2a(β+1) + C2k1+2β)uT u
+∼
+�
+β
+β − a(β + 1)
+�2
+nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.
+□
+Lemma A.16. For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞,
+E
+�
+⟨M(u), N(u)⟩n
+�
+∼
+β
+β − a(β + 1) · Γ(β + 1)Γ(a(β + 1) + 1)
+(1 − a)(β + 1)
+n(1−a)(β+1)uT u
+− (C1n−(1−a)(β+1) + C2n(1−a)(β+1)−1)uT u.
+Proof. By (3.7) and Lemma A.2, for all test vector u ∈ Rd
+∆Ln+1(u) =
+�
+βµ−1
+n+1
+β − a(β + 1)
+�T
+ϵn+1(u),
+
+34
+JIAMING CHEN AND LUCILE LAULIN
+and therefore,
+⟨M(u), N(u)⟩n =
+n
+�
+k=1
+β
+β − a(β + 1)akµ−1
+k E
+�
+ϵk(u)ϵk(u)T |Fk−1
+�
+.
+Taking the trace will give us
+Tr⟨M, N⟩n =
+β
+β − a(β + 1)
+n
+�
+k=1
+akµk −
+β
+β − a(β + 1)
+n
+�
+k=1
+akµ−1
+k (γk − 1)2∥Yk∥2.
+Taking the expectation and using Lemma A.14 completes the proof.
+□
+A.4. Moderate deviations.
+Lemma A.17. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+��∆M j
+n
+�� ≤
+�
+a(β + 1) + 1
+�
+anµn
+for all
+n ∈ N.
+(A.36)
+Proof. By (2.3) and (3.1),
+∆M j
+n = anY j
+n − an−1Y j
+n−1 = anµnXj
+n − (an − an−1)
+n−1
+�
+k=1
+µkXj
+k.
+Since ∥Xk∥ = 1 for eack k ≤ n, then by (2.4),
+��∆M j
+n
+�� ≤ anµn + (n − 1)(an−1 − an)µn−1 ≤ anµn + a(β + 1)anµn.
+And the assertion is verified.
+□
+Lemma A.18. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+��∆N j
+n
+�� ≤ 2a(β + 1) +
+β
+β − a(β + 1)
+for all
+n ∈ N.
+Proof. By (2.3) and (3.6),
+∆N j
+n =
+βµ−1
+n+1
+β − a(β + 1)ϵj
+n+1 =
+βµ−1
+n+1
+β − a(β + 1) ·
+�
+µn+1Xj
+n+1 + (1 − γn)
+n
+�
+k=1
+Xj
+kµk
+�
+.
+Taking absolute value on both sides, and the assertion is verified.
+□
+Lemma A.19. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+����
+1
+√wn
+∆M j
+k
+���� ≤
+�
+a(β + 1) + 1
+�anµn
+√wn
+for each
+1 ≤ k ≤ n,
+(A.37)
+and in the diffusive and critical regime,
+����
+1
+wn
+⟨M j⟩n − 1
+���� ≤
+�
+�
+�
+C · n−1
+when
+a < 1 −
+1
+2(β+1)
+C · (log n)−1
+when
+a = 1 −
+1
+2(β+1).
+Proof. Dividing by √wn from both sides of (A.36), we get (A.37). Moreover, by (A.9),
+��⟨M j⟩n − wn
+�� ≤
+n
+�
+k=1
+(γk − 1)2a2
+k+1∥Yk∥2 ≤ C
+n
+�
+k=1
+wk
+k2 .
+Dividing both sides by wn and following (3.3), (3.4), the assertion is verified.
+□
+Lemma A.20. For all p ∈ (0, 1) and for all j = 1, . . . , d,
+����
+anµn
+√wn
+∆N j
+k
+���� ≤
+�
+2a(β + 1) +
+β
+β − a(β + 1)
+�anµn
+√wn
+for each
+1 ≤ k ≤ n,
+(A.38)
+
+MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK
+35
+and in both the diffusive and critical regime,
+����
+a2
+nµ2
+n
+wn
+⟨N j⟩n − 1
+���� ≤
+�
+�
+�
+C · n−2(1−a)(β+1)
+when
+a < 1 −
+1
+2(β+1)
+C · (n log n)−1
+when
+a = 1 −
+1
+2(β+1).
+Proof. Dividing by √wn and multiplied by anµu from both sides of (A.18), we get (A.38). Then,
+by (A.10), we make use of the estimates and the inequalities hold.
+□
+Denote by Φ(·) := (2π)−1/2 � ·
+−∞ e−t2/2 dt the cumulative distribution of the standard normal
+random variable. The following lemmas are straightforward derivations from [19, Theorem 1], see
+also [22].
+Lemma A.21. There exists an absolute constant α′(p, β) > 0 depending only on p, β such that
+for all j = 1, . . . , d and all 0 ≤ x ≤ α′(p, β) · n−1/2, in the diffusive and critical regime,
+P(M j
+n/√wn ≥ x)
+1 − Φ(x)
+= P(M j
+n/√wn ≤ −x)
+1 − Φ(−x)
+=
+�
+�
+�
+C · exp
+�
+x3
+√n + x2
+n +
+1
+√n(1 + 1
+2 log n)(1 + x)
+�
+when
+a < 1 −
+1
+2(β+1)
+C · exp
+�
+x3
+√n +
+x2
+log n + (
+1
+√log n +
+1
+2√n log n)(1 + x)
+�
+when
+a = 1 −
+1
+2(β+1).
+Lemma A.22. There exists an absolute constant α′′(p, β) > 0 depending only on p, β such that
+for all j = 1, . . . , d and all 0 ≤ x ≤ α′′(p, β) · n−1/2, in the diffusive and critical regime,
+P(anµnN j
+n/√wn ≥ x)
+1 − Φ(x)
+= P(anµnN j
+n/√wn ≤ −x)
+1 − Φ(−x)
+=
+�
+�
+�
+C · exp
+�
+x3
+√n +
+x2
+n2(1−a)(β+1) +
+1
+√n(n1/2−(1−a)(β+1) + 1
+2 log n)(1 + x)
+�
+when
+a < 1 −
+1
+2(β+1)
+C · exp
+�
+x3
+√n +
+x2
+n log n + (
+1
+√n log n +
+1
+2√n log n)(1 + x)
+�
+when
+a = 1 −
+1
+2(β+1).
+Acknowledgements. The authors wish to thank Jean Bertoin and Pierre Tarres for numerous
+discussions and insightful comments.
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+Departement Mathematik, ETH Z¨urich
+Current address: 101, R¨amistrasse, CH-8092 Z¨urich, Switzerland
+Email address: jiamchen@student.ethz.ch
+Laboratoire de Math´ematiques Jean Leray, Nantes Universit´e
+Current address: 2 Chem. de la Houssini`ere, 44322 Nantes, France
+Email address: lucile.laulin@math.cnrs.fr
+
diff --git a/-NFAT4oBgHgl3EQfqB2B/content/tmp_files/load_file.txt b/-NFAT4oBgHgl3EQfqB2B/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1637 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf,len=1636
+page_content='ANALYSIS OF THE SMOOTHLY AMNESIA-REINFORCED  MULTIDIMENSIONAL ELEPHANT RANDOM WALK  JIAMING CHEN AND LUCILE LAULIN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant  random walk (MARW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The scaling limit of the MARW is shown to exist in the diffusive, critical  and superdiffusive regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We also establish the almost sure convergence in all of the three  regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The quadratic strong law is displayed in the diffusive regime as well as in the critical  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The mean square convergence towards a non-Gaussian random variable is established  in the superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Similar results for the barycenter process are also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Finally,  the last two Sections are devoted to a discussion of the convergence velocity of the mean square  displacement and the Cram´er moderate deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The amnesia-reinforced elephant random walk  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  A correlated martingale approach  6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Scaling limit and convergence  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Scaling limit of the barycenter process  13  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Velocity of quadratic mean displacement  21  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Cram´er moderate deviations  23  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Technical Lemmas  24  References  35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Introduction  The study of reinforced processes and reinforced random walks has known a growing interest  over the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In particular, random walks on graphs, or more precisely edge [37] or vertex  [39] reinforced random walks, have been the subject of a great number of contributions, see also  [1, 12, 27] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The insight of introducing reinforcement mechanisms to  stochastic processes has also shed light on more applied models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In [30], the adaptive strategy of  an agent who plays a two-armed bandit machine was described as a self-reinforced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The philosophy of stochastic reinforcement has also been discussed in the topics of evolutionary  ecology [4] and machine learning theory [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Another manifestation of reinforced P´olya urn models  on financial economics can be found in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We also refer the readers to [38] for a comprehensive  and extensive survey on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The Elephant Random Walk (ERW) is a discrete-time random walk, introduced by Sch¨utz and  Trimper [40] in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It was referred to as the ERW in allusion to the traditional saying that  elephants can always remember anywhere they have been.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' As it was pointed out [12] by Bertoin  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' 60G50, 60G42, 62M09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Reinforced random walk, scaling limit, Cram´er moderate deviation, martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='08644v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='PR]  20 Jan 2023  2  JIAMING CHEN AND LUCILE LAULIN  (a) Diffusive regime  (b) Critical regime  (c) Superdiffusive regime  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The two-dimensional ERW with amnesia (in blue) and its barycenter  (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  who relied on K¨ursten’s work [29], the ERW is a special case of step-reinforced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In  fact, the ERW is reinforced because its behavior is influenced by its past : the ERW may have  a tendency to do the same thing over and over, or on the contrary, it may try to compensate its  previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This different types of behavior, here-called regimes, are ruled by the memory  parameter p and it is well-known that the ERW shows three regimes of behavior and that the  critical value is p = 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The ERW in dimension d = 1 has received a lot of attention from mathematicians and physicists  over the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The almost sure convergence and the asymptotic normality of the  position of the ERW were established in the diffusive regime p < 3/4 and the critical regime  p = 3/4, see [3, 9, 16] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the superdiffusive regime p > 3/4, Bercu  [5] proved that the limit of the position of the ERW is not Gaussian and Kubota and Takei [28]  showed that the fluctuation of the ERW around this limit is Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' To obtain those asymptotics,  various approaches have been followed : Baur and Bertoin [3] went with the connection to P´olya-  type urns while martingales were used by Bercu [5] and Coletti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' [16] and the construction of  random trees with Bernoulli percolation have been explicited by K¨ursten [29] and Businger [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Other quantities of interest regarding the ERW have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For example, Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [20] provided the Cramer moderate deviations associated with the ERW in dimension 1 and, more  recently, Hayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' [26] studied the rate of quadratic mean displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Bercu and Laulin [9] introduced the multidimensional ERW (MERW), where d ≥ 1, and estab-  lished the natural extensions of the results [5] in dimension d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then, they investigated the  center of mass of the MERW [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In both papers, they extensively used a martingale approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Bertenghi [10] made use of the connection to P´olya-type urns in order to establish functional  results for the MERW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Finally, the ERW with changing memory has also been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The ERW with linearly  reinforced memory has been studied by Baur [1] via the urn approach, and Laulin [31] using  martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Gut and Stadm¨uller [25] proposed an amnesic ERW where the elephant could stop  and only remember the first (and second) step it tooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' They also investigated the case where  the elephant only remembered a fixed or time-evolving portion of its past (recent or distant)  [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the recent work [32], Laulin introduced smooth amnesia to the memory of the ERW and  established the asymptotic behavior of this new process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The idea of our paper is to generalise the work [32] in dimension 1 to the dimension d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  In other words, we introduce smooth amnesia to the memory of the multidimensional elephant  random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  40  19  30  20  10  0  10  20  0  10  20  30  40  50  60400  300  200  100  0  0  50  100  150  200  250  35060  50  40  30  20  10  0  0  200  400  600  800MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  3  Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In Section 2, we introduce the basic setting of the elephant  random walk (Sn)n∈N placed under an amnesia reinforcement mechanism, which is controlled  by the memory sequence (βn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This type of multidimensional reinforced random walked is  named as the multidimensional amnesia-reinforced elephant random walk (MARW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Similar to  the ERW with the amnesia reinforcement, the MARW also admits a martingale structure, which  is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Unlike the usual ERW, the additional amnesia-reinforcement induces  two discrete-time martingales, instead of a single martingale, which are strongly correlated in a  nontrivial fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Such strong correlation of martingales will eventually pose some computational  difficulties when we analyze the limiting behavior of the MARW in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For instance, when  we compute the pointwise limit and the scaling limit of (Sn)n∈N in the diffusive regime, the two  strongly correlated martingales have to be dealt with separately, see [8, 31, 32] for the same  methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  As a courtesy to our readers, we give a preview of some of our main results, whose proofs will  be deferred to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2, and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the diffusive regime, we have the  almost sure convergence,  1  nSn → 0  as  n → ∞  P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Another logarithmic scaling to the MARW yields the quadratic strong law,  1  log n  n  �  k=1  SkST  k  k2  → C(p, (βn)n∈N) · 1  dId  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  where the constant C(p, (βn)n∈N) > 0 depends only on the parameter p and the control sequence  (βn)n∈N of the amnesia-reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Using square-root scaling factor, we observe that the  MARW also admits a scaling limit in the diffusive regime, or convergence in distribution, in the  Skorokhod space D(R+) of c`adl`ag functions, in the sense that  � 1  √nS⌊nt⌋, t ≥ 0  �  =⇒  �  Wt, t ≥ 0  �  where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with  covariance structure given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  It is also of interest to look at the barycenter process (Gn)n∈N of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Its definition as  well as its limiting behavior are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Similar to the discussion of the MARW,  we obtain its pointwise convergence, quadratic strong law, and its scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In particular,  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5 states that the barycenter process admits a scaling limit at the diffusive regime, or  convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that  � 1  √nG⌊nt⌋, t ≥ 0  �  =⇒  �  1  �  0  Wtr dr, t ≥ 0  �  where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3 with  its covariance structure defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  A natural question to ask is how fast the limiting Theorems in Section 4 are carried on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Section  6 provides a quantitative estimate on the mean square convergence velocity of the pointwise limit,  quadratic strong law, and the scaling limit of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It should be possible to derive similar  convergence velocity to the barycenter process, which is not computed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In Section  7, we end this work with a discussion on the Cram´er moderate deviations of the MARW in the  diffusive and critical regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' As a preview of our result in this Section, let (ϑn)n∈N ⊆ R be a  non-decreasing sequence so that ϑn/√n → 0 as n → ∞, and wn the sequence with asymptotic  4  JIAMING CHEN AND LUCILE LAULIN  behavior described in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Take any non-empty Borel set B ⊆ Rd, then we have  −  inf  x∈int B  1  2∥x∥2 ≤ lim inf  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≤ lim sup  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≤ − inf  x∈cl B  1  2∥x∥2,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This is the  Cram´er moderate deviations for the MARW in the diffusive and critical regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Moreover, we chose to postpone some technicalities regarding the analysis of the random walk  to the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' That way, the reader can focus on the main Theorems and the ideas of their  proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' However, some analogous technicalities are displayed in the proof of the Theorems on the  barycenter such that the reader can also have a complete overview of the work needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Other probabilistic aspects of interest to the MARW include the statistical inference and an  analysis on the Fisher information, see [7], as well as the Wasserstein distance of the reinforced  random walk, see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Perturbations of the amnesia intensity and its stability for the MARW is  also of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A similar topic for another type of stochastic process, the Schramn-  Loewner evolution, has been considered in [2, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The transience and recurrence property of the  MARW remains unknown, to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Readers are referred to [11, 20] for an  exposition on the ERW without the amnesia reinforcement mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The amnesia-reinforced elephant random walk  To begin with, let us properly introduce the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It is the natural extension to higher  dimensions of the one-dimensional MARW, defined in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For an arbitrarily given dimension  d ≥ 1, let (Sn)n∈N be a (reinforced) random walk on Zd starting from the origin at time n = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' At time n = 1, the reinforced random walk moves to one of the 2d nearest-neighbors  with equal probability 1/2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' After that, at time n ≥ 1, the reinforced random walk chooses at  random an integer 1 ≤ k ≤ n among the past times and performs the same step with probabily p,  or goes in any of the 2d − 1 other directions with probability (1 − p)/(2d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This random walk  possesses the amnesia property, in the sense that it remembers its most recent past steps better  than its remote past steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Colloquially, this random walk has higher probability to choose its  recent steps than its earlier steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  From a mathematical perspective, the position of this reinforced random walk at time n+1 ≥ 1  is given by  Sn+1 = Sn + Xn+1  with Xn+1 being defined as the step of this random walk at time n + 1, satisfying  Xn+1 = An+1Xβn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Here An+1 is a random d × d matrix given by  P(An = +Id) = p,  and, for all 1 ≤ k ≤ d − 1,  P(An = −Id) = P(An = +Jk  d ) = P(An = −Jk  d ) = 1 − p  2d − 1  where Id is the identity matrix of order d, Id = (δi,j)d and Jd = C(0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , 0) is the circulant  matrix of order d such that J = (δi+1,j)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It is easy to observe that the fixed permutation matrix  Jd satisfied Jd  d = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The distribution of the memory βn of the reinforced random walk is such  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  5  that the probability of choosing a fixed past time k ∈ N decays approximately with rate kβ/nβ+1,  where β ≥ 0 is the amnesia parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (a) n = 10  (b) n = 100  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Evolution of the distribution of the memory β depending on the value  of β and the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  To be precise, this random walk chooses βn+1 according to  P  �  βn+1 = k  �  = (β + 1)Γ(β + k)Γ(n)  Γ(k)Γ(β + n + 1)  = β + 1  n µk  µn+1  for all  1 ≤ k ≤ n,  where  µn =  n−1  �  k=1  �  1 + β  k  �  =  Γ(β + n)  Γ(n)Γ(β + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  (a) d = 1  (b) d = 2  (c) d = 3  (d) d = 10  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Competition between the dimension and the amnesia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Figure 3 aims to give a better understanding on how amnesia affects the MARW in various  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The horizontal axis corresponds to p (from 0 to 1) and the vertical axis corresponds to  β (from 0 to 10, arbitrary chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The diffusive regime, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' when p < 4dβ+2d+1  4d(β+1)  or a < 1−  1  2(β+1),  is in blue while the superdiffusive regime is in red, see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1 for the definition of the regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  One can observe that when the amnesia parameter β grows, the superdiffusive regime tends to be  less represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It should also be noted that when the dimension grows the superdiffusive regime  is more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence, the amnesia is somehow leading the MARW to a behavior closer to  the one in dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' When β vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' β = 0, the MARW reduces to the multidimensional  elephant random walk (MERW) introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The two random variables An and βn are constructed to be conditionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' At each  time n, define the σ-algebra Fn = σ(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then (Fn)n∈N is a discrete-time filtration to  which the MARW is clearly adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  β= 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5  β= 1  β= 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4  β= 5  β= 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  2  4  9  00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10  β= 0  β= 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='08  β= 2  β= 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='06  β= 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='04-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='00  0  20  40  60  80  10010  8 -  6  4  2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  p10  8 -  6  B  4  +0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  p10  8 -  6  B  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  p10  8 -  6  B  4  +0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='0  p6  JIAMING CHEN AND LUCILE LAULIN  Since An and βn are conditionally independent, we clearly have  E  �  Xn+1|Fn  �  = E  �  An  �  E  �  Xβn+1|Fn  �  = 2dp − 1  2d − 1 E  �  n  �  k=1  Xk1{βn+1=k}|Fn  �  = 2dp − 1  2d − 1 · β + 1  nµn+1  n  �  k=1  µkXk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2)  We further denote  a = 2dp − 1  2d − 1  and  Yn =  n  �  k=1  µkXk  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3)  such that  E  �  Yn+1|Fn  �  =  �  1 + a(β + 1)  n  �  Yn = γnYn  with γn = 1 + a(β + 1)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hereafter, for each n ≥ 1, let  an =  n−1  �  k=1  γ−1  k  = Γ(n)Γ(a(β + 1) + 1)  Γ(a(β + 1) + n)  and  wn =  n  �  k=1  (akµk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4)  From a Gamma function estimate, also see in [31], we have that  na(β+1)an → Γ(a(β + 1) + 1)  as  n → ∞  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5)  and  n−βµn → Γ(β + 1)−1  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A correlated martingale approach  Define the following two Rd-valued processes by  Mn = anYn  and  Nn = Sn +  a(β + 1)  β − a(β + 1)µ−1  n Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The Rd-valued processes (Mn)n∈N and (Nn)n∈N defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) are locally  square-integrable martingales adapted to (Fn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Since, both Mn and Nn are finite sums for each n ≥ 1, the square-integrability and adapt-  ness are immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4), we have  E  �  Mn+1|Fn  �  = anγ−1  n Yn + anµnγ−1  n E  �  Xn+1|Fn  �  = anYn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  And by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2), we have  E  �  Nn+1|Fn  �  = E  �  Sn+1 +  a(β + 1)  β − a(β + 1)µ−1  n+1Yn+1|Fn  �  = Sn +  a(β + 1)  β − a(β + 1)µ−1  n Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Notice that via introducing the martingales (Mn)n∈N and (Nn)n∈N, we can write Sn as  Sn = Nn −  a(β + 1)  β − a(β + 1)(anµn)−1Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2)  This writing is the key on which rely all of our analysis and our martingale approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Moreover, the asymptotic behavior of (Mn)n∈N is closely related to wn defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In fact,  we have the following asymptotic result, which states the three regimes of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the diffusive regime when p < 4dβ+2d+1  4d(β+1)  or a < 1 −  1  2(β+1), we have  wn  n1−2(a(β+1)−β) → l(β)  as  n → ∞  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3)  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  7  with  l(β) =  1  1 + 2(β − a(β + 1))  �Γ(a(β + 1) + 1)  Γ(β + 1)  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  In the critical regime when p = 4dβ+2d+1  4d(β+1)  or a = 1 −  1  2(β+1), we have  wn  log n →  �Γ(β + 1 + 1  2)  Γ(β + 1)  �2  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4)  In the superdiffusive regime when p > 4dβ+2d+1  4d(β+1)  or a > 1 −  1  2(β+1), we have  wn →  ∞  �  k=1  �Γ(a(β + 1) + 1)Γ(β + k)  Γ(a(β + 1) + k)Γ(β + 1)  �2  < ∞  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5)  In order to investigate the asymptotic behavior of (Sn)n∈N, we first introduce an arbitrarily  fixed test non-zero vector u ∈ Rd and we define  Mn(u) = uT Mn  and  Nn(u) = uT Nn  for each  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  It is then clear that (Mn(u))n∈N (Nn(u))n∈N are real-valued locally square-integrable martingales  for each fixed u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  We further infer that (Sn(u))n∈N satisfies an equation analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In  this setting, we have reduced the multidimensional martingales to real-valued martingales without  loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This technique greatly simplifies our martingale analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' From now on, we fix  the test vector u ∈ Rd and we introduce the two-dimensional martingale (Ln(u))n∈N defined as  Ln(u) =  �  Nn(u)  Mn(u)  �  for each  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6)  Denote the martingale increment ϵn+1 = Yn+1 − γnYn for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then (ϵn)n∈N satisfies the  martingale difference relation E[ϵn+1|Fn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We obtain that  ∆Ln+1(u) = Ln+1(u) − Ln(u) =  �  Sn+1(u) − Sn(u) +  a(β+1)  β−a(β+1)  �  µ−1  n+1Yn+1(u) − µ−1  n Yn(u)  �  an+1Yn+1(u) − anYn(u)  �  =  �  βµ−1  n+1  β−a(β+1)  �  µn+1Xn+1(u) − (γn − 1)Yn(u)  �  an+1ϵn+1(u)  �  =  �  βµ−1  n+1  β−a(β+1)  an+1  �  ϵn+1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Scaling limit and convergence  In this section, we discuss the scaling limit as well as the almost sure convergence in the  diffusive, critical and the superdiffusive regimes, depending on the value of p with respect to  (4dβ +2d+1)/(4d(β +1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We also give the quadratic strong law in the diffusive regime as well as  in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Afterwards, the mean square convergence is established in the superdiffusive  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the almost sure convergence  1  nSn → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  8  JIAMING CHEN AND LUCILE LAULIN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have from [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15] again that, for all γ > 0,  ∥Mn∥2  λmax⟨M⟩n  = o  ��  log Tr⟨M⟩n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  From equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9) and the fact that λmax⟨M⟩n ≤ Tr⟨M⟩n ≤ wn, we get  ∥Mn∥2 = o  �  wn  �  log wn  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2)  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3), we observe  ∥Mn∥2 = o  �  n1−2(a(β+1)−β)�  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since Mn = anYn, we have from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6)  ∥Yn∥2  (nµn+1)2 = o  �  n−1�  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  which implies  Yn  nµn+1  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10) and [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15] again, we find that  ∥Nn∥2 = o  �  n  �  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3)  Moreover, we obtain from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2)  1  n2  ����Sn +  a(β + 1)  (β − a(β + 1))µn+1  Yn  ����  2  = o  �  n−1�  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence, we conclude that  Sn  n +  a(β + 1)  β − a(β + 1) ·  Yn  nµn+1  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the quadratic strong law  1  log n  n  �  k=1  SkST  k  k2  →  2β + 1 − a  (1 − a)(1 − 2(a(β + 1) − β)) · 1  dId  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We will check that all the conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3] are satisfied, see also [14, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) is satisfied thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4 while the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) directly follows  from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5 and the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) is exactly the statement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Therefore,  1  log  �  det V −1  n  �2  n  �  k=1  �(det Vk)2 − (det Vk+1)2  (det Vk)2  �  VkLk(u)Lk(u)T V T  k → 1  duT uVt=1  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the one hand, we have from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='24) that  1  log n  n  �  k=1  �(det Vk)2 − (det Vk+1)2  (det Vk)2  �  VkLk(u)Lk(u)T V T  k → 2(1 − a)(β + 1)  d  uT uVt=1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='24), we have  n  �(det Vn)2 − (det Vn+1)2  (det Vn)2  �  → 2(1 − a)(β + 1)  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Finally, we obtain from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) that  1  log n  n  �  k=1  uT SkST  k u  k2  =  1  log n  n  �  k=1  vT VkLk(u)Lk(u)T V T  k v  k  → vT Vt=1v · 1  duT u  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5)  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  9  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The MARW admits a scaling limit at the diffusive regime, or convergence in  distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that  � 1  √nS⌊nt⌋, t ≥ 0  �  =⇒  �  Wt, t ≥ 0  �  where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process such that W0 = 0 and with  covariance  E  �  WsW T  t  �  =  a(β + 1)(1 − a) + aβ  (2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)s  � t  s  �a−β(1−a)  1  dId  +  β  (β(1 − a) − a)(1 − a)s · 1  dId  for all  0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We will check that all the conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2] are satisfied, see also [14, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) is satisfied thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4 while the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) directly follows  from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5 and the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) is exactly the statement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Consequently,  we have the convergence in distribution in the Skorokhod space D(R+) such that  �  VnL⌊nt⌋(u), t ≥ 0  �  =⇒  �  Wt(u), t ≥ 0  �  where (Wt(u))t≥0 is a continuous R2-valued centered Gaussian process such that W0 = 0 and with  covariance  E  �  Ws(u)Wt(u)T �  = 1  duT uVs  for all  0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2), we see that S⌊nt⌋(u) is asymptotically equivalent to  N⌊nt⌋(u) + tβ−a(β+1)  a(β + 1)  β − a(β + 1)(anµn)−1M⌊nt⌋(u)  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Multiplying on the left side by vt = (1, ta(β+1)−β)T , we obtain  � 1  √nS⌊nt⌋(u), t ≥ 0  �  =⇒  �  Wt(u), t ≥ 0  �  with Wt(u) = vT  t Wt(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hereafter, when 0 ≤ s ≤ t < ∞, we have the covariance  E  �  Ws(u)Wt(u)T �  = vT  s E  �  Ws(u)Wt(u)T �  vt = 1  d(uT u)vT  s Vsvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7)  Solving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7), we have  E  �  WsW T  t  �  = 1  dvT  s Vsvt  for all  0 ≤ s ≤ t < ∞  and the assertion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the almost sure convergence  1  √n log nSn → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We still have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) such that  ∥Mn∥2 = o  �  wn  �  log wn  �1+γ�  for all  γ > 0  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  However, in the critical regime, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) rather than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3), and  wn  log n →  �Γ(β + 1 + 1  2)  Γ(β + 1)  �2  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  10  JIAMING CHEN AND LUCILE LAULIN  Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), and since Mn = anYn, we observe for all γ > 0 that  ∥Yn∥2  n(log n)2µ2n  = o  �  (log n)−1�  log log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  In this regard  Yn  √n log nµn  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8)  Similarly, we still have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10) and  ∥Nn∥2 = o  �  n  �  log n  �1+γ�  for all  γ > 0  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then  ∥Nn∥2  n(log n)2 = o  �  (log n)γ−1�  for all  γ ∈ (0, 1)  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  and therefore  Nn  √n log n → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2), we can hereafter conclude that  Sn  √n log n +  a(β + 1)  β − a(β + 1) ·  Yn  √n log nµn  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Combining with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8), the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the quadratic strong law  1  log log n  n  �  k=1  SkST  k  (k log k)2 → (2β + 1)2 · 1  dId  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We will check that all the conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The condition  (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) is satisfied thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8 while the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) directly follows from Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9 and the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) is exactly the statement of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Therefore,  1  log  �  det W −1  n  �2  n  �  k=1  �(det Wk)2 − (det Wk+1)2  (det Wk)2  �  WkLk(u)Lk(u)T W T  k → 1  duT uW  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the one hand, we have from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='34)  1  log log n  n  �  k=1  �(det Wk)2 − (det Wk+1)2  (det Wk)2  �  WkLk(u)Lk(u)T W T  k → 1  duT uW  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='33), we have  n log n  �(det Wk)2 − (det Wk+1)2  (det Wk)2  �  → (2β + 1)2  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, we obtain from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9) that  1  log log n  n  �  k=1  uT SkST  k u  (k log k)2 =  1  log log n  n  �  k=1  wT WkLk(u)Lk(u)T W T  k w  k log k  → (2β + 1)2  d  uT u  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The MARW admits a scaling limit at the critical regime, or convergence in dis-  tribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that  �  1  �  nt log n  S⌊nt⌋, t ≥ 0  �  =⇒  �  (2β + 1)Bt, t ≥ 0  �  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  11  where (Bt)t≥0 is a continuous d-dimensional canonical Brownian motion with covariance  E  �  BsBT  t  �  = s · 1  dId  for all  0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We will check that all the three conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2] are satisfied, see also [42,  Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' First of all, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7) we know that  w−1/2  n  ⟨M(u)⟩⌊nt⌋w−1/2  n  → t  d · uT u  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11)  Hence the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Notice that  ⌊nt⌋  �  k=1  1  wn  E  �  ∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1  �  ≤  ⌊nt⌋  �  k=1  �w⌊nt⌋  wn  �2  1  ϵ2w2  ⌊nt⌋  E  �  ∆Mk(u)4|Fk−1  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12)  since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21), we observe that  ⌊nt⌋  �  k=1  ��∆Mk(u)4�� ≤ C1(β)∥u∥4  ⌊nt⌋  �  k=1  (akµk)4 ≤ C2(β)∥u∥4  ⌊nt⌋  �  k=1  1  k2  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13)  with constants C1(β), C2(β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Therefore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13), we have  ⌊nt⌋  �  k=1  1  wn  E  �  ∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1  �  ≤ C3(β)∥u∥4 · t2  ϵ2 ·  1  nt(log nt)2  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Simplifying the above expression, we obtain  ⌊nt⌋  �  k=1  1  wn  E  �  ∆Mk(u)21{|∆Mk(u)|≥ϵ√wk}|Fk−1  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14)  Then the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2), or the Lindeberg condition, is satisfied by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In this particular case  at critical regime, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11) implies that the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence  �  1  √wn  M⌊nt⌋(u), t ≥ 0  �  =⇒  �  Bt(u), t ≥ 0  �  where (Bt(u))t≥0 is a continuous real-valued centered Gaussian process such that B0(u) = 0 and  with covariance  E  �  Bs(u)Bt(u)  �  = s  d · uT u  for all  0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  In the critical regime, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) we can write  S⌊nt⌋(u) = N⌊nt⌋(u) + (2β + 1) M⌊nt⌋(u)  a⌊nt⌋µ⌊nt⌋  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15)  From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8) we know that  ⟨N(u)⟩⌊nt⌋  nt log n  → 0  and  N⌊nt⌋(u)  �  nt log n  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16)  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) again, we conclude that  �  1  �  nt log n  S⌊nt⌋(u), t ≥ 0  �  =⇒  �  (2β + 1)Bt(u), t ≥ 0  �  with  E  �  Bs(u)Bt(u)  �  = s · uT u  d  for all  0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17)  Solving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17), we get  E  �  BsBT  t  �  = s · 1  dId  for all  0 ≤ s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  12  JIAMING CHEN AND LUCILE LAULIN  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the almost sure convergence  1  na(β+1)−β Sn → Lβ  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  where the limiting Lβ is an Rd-valued random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In fact, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8 below, we will see the random vector Lβ is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7), in the superdiffusive regime, we have  Tr⟨M⟩n ≤ wn ≤  ∞  �  k=1  �Γ(a(β + 1) + 1)Γ(β + k)  Γ(a(β + 1) + k)Γ(β + 1)  �2  < ∞  for all  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15], this leads to  Mn → M  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  with  M =  ∞  �  k=1  akϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1), Mn = anYn, and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), we observe that  Yn  na(β+1) →  1  Γ(a(β + 1) + 1)M  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18)  Moreover, equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) still holds and, as 2a(β + 1) > 2β + 1 in the superdiffusive regime, we  find that  1  n2(a(β+1)−β)  ����Sn +  a(β + 1)  (β − a(β + 1))µn+1  Yn  ����  2  = o  �  n−(1−2a(β+1)+2β)�  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), we obtain  Sn  na(β+1)−β +  a(β + 1)  β − a(β + 1) · Γ(β + 1)Yn  na(β+1)  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19)  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18), it yields  Sn  na(β+1)−β → Lβ  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  where  Lβ =  a(β + 1)  a(β + 1) − β ·  Γ(β + 1)  Γ(a(β + 1) + 1)M  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20)  and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the following mean square convergence  E  �����  1  na(β+1)−β Sn − Lβ  ����  2�  → 0  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For each test vector u ∈ Rd, we have  E  �  Mn(u)2�  = E  �  ⟨M(u)⟩n  �  ≤ 1  dwnuT u  for all  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), we obtain  sup  n≥1  E  �  Mn(u)2�  < ∞  which implies that (Mn(u))n∈N is a martingale bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Therefore  E  �  |Mn(u) − M(u)|2�  → 0  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22)  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  13  Moreover, on the one hand (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22) together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18) implies that  E  �����  1  na(β+1) Yn(u) − Y (u)  ����  2�  → 0  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='23)  On the other hand, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8) we know that  E  �  Nn(u)2�  = E  �  ⟨N(u)⟩n  �  ≤ 1  d  �  β  β − a(β + 1)  �2  nuT u  for all  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since a(β + 1) > β + 1  2 in the superdiffusive regime, we have  E  �����  1  na(β+1)−β Nn(u)  ����  2�  → 0  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='24)  The proof is complete by combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The expected value of Lβ is  E  �  Lβ  �  = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='25)  whereas its quadratic deviation is  E  �  LβLT  β  �  =  �  a(β + 1)  β − a(β + 1)  �2 Γ(β + 1)2Γ(2(a − 1)(β + 1) + 1)  Γ((2a − 1)(β + 1) + 1)2  1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='26)  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The MARW admits a scaling limit at the superdiffusive regime, or convergence in  distribution, in the Skorokhod space D(R+) of c`adl`ag functions, in the sense that  �  1  na(β+1)−β S⌊nt⌋, t ≥ 0  �  =⇒  �  Qt, t ≥ 0  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='27)  with the limiting Qt = ta(β+1)−βLβ for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all t ≥ 0 and from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19), we observe that  S⌊nt⌋  ⌊nt⌋a(β+1)−β +  a(β + 1)  β − a(β + 1) ·  Y⌊nt⌋  ⌊nt⌋a(β+1) → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  which implies  1  na(β+1)−β Sn → ta(β+1)−βLβ  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='28)  The P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='28)holds in all finite-dimensional distributions which characterizes  the Skorokhod space topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='27) and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Scaling limit of the barycenter process  The study of the scaling limit of the MARW (Sn)n∈N gives us some information on its asymptotic  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Nonetheless, to understand its pathwise geometric features, we need to discuss its  barycenter, or center of mass process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Such topics have been raised and discussed in [36, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In  this Section, we turn our attention to the above-mentioned barycenter process (Gn)n∈N defined  by  Gn := 1  n  n  �  k=1  Sk  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  Our work contains the discussion on the scaling limit and the almost sure convergence in the  diffusive, critical and superdiffusive regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The quadratic strong law in the diffusive and crit-  ical regimes is also discussed while the mean square convergence in the superdiffusive regime is  established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  14  JIAMING CHEN AND LUCILE LAULIN  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Almost sure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The barycenter process was discussed in [8] for the elephant  random walk in dimension d, which is a special case of the process we study here when β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We  first begin with the almost sure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the almost sure convergence, in the diffusive regime,  1  nGn → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2)  while in the critical regime,  1  √n log nGn → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3)  and, in the superdiffusive regime,  1  na(β+1)−β Gn →  1  1 + a(β + 1) − β Lβ  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4)  where Lβ was characterized in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the diffusive regime, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) we observe that  1  nGn =  n  �  k=1  k  n2 · 1  k Sk =  n  �  k=1  1  k Ska′  n,k  with  a′  n,k = k  n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since �n  k=1 a′  n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1, from Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12 we can conclude that  1  nGn =  n  �  k=1  1  k Ska′  n,k → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the critical regime, we have from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) that  1  √n log nGn =  1  n3/2 log n  n  �  k=1  Sk =  n  �  k=1  1  √  k log k  Ska′′  n,k  with  a′′  n,k = k1/2 log k  n3/2 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since �n  k=1 a′′  n,k ≤ 1 for all n ∈ N and the almost sure convergence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4 holds, we get  from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12 hat  1  √n log nGn =  n  �  k=1  1  √  k log k  Ska′′  n,k → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  and we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Finally, in the superdiffusive regime, we also get from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) that  1  na(β+1)−β Gn =  1  n1+a(β+1)−β  n  �  k=1  Sn =  n  �  k=1  1  ka(β+1)−β Ska′′′  n,k  with  a′′′  n,k =  ka(β+1)−β  n1+a(β+1)−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since  n  �  k=1  a′′′  n,k →  1  1 + a(β + 1) − β  as  n → ∞  by a simple calculation, and because of the almost sure convergence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7, we can  conclude using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13  1  na(β+1)−β Gn →  1  1 + a(β + 1) − β Lβ  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Quadratic strong law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  15  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the diffusive regime, we have the quadratic strong law  1  log n  n  �  k=1  GkGT  k  k2  → 4I(a, β) · 1  dId  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  where I(a, β) is given explicitly  I(a, β) =  1  Γ(a(β + 1) + 1)2Γ(β + 1)2 ·  2a2(1 − a)(β + 1)3  3(β − a(β + 1))2(1 − a(β + 1) + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We will check that all the three conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Looking back  to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1), we observe that  Gn = 1  n  n  �  k=1  Nk − 1  n  a(β + 1)  β − a(β + 1)  n  �  k=1  1  akµk  Mk = 1  n  n  �  k=1  Nk − 1  n  a(β + 1)  β − a(β + 1)  n  �  k=1  1  akµk  k  �  l=1  alϵl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, changing the order of summation, we have  Gn = 1  n  n  �  k=1  Nk − 1  n  a(β + 1)  β − a(β + 1)  n  �  k=1  akϵk  n  �  l=k  1  alϵl  = 1  n  n  �  k=1  Nk − 1  n  a(β + 1)  β − a(β + 1)  n  �  k=1  ak(δn − δk−1)ϵk  where we define δn = �n  k=1(akµk)−1 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Moreover, we denote  Zn =  n  �  k=1  Nk −  a(β + 1)  β − a(β + 1)  n  �  k=1  akδk−1ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  such that we have  Gn = 1  nZn − δn  n ·  a(β + 1)  β − a(β + 1)  n  �  k=1  akϵk = 1  n  �  Zn −  a(β + 1)  β − a(β + 1)δnMn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  For a fixed text vector u ∈ Rd, we define  Hn(u) =  �  Zn(u)  Mn(u)  �  for all  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5)  which implies  ∆Hn(u) = Hn+1(u) − Hn(u) =  �  Nn+1(u)ϵn+1(u)−1 −  a(β+1)  β−a(β+1)an+1δn  an+1  �  ϵn+1(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, let  Vn =  1  n3/2  �  1  0  0  a(β+1)  β−a(β+1)δn  �  and  v =  �  1  −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then it is immediate that  vT VnHn(u) =  1  √nGn  for all  n ∈ N  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6)  and that  lim  n→∞ Vn⟨H(u)⟩nV T  n = lim  n→∞  1  n3  �  1  −1  −1  1  � n−1  �  k=1  �  a(β + 1)  β − a(β + 1)  �2  δ2  ka2  k+1E  �  ϵk+1(u)2|Fk  �  = lim  n→∞  1  n3 ·  a2(1 − a)(β + 1)3uT u  d(β − a(β + 1))2(1 − a(β + 1) + β)  �  1  −1  −1  1  � n−1  �  k=1  δ2  ka2  k+1µ2  k+1  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  16  JIAMING CHEN AND LUCILE LAULIN  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), we know that  n−(1+a(β+1)−β)δn →  1  1 + a(β + 1) − β ·  1  Γ(a(β + 1) + 1)Γ(β + 1)  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence the above calculation leads us to  Vn⟨H(u)⟩nV T  n → I(a, β)uT u · 1  d  �  1  −1  −1  1  �  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7)  where  I(a, β) =  1  1 − 2(a(β + 1) − β) ·  a2(1 − a)(β + 1)3  (β − a(β + 1))2(1 − a(β + 1) + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8)  Consequently, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7) ensures that the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Notice that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1),  there exists some constant C1(a, β) > 0 and similarly, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22), there exists some  other constant C2(a, β) > 0 such that  ∥Nn∥2 ≤ C1(a, β)n2  and  a2  kϵk(u)2 ≤ C2(a, β)n2δ−2  n  for all  1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Moreover, notice that for all 1 ≤ k ≤ n,  Vn∆Hk(u) =  1  n3/2  �  Nk(u)ϵk(u)−1 −  a(β+1)  β−a(β+1)akδk−1  a(β+1)  β−a(β+1)akδn  �  ϵk(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence, for all 1 ≤ k ≤ n, we observe that  ∥Vn∆Hk(u)∥2 ≤ 4a2  k  n3  �  a(β + 1)  β − a(β + 1)  �2��β − a(β + 1)  aka(β + 1)  Nk(u)  ϵk(u)  �2  + δ2  k−1 + δ2  n  �  ϵk(u)2 ≤ C(a, β)  n  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9)  for some constant C(a, β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Consequently, we  n  �  k=1  E  �  ∥Vn∆Hk(u)∥4�  ≤ 1  nC(a, β) → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  since, for all ϵ > 0,  n  �  k=1  E  �  ∥Vn∆Hk(u)∥21{∥Vn∆Hk(u)∥>ϵ}|Fk−1  �  ≤ 1  ϵ2  n  �  k=1  E  �  ∥Vn∆Hk(u)∥4�  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10)  Then the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2), or the Lindeberg condition, is satisfied by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hereafter, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), and by the definition of δn, we know there exists some constant C′(a, β) ̸= 0 such that  log  �  det V −1  n  �2  log n  → C′(a, β)  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  This ensures that there exists some other constant C′′(a, β) > 0 such that  ∞  �  n=1  1  �  log  �  det V −1  n  �2�2 E  �  ∥Vn∆Hn(u)∥4|Fn−1  �  ≤ C2(a, β)  ∞  �  n=1  1  (log n)2 E  �  ∥Vn∆Hn(u)∥4|Fn−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Finally, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9) leads to  ∞  �  n=1  1  (log n)2 ∥Vn∆Hn(u)∥4 ≤ C(a, β)  ∞  �  n=1  1  (n log n)2 < ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  17  for some constant C(a, β) > 0 depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) is satisfied by  combining the above with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the one hand,  1  log  �  det V −1  n  �2  n  �  k=1  �(det Vk)2 − (det Vk+1)2  (det Vk)2  �  VkHk(u)Hk(u)T V T  k → 1  duT uV  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' where  V =  �  1  −1  −1  1  �  I(a, β)  and I(a, β) has been specified in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then, we have  1  log n  n  �  k=1  �(det Vk)2 − (det Vk+1)2  (det Vk)2  �  VkHk(u)Hk(u)T V T  k → 4 − 2(a(β + 1) − β)  d  uT uV  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' since  log n  log  �  det V −1  n  �2 → 4 − 2(a(β + 1) − β)  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  On the other hand, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), we have  n  �(det Vn)2 − (det Vn+1)2  (det Vn)2  �  → 4 − 2  �  a(β + 1) − β  �  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11), we observe that  1  log n  n  �  k=1  uT GkGT  k u  k2  =  1  log n  n  �  k=1  vT VkHk(u)Hk(u)T V T  k v  k  → vT V v · 1  duT u  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the critical regime, we have the quadratic strong law  1  log log n  n  �  k=1  GkGT  k  (k log k)2 → 4(2β + 1)2  9  1  dId  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We will check that all the three conditions of [32, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2] are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Denote  Wn =  1  n√n log n  �  1  0  0  a(β+1)  β−a(β+1)δn  �  and  w =  �  1  −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, for H defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), it is clear that  wT WnHn(u) =  1  √n log nGn  for all  n ∈ N  and that  lim  n→∞ Wn⟨H(u)⟩nW T  n = lim  n→∞  1  n3 log n  �  1  −1  −1  1  � n−1  �  k=1  (2β + 1)2δ2  ka2  k+1E  �  ϵk+1(u)2|Fk  �  = lim  n→∞  (2β + 1)2  n3 log n · uT u  d  �  1  −1  −1  1  � n−1  �  k=1  δ2  ka2  k+1µ2  k+1  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), we know that  n−3/2δn → 2  3 ·  Γ(β + 1)  Γ(β + 1 + 1  2)  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  18  JIAMING CHEN AND LUCILE LAULIN  Hence, the above calculation leads us to  Wn⟨H(u)⟩nW T  n → I(β)uT u · 1  d  �  1  −1  −1  1  �  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  with  I(β) = 4(2β + 1)2  9  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12)  Consequently, the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) is satisfied thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Notice that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1),  there exists some constant C1(β) > 0 and similarly, there exists some constant C2(β) > 0 such  that  ∥Nn∥2 ≤ C1(β)n2  and  a2  kϵk(u)2 ≤ C2(β)n2δ−2  n log n  for all  1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, notice for all 1 ≤ k ≤ n that  Wn∆Hk(u) =  1  n√n log n  �  Nk(u)ϵk(u)−1 −  a(β+1)  β−a(β+1)akδk−1  a(β+1)  β−a(β+1)akδn  �  ϵk(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The ensures that, for all 1 ≤ k ≤ n,  ∥Wn∆Hk(u)∥2 ≤  4a2  k  n3 log n(2β + 1)2  ��  (2β + 1)−2 Nk(u)  ϵk(u)  �2 + δ2  k−1 + δ2  n  �  ϵk(u)2 ≤ C(β)  n  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13)  for some constant C(β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence,  n  �  k=1  E  �  ∥Wn∆Hk(u)∥4�  ≤ 1  nC(β) → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  since, for all ϵ > 0,  n  �  k=1  E  �  ∥Wn∆Hk(u)∥21{∥Wn∆Hk(u)∥>ϵ}|Fk−1  �  ≤ 1  ϵ2  n  �  k=1  E  �  ∥Wn∆Hk(u)∥4�  → 0  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14)  Therefore, the condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2), or the Lindeberg condition, is satisfied using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hereafter, we  know that  log  �  det W −1  n  �2  log log n  → 4  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  This ensures that there exists some constant C2(β) > 0 such that  ∞  �  n=1  1  �  log  �  det W −1  n  �2�2 E  �  ∥Wn∆Hn(u)∥4|Fn−1  �  ≤  ∞  �  n=1  C2(β)  (log log n)2 E  �  ∥Wn∆Hn(u)∥4|Fn−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15)  We get from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13) that  ∞  �  n=1  1  (log log n)2 ∥Wn∆Hn(u)∥4 ≤ C(β)  ∞  �  n=1  1  (n log n log log n)2 < ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  for some constant C(β) > 0 depending only onβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The condition (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) is satisfied using the above  together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then,  1  log  �  det W −1  n  �2  n  �  k=1  �(det Wk)2 − (det Wk+1)2  (det Wk)2  �  WkHk(u)Hk(u)T W T  k → 1  duT uW  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' where  W = 4(2β + 1)2  9  �  1  −1  −1  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  19  Furthermore, on the one hand we have  1  log log n  n  �  k=1  �(det Wk)2 − (det Wk+1)2  (det Wk)2  �  WkHk(u)Hk(u)T W T  k → 1  duT uW  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' since  log log n  log  �  det W −1  n  �2 → 1  4  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  On the other hand, we have  n log n  �(det Wn)2 − (det Wn+1)2  (det Wn)2  �  → 1  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11), we observe that  1  log log n  n  �  k=1  uT GkGT  k u  (k log k)2 =  1  log log n  n  �  k=1  wT WkHk(u)Hk(u)T W T  k w  4k log k  → wT Ww · 1  4duT u  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Since u ∈ Rd is arbitrary, the assertion follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the superdiffusive regime, we have the mean square convergence, given by  E  �����  1  na(β+1)−β Gn −  1  1 + a(β + 1) − β Lβ  ����  2�  → 0  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all test vector u ∈ Rd, it is immediate that  E  �����  1  na(β+1)−β Gn(u) −  1  1 + a(β + 1) − β Lβ(u)  ����  2�  ≤ 2E  �����  1  n1+a(β+1)−β Zn(u)  ����  2�  + 2E  �����  1  n1+a(β+1)−β ·  a(β + 1)  a(β + 1) − β δnMn −  1  1 + a(β + 1) − β Lβ  ����  2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18)  By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7), the second term converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Looking back to the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18),  we observe  E  �����  1  n1+a(β+1)−β Zn(u)  ����  2�  ≤  4  n1+2(a(β+1)−β)  n  �  k=1  E  �  Nk(u)2�  +  4  n1+2(a(β+1)−β)  �  a(β + 1)  a(β + 1) − β  �2  E  ������  n  �  k=1  akδk−1ϵk(u)  �����  2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19)  The first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19) converges to zero because E[Nk(u)] ≤ (uT u)n for all 1 ≤ k ≤ n, and  moreover, in the superdiffusive regime we have a(β +1) > β +1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19) also  converges to zero because  n−(1+a(β+1)−β)δn →  1  1 + a(β + 1) − β ·  1  Γ(1 + a(β + 1))Γ(β + 1)  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Finally, using the above and that M(u) = �∞  k=1 akϵk(u) is bounded in L2, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Scaling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The barycenter process admits a scaling limit at the diffusive regime, or conver-  gence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that  � 1  √nG⌊nt⌋, t ≥ 0  �  =⇒  �  1  �  0  Wtr dr, t ≥ 0  �  20  JIAMING CHEN AND LUCILE LAULIN  where (Wt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3 with its  covariance defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In particular,  E  ��  1  �  0  Wsv dv  ��  1  �  0  Wtu du  �T �  =  β  3(β(1 − a) − a)(1 − a)s · 1  dId  +  2(a(β + 1)(1 − a) + aβ)  3(2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a)(1 + (1 − a)(β + 1))ta−β(1−a)s1−a+β(1−a) · 1  dId  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20)  for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' An easy calculation leads to  lim  n→∞  1  √nG⌊nt⌋ = lim  n→∞  1  �  0  1  √nS⌊ntr⌋ dr =⇒  1  �  0  Wtr dr  which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then, the last convergence  in law is due to the functional central limit Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3, with (Wt)t≥0 defined there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence, the  barycenter process (Gn)n∈N admits a Gaussian scaling limit in the diffusive regime as well, with  covariance  E  ��  1  �  0  Wsv dv  ��  1  �  0  Wtu du  �T �  = 2  1  �  0  u  �  0  E  �  WsvW T  tu  �  dv du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20) and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The barycenter process admits a scaling limit at the critical regime, or convergence  in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that  �  1  �  nt log n  G⌊nt⌋, t ≥ 0  �  =⇒  �  1  �  0  (2β + 1)Btr dr, t ≥ 0  �  where (Bt)t≥0 is a continuous Rd-valued centered Gaussian process define in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6 with its  covariance defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For each r ∈ [0, 1], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11) implies that  lim  n→∞  1  �  nt log n  M⌊ntr⌋(u)  a⌊ntr⌋µ⌊ntr⌋  = lim  n→∞  1  �  nt log n  �  ntr(log n + r  t log r)  �1/2Btr(u)  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  for all u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Moreover, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16) yields  lim  n→∞  1  �  nt log n  N⌊ntr⌋(u) = lim  n→∞ r1/2 ·  1  �  ntr log n  N⌊ntr⌋(u) = 0  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  for all u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15), we have  �  1  �  nt log n  S⌊ntr⌋(u), t ≥ 0  �  =⇒  �  (2β + 1)Btr(u), t ≥ 0  �  for all u ∈ Rd and r ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence, we use again  lim  n→∞  1  �  nt log n  G⌊nt⌋ = lim  n→∞  1  �  0  1  �  nt log n  S⌊ntr⌋ dr =⇒  1  �  0  (2β + 1)Btr dr  and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  21  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The barycenter process admits a scaling limit at the superdiffusive regime, or  convergence in distribution, in the Skorokhod space D([0, 1]) of c`adl`ag functions, such that  �  1  na(β+1)−β G⌊nt⌋, t ≥ 0  �  =⇒  �  1  �  0  Qtr dr, t ≥ 0  �  with the covariance specified in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and the limiting Lβ characterized in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8 and Qt =  ta(β+1)−βLβ characterized in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Again, we find that  lim  n→∞  1  na(β+1)−β G⌊nt⌋ =  1  �  0  1  na(β+1)−β S⌊ntr⌋ dr =⇒  1  �  0  Qtr dr  which ensures that G⌊nt⌋ is a continuous function of S⌊ntr⌋ in D([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then, the last convergence  in law is due to the functional central limit Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence the barycenter process (Gn)n∈N  admits a non-degenerate scaling limit in the superdiffusive regime as well, with covariance  E  ��  1  �  0  Qsv dv  ��  1  �  0  Qtu du  �T �  = 2  1  �  0  u  �  0  E  �  QsvQT  tu  �  dv du = ta(β+1)−βsa(β+1)−β  (1 + a(β + 1) − β)2 E  �  LβLT  β  �  = ta(β+1)−βsa(β+1)−β  (1 + a(β + 1) − β)2  �  a(β + 1)  β − a(β + 1)  �2 Γ(2(a − 1)(β + 1) + 1)  Γ((2a − 1)(β + 1) + 1)2 · 1  dId  for all 0 ≤ s ≤ t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Velocity of quadratic mean displacement  In this Section, we investigate the velocity of the mean square displacement of the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  This quantitative estimates give us the information on how fast the limit Theorems in Section 4  are carried on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Similar convergence velocities have been discussed in [20, 26], where the authors  analyzed the convergence velocity of the moments of a one-dimensional elephant random walk of  all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the superdiffusive regime, the convergence velocity was discussed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Here, only  the rate of quadratic moment convergence for the MARW in all of the three (diffusive, critical,  and superdiffusive) regimes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Following the limit Theorems in Section 4, we expect the asymptotic behavior of the mean  square displacement is as follows,  E  �  SnST  n  �  ∼  �  �  �  �  �  �  �  �  �  �  �  n ·  (a−2β)(1−a)(β+1)+β(a+1)  (2(β+1)(1−a)−1)(a−β(1−a))(1−a) · 1  dId  when  a < 1 −  1  2(β+1)  n log n · (2β + 1)2 · 1  dId  when  a = 1 −  1  2(β+1)  n2(a(β+1)−β) ·  �  a(β+1)  β−a(β+1)  �2  Γ(2(a−1)(β+1)+1)  Γ((2a−1)(β+1)+1)2 · 1  dId  when  a > 1 −  1  2(β+1),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  where the notation ∼ indicates two sequences an ∼ bn if and only if an/bn → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The aim of this Section is not only to show that the above estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) are valid, but also  to investigate the exact velocity of their convergence in the diffusive and critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p < (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,  1  nE  �  SnST  n  �  −  (a − 2β)(1 − a)(β + 1) + β(a + 1)  (2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1  dId  ∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  22  JIAMING CHEN AND LUCILE LAULIN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Take the vector v = (1, −1)T and Vn ∈ R2×2 as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then,  1  √nSn(u) = vT VnLn(u),  where Ln(u) = (Nn(u), Mn(u))T is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In particular,  1  nuT E  �  SnST  n  �  u = vT VnE  �  Ln(u)Ln(u)T �  V T  n v  = vT VnE  � �  E  �  Nn(u)2�  E  �  Nn(u)Mn(u)  �  E  �  Mn(u)Nn(u)  �  E  �  Mn(u)2�  � �  V T  n v  = vT VnE  � �  E  �  ⟨N(u)⟩n  �  E  �  ⟨N(u), M(u)⟩n  �  E  �  ⟨M(u), N(u)⟩n  �  E  �  ⟨M(u)⟩n  �  � �  V T  n v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Therefore,  1  nuT E  �  SnST  n  �  u = 1  nE  �  ⟨N(u)⟩n  �  +  1  na2nµ2n  �  a(β + 1)  β − a(β + 1)  �2  E  �  ⟨M(u)⟩n  �  −  2  nanµn  �  a(β + 1)  β − a(β + 1)  �  E  �  ⟨M(u), N(u)⟩n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since the test vector u ∈ Rd is taken arbitrarily, we get from Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16 that  1  nE  �  SnST  n  �  −  (a − 2β)(1 − a)(β + 1) + β(a + 1)  (2(β + 1)(1 − a) − 1)(a − β(1 − a))(1 − a) · 1  dId  ∼ −(C1n−2(1−a)(β+1) + C2n−1) · 1  dId  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' When p = (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,  1  n log nE  �  SnST  n  �  − (2β + 1)2 · 1  dId ∼ −(C1(log n)−1 + C2n−1) · 1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Take w = (1, −1)T and Wn ∈ R2×2 as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then  1  √n log nSn(u) = wT WnLn(u) as in  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='29) for all u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In particular,  1  n log nuT E  �  SnST  n  �  u = wT WnE  �  Ln(u)Ln(u)T �  W T  n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence,  1  n log nuT E  �  SnST  n  �  u = wT WnE  � �  E  �  ⟨N(u)⟩n  �  E  �  ⟨N(u), M(u)⟩n  �  E  �  ⟨M(u), N(u)⟩n  �  E  �  ⟨M(u)⟩n  �  � �  W T  n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Therefore, we get by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) as n → ∞,  1  n log nuT E  �  SnST  n  �  u =  1  n log n  �  E  �  ⟨N(u)⟩n  �  + (2β + 1)2  a2nµ2n  E  �  ⟨M(u)⟩n  ��  ,  which implies  1  n log nE  �  SnST  n  �  − (2β + 1)2 · 1  dId ∼ −(C1(log n)−1 + C2n−1) · 1  dId  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Superdiffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  23  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' When p > (4dβ + 2d + 1)/4d(β + 1), we have, as n → ∞,  1  n2(a(β+1)−β) E  �  SnST  n  �  −  �  a(β + 1)  β − a(β + 1)  �2 Γ(2(a − 1)(β + 1) + 1)  Γ((2a − 1)(β + 1) + 1)2 · 1  dId  ∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Similar to previous computations for the diffusive regime, we have for all u ∈ Rd,  1  n2(a(β+1)−β) uT E  �  SnST  n  �  u =  1  n2(a(β+1)−β) E  �  ⟨N(u)⟩n  �  +  1  n2(a(β+1)−β)a2nµ2n  �  a(β + 1)  β − a(β + 1)  �2  E  �  ⟨M(u)⟩n  �  −  2  n2(a(β+1)−β)anµn  �  a(β + 1)  β − a(β + 1)  �  E  �  ⟨M(u), N(u)⟩n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and since u ∈ Rd is arbitrary,  1  n2(a(β+1)−β) E  �  SnST  n  �  −  �  a(β + 1)  β − a(β + 1)  �2 Γ(2(a − 1)(β + 1) + 1)  Γ((2a − 1)(β + 1) + 1)2 · 1  dId  ∼ −(C1n−4(a(β+1)−β)+1 + C2n−2(a(β+1)−β))  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Cram´er moderate deviations  In this Section, we discuss the Cram´er moderate deviations for the multidimensional reinforced  random walk (Sn)n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The similar statistical quantity as well as the Berry-Esseen bound for the  one-dimensional elephant random walk (ERW) without amnesia-reinforcement has been given in  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Our derivation of Cram´er moderate deviations for the MARW does not rely on a Berry-  Esseen bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The discussion of such statistical quantities is expected to reveal the transience  property and the central limit Theorems for the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For this direction, readers are refereed  to [3, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Thanks to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22, we can properly state the Cram´er moderate  deviations principles for the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the diffusive and critical regimes, we have the following Cram´er moderate  deviations for the MARW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Let (ϑn)n∈N ⊆ R be a non-decreasing sequence so that ϑn/√n → 0 as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Take any non-empty Borel set B ⊆ Rd, then we have  −  inf  x∈int B  1  2∥x∥2 ≤ lim inf  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≤ lim sup  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≤ − inf  x∈cl B  1  2∥x∥2,  where int B and cl B denote the interior and the closure of B ⊆ Rd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Our proof will only present the Cram´er moderate deviations for the MARW in the diffusive  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The same property for the critical regime follows from exactly the same steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' First, take  xB = infx∈B ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then it is obvious that infx∈cl B ∥x∥ ≤ xB and infx∈cl B ∥x∥2/2 ≤ x2  B/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Henceforth,  P  �anµnSn  ϑn√wn  ∈ B  �  ≤  d  �  j=1  P  �����  anµnSj  n  √wn  ���� ≥ ϑnxB  d  �  ≤  �  1 − Φ(ϑnxB)  �  F(B, ϑ, n),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  24  JIAMING CHEN AND LUCILE LAULIN  where we write  F(B, ϑ, n) := 2Cd · exp  �  1  √n  � ϑnxB  2d  �3 + 1  n  � ϑnxB  2d  �2 +  1  √n(1 + 1  2 log n)(1 + ϑnxB  2d )  �  + 2Cd · exp  �  1  √n  � ϑnxB  2d  �3 +  1  n2(1−a)(β+1)  � ϑnxB  2d  �2 +  1  √n(1 + 1  2 log n)(n1/2−(1−a)(β+1) + ϑnxB  2d )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence,  lim sup  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≤ −1  2x2  B ≤ − inf  x∈cl B  1  2∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  To achieve the asymptotic lower bound, we first notice that this assertion automatically holds if  int B = ∅, whence − infx∈∅ ∥x∥2/2 = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Consequently, we assume that int B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Notice that  int B is open in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence, for all ϵ∗ > 0 sufficiently small, we could find x∗ ∈ int B with  0 < 1  2∥x∗∥2 <  inf  x∈int B  1  2∥x∥2 + ϵ∗  and  0 < min  ���xj  ∗  �� : 1 ≤ j ≤ d  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Choose ϵ∗∗ sufficient small such that 0 < ϵ∗∗ <  ���xj  ∗  ��� for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then,  U(x∗, ϵ∗∗) ⊆ int B ⊆ B,  where  U(x∗, ϵ∗∗) :=  �  x ∈ Rd :  ��xj − xj  ∗  �� < ϵ∗∗ for all j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  On the other hand,  P  �anµnSn  ϑn√wn  ∈ B  �  ≥ P  �anµnSn  √wn  ∈ ϑn · U(x∗, ϵ∗∗)  �  ≥  d  �  j=1  P  �  ϑn(xj  ∗ + ϵ∗∗) ≥ anµnSj  n  √wn  ≥ ϑn(xj  ∗ − ϵ∗∗)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22, we know that  lim  n→∞ P  �anµnSj  n  √wn  ≥ ϑn(xj  ∗ + ϵ∗∗)  ��  P  �anµnSj  n  √wn  ≥ ϑn(xj  ∗ − ϵ∗∗)  �  = 0  for each  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Similar to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1),  lim inf  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≥ −1  2∥x∗ − ϵ∗∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Letting ϵ∗∗ → 0, we observe that  lim inf  n→∞ ϑ−2  n log P  �anµnSn  ϑn√wn  ∈ B  �  ≥ −1  2∥x∗∥2 ≥ −  inf  x∈int B  1  2∥x∥2 − ϵ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since ϵ∗ > 0 was take arbitrarily, letting ϵ∗ → 0, we verify the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Technical Lemmas  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Asymptotics of the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We start by introducing the following processes that are  of great influence on the behavior of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Let (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , ed) denote a canonical  Euclidean basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For each n ∈ N and 1 ≤ j ≤ d, define  N X  n (j) =  n  �  k=1  1{Xj  k̸=0}µk  and  Σn =  d  �  j=1  N X  n (j)ejeT  j ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1)  such that (Σn)n∈N is a matrix-valued process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the following almost sure convergence in the three regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  1  nµn+1  Σn →  1  d(β + 1)Id  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2)  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  25  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For each n ∈ N and 1 ≤ j ≤ d, define  ΛX  n (j) = N X  n (j)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3)  It follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) that  ΛX  n+1(j) =  n  n + 1ΛX  n (j) +  1  n + 11{Xj  n+1̸=0}µn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Moreover, we observe thanks to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12) that  ΛX  n+1(j) =  n  n + 1 · γnΛX  n (j) +  1  n + 11{Xj  n+1̸=0}µn+1 − a(β + 1)  n + 1 ΛX  n (j)  =  n  n + 1 · γnΛX  n (j) + µn+1  n  δX  n+1(j) + (1 − a)µn+1  d(n + 1)  with  δX  n+1(j) = 1{Xj  n+1̸=0} − P  �  Xj  n+1 ̸= 0|Fn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4) we know  ΛX  n (j) =  1  nan  �  ΛX  1 (j) + 1 − a  d  n  �  k=2  akµk + HX  n (j)  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4)  with  HX  n (j) =  n  �  k=2  akµkδX  k (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  It is clear that for a fixed 1 ≤ j ≤ d, the real-valued process (HX  n (j))n∈N is locally square-integrable  since it is a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Afterwards, this process appears to be a martingale adapted to (Fn)n∈N  because (δX  n (j))n∈N satisfied the martingale difference relation E[δX  n+1(j)|Fn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It is obvious  that  ⟨HX(j)⟩n ≤ wn =  n  �  k=1  (akµk)2  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence, we get by [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15] that for all γ > 0  HX  n (j)2  ⟨HX(j)⟩n  = o  ��  log⟨HX(j)⟩n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5)  Since ⟨HX(j)⟩n ≤ wn and by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5), we obtain that  HX  n (j)2 = o  �  wn  �  log wn  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  In the diffusive regime, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3), we have  HX  n (j)2 = o  �  n1−2(a(β+1)−β)�  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), we observe that  � HX  n (j)  nanµn+1  �2  = o  �  n−1�  log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence  HX  n (j)  nanµn+1  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6) again, we observe further  1  nanµn+1  n  �  k=1  akµk →  1  (1 − a)(β + 1)  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6)  26  JIAMING CHEN AND LUCILE LAULIN  Hence, we have  µ−1  n+1ΛX  n (j) →→  1  β + 1  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4), we can then conclude that  1  nµn+1  Σn →  1  d(β + 1)Id  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the critical regime, where a = 1 −  1  2(β+1), we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4))  HX  n (j)2 = o  �  log n  �  log log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence  � HX  n (j)  nanµn+1  �2  = o  �  n−1 log n  �  log log n  �1+γ�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  which implies that  HX  n (j)  nanµn+1  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Similar to the convergence in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), in the critical regime, we observe  1  nanµn+1  n  �  k=1  akµk → 1  2  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence, we conclude that  µ−1  n+1ΛX  n (j) →  1  d(β + 1)  and  1  nµn+1  Σn →  1  d(β + 1)Id  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  which proves (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' In the superdiffusive regime, we have  HX  n (j)2 = o  �  1  �  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  and then  � HX  n (j)  nanµn+1  �2  = o  �  n−2(1−a)(β+1)�  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  which implies  HX  n (j)  nanµn+1  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  We can similarly show that  µ−1  n+1ΛX  n (j) →  1  β + 1  as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  which then ensures that  1  nµn+1  Σn →  1  d(β + 1)Id  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Consequently, the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  The next result follows directly from the definition of Mn and Nn  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the following formulas for the predictable matrix-valued quadratic varia-  tions  ⟨M⟩n = (a1µ1)2E  �  X1XT  1  �  +  n−1  �  k=1  a(β + 1)  ka−2  k+1  µk+1Σk + 1 − a  da−2  k+1  µ2  k+1Id −  �γk − 1  a−1  k+1  �2  YkY T  k ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7)  and  ⟨N⟩n =  �  β  β − a(β + 1)  �2  E  �  X1XT  1  �  +  n−1  �  k=1  a(β + 1)  kµk+1  Σk + 1 − a  d  Id −  �γk − 1  µk+1  �2  YkY T  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8)  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  27  In particular, we have  Tr⟨M⟩n = wn −  n  �  k=1  (γk − 1)2a2  k+1∥Yk∥2,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9)  and  Tr⟨N⟩n =  �  β  β − a(β + 1)  �2  n −  n−1  �  k=1  �a(β + 1)  kµk+1  �2  ∥Yk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10)  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We have the following estimate for the matrix-valued conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  E  �  ϵn+1ϵT  n+1|Fn  �  = a(β + 1)  n  µn+1Σn + 1 − a  d  µ2  n+1Id − (γn − 1)2YnY T  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  And as a consequence  E  �  ∥ϵn+1∥2|Fn  �  = µ2  n+1 − (γn − 1)2∥Yn∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Observe that  E  �  ϵn+1ϵT  n+1|Fn  �  = E  �  Yn+1Y T  n+1|Fn  �  − γ2  nYnY T  n  with  E  �  Yn+1Y T  n+1|Fn  �  = YnY T  n + 2µn+1YnE  �  XT  n+1|Fn  �  + µ2  n+1E  �  Xn+1XT  n+1|Fn  �  =  �  1 + 2a(β + 1)  n  �  YnY T  n + µ2  n+1E  �  Xn+1XT  n+1|Fn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11)  For all k ≥ 1, we know that XkXT  k = �d  j=1 1{Xj  k̸=0}ejeT  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then  P  �  Xj  n+1 ̸= 0|Fn  �  =  n  �  k=1  P  �  βn+1 = k  �  P  �  (AnXk)j ̸= 0|Fn  �  =  n  �  k=1  1{Xj  k̸=0}P  �  An = ±Id  �  (β + 1)µk  nµn+1  +  n  �  k=1  �  1 − 1{Xj  k̸=0}  �  P  �  An = ±Jd  �  (β + 1)µk  nµn+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence  P  �  Xj  n+1 ̸= 0|Fn  �  = β + 1  nµn+1 �  P  �  An = +Id  �  − P  �  An = +Jd  ��  N X  n (j) + 2P  �  An = +Jd  �  = a(β + 1)  nµn+1  N X  n (j) + 1 − a  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12)  Therefore  E  �  Xn+1XT  n+1|Fn  �  =  d  �  j=1  P  �  Xj  n+1 ̸= 0|Fn  �  ejeT  j = a(β + 1)  nµn+1  Σn + 1 − a  d  Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13)  And from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13) we can conclude that  E  �  ϵn+1ϵT  n+1|Fn  �  = E  �  Yn+1Y T  n+1|Fn  �  − γ2  nYnY T  n  =  �  1 + 2a(β + 1)  n  �  YnY T  n + a(β + 1)  n  µn+1Σn + 1 − a  d  µ2  n+1Id − γ2  nYnY T  n  = a(β + 1)  n  µn+1Σn + 1 − a  d  µ2  n+1Id − (γn − 1)2YnY T  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14)  On the other hand  Tr(Σn) = nµn+1  β + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15)  Taking traces in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14) and by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15), we have  E  �  ∥ϵn+1∥2|Fn  �  = µ2  n+1 − (γn − 1)2∥Yn∥2  which ensures that the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  28  JIAMING CHEN AND LUCILE LAULIN  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Scaling limits of the random walk and the barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For each n ∈ N and test vector u ∈ Rd, let  Vn =  1  √n  �  1  0  0  a(β+1)  β−a(β+1)(anµn)−1  �  and  v =  �  1  −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16)  Then  vT VnLn(u) =  1  √nSn(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17)  And for all t ≥ 0, we have  Vn⟨L(u)⟩⌊nt⌋V T  n → uT u  d Vt  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18)  where  Vt =  1  (β − a(β + 1))2  �  β2t  aβ  1−at1+β−a(β+1)  aβ  1−at1+β−a(β+1)  a2(β+1)2  1−2a(β+1)+2β t1+2β−2a(β+1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3 and the fact that ⟨M(u)⟩n = uT ⟨M⟩nu, we see that  ⟨M(u)⟩⌊nt⌋ = a2  1µ2  1uT E  �  X1XT  1  �  u  +  ⌊nt⌋−1  �  k=1  a(β + 1)  k  a2  k+1µk+1uT Σku + 1 − a  d  a2  k+1µ2  k+1uT u − (γk − 1)2a2  k+1uT YkY T  k u  and  ⟨N(u)⟩⌊nt⌋ =  �  β  β − a(β + 1)  �2  uT E  �  X1XT  1  �  u  +  �  β  β − a(β + 1)  �2 ⌊nt⌋−1  �  k=1  a(β + 1)  kµk+1  uT Σku + 1 − a  d  uT u −  �γk − 1  µk+1  �2  uT YkY T  k u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Using a similar token and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1, we can work out the off-diagonal entries in ⟨L(u)⟩⌊nt⌋, and  we obtain that  lim  n→∞ Vn⟨L(u)⟩⌊nt⌋V T  n  = lim  n→∞  uT u  nd(β − a(β + 1))2  �  �  �  β2⌊nt⌋  a(β+1)β  anµn  �⌊nt⌋−1  k=0  ak+1µk+1  a(β+1)β  anµn  �⌊nt⌋−1  k=0  ak+1µk+1  �  a(β+1)  anµn  �2 �⌊nt⌋−1  k=0  (ak+1µk+1)2  �  �  �  =  uT u  d(β − a(β + 1))2  �  β2t  aβ  1−at1−(a(β+1)−β)  aβ  1−at1−(a(β+1)−β)  a2(β+1)2  1−2(a(β+1)−β)t1−2(a(β+1)−β)  �  = uT u  d Vt  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  where the last equality is due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Thus, it implies that  1  nanµn  n  �  k=1  akµk →  1  1 − (a(β + 1) − β)  and  1  n(anµn)2  n  �  k=1  (akµk)2 →  1  1 − 2(a(β + 1) − β)  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence, equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18) holds and the assertion is then verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The MARW satisfies the Lindeberg condition in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' That is, for  all t ≥ 0 and all ϵ > 0,  ⌊nt⌋  �  k=1  E  �  ∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  29  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the one hand, it is easy to compute from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16) that, for all 1 ≤ k ≤ n,  Vn∆Lk(u) =  1  √n(β − a(β + 1))µn  �  β µn  µk  a ak  an  �  ϵk(u)  which implies  ∥Vn∆Lk(u)∥2 =  1  n(β − a(β + 1))2  �β2  µ2  k  +  a2a2  k  (anµn)2  �  ϵk(u)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence  ∥Vn∆Lk(u)∥4 ≤  2  n2(β − a(β + 1))4  �β4  µ4  k  +  a4a4  k  (anµn)4  �  ϵk(u)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20)  On the other hand, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) we observe that  1  na2n  n  �  k=1  a2  k ≤ C1(a, β)−1  and  1  na4n  n  �  k=1  a4  k ≤ C2(a, β)−1  for all  n ∈ N  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21)  and where C1(a, β), C2(a, β) > 0 are constants depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Moreover, we get that  sup  1≤k≤n  |ϵk(u)| ≤  sup  1≤k≤n  ∥ϵk∥∥u∥ ≤  sup  1≤k≤n  (β + 2)µk∥u∥ ≤ (β + 2)µn∥u∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22)  Hence, we deduce from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22)  n  �  k=1  ∥Vn∆Lk(u)∥4 ≤  2  n2(β − a(β + 1))4  ��  β(β + 2)  �4∥u∥4 +  �  a(β + 2)  �4∥u∥4  C2(a, β)  �  → 0  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='23)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This implies that  n  �  k=1  E  �  ∥Vn∆Lk(u)∥4|Fk−1  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Therefore, for all ϵ > 0, we obtain  n  �  k=1  E  �  ∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1  �  ≤ 1  ϵ2  n  �  k=1  E  �  ∥Vn∆Lk(u)∥4|Fk−1  �  → 0  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This yields finally  ⌊nt⌋  �  k=1  E  �  ∥Vn∆Lk(u)∥21{∥VnLk(u)∥2>ϵ}|Fk−1  �  ≤ 1  ϵ2  ⌊nt⌋  �  k=1  E  ����(VnV −1  ⌊nt⌋)V⌊nt⌋∆Lk(u)  ���  4  |Fk−1  �  → 0  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' since VnV −1  ⌊nt⌋ converges as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The deterministic matrix Vt defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19) can be rewritten as  Vt = tα1K1 + tα2K2 + · · · + tαqKq  with q ∈ N, αj > 0 and each Kj is a symmetric matrix for all 1 ≤ j ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A direct computation analoguous to the one in [32] shows that Vt = tα1K1+tα2K2+tα3K3,  where  α1 = 1,  α2 = 1 − a(β + 1) > 0,  α3 = 1 − 2a(β + 1) > 0  since a < 1 −  1  2(β+1) is in the diffusive regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Moreover  K1 =  β2  (a(β + 1) − β)2  �  1  0  0  0  �  ,  K2 =  aβ  (1 − a)(a(β + 1) − β)2  �  0  1  1  0  �  ,  K3 =  a2(β + 1)2  (1 − 2a(β + 1) + 2β)(a(β + 1) − β)2  �  0  0  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  30  JIAMING CHEN AND LUCILE LAULIN  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Given the matrix-valued process (Vn)n∈N define in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16), we have  ∞  �  n=1  1  �  log  �  det V −1  n  �2�2 E  �  ∥Vn∆Ln(u)∥4|Fn−1  �  < ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16), it is immediate that  det V −1  n  = β − a(β + 1)  a(β + 1)  nanµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='24)  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), we obtain  log  �  det V −1  n  �2  log n  → 2(1 − a)(β + 1)  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='25)  Hence there exists a constant C(a, β) > 0 depending only on a and β such that  ∞  �  n=1  1  �  log  �  det V −1  n  �2�2 E  �  ∥Vn∆Ln(u)∥4|Fn−1  �  ≤ C(a, β)  ∞  �  n=1  1  (log n)2 E  �  ∥Vn∆Ln(u)∥4|Fn−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='26)  Hereafter, equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='23) together imply that  ∞  �  n=1  1  (log n)2 ∥Vn∆Ln(u)∥4 ≤ C′(a, β)  ∞  �  n=1  1  (n log n)2 < ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='27)  for some other constant C′(a, β) > 0 depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Consequently, equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='27)  together (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='26) ensures that the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For each n ∈ N and test vector u ∈ Rd, let  Wn =  1  √n log n  �  1  0  0  2β+1  anµn  �  and  w =  �  1  −1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='28)  Then for all t ≥ 0, we have  wT WnLn(u) =  1  √n log nSn(u)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='29)  and  Wn⟨L(u)⟩nW T  n → uT u  d W  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  where  Wt = (2β + 1)2  �  0  0  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='30)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' It is clear that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='29) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Using a similar token than for the proof Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4, we have  lim  n→∞ Wn⟨L(u)⟩nW T  n  = lim  n→∞  4uT u  (n log n)d  �  �  �  β2n  β(β+ 1  2 )  anµn  �n−1  k=0 ak+1µk+1  β(β+ 1  2 )  anµn  �n−1  k=0 ak+1µk+1  �  β+ 1  2  anµn  �2 �n−1  k=0(ak+1µk+1)2  �  �  �  = 4uT u  d  �  0  0  0  �  β + 1  2  �2  �  = uT u  d W  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  31  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The MARW satisfies the Lindeberg condition in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' That is, for all  t ≥ 0 and all ϵ > 0, given the (Wn)n∈N defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16), it satisfies  n  �  k=1  E  �  ∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' We state that equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21) remain true with Vn replaced by Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' More  precisely, they can be rewritten as  ∥Wn∆Lk(u)∥4 ≤  32  (n log n)2  �β4  µ4  k  +  a4a4  k  (anµn)4  �  ϵk(u)4  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='31)  and  1  na4n  n  �  k=1  a4  k ≤ C(a, β)−1  for all  n ∈ N  where C(a, β) > 0 is a constant depending only on t, a, and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Since (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22) is not affected by  switching regimes, we have that  n  �  k=1  ∥Wn∆Lk(u)∥4 ≤  32  (n log n)2  ��  β(β + 2)  �4∥u∥4 +  �  a(β + 2)  �4∥u∥4  C(t, a, β)  �  → 0  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='32)  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' This implies  n  �  k=1  E  �  ∥Wn∆Lk(u)∥4|Fk−1  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Therefore, for all ϵ > 0, we obtain  n  �  k=1  E  �  ∥Wn∆Lk(u)∥21{∥WnLk(u)∥2>ϵ}|Fk−1  �  ≤ 1  ϵ2  n  �  k=1  E  �  ∥Wn∆Lk(u)∥4|Fk−1  �  → 0  as n → ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Given the matrix-valued sequence (Wn)n∈N define in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='28), we have  ∞  �  n=1  1  �  log  �  det W −1  n  �2�2 E  �  ∥Wn∆Ln(u)∥4|Fn−1  �  < ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='28), it is immediate that  det W −1  n  =  1  2β + 1  �  n log n · anµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='33)  Then, we obtain by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6) that  log  �  det W −1  n  �2  log log n  → 1  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='34)  Hence, there exists a constant C(a, β) > 0 depending only on a and β such that  ∞  �  n=1  1  �  log  �  det W −1  n  �2�2 E  �  ∥Wn∆Ln(u)∥4|Fn−1  �  ≤  ∞  �  n=1  C(a, β)  (log log n)2 E  �  ∥Wn∆Ln(u)∥4|Fn−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='35)  Hereafter, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='31) together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='32) imply that  ∞  �  n=1  1  (log log n)2 ∥Wn∆Ln(u)∥4 ≤ C′(a, β)  ∞  �  n=1  1  (n log n log log n)2 < ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  for some other constant C′(a, β) > 0 depending only on a and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Finally, using the above equation  together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='35) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  32  JIAMING CHEN AND LUCILE LAULIN  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Fix the test vector u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The growth rate of the compensator of the partial sum  of (Nn(u)2)n∈N is less than cubic growth, in the sense that  1  n3  n−1  �  k=1  E  �  Nk+1(u)2|Fn  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The law of iterated expectations and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='8) yields  1  nE  �  E  �  Nn+1(u)2|Fn  ��  = 1  nE  �  ⟨N(u)⟩n  �  →  �  β  β − a(β + 1)  �2  uT u  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  The strong law of large numbers then yields  1  n  n−1  �  k=1  1  k E  �  Nk+1(u)2|Fk  �  →  �  β  β − a(β + 1)  �2  uT u  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence  1  n3  n−1  �  k=1  E  �  Nk+1(u)2|Fn  �  ≤ 1  n2  n−1  �  k=1  1  k E  �  Nk+1(u)2|Fk  �  → 0  as  n → ∞  P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The barycenter process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For the following Toeplitz Lemmas, see [18] and [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' [33, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1 Part I] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers  such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N  �kn  k=1 |an,k| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Let (xn)n∈N  be a real sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' If xn → 0 as n → ∞, then �kn  k=1 an,kxk → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' [33, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1 Part II] Let (an,k)1≤k≤kn, n∈N be a double array of real numbers  such that for all k ≥ 1, we have an,k → 0 as n → ∞ and supn∈N  �kn  k=1 |an,k| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Let (xn)n∈N  be a real sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' If xn → x as n → ∞ with x ∈ R and �kn  k=1 an,k = 1, then �kn  k=1 an,kxk → x  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Quadratic rate estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Our first result is about the convergence rate of the process  (Yn)n∈N defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1), then we have, as n → ∞,  E[YnY T  n ] ∼  n2a(β+1)  Γ(1 + 2a(β + 1)) · 1  dId +  n1+2β  Γ(β + 1)2(1 + 2β − 2a(β + 1))(β + 1) · 1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='11) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='13), we see  E  �  Yn+1Y T  n+1|Fn  �  =  �  1 + 2a(β + 1)  n  �  YnY T  n + µ2  n+1  �a(β + 1)  nµn+1  Σn + 1 − a  d  Id  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Then, remember that  E  �  Σn  �  =  d  �  j=1  E  �  N X  n (j)  �  ejeT  j =  d  �  j=1  n  �  k=1  P  �  Xj  k ̸= 0  �  µk · ejeT  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1 yields E[(nµn+1)−1Σn] ∼ (β + 1)−1 · 1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hence,  E  �  Yn+1Y T  n+1  �  ∼  �  1 + 2a(β + 1)  n  �  E  �  YnY T  n  �  + µ2  n+1  β + 1 · 1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  33  A recursive argument then gives  E  �  YnY T  n  �  ∼  Γ(n + 2a(β + 1))  Γ(n)Γ(1 + 2a(β + 1))E  �  Y1Y T  1  �  +  n−1  �  j=1  µ2  j  β + 1 ·  �n−1  k=1(1 + k−12a(β + 1))  �j−1  k=1(1 + k−12a(β + 1))  1  dId  ∼  Γ(n + 2a(β + 1))  Γ(n)Γ(1 + 2a(β + 1)) · 1  dId +  n−1  �  j=1  µ2  j  β + 1 · Γ(n + 2a(β + 1))Γ(j)  Γ(j + 2a(β + 1))Γ(n) · 1  dId.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Employing the asymptotics in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6), the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  The process Yn = �n  k=1 µkXk differs from Sn by a multiplicative factor at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' When  there is no amnesia, the asymptotics of these two processes coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' However, when β ≥ 0, we  have to treat the general case in another way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞,  E  �  ⟨M(u)⟩n  �  ∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u,  and  E  �  ⟨N(u)⟩n  �  ∼  �  β  β − a(β + 1)  �2  nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2  E  �  ⟨M(u)⟩n  �  = E  �  Tr⟨M⟩n  �  uT u = wnuT u −  n  �  k=1  (γk − 1)2a2  k+1uT E  �  YkY T  k  �  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14 and a finite summation,  E  �  ⟨M(u)⟩n  �  ∼ wnuT u −  n−1  �  k=1  a2(β + 1)2  k2  (k + 1)−2a(β+1)(C1k2a(β+1) + C2k1+2β)uT u  ∼ wnuT u − (C1n−1 + C2n−2(a(β+1)−β))uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Similarly,  E  �  ⟨N(u)⟩n  �  = E  �  Tr⟨N⟩n  �  uT u =  �  β  β − a(β + 1)  �2  nuT u −  n−1  �  k=1  a2(β + 1)2  k2  µ−2  k+1uT E  �  YkY T  k  �  u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Hence, using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14 again, we observe  E  �  ⟨N(u)⟩n  �  ∼  �  β  β − a(β + 1)  �2  nuT u −  n−1  �  k=1  a2(β + 1)2  k2  (k + 1)−2β(C1k2a(β+1) + C2k1+2β)uT u  ∼  �  β  β − a(β + 1)  �2  nuT u − (C1n1−2(1−a)(β+1) + C2)uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1) and test vector u ∈ Rd, we have, as n → ∞,  E  �  ⟨M(u), N(u)⟩n  �  ∼  β  β − a(β + 1) · Γ(β + 1)Γ(a(β + 1) + 1)  (1 − a)(β + 1)  n(1−a)(β+1)uT u  − (C1n−(1−a)(β+1) + C2n(1−a)(β+1)−1)uT u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='7) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='2, for all test vector u ∈ Rd  ∆Ln+1(u) =  �  βµ−1  n+1  β − a(β + 1)  �T  ϵn+1(u),  34  JIAMING CHEN AND LUCILE LAULIN  and therefore,  ⟨M(u), N(u)⟩n =  n  �  k=1  β  β − a(β + 1)akµ−1  k E  �  ϵk(u)ϵk(u)T |Fk−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Taking the trace will give us  Tr⟨M, N⟩n =  β  β − a(β + 1)  n  �  k=1  akµk −  β  β − a(β + 1)  n  �  k=1  akµ−1  k (γk − 1)2∥Yk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Taking the expectation and using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='14 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Moderate deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d,  ��∆M j  n  �� ≤  �  a(β + 1) + 1  �  anµn  for all  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='36)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='1),  ∆M j  n = anY j  n − an−1Y j  n−1 = anµnXj  n − (an − an−1)  n−1  �  k=1  µkXj  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Since ∥Xk∥ = 1 for eack k ≤ n, then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4),  ��∆M j  n  �� ≤ anµn + (n − 1)(an−1 − an)µn−1 ≤ anµn + a(β + 1)anµn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  And the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d,  ��∆N j  n  �� ≤ 2a(β + 1) +  β  β − a(β + 1)  for all  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='6),  ∆N j  n =  βµ−1  n+1  β − a(β + 1)ϵj  n+1 =  βµ−1  n+1  β − a(β + 1) ·  �  µn+1Xj  n+1 + (1 − γn)  n  �  k=1  Xj  kµk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Taking absolute value on both sides, and the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d,  ����  1  √wn  ∆M j  k  ���� ≤  �  a(β + 1) + 1  �anµn  √wn  for each  1 ≤ k ≤ n,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='37)  and in the diffusive and critical regime,  ����  1  wn  ⟨M j⟩n − 1  ���� ≤  �  �  �  C · n−1  when  a < 1 −  1  2(β+1)  C · (log n)−1  when  a = 1 −  1  2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Dividing by √wn from both sides of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='36), we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Moreover, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='9),  ��⟨M j⟩n − wn  �� ≤  n  �  k=1  (γk − 1)2a2  k+1∥Yk∥2 ≤ C  n  �  k=1  wk  k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Dividing both sides by wn and following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='4), the assertion is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' For all p ∈ (0, 1) and for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d,  ����  anµn  √wn  ∆N j  k  ���� ≤  �  2a(β + 1) +  β  β − a(β + 1)  �anµn  √wn  for each  1 ≤ k ≤ n,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='38)  MULTIDIMENSIONAL AMNESIA-REINFORCED ELEPHANT RANDOM WALK  35  and in both the diffusive and critical regime,  ����  a2  nµ2  n  wn  ⟨N j⟩n − 1  ���� ≤  �  �  �  C · n−2(1−a)(β+1)  when  a < 1 −  1  2(β+1)  C · (n log n)−1  when  a = 1 −  1  2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Dividing by √wn and multiplied by anµu from both sides of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='18), we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Then,  by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='10), we make use of the estimates and the inequalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  □  Denote by Φ(·) := (2π)−1/2 � ·  −∞ e−t2/2 dt the cumulative distribution of the standard normal  random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The following lemmas are straightforward derivations from [19, Theorem 1], see  also [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' There exists an absolute constant α′(p, β) > 0 depending only on p, β such that  for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d and all 0 ≤ x ≤ α′(p, β) · n−1/2, in the diffusive and critical regime,  P(M j  n/√wn ≥ x)  1 − Φ(x)  = P(M j  n/√wn ≤ −x)  1 − Φ(−x)  =  �  �  �  C · exp  �  x3  √n + x2  n +  1  √n(1 + 1  2 log n)(1 + x)  �  when  a < 1 −  1  2(β+1)  C · exp  �  x3  √n +  x2  log n + (  1  √log n +  1  2√n log n)(1 + x)  �  when  a = 1 −  1  2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' There exists an absolute constant α′′(p, β) > 0 depending only on p, β such that  for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' , d and all 0 ≤ x ≤ α′′(p, β) · n−1/2, in the diffusive and critical regime,  P(anµnN j  n/√wn ≥ x)  1 − Φ(x)  = P(anµnN j  n/√wn ≤ −x)  1 − Φ(−x)  =  �  �  �  C · exp  �  x3  √n +  x2  n2(1−a)(β+1) +  1  √n(n1/2−(1−a)(β+1) + 1  2 log n)(1 + x)  �  when  a < 1 −  1  2(β+1)  C · exp  �  x3  √n +  x2  n log n + (  1  √n log n +  1  2√n log n)(1 + x)  �  when  a = 1 −  1  2(β+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' The authors wish to thank Jean Bertoin and Pierre Tarres for numerous  discussions and insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  References  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' On a class of random walks with reinforced memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Bernard, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Kyt¨ol¨a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martin-  gales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content='  [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Baur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Bertoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Elephant random walks and their connection to P´olya-type urns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Generalized urn models of evolutionary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=',  14 (3): 1455–1478, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Bercu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A martingale approach for the elephant random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Chabanol, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Ruch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Hypergeometric identities arising from the elephant random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Bercu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Laulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' How to estimate the memory of the elephant random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=',  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Laulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the center of mass of the elephant random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Sto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Proces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Laulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' On the Multi-dimensional Elephant Random Walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Functional limit Theorems for the multi-dimensional elephant random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Sto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Sto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Proces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=', 129 (11): 4663–4686, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Marcaccioli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Livan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A P´olya urn approach to information filtering in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=',  10, 745 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' McRedmond, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+page_content=' The convex hull of a planar random walk: perimeter, diameter, and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' 23: 1–24, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [37] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Merkl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Rolles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Linearly edge-reinforced random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' IMS Lect Note - Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Sto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=', 48:  66–77, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Pemantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' A survey of random processes with reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Surveys 4 (2007), 1–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Pemantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Vertex-reinforced random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=', 92: 117–136, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [40] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Sch¨utz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Trimper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Elephants can always remember:  Exact long-range memory effects in a non-  markovian random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Physical review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' E 70, 045101 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Sheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Arzela-Ascoli’s Theorem and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=', 2022100209, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Touati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Sur la convergence en loi fonctionnelle de suites de semimartingales vers un m´elange de mouvements  browniens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Veroyatnost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' i Primenen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=', 36 (4): 744–763, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Wade, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Convex hulls of planar random walks with drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=', 143 (1): 433–445,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='  Departement Mathematik, ETH Z¨urich  Current address: 101, R¨amistrasse, CH-8092 Z¨urich, Switzerland  Email address: jiamchen@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='ch  Laboratoire de Math´ematiques Jean Leray, Nantes Universit´e  Current address: 2 Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content=' de la Houssini`ere, 44322 Nantes, France  Email address: lucile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='laulin@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
+page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFAT4oBgHgl3EQfqB2B/content/2301.08644v1.pdf'}
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+A compositional account of motifs, mechanisms, and dynamics in
+biochemical regulatory networks
+Rebekah Aduddell
+James Fairbanks
+Amit Kumar
+Pablo S. Ocal
+Evan Patterson
+Brandon T. Shapiro
+Abstract
+Regulatory networks depict promoting or inhibiting interactions between molecules in a
+biochemical system. We introduce a category-theoretic formalism for regulatory networks, using
+signed graphs to model the networks and signed functors to describe occurrences of one network
+in another, especially occurrences of network motifs. With this foundation, we establish functorial
+mappings between regulatory networks and other mathematical models in biochemistry. We
+construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory
+networks, enabling us to precisely define when a reaction network could be a physical mechanism
+underlying a regulatory network. Turning to quantitative models, we associate a regulatory
+network with a Lotka-Volterra system of differential equations, defining a functor from the
+category of signed graphs to a category of parameterized dynamical systems. We extend this
+result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not
+only inclusions and collapsings of regulatory networks, but also the process of building up complex
+regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured
+cospans to produce a lax double functor from the double category of open signed graphs to that
+of open parameterized dynamical systems. Throughout the paper, we ground the categorical
+formalism in examples inspired by systems biology.
+1
+Introduction
+The genes, proteins, and RNA molecules that comprise living cells interact in complex, varied ways
+to sustain the cell throughout its lifecycle and respond to changes in its environment. Intensive
+experimental study of these interactions is distilled in an idealized form as regulatory networks,
+a kind of directed graph in which vertices represent molecules and edges represent interactions
+between molecules (Figure 1.1). The edges are labeled with a positive or negative sign according to
+whether the interaction is activating or inhibiting. Regulatory networks are the subject of a large
+body of experimental and theoretical work, notably reviewed by Alon [Alo07; Alo19] and Tyson
+et al [TN10; TLK19] among others. Particular attention has been paid to network motifs [Alo07;
+TN10], the simple but functionally meaningful patterns that recur frequently in regulatory networks,
+and to various quantitative dynamics [TLK19] that can be assigned to the networks.
+Although regulatory networks are simple enough to define mathematically—we shall define them
+to be directed graphs, possibly with multiple edges and loops, whose edges are assigned a positive
+or negative sign—important scientific concepts involving them, such as occurrences of motifs in
+networks and biochemical mechanisms generating networks, are often treated imprecisely. Likewise
+for relationships between regulatory networks and other mathematical models in biochemistry,
+particularly dynamical models based on ordinary or stochastic differential equations. Hence a first
+aim of this paper is to put certain concepts and relations concerning regulatory networks on a firm
+mathematical footing. To do so, we will use methods from category theory.
+1
+arXiv:2301.01445v1  [q-bio.MN]  4 Jan 2023
+
+Ash1
+Cdk1/ClbS
+Sld2
+Figure 1.1: A small biochemical regulatory network: regulation of Sld2 by Cdk1 or ClbS with Ash1 as a
+predicted transcription factor. Adapted from Csikász-Nagy et al [Csi+09, Figure 3C].
+Category theory, in both the small and the large, is a natural tool for this study. In saying that
+a motif occurs in a network, one should allow for the possibility that the occurrence is indirect,
+involving a sequence of appropriately signed interactions. For example, positive autoregulation
+can occur directly but also indirectly through a double-negative feedback loop. Since a small
+category is exactly a graph in which consecutive edges can be composed, subject to certain laws,
+regulatory networks should be viewed not only as signed graphs (Section 2.1) but also as signed
+categories freely generated by those (Section 2.2). Sign-preserving functors, unlike sign-preserving
+graph homomorphisms, can express indirect occurrences and are in this sense a better notion of
+morphism for regulatory networks. Here we are doing category theory in the small, using categories
+as algebraic structures comparable to familiar ones like graphs, groups, and monoids.
+Having laid these foundations for regulatory networks, we turn to category theory in the large, a
+mathematical theory of structure well suited to describe the passages between regulatory networks
+and other mathematical models of biochemical systems. Formally speaking, these passages are
+functors into or out of the category of regulatory networks. Making a functor is significantly stronger
+than making an objects-only mapping, as is typically done in the literature, since if morphisms
+of signed graphs formalize relationships between different regulatory networks, then functorality
+requires that these relationships be transported to or from other models of interest. By contrast,
+an objects-only mapping is, abstractly speaking, entirely unconstrained and so is capable of acting
+highly irregularly across different models of a given class. Functorality thus serves as a kind of
+safeguard for model transformation: it does not, on its own, ensure that a transformation makes
+good scientific sense but it does impose nontrivial logical constraints and coherences.
+A first illustration of this principle is the connection between regulatory networks and biochemical
+reaction networks (Section 2.3). When modeling the complex biochemical systems that constitute a
+living cell, it is often practically necessary to abstract away certain details of the underlying chemical
+processes. Regulatory networks generally do not capture all the species or reactions involved in
+a given system; nor can they capture multispecies reactions faithfully because they describe only
+pairwise interactions. Given that regulatory networks are, to some degree, phenomenological models,
+it is natural to ask whether a given network could arise as a summary of a specific chemical process.
+The latter are described by biochemical reaction networks, graph-like structures allowing reactions
+or transitions with multiple inputs and outputs. Inspired by graphical syntax from systems biology
+[Voi00; Voi13], we formalize reaction networks as “Petri nets with links,” and we construct a functor
+from the category of Petri nets with signed links to the category of signed graphs. This functor
+enables us to propose a formal definition for when a reaction network could be a mechanism for a
+regulatory network, a concept that is rarely if ever treated in a precise way.
+This concludes the content of Section 2. In Section 3, we turn from qualitative to quantitative
+analysis, seeking a functorial assignment of continuous dynamics to regulatory networks. Although
+rarely made explicitly functorial, systematic ways to formulate a model belonging to a mathematically
+homogeneous class of models are ubiquitous in science. Voit calls these “canonical representations”
+2
+
+or “canonical models,”1 and identifies Lotka-Volterra models and BST models/S-systems as two
+prominent examples in biology [Voi13, §3]. Reflecting their phenomenological status, regulatory
+networks do not admit a single, obvious dynamical interpretation, and so a wide variety of dynamical
+models have been considered, spanning the discrete and continuous, deterministic and stochastic
+[TLK19]. We consider the Lotka-Volterra systems of ordinary differential equations. While not
+necessarily the most biologically plausible, it is among the simplest continuous models and hence a
+natural place to begin a functorial study.
+A Lotka-Volterra system of equations has the form
+˙xi = ρi xi +
+n
+�
+j=1
+βi,j xi xj,
+i = 1, . . . , n.
+or equivalently, has logarithmic derivatives that are affine functions of the state variables:
+d
+dt[log xi(t)] = ρi +
+n
+�
+j=1
+βi,j xj(t),
+i = 1, . . . , n.
+The coefficients ρi specify baseline rates of growth or decay, according to their sign, and the
+coefficients βi,j rates of activation or inhibition, according to their sign. We construct a functor
+that sends a signed graph (regulatory network) to a Lotka-Volterra model suitably constraining the
+signs of the rate coefficients (Section 3.2). As a prerequisite, we define a category of parameterized
+dynamical systems (Section 3.1), a construction of intrinsic interest that is by no means confined to
+Lotka-Volterra dynamics.
+In order to comprehend complex biological systems, we must decompose them into small, readily
+understandable pieces and then compose them back together to reproduce the behavior of the
+original system. This is the mantra of systems biology, which stresses that compositionality is no
+less important than reductionism in biology. With this motivation, a secondary aim of this paper
+is to extend the above constructions from closed systems to open ones, which can be composed
+together by gluing them along their interfaces. Mathematically, we pass from categories to double
+categories [Gra19], two-dimensional categorical structures in which the usual morphisms of systems
+compose along one direction (by convention, the “vertical” one) and open systems compose along
+the other direction (the “horizontal” one). Among other results, we show that the Lotka-Volterra
+dynamics functor extends to a lax double functor from the double category of open signed graphs
+to a double category of open parameterized dynamical systems (Section 3.3).
+The mathematics developed here is motivated by biochemistry but need not be restricted to it.
+Famously, Lotka-Volterra systems originated in ecology to model predator-prey dynamics [Lot25].
+Regulatory networks and Lotka-Volterra systems can be used as generic models of entities that
+“regulate” each other in some manner, be it at the scale of individual cells or animal ecosystems.
+Regulatory networks are highly reminiscent of the causal loop diagrams in system dynamics [Ste00,
+Chapter 5], where the latter explicitly label feedback loops and their polarities.
+The language of category theory is indispensable to this work but the level of knowledge assumed
+by the reader is not constant. We assume throughout that the reader is familiar with the basic
+notions of category theory, such as categories, functors, and natural transformations. Our main
+reference for facts about category theory is Riehl’s text [Rie16], although there are many others.
+In the definitions and theorems, we have tried to minimize the technical level and explicate the
+statements in concrete terms. In the proofs, we have aimed for efficiency and freely use concepts
+1This usage of “canonical” should not be confused with the unrelated, in fact incompatible, meaning of “canonical”
+in pure mathematics, especially category theory.
+3
+
+and results from the literature that do not appear in the main text. The reader can omit the proofs
+without disrupting the continuity of the paper.
+2
+Qualitative analysis: motifs and mechanisms
+2.1
+Regulatory networks as signed graphs
+To begin, we clarify the notion of graph to be used throughout in this paper. The following definition
+is standard among category theorists. In other fields, it might be called a “directed multigraph,”
+but we will call it simply a “graph.”
+Definition 2.1 (Graphs). The schema for graphs is the category Sch(Graph) freely generated
+by two parallel morphisms:
+V
+E
+src
+tgt
+.
+A graph is a functor X : Sch(Graph) → Set. A graph homomorphism from a graph X to another
+graph Y is a natural transformation φ : X → Y . Graphs and graph homomorphisms form the
+category Graph.
+To restate the definition in explicit terms, a graph X consists of
+• a set X(V ) of vertices;
+• a set X(E) of edges; and
+• functions X(src), X(tgt) : X(E) → X(V ), assigning to each edge its source and target.
+A graph homomorphism φ : X → Y consists of a function φV : X(V ) → Y (V ), the vertex map,
+and another function φE : X(E) → Y (E), the edge map. These maps must preserve sources and
+targets, meaning that the following squares commute:
+X(E)
+X(V )
+Y (E)
+Y (V )
+X(src)
+φE
+Y (src)
+φV
+X(E)
+X(V )
+Y (E)
+Y (V )
+X(tgt)
+φE
+Y (tgt)
+φV .
+We now turn to the main notion of this section, signed graph. Write Sgn for the set of (nonzero)
+signs, whose two elements may be denoted {1, −1} or {+, −}. The set of signs is an abelian group,
+isomorphic to the cyclic group Z2, under the usual multiplication.
+Definition 2.2 (Signed graphs). The category of signed graphs is the slice category
+SgnGraph := Graph/Sgn,
+where, by abuse of notation, Sgn is regarded as a graph with one vertex and two loops.
+Unpacking the definition, a signed graph is seen to be a graph X equipped with a function
+X(sgn) : X(E) → Sgn that assigns a sign to each edge. Given signed graphs X and Y , a morphism
+of signed graphs from X to Y is a graph homomorphism φ that preserves signs, meaning that
+the following triangle commutes:
+X(E)
+Y (E)
+Sgn
+X(sgn)
+Y (sgn)
+φE
+.
+4
+
+Signed graphs are a mathematical description of the regulatory networks studied in systems
+biology [Alo07; TN10].
+For the purposes of this paper, we will simply define a regulatory
+network to be a signed graph. The vertices of the graph represent the components of the network,
+which could be proteins, genes, or RNA molecules. Signed edges represent interactions between
+components, where the source has the effect of either activating/promoting the target (positive sign)
+or inhibiting/repressing it (negative sign). As is customary, we denote activation interactions by
+arrows with pointed heads (−→) and inhibition interactions by arrows with flat heads (−−⊣). For
+instance, the two drawings
+x
+y
++
+−
+↭
+x
+y
+represent the same network, a negative feedback loop in which x activates y, which in turn inhibits
+x [TN10, Scheme 1, Motif B].
+In the literature [TN10], regulatory networks are often modeled as sign-valued matrices. This
+approach is a special case of ours in that an n-by-n matrix valued in {+1, −1, 0} can be interpreted
+as a simple signed graph on n vertices, with signed edges defined by the nonzero matrix elements.
+Unlike the matricial formalism, our formalism allows multiple edges between the same pair of edges,
+which can model multiple interactions based on different mechanisms. Allowing multiple edges and
+self-loops also ensures that graphs and signed graphs form well behaved categories, as the following
+proposition shows.
+Proposition 2.3. The category of signed graphs is complete (has all limits) and cocomplete (has
+all colimits).
+Proof. Because Graph is a copresheaf category, it is complete and cocomplete [Rie16, Proposition
+3.3.9]. The slice category SgnGraph = Graph/Sgn is hence also complete and cocomplete [Rie16,
+Proposition 3.5.5]; alternatively, this follows because slices of copresheaf categories are again
+(equivalent to) copresheaf categories [Str00, Remark p. 303].
+Colimits of signed graphs can be used to construct a category, or rather a double category, of
+open signed graphs. Composition of open signed graphs formalizes the process of building large
+regulatory networks from smaller pieces, including network motifs.
+Proposition 2.4 (Open signed graphs). There is a symmetric monoidal double category of open
+signed graphs, Open(SgnGraph), having
+• as objects, sets A, B, C, . . . ;
+• as vertical arrows, functions f : A → B;
+• as horizontal arrows, open signed graphs, which consist of a signed graph X together with
+a cospan of sets A0
+ℓ0
+−→ X(V )
+ℓ1
+←− A1;
+• as cells, morphisms of open signed graphs (X, ℓ0, ℓ1) → (Y, m0, m1), which consist of a
+map of signed graphs φ : X → Y along with functions fi : Ai → Bi, i = 0, 1, making the
+following diagram commute:
+A0
+X(V )
+A1
+B0
+Y (V )
+B1
+ℓ0
+f0
+ℓ1
+φV
+m0
+f1
+m1
+.
+5
+
+Vertical composition is by composition in Set and in SgnGraph. Horizontal composition and monoidal
+products are by pushouts and coproducts in SgnGraph, respectively, viewing the sets in the feet of the
+cospans as discrete signed graphs.
+Proof. To construct this symmetric monoidal double category, we use the method of structured
+cospans [FS07] in its double-categorical form [BC20]. The categories of sets and of signed graphs
+are related by an adjoint pair of functors
+Set
+SgnGraph
+Disc
+evV
+⊣
+.
+Here evV : SgnGraph → Set is the evaluation at V functor, sending a signed graph X to its set of
+vertices X(V ) and a morphism of signed graphs φ to its vertex map φV , and Disc : Set → SgnGraph
+is the discrete signed graph functor, sending a set A to the signed graph with vertex set A and no
+edges. We obtain a symmetric monoidal double category of open signed graphs as the L-structured
+cospans for the functor L := Disc : Set → SgnGraph [BC20, Theorems 2.3 and 3.9].
+To show that this symmetric monoidal double category is the same one in the proposition
+statement, suppose that L ⊣ R : A → X is an adjoint pair of functors, where in our application
+L = Disc and R = evV . By the defining bijection of an adjunction, L-structured cospans, i.e.,
+objects A0 and A1 in A together with a cospan L(A0) → X ← L(A1) in X, correspond exactly
+to “R-decorated cospans,” i.e., an object X in X together with a cospan A0 → R(X) ← A1 in A.
+Furthermore, by the naturality of this bijection [Rie16, Lemma 4.1.3], morphisms of L-structured
+and R-decorated cospans
+L(A0)
+X
+L(A1)
+L(B0)
+Y
+L(B1)
+L(f0)
+φ
+L(f1)
+↭
+A0
+R(X)
+A1
+B0
+R(Y )
+B1
+f0
+R(φ)
+f1
+related by the adjunction are equivalent in that one diagram commutes if and only if the other does.
+We will tacitly reuse this reasoning in future constructions, such as Proposition 2.8 below.
+A morphism of signed graphs can do two things. Most obviously, it can pick out a signed graph
+as a subobject of another one, via a sign-preserving subgraph embedding. A signed graph morphism
+can also collapse multiple vertices onto a single vertex, and multiple edges onto a single edge with
+the same sign, in the fairly restrictive sense permitted by a graph homomorphism. To illustrate,
+consider the following morphism inspired by Alon’s review [Alo07, Figure 5].
+argR
+argCBH
+argD
+argE
+argF
+argI
+−→
+argR
+arg∗
+(2.1)
+The network in the domain is a “single-input module” in the arginine biosynthesis system, in which
+the regulator argR represses five different enzymes (argCHB, argD, etc.) involved in producing
+arginine. The morphism above forgets the distinction between these enzymes, collapsing them
+into a catch-all entity labeled “arg∗”. These two functions—embedding and collapsing—are all
+that a signed graph morphism can do, because any such morphism factors essentially uniquely as
+6
+
+an epimorphism (morphism with surjective vertex and edge maps) followed by a monomorphism
+(morphism with injective vertex and edge maps), using the epi-mono factorization available in any
+copresheaf category, or more generally in any topos [MM94, §IV.6].
+2.2
+Refining regulatory networks using signed categories and functors
+While morphisms of signed graph have their uses, they do not capture the important idea of refining
+regulatory networks, in which an interaction in one network is realized as a composite of several
+interactions in another. To express refinement, we must generalize our notion of morphism from
+graph homomorphisms to functors. This, in turn, requires the concept of a signed category.
+Definition 2.5 (Signed categories). The category of signed categories is the slice category
+SgnCat := Cat/Sgn,
+where Cat is the category of small categories and the group of signs, Sgn, is regarded as a category
+with one object and two morphisms.
+Unpacking the definition, a signed category is a category C in which every morphism f is
+assigned a sign sgn(f) ∈ {1, −1} in a functorial way, meaning that
+sgn(x0
+f1
+−→ x1
+f2
+−→ · · ·
+fn
+−→ xn) =
+n
+�
+i=1
+sgn(fi)
+for every n ≥ 0 and every sequence of composable morphisms f1, . . . , fn. In particular (n = 0), the
+identity morphisms have positive sign. A morphism of signed categories, or signed functor,
+is a functor F : C → D between signed categories that preserves the signs, meaning that
+sgnD(F(f)) = sgnC(f)
+for every morphism f in C.
+Since our aim is to have a more flexible notion of morphism between signed graphs, we will
+mostly restrict ourselves to those signed categories that are freely generated by a signed graph. The
+free signed category or signed path category functor
+Path : SgnGraph → SgnCat
+sends a signed graph X to the signed category Path(X) having
+• as objects, the vertices of X;
+• as morphisms from x to y, the paths in X from x to y, whose sign is defined to be the product
+of the signs of the edges comprising the path.
+Composition of paths is by concatenation, which clearly preserves the sign. The identity morphism
+at x is the empty path at x, which has positive sign. By convention, if X and Y are signed graphs,
+we say that a signed functor from X to Y is a signed functor F : Path(X) → Path(Y ) between
+the corresponding signed path categories. Since the morphisms of Path(X) are freely generated
+by the edges in X, a signed functor from X to Y is uniquely determined by a morphism of signed
+graphs from X to the underlying signed graph of Path(Y ). This means that each edge in X is
+sent to an appropriately signed path of edges in Y , which can be regarded as a refinement of the
+relationship that the edge represents.
+7
+
+Motif
+Generic instance
+Positive autoregulation
+L+ :=
+�
+•
+�
+Negative autoregulation
+L− :=
+�
+•
+�
+Coherent feedforward loop
+I++ :=
+�
+•
+•
+�
+Incoherent feedforward loop
+I± :=
+�
+•
+•
+�
+Positive feedback loop
+L++ :=
+�
+•
+•
+�
+Negative feedback loop
+L± :=
+�
+•
+•
+�
+Double-negative feedback loop
+L−− :=
+�•
+•
+�
+Table 2.1: Common motifs in biochemical regulation networks [Alo07; TN10]
+We now have a precise language with which to classify network motifs and their occurrences.
+As a first example, Alon identifies four types of incoherent feedforward loop (FFL) involving three
+components,
+x
+x
+x
+x
+y
+y
+y
+y
+z
+z
+z
+z
+,
+those of type 1, 2, 3, and 4, respectively [Alo07, Figure 2a]. Besides having three components, what
+these motifs have in common is that there exists a signed functor into each of them from the signed
+graph I± :=
+�
+•
+•
+�
+having two parallel arrows of opposite sign. The network I± is thus the
+“generic” incoherent feedforward loop, in the sense that signed functors out of it refine the pattern
+in specific ways. A similar situation holds for other common network motifs (Table 2.1), which
+motivates the following definition.
+Definition 2.6 (Motif instance). Given a signed graph A, regarded as a motif, an instance or
+occurrence of the motif A in a network X is a monic signed functor A ↣ X.
+Note that a signed functor is a monomorphism exactly when the functor is an embedding of
+categories, i.e., an injective-on-objects, faithful functor. Requiring the functor in the definition to
+be monic excludes “degenerate instances” of motifs where vertices or edges are identified.
+Now, should the incoherent FFL be regarded as a network motif, or is it the more specific types,
+such as the incoherent FFL of type 1, that are motifs? From our point of view, they are all equally
+motifs but they have different degrees of specificity, and the functorial language clarifies how motifs
+are iteratively refined. Specifically, an instance of an incoherent FFL of type 1 in a network X
+also gives an instance of an incoherent FFL in X (of unspecified type), simply by composing the
+monomorphisms involved:
+I± ∼=
+�
+x
+z
+�
+↣
+�
+x
+y
+z
+�
+↣
+X.
+8
+
+Similarly, in the notation of Table 2.1, any instance of double-negative feedback (L−−) also gives an
+instance of positive autoregulation (L+) [CP09], via the monomorphism L+ ↣ L−− that sends the
+positive loop to the double-negative 2-cycle.
+For any choice of motif A, the mapping that sends a regulatory network X to the set of
+occurrences of A in X is a functor
+HomSgnCatm(Path(A), Path(−)) : SgnGraphm → Set,
+where SgnGraphm and SgnCatm denote the wide subcategories of monomorphisms in SgnGraph and
+SgnCat, respectively. This functor is almost, but not quite, representable, due to the distinction
+between signed graphs and signed categories. More importantly, the existence of this functor means
+that a monomorphism between regulatory networks induces a map between instances of A, for any
+motif A.
+We now extend the construction of open signed graphs to open signed categories.
+Proposition 2.7. The category of signed categories is complete and cocomplete.
+Proof. Because the category Cat is complete and cocomplete [Rie16, Proposition 3.5.6], its slice
+SgnCat = Cat/Sgn is also [Rie16, Proposition 3.5.5].
+Proposition 2.8 (Open signed categories). There is a symmetric monoidal double category of open
+signed categories, Open(SgnCat), having
+• as objects, sets A, B, C, . . . ;
+• as vertical arrows, functions f : A → B;
+• as horizontal arrows, open signed categories, which consist of a signed category C together
+with a cospan of sets A0
+ℓ0
+−→ Ob(C)
+ℓ1
+←− A1;
+• as cells, morphisms of open signed categories (C, ℓ0, ℓ1) → (D, m0, m1), which consist of
+a signed functor F : C → D along with functions fi : Ai → Bi, i = 0, 1, making the diagram
+commute:
+A0
+Ob(C)
+A1
+B0
+Ob(D)
+B1
+ℓ0
+f0
+ℓ1
+Ob(F)
+m0
+f1
+m1
+.
+Vertical composition is by composition in Set and in SgnCat. Horizontal composition and monoidal
+products are by pushouts and coproducts in SgnCat, respectively, viewing the sets in the feet of
+cospans as discrete signed categories.
+Moreover, the signed path category functor extends to a symmetric monoidal double functor
+Path : Open(SgnGraph) → Open(SgnCat).
+Proof. We take Open(SgnCat) to be the symmetric monoidal double category of L′-structured
+cospans for the functor L′ := Disc : Set → SgnCat involved the composite adjunction
+Set
+SgnCat
+=
+Set
+SgnGraph
+SgnCat
+Disc
+Ob
+Disc
+evV
+U
+Path
+⊣
+⊣
+⊣
+.
+On the right hand side, the first adjunction was already used in the proof of Proposition 2.4, and
+the second adjunction is the free-forgetful adjunction between signed graphs and signed categories.
+9
+
+To prove the last statement, we notice that all functors involved in the commutative square
+Set
+SgnGraph
+Set
+SgnCat
+L=Disc
+L′=Disc
+Path
+are left adjoints, hence preserve colimits. We can therefore appeal to [BC20, Theorem 4.3] to obtain
+a symmetric monoidal double functor
+Open(SgnGraph) ∼= LCsp(SgnGraph) → L′Csp(SgnCat) ∼= Open(SgnCat).
+2.3
+Mechanistic models as Petri nets with links
+However challenging they may be to identify through experiments and data analysis, regulatory
+networks still only summarize how the components of a complex biochemical system interact.
+Regulatory networks typically include only a subset of the system’s components, and they do not
+model individual reactions and processes, only pairwise promoting or inhibiting interactions between
+components. In this sense, regulatory networks are not fully mechanistic models, even if they have
+a stronger causal interpretation than, say, a correlation matrix.
+By contrast, mechanistic models in biochemistry model individual reactions, which requires a
+different formalism. Pictures like the following, adapted from Voit’s review [Voi13, Figure 4], are
+common in systems biology.
+A
+B
+D
+C
+−
+(2.2)
+This diagram possesses two distinctive features. First, directed hyperedges represent reactions
+having a number of inputs or outputs different than one. There are, for example, hyperedges from
+B and C to D, from nothing to A (an inflow), and from D to nothing (an outflow). If, in lieu of
+hyperedges, we introduce a second type of vertex, we obtain a structure similar to a Petri net
+A
+B
+C
+D
+−
+(2.3)
+but with the second distinctive feature of having signed links from the first type of vertices (species)
+to the second type (transitions), whose signs indicate promotion or inhibition.
+In this section, we explain how a Petri net with signed links can provide a mechanism for a
+regulatory network. This involves constructing a functor from Petri nets with signed links to signed
+graphs, approximating the former as the latter. As a prerequisite, we need a rigorous definition of a
+Petri net with links, which seems to be absent from the literature.
+10
+
+Definition 2.9 (Petri net with links). The schema for Petri nets with links is the category
+Sch(LPetri) freely generated by these objects and morphisms:
+I
+S
+O
+T
+L
+srcL
+tgtL
+srcI
+tgtI
+tgtO
+srcO
+.
+A Petri nets with links is a functor P : Sch(LPetri) → Set, and a morphism of these is a natural
+transformation. A morphism φ : P → Q preserves arities if the naturality squares associated
+with the morphisms I → T and O → T are also pullback squares:
+P(I)
+P(T)
+Q(I)
+Q(T)
+P(tgtI)
+φI
+φT
+P(tgtI)
+⌟
+P(O)
+P(T)
+Q(O)
+Q(T)
+P(srcO)
+φO
+φT
+P(srcO)
+⌟
+.
+Petri nets with links and their morphisms form the category LPetri.
+To explicate the definition, a Petri net with links P consists of
+• a set P(S) of species;
+• a set P(T) of transitions;
+• a set P(I) of input arcs going from species to transitions, via maps P(srcI) : P(I) → P(S)
+and P(tgtI) : P(I) → P(T);
+• a set P(O) of output arcs going from transitions to species, via maps P(srcO) : P(O) → P(T)
+and P(tgtO) : P(O) → P(S); and finally
+• a set P(L) of links going from species to transitions, via maps P(srcL) : P(L) → P(S) and
+P(tgtL) : P(L) → P(T).
+The property of preserving arities, called “etale” by Kock [Koc22, §3.4], means that a morphism
+φ : P → Q of Petri nets with links preserves the input and output arities of all transitions. Namely,
+for each transition t in the net P the map φI : P(I) → Q(I) restricts to a bijection between the
+input arcs to t and to φT (t), and similarly the map φO : P(O) → Q(O) restricts to a bijection
+between the output arcs from t and from φT (t). This property seems appropriate for many purposes,
+including in biochemistry, but for mathematical convenience we will not always assume it.
+Remark 2.10 (Related literature). Our definition of a Petri net with links, while apparently novel,
+is the obvious joint generalization of two existing concepts. Kock has described Petri nets as
+copresheaves on a category with objects S, T, I, O [Koc22], calling them whole-grain Petri nets to
+distinguish them from classical Petri nets, whose semantics are subtly different [Bae+21]. Meanwhile,
+the concept of a link is essential to stock and flow diagrams, originating in the field of system
+dynamics [For61; Ste00] and recently given a rigorous categorical account [Bae+22]. We also note
+that the Petri nets with catalysts proposed by Baez, Foley, and Moeller [BFM19] differ significantly
+from Petri nets with links: the former fix a subset of the species to be catalysts throughout the net,
+whereas the latter uses links to make catalyzation specific to individual reactions.
+11
+
+Remark 2.11 (Petri nets as typed graphs). Like bare Petri nets, Petri nets with links can be described
+as graphs with two types of vertices. To see this, take the graph
+TLPetri :=
+�
+�
+�
+�
+�
+�
+�
+S
+T
+I
+L
+O
+�
+�
+�
+�
+�
+�
+�
+with vertices S and T and edges I, O, and L. The category of Petri nets with links and natural
+transformations is isomorphic to the slice category Graph/TLPetri. Moreover, the schema category
+Sch(LPetri) is isomorphic to the category of elements of the functor TLPetri : Sch(Graph) → Set,
+exemplifying a general fact about slices of copresheaf categories [Str00, Remark p. 303].
+Petri nets with signed links are defined analogously to signed graphs (Definition 2.2).
+Definition 2.12 (Petri nets with signed links). The category of Petri nets with signed links
+is the slice category
+SgnPetri := LPetri/PSgn,
+where PSgn is the Petri net with links having one species, one transition, one input arc, one output
+arc, and two links, namely the elements of Sgn.
+Incidentally, the morphism P → PSgn defining a Petri net with signed links does not preserve
+arities unless every transition in P has exactly one input and one output.
+We now turn to the main construction of this section, a functor that “approximates” a Petri net
+with signed links as a signed graph. On the example in Equations (2.2) and (2.3), this functor gives
+the signed graph
+A
+B
+D
+C
+.
+(2.4)
+In general, the resulting signed graph has, as vertices, the Petri net’s species and has signed edges
+for each of the four cases:
+(a) for every input-output pair to a transition, a positive edge from input to output;
+(b) for every input to a transition, a negative self loop, representing consumption by the reaction;
+(c) for every signed link, an edge of opposite sign going from the linked species to each input to
+the linked transition;
+(d) for every signed link, an edge of equal sign going from the linked species to each output from
+the linked transition.
+All four cases are visible in the example of Equation (2.4). Set-theoretically, each of these cases is
+the result of a conjunctive query, or equivalently of a representable functor
+HomLPetri(P, −) : LPetri → Set
+associated with a particular Petri net with links P, the generic instance for that query. The four
+generic instances we need are shown in Figure 2.1. Their sum is a disjoint union of conjunctive
+queries, or duc-query for short.
+12
+
+(a) Input-output pair to tran-
+sition
+(b) Input to transition
+(c) Input to transition with
+incident link
+(d) Output from transition
+with incident link
+Figure 2.1: Four different Petri nets with links. For each of these instances P, evaluating the representable
+functor HomLPetri(P, −) : LPetri → Set gives the edges for one case in the case analysis that defines the functor
+from Petri nets with signed links to signed graphs (Proposition 2.13).
+To make the construction just sketched on objects fully precise and functorial, we use Spivak’s
+theory of functorial data migration based on duc-queries [Spi21]. In order to apply it, we fully
+schematize the definitions of signed graphs and Petri nets with signed links.
+The schema for signed graphs is the category Sch(SgnGraph) freely generated by these objects
+and morphisms:
+V
+E
+A
+neg
+src
+tgt
+sgn
+.
+A signed graph as in Definition 2.2 is equivalent to a functor X : Sch(SgnGraph) → Set such that
+X(A) = Sgn, the set of signs, and X(neg) : Sgn → Sgn is negation (i.e., multiplication by −1). Note
+that negation is not needed to define the data of a signed graph but is relevant to the data migration.
+A morphism of signed graphs X → Y is a natural transformation φ : X → Y whose component at
+A is the identity function, φA = 1Sgn. We thus obtain a category isomorphic to SgnGraph.
+Similarly, the schema for Petri nets with signed links is the category Sch(SgnPetri) freely
+generated by:
+S
+I
+O
+L
+A
+T
+neg
+srcL
+tgtL
+srcI
+tgtI
+tgtO
+srcO
+sgn
+one
+.
+A Petri net with signed links, as in Definition 2.12, is equivalent to a functor P : Sch(SgnPetri) → Set
+such that P(A) = Sgn, the map P(neg) : Sgn → Sgn is negation, and P(one) : P(T) → Sgn is the
+constant map at +1. Again, these maps are needed for data migration, not for the data itself. A
+morphism of Petri nets with signed links is a natural transformation φ : P → Q such φA = 1Sgn,
+yielding a category isomorphic to SgnPetri.
+Proposition 2.13 (Regulatory net induced by Petri net). A functor
+Net : SgnPetri → SgnGraph
+13
+
+is specified by the following functor from Sch(SgnGraph) to the category of duc-queries on Sch(SgnPetri):
+Sch(SgnGraph) → ⨿
+��
+SetSch(SgnPetri)�op�
+V �→ S
+E �→ I ×T O + I + I ×T L + O ×T L
+A �→ A
+src �→ [srcI ◦πI, srcI, srcL ◦πL, srcL ◦πL]
+tgt �→ [tgtO ◦πO, srcI, srcI ◦πI, tgtO ◦πO]
+sgn �→ [one ◦πT , neg ◦ one ◦ tgtI, neg ◦ sgn ◦πL, sgn ◦πL]
+neg �→ neg .
+(2.5)
+Proof. We will define the functor Net : SgnPetri → SgnGraph as the restriction of a functor
+SetD → SetC between the categories of copresheaves on D := Sch(SgnPetri) and C := Sch(SgnGraph).
+In fact, the functor SetD → SetC will be of the special kind known as a parametric right adjoint
+[Str00, Definition p. 311].
+According to the theory of data migration [Spi21, Corollary 2.3.6], giving a parametric right
+adjoint SetD → SetC is equivalent to giving a functor from C to ⨿((SetD)op), the free coproduct
+completion of the free limit completion of D. Our functor C → ⨿((SetD)op) is defined by Equa-
+tion (2.5). The assignment of E ∈ C can also be described as the sum of the four representables
+associated with the Petri nets with links in Figure 2.1.
+Finally, the assignments A �→ A and neg �→ neg ensure that if P is a copresheaf on D with
+P(A) = Sgn and P(neg) is negation, then applying this parametric right adjoint functor to P yields
+a copresheaf X on C where again X(A) = Sgn and X(neg) is negation. Thus, this functor between
+copresheaf categories restricts to a functor SgnPetri → SgnGraph as claimed.
+With this construction, we can give a formal account of what it means to have a mechanistic
+model for a regulatory network.
+Definition 2.14 (Mechanism). A mechanistic model for a regulatory network X is a Petri
+net with signed links P together with an occurrence of X in Net(P), i.e., a monic signed functor
+X ↣ Net(P).
+For example, the Petri net with signed links in Equation (2.3) is a mechanistic model for a
+regulatory network in which A and D participate in a positive feedback loop:
+A
+D .
+3
+Quantitative analysis: parameters and dynamics
+3.1
+Parameterized dynamical systems
+Pioneering the idea of functorial semantics for scientific models, Baez and Pollard extended the
+mass-action model of reaction networks to a functor from the category of Petri nets with rates
+into a category of dynamical systems [BP17]. In this picture, the reaction rate coefficients are
+known constants associated with the reaction network. In practice, however, rate coefficients are
+often unknown and must be extracted from existing literature or estimated from experimental data.
+We therefore change our perspective slightly and consider dynamical systems not in isolation but
+as parameterized families. This shift also turns out to have formal advantages: the category of
+14
+
+parameterized dynamical systems is better behaved than the category of dynamical systems, which
+has too few morphisms.
+To begin, we recall the dynamics functor, nearly identical to Baez-Pollard’s [BP17, Lemma 15]:
+Lemma 3.1 (Dynamics). There is a functor Dynam : FinSet → VectR that sends
+• a finite set S to the vector space of algebraic vector fields v : RS → RS, where algebraic
+means that the components of the vector field are polynomials in the state variables;
+• a function f : S → S′ between finite sets to the linear transformation
+(v : RS → RS)
+�→
+(f∗ ◦ v ◦ f∗ : RS′ → RS′),
+where the linear map f∗ : RS′ → RS is the pullback along f
+f∗(x′)(i) := x′(f(i)),
+x′ ∈ RS′, i ∈ S,
+and the linear map f∗ : RS → RS′ is the pushforward along f
+f∗(x)(i′) :=
+�
+i∈f−1(i′)
+x(i),
+x ∈ RS, i′ ∈ S′.
+Proof. The functor Dynam : FinSet → VectR can be constructed as the composite
+FinSet
+⟨D,F⟩
+−−−−→ Vectop
+R × VectR
+Poly(−,−)
+−−−−−−→ VectR,
+where F : FinSet → VectR is the free vector space functor (restricted to finite sets); D : FinSetop →
+VectR is the dual vector space functor (restricted to F), whose underlying set-valued functor is
+VectR(F(−), R) ∼= Set(−, R) : FinSetop → Set;
+and Poly(−, −) is the VectR-enriched hom-functor that sends a pair of vector spaces to the vector
+space of polynomial maps between them.2
+The dynamics functor is the same one studied by Baez and Pollard except that we take the
+vector space, rather than merely the set, of vector fields. That is because we are interested in
+linearly parameterized dynamical systems. In calling the functor “dynamics,” we implictly identify a
+vector field with the differentiable dynamical system that it generates. This common practice is not
+entirely innocent since even when a system of differential equations depends linearly on parameters,
+its solutions rarely do. We also note that the restriction to algebraic vector fields, as opposed to
+smooth or even just continuous ones, is inessential but suffices for us and agrees with Baez-Pollard.
+The dynamics functor is the main building block in constructing a category of parameterized
+dynamical systems.
+Definition 3.2 (Linear parameterizations). The category of linearly parameterized dynami-
+cal systems is the comma category
+Para(Dynam) := F/ Dynam,
+where F : FinSet → VectR, X �→ RX is the free vector space functor restricted to finite sets.
+2For a coordinate-free description of polynomial maps between vector spaces, see [Car71, §1.6].
+15
+
+So, by definition, a linearly parameterized dynamical system consists of a finite set P, the
+parameter variables, and a finite set S, the state variables, together with a linear map
+v : RP → Dynam(S)
+sending each choice of parameters θ ∈ RP to an algebraic vector field v(θ). In more conventional
+notation, we can write v(x; θ) := v(θ)(x) for x ∈ RS and θ ∈ RP . A morphism (P, S, v) → (P ′, S′, v′)
+of linearly parameterized dynamical systems is a pair of functions q : P → P ′ and f : S → S′
+making the square
+RP
+Dynam(S)
+RP ′
+Dynam(S′)
+v
+f∗◦(−)◦f∗
+v′
+q∗
+(3.1)
+commute, or equivalently making the equation
+f∗(v(f∗(x′); θ)) = v′(x′; q∗(θ))
+hold for all x′ ∈ RS′ and θ ∈ RP .
+While certainly not all dynamical models depend linearly on their parameters, a great many
+of them do, including several important canonical models in biology and chemistry. The law of
+mass action defines dynamical systems that depend linearly on the rate coefficients. The generalized
+Lotka-Volterra equations, studied in the next section, are linear in the rate and affinity parameters.
+Of course, the mass-action and Lotka-Volterra equations are nonlinear ODEs; linearity of a vector
+field in state or in parameters are separate matters. Nevertheless, even for nonlinear models such
+as Lotka-Volterra, linearity in parameters is a useful assumption that aides in the identifiability
+analysis of the model [SRS14, §5].
+To express important physical constraints and to define a semantics for signed graphs, we will
+restrict the dynamical system and its parameters to be nonnegative. This is straightforward enough
+but requires a bit of additional formalism.
+Write R+ := {x ∈ R : x ≥ 0} for the semiring of nonnegative real numbers. A module over R+
+is called a conical space, and the category of conical spaces and conic-linear (R+-linear) maps is
+denoted Con := ModR+. A conical space is a structure in which one can take linear combinations
+with nonnegative real coefficients, just as a real vector space (R-module) is a structure in which
+one can take linear combinations with arbitrary real coefficients. Any convex cone in a real vector
+space is a conical space. Our main example is the nonnegative orthant of RS for some set S: the
+function space RS
++ := {x : S → R+}, with conical combinations taken pointwise. A real vector space
+can itself be regarded as a conical space; more precisely, the inclusion of semirings R+ �→ R induces
+a forgetful functor VectR → Con by restriction of scalars.
+Recall that a dynamical system is nonnegative if whenever the initial condition is in the
+nonnegative orthant, its trajectory always remains in the nonnegative orthant. A dynamical system
+of form ˙x = v(x) is nonnegative if and only if vi(x) ≥ 0 whenever x ≥ 0 componentwise and
+xi = 0 [HCH10, Proposition 2.1], in which case the vector field v is called essentially nonnegative
+[HCH10, Definition 2.1]. Using this criterion, it is easy to see that a reaction network with mass-
+action kinetics is nonnegative assuming the rate constants are nonnegative, as is a Lotka-Volterra
+system for any choice of parameters. Hence both systems satisfy the obvious physical constraint
+that no species should have negative concentration or population.
+Lemma 3.3 (Nonnegative dynamics). There is a functor Dynam+ : FinSet → Con that sends a
+finite set S to the conical space of essentially nonnegative, algebraic vector fields v : RS → RS and
+sends a function f : S → S′ to the transformation v �→ f∗ ◦ v ◦ f∗.
+16
+
+Proof. The proof is similar to that of Lemma 3.1. It is clear that the essentially nonnegative functions
+are stable under pointwise conical combinations, hence form a conical space. (They are, of course, not
+stable under arbitrary linear combinations.) We just need to check that if v : RS → RS is essentially
+nonnegative, then so is the transformed vector field f∗ ◦ v ◦ f∗ : RS′ → RS′. Fix x′ ∈ RS′
++ and i′ ∈ S′,
+and suppose that x′(i′) = 0. For every i ∈ f−1(i′), we have f∗(x′)(i) = x′(f(i)) = x′(i′) = 0 and so
+v(f∗(x′))(i) ≥ 0, whence the inequality of essential nonnegativity follows:
+(f∗ ◦ v ◦ f∗)(x′)(i′) =
+�
+i∈f−1(i′)
+v(f∗(x′))(i) ≥ 0.
+We now define the conical analogue of linearly parameterized dynamical systems.
+Definition 3.4 (Conical parameterizations). The category of conically parameterized non-
+negative dynamical systems is the comma category
+Para(Dynam+) := F+/ Dynam+ .
+where F+ : FinSet → Con, X �→ RX
++ is the free conical space functor restricted to finite sets.
+So, a conically parameterized nonnegative dynamical system consists of finite sets P and S
+together with a conic-linear map
+v : RP
++ → Dynam+(S).
+Proposition 3.5 (Colimits of parameterized dynamical systems). The categories of linearly and
+conically parameterized dynamical systems are finitely cocomplete. Moreover, these finite colimits
+are computed by colimits in FinSet of the parameter variables and of the state variables.
+Proof. The category FinSet has finite colimits and the functors F : FinSet → VectR and F+ :
+FinSet → Con preserve finite colimits, since they are composites of the inclusion FinSet �→ Set with
+the left adjoints
+Set
+VectR
+F
+U
+⊣
+and
+Set
+Con
+F+
+U+
+⊣
+to the underlying set functors on vector spaces and conical spaces. By Lemma 3.6 below, the comma
+categories Para(Dynam) = F/ Dynam and Para(Dynam+) = F+/ Dynam+ have finite colimits,
+which are preserved by the projection functors onto FinSet.
+To illustrate, we describe the initial object and binary coproducts in Para(Dynam). The initial
+linearly parameterized dynamical system has no parameter variables, no state variables, and the
+unique (trivial) vector field on the zero vector space. The coproduct of two linearly parameterized
+dynamical systems (P1, S1, v1) and (P2, S2, v2) has parameter variables P1 + P2, state variables
+S1 + S2, and parameterized vector field
+RP1+P2 ∼= RP1 ⊕ RP2
+v1⊕v2
+−−−−→ Dynam(S1) ⊕ Dynam(S2)
+[Dynam(ι1),Dynam(ι2))]
+−−−−−−−−−−−−−−−→ Dynam(S1 + S2),
+where ι1 : S1 → S1 +S2 and ι2 : S2 → S1 +S2 are the canonical inclusions. In conventional notation,
+the coproduct system has parameterized vector field
+v
+��
+x1
+x2
+�
+;
+�
+θ1
+θ2
+��
+=
+�
+v1(x1; θ1)
+v2(x2; θ2)
+�
+.
+The proof of Proposition 3.5, as well as of Theorems 3.7 and 3.8 below, depends on the following
+technical lemma about comma categories, which the reader can omit without loss of continuity.
+17
+
+Lemma 3.6 (Colimits in comma categories). Let C0
+F0
+−→ C
+F1
+←− C1 be a cospan of categories such
+that C0 and C1 have colimits of shape J and F0 preserves J-shaped colimits. Then the comma category
+F0/F1 also has J-shaped colimits, and the projection functors πi : F0/F1 → Ci, i = 0, 1, preserve
+those colimits.
+Furthermore, a functor G : X → F0/F1 into the comma category preserves J-shaped colimits
+whenever the associated functors Gi := πi ◦ G : X → Ci, i = 0, 1, do so.
+Proof. Colimits in the comma category F0/F1 are constructed in [RB88, §5.2]. To make the rest of
+the proof self-contained, we recall the construction here.
+By the universal property of the comma category, a diagram D : J → F0/F1 is equivalent to
+diagrams Di := πi ◦ D : J → Ci, i = 0, 1, along with a natural transformation ⃗D : F0 ◦ D0 ⇒ F1 ◦ D1.
+Let (ci, λi) be a colimit cocone for the diagram Di in Ci, having legs Di(j)
+λi
+j
+−→ ci for each j ∈ J.
+The family of morphisms
+F0(D0(j))
+⃗Dj
+−−→ F1(D1(j))
+F1(λ1
+j)
+−−−−→ F1(c1),
+j ∈ J,
+is then a cocone under F0 ◦ D0. Since F0 preserves J-shaped limits, (F0(c0), F0 ∗ λ0) is a colimit
+cocone for F0 ◦ D0, so by its universal property, there exists a unique morphism f : F0(c0) → F1(c1)
+making the squares commute:
+F0(D0(j))
+F1(D1(j))
+F0(c0)
+F1(c1)
+⃗Dj
+f
+F0(λ0
+j)
+F1(λ1
+j) ,
+j ∈ J.
+Setting λ := (λ0
+j, λ1
+j)j∈J, the cocone ((c0, c1, f), λ) can be shown to be a colimit of the diagram D.
+To prove the last statement about colimit preservation, let D : J → X be a diagram with colimit
+cocone (x, λ), having legs Dj
+λj
+−→ y for j ∈ J. We must show that its image cocone (G(x), G ∗ λ) is
+a colimit of the diagram G ◦ D in F0/F1. By the universal property of the comma category, the
+functor G : X → F0/F1 is equivalent to the functors Gi : X → Ci, i = 0, 1, along with a natural
+transformation ⃗G : F0 ◦ G0 ⇒ F1 ◦ G1. The image cocone (G(x), G ∗ λ) then amounts to cocones
+(G0(x), G0 ∗ λ) and (G1(x), G1 ∗ λ), which by hypothesis are colimits of the diagrams G0 ◦ D and
+G1 ◦ D in C0 and C1, together with a family of commutative squares in C:
+F0(G0(Dj))
+F1(G1(Dj))
+F0(G0(x))
+F1(G1(x))
+⃗GDj
+F0(G0(λj))
+⃗Gx
+F1(G1(λj)) ,
+j ∈ J.
+But a morphism ⃗Gx making all these squares commute is already uniquely determined by the
+universal property of the colimit cocone (F0(G0(x)), F0 ∗ G0 ∗ λ) for the diagram F0 ◦ G0 ◦ D, using
+the hypothesis that F0 preserves J-shaped colimits. Indeed, this is precisely how one constructs the
+colimit of the diagram G ◦ D in F0/F1, as shown above. It follows that (G(x), G ∗ λ) is a colimit
+cocone for G ◦ D.
+18
+
+3.2
+The Lotka-Volterra dynamical model
+A Lotka-Volterra system with n species has, using matrix notation, the vector field
+v(x; ρ, β) := x ⊙ (ρ + βx) = diag(x)(ρ + βx)
+with state vector x ∈ Rn and arbitrary real-valued parameters ρ ∈ Rn and β ∈ Rn×n [SMH18, §2.2].
+In coordinates, the vector field is
+vi(x; ρ, β) = xi
+�
+�ρi +
+n
+�
+j=1
+βi,jxj
+�
+� = ρixi +
+n
+�
+j=1
+βi,jxixj,
+i = 1, . . . , n.
+The parameter ρi sets the baseline rate of growth (when positive) or decay (when negative) for
+species i, whereas βi,j defines a promoting (when positive) or inhibiting (when negative) effect of
+species j on species i. In typical applications the signs of the parameters are fixed and known in
+advance of any data. For example, in the famous predator-prey Lotka-Volterra system
+˙x = ax − bxy
+˙y = dxy − cy,
+with prey x and predators y, the parameters ρ =
+�
+a
+−c
+�
+and β =
+�
+0
+−b
+d
+0
+�
+are specified by nonnegative
+real numbers a, b, c, d ≥ 0.
+In this section, we define quantitative semantics for graphs and signed graphs using the Lotka-
+Volterra dynamical model. To illustrate the main ideas, we first construct a functor from finite
+graphs (Definition 2.1) to linearly parameterized dynamical systems (Definition 3.2), giving a
+semantics for unlabeled graphs. It is more useful to have a semantics for regulatory networks, which
+we have defined to be signed graphs. We therefore construct a second functor from finite signed
+graphs (Definition 2.2) to conically parameterized nonnegative dynamical systems (Definition 3.4).
+Recall that a graph is finite if its vertex and edge sets are both finite. Let FinGraph denote the
+full subcategory of Graph spanned by finite graphs.
+Theorem 3.7 (Lotka-Volterra model for finite graphs). There is a functor
+LV : FinGraph → Para(Dynam)
+that sends
+• a finite graph X to the linearly parameterized dynamical system with parameter variables
+P := X(V ) + X(E), state variables S := X(V ), and algebraic vector field3
+v(x; ρ, β)(i) := ρ(i) x(i) +
+�
+(e:i′→i)∈X
+β(e) x(i′) x(i),
+x ∈ RX(V ), i ∈ X(V ),
+parameterized by vectors ρ ∈ RX(V ) and β ∈ RX(E);
+• a graph homomorphism φ : X → Y to a morphism of systems with parameter variable map
+φV + φE : X(V ) + X(E) → Y (V ) + Y (E) and state variable map φV : X(V ) → Y (V ).
+Moreover, the functor LV preserves finite colimits.
+3For a fixed graph X and vertex i ∈ X(V ), the notation (e : i′ → i) ∈ X means any edge e ∈ X(tgt)−1(i) incoming
+to i, whose source i′ = X(src)(e) varies with e.
+19
+
+Proof. By the universal property of the comma category Para(Dynam) = F/ Dynam, to give a
+functor LV : FinGraph → Para(Dynam) is to give a pair of functors LV0, LV1 : FinGraph → FinSet
+along with a natural transformation
+⃗
+LV : (F ◦ LV0) ⇒ (Dynam ◦ LV1) : FinGraph → VectR.
+We set LV0(X) := X(V ) + X(E) and LV1(X) := X(V ). Using the universal property of the
+coproduct in VectR, the components
+⃗
+LVX : RX(V ) ⊕ RX(E) ∼= RX(V )+X(E) → Dynam(X(V )).
+of the transformation ⃗
+LV themselves decompose into two parts, call them
+v0
+X := ⃗
+LV
+0
+X : RX(V ) → Dynam(X(V ))
+and
+v1
+X := ⃗
+LV
+1
+X : RX(E) → Dynam(X(V )).
+We define these to be
+v0
+X(x; ρ)(i) := ρ(i) x(i)
+and
+v1
+X(x; β)(i) :=
+�
+e∈X(tgt)−1(i)
+β(e) x(X(src)(e)) x(i).
+Putting the pieces back together reproduces the first statement of the theorem. We just need to
+check that the transformation ⃗
+LV is, in fact, natural.
+Given a graph homomorphism φ : X → Y , the naturality square for the transformation ⃗
+LV is
+RX(V )+X(E)
+Dynam(X(V ))
+RY (V )+Y (E)
+Dynam(Y (V ))
+⃗
+LVX
+(φV +φE)∗
+(φV )∗◦(−)◦(φV )∗
+⃗
+LVY
+,
+(3.2)
+which decomposes into two squares,
+RX(V )
+Dynam(X(V ))
+RY (V )
+Dynam(Y (V ))
+v0
+X
+(φV )∗
+(φV )∗◦(−)◦(φV )∗
+v0
+Y
+and
+RX(E)
+Dynam(X(V ))
+RY (E)
+Dynam(Y (V ))
+v1
+X
+(φE)∗
+(φV )∗◦(−)◦(φV )∗
+v1
+Y
+.
+Let us check that both squares commute. For the first, we have
+(φV )∗(v0
+X(φ∗
+V (y); ρ))(j) =
+�
+i∈φ−1
+V (j)
+v0
+X(y ◦ φV ; ρ)(i) =
+�
+i∈φ−1
+V (j)
+ρ(i) y(φV (i))
+=
+�
+�
+�
+�
+i∈φ−1
+V (j)
+ρ(i)
+�
+�
+� y(j) = v0
+Y (y; (φV )∗(ρ))(j)
+20
+
+for all y ∈ RY (V ), ρ ∈ RX(V ), and j ∈ Y (V ). For the second, we have
+(φV )∗(v1
+X(φ∗
+V (y); β))(j) =
+�
+i∈φ−1
+V (j)
+v1
+X(y ◦ φV ; β)(i)
+=
+�
+i∈φ−1
+V (j)
+�
+e∈X(tgt)−1(i)
+β(e) y(φV (X(src)(e))) y(j)
+=
+�
+f∈Y (tgt)−1(j)
+�
+e∈φ−1
+E (f)
+β(e) y(Y (src)(φE(e))) y(j)
+=
+�
+f∈Y (tgt)−1(j)
+�
+�
+�
+�
+e∈φ−1
+E (f)
+β(e)
+�
+�
+� y(Y (src)(f))y(j)
+= v1
+Y (y; (φE)∗(β))(j),
+for all y ∈ RY (V ), β ∈ RX(E), and j ∈ Y (V ). When exchanging the order of the summations we
+have used the facts that the graph homomorphism φ : X → Y preserves sources and targets, the
+latter in its contravariant form
+X(E)
+X(V )
+Y (E)
+Y (V )
+X(tgt)
+φE
+Y (tgt)
+φV
+⇝
+P(Y (V ))
+P(X(V ))
+P(Y (E))
+P(X(E))
+X(tgt)−1
+φ−1
+E
+Y (tgt)−1
+φ−1
+V
+,
+where P(S) denotes the power set of a set S.
+Finally, we must verify that the functor LV preserves finite colimits. By Lemma 3.6, that
+happens provided both functors LV0, LV1 : FinGraph → FinSet preserve finite colimits. The functor
+LV1 = evV is an evaluation functor on a copresheaf category, hence preserves colimits [Rie16,
+Proposition 3.3.9].
+Since coproducts commute with colimits, the pointwise coproduct of two
+evaluation functors
+LV0 =
+�FinGraph
+⟨evV ,evE⟩
+−−−−−−→ FinSet × FinSet +
+−→ FinSet
+�
+also preserves colimits. This completes the proof.
+A quantitative semantics for signed graphs can be defined similarly, subject to a caveat about
+the vertex parameters. Our notion of signed graph, designed to capture regulatory networks as
+studied in the biochemistry literature, attaches signs only to edges. We are thus led to assume that,
+in the Lotka-Volterra dynamical model, all species have baseline rates of decay rather than growth.
+This assumption is generally valid for protein regulatory networks, but not for gene regulatory
+networks in which mediating proteins are ignored [TN10], nor for predator-prey models in ecology.
+More flexible approaches are certainly possible. It would be straightforward to attach signs to
+vertices as well as edges and use them in the quantitative semantics. Alternatively, at the expense
+of a more cumbersome formalism, one could define dynamical systems with mixed linear-conical
+parameterizations, allowing the vertex parameters to be arbitrary reals while the edge parameters
+are constrained to be nonnegative.4 For simplicity and uniformity of presentation, we do not describe
+these extensions further.
+Let FinSgnGraph denote the full subcategory of SgnGraph spanned by finite signed graphs.
+4Similar mixed parameterizations are a practical necessity for parametric statistical models, studied in detail in
+one author’s PhD thesis [Pat20].
+21
+
+Theorem 3.8 (Lotka-Volterra model for finite signed graphs). There is a functor
+LV : FinSgnGraph → Para(Dynam+)
+that sends a finite signed graph X to the conically parameterized nonnegative dynamical system with
+parameter variables P := X(V ) + X(E), state variables S := X(V ), and essentially nonnegative,
+algebraic vector field
+v(x; ρ, β)(i) := −ρ(i) x(i) +
+�
+(e:i′→i)∈X
+X(sgn)(e) β(e) x(i′) x(i),
+x ∈ RX(V ), i ∈ X(V ),
+parameterized by ρ ∈ RX(V )
++
+and β ∈ RX(E)
++
+. Moreover, the functor LV preserves finite colimits.
+Proof. Similarly to the previous proof, the functor LV : FinSgnGraph → Para(Dynam+) is defined
+by functors LV0, LV1 : FinSgnGraph → FinSet along with a natural transformation
+⃗
+LV : F+ ◦ LV0 ⇒ Dynam+ ◦ LV1 : FinSgnGraph → Con,
+now having components ⃗
+LVX given by the copairing of
+v0
+X := ⃗
+LV
+0
+X : RX(V )
++
+→ Dynam+(X(V ))
+and
+v1
+X := ⃗
+LV
+1
+X : RX(E)
++
+→ Dynam+(X(V )),
+where we define
+v0
+X(x; ρ)(i) := −ρ(i) x(i)
+and
+v1
+X(x; β)(i) :=
+�
+e∈X(tgt)−1(i)
+X(sgn)(e) β(e) x(X(src)(e)) x(i).
+The proof of naturality is essentially the same as before, using the crucial additional fact that
+morphisms of signed graphs preserve signs. The proof that the functor LV preserves finite colimits
+is unchanged.
+To exemplify the theorem, let us see how the Lotka-Volterra dynamics functor acts on a
+monomorphism and on an epimorphism of signed graphs.
+In order to compare the dynamics of two species A and B involved in a negative feedback loop
+versus A and B in isolation, we take the inclusion of signed graphs
+A
+B
+A
+B
+ι
+Labeling the edges in the feedback loop as AB and BA, the morphism LV(ι) sends the conically
+parameterized dynamical system
+�
+vA(x; ρ) = −ρA xA
+vB(x; ρ) = −ρB xB
+,
+ρ ∈ R{A,B}
++
+,
+to the parameterized dynamical system
+�
+vA(x; ρ, β) = −ρA xA − βBA xB xA
+vB(x; ρ, β) = −ρB xB + βAB xA xB
+,
+ρ ∈ R{A,B}
++
+, β ∈ R{AB,BA}
++
+,
+by setting the latter’s interaction coefficients to zero: βAB = βBA = 0.
+This formalizes the
+commonsense fact that the first system is a degenerate case of the second.
+22
+
+For a more interesting example, we return to the projection map between regulatory networks
+given by Equation (2.1) of Section 2.1, inspired by the arginine biosynthesis system. Call this
+projection map p, and abbreviate the regulator molecule as R and the enzymes as S := {C, D, E, F, I}.
+The morphism LV(p) sends the parameterized dynamical system
+�
+�
+�
+�
+�
+�
+�
+vR(x; ρ, β) = −ρR xR − βR x2
+R
+vC(x; ρ, β) = −ρC xC − βC xR xC
+vD(x; ρ, β) = −ρD xD − βD xR xD
+vE(x; ρ, β) = −ρE xE − βE xR xE
+vF (x; ρ, β) = −ρF xF − βF xR xF
+vI(x; ρ, β) = −ρI xI − βI xR xI
+with state variables {R} + S and parameters ρ, β ∈ R{R}+S
++
+to the parameterized dynamical system
+�
+vR(x; ρ, β) = −ρR xR − βR x2
+R
+v∗(x; ρ, β) = −ρ∗ x∗ − β∗ xR x∗
+with state variables {R, ∗} and parameters ρ, β ∈ R{R,∗}
++
+, in two different but equivalent ways. The
+first way sets the latter system’s coefficients equal to sums of the former’s coefficients, namely
+ρ∗ =
+�
+i∈S
+ρi
+and
+β∗ =
+�
+i∈S
+βi.
+The second way substitutes x∗ for each xi, i ∈ S, in the first system and then takes the vector field
+v∗ to be the sum of the vi’s, i ∈ S, with these substitutions. The equivalence of these operations
+is precisely the condition for LV(p) to be a morphism of parameterized dynamical systems, cf.
+Equations (3.1) and (3.2).
+3.3
+Composing Lotka-Volterra models
+To complete this part of the story, we extend the Lotka-Volterra dynamics functors between
+graphs and parameterized dynamical systems, constructed in Theorems 3.7 and 3.8, to double
+functors between open graphs and open parameterized dynamical systems. We begin by making
+parameterized dynamical systems into open systems.
+Proposition 3.9 (Open parameterized dynamical systems). There is a symmetric monoidal double
+category of open linearly parameterized dynamical systems, Open(Para(Dynam)), having
+• as objects, finite sets A, A′, . . . ;
+• as vertical arrows, functions f : A → A′;
+• as horizontal arrows, open linearly parameterized dynamical systems, which consist of
+a linearly parameterized dynamical system (P, S, v : RP → Dynam(S)) along with a cospan
+A0
+ℓ0
+−→ S
+ℓ1
+←− A1 whose apex is the set S of state variables;
+• as cells, morphisms of such open systems (P, S, v, ℓ0, ℓ1) → (P ′, S′, v′, ℓ′
+0, ℓ′
+1), which
+consist of a morphism (q, f) : (P, S, v) → (P ′, S′, v′) between linearly parameterized dynamical
+systems along with functions f0 : A0 → A′
+0 and f1 : A1 → A′
+1 making the diagram commute:
+A0
+S
+A1
+A′
+0
+S′
+A′
+1
+ℓ0
+ℓ1
+f0
+f
+ℓ′
+0
+f1
+ℓ′
+1
+.
+23
+
+Vertical composition is by composition in FinSet and in Para(Dynam). Horizontal composition and
+monoidal products are by pushouts and coproducts in Para(Dynam), respectively, interpreting the
+finite sets in the feet of the cospans as linearly parameterized dynamical systems with no parameter
+variables and identically zero vector fields.
+Similarly, there is a symmetric monoidal double category Open(Para(Dynam+)) of open conically
+parameterized nonnegative dynamical systems.
+Proof. We perform the construction for linearly parameterized dynamical systems. The construction
+for conically parameterized nonnegative dynamical systems is perfectly analogous, replacing R with
+R+ and vector spaces with conical spaces.
+The projection functor πS : Para(Dynam) → FinSet, (P, S, v) �→ S that sends a linearly parame-
+terized dynamical systems to its set of state variables has a left adjoint Z : FinSet → Para(Dynam)
+that sends a finite set S to the system (∅, S, 0) with empty set of parameter variables. By linearity, its
+parameterized vector field 0 ∼= R∅ → Dynam(S) is necessarily the zero vector field. This indeed gives
+an adjunction Z ⊣ πS, because to any function f : S → S′ and linearly parameterized dynamical
+system (P ′, S′, v′) there corresponds a unique morphism (0P ′, f) : Z(S) → (P ′, S′, v′), where the
+required square
+0
+Dynam(S)
+RP ′
+Dynam(S′)
+f∗◦(−)◦f∗
+v′
+commutes trivially, since the zero vector space is initial in VectR.
+Since Para(Dynam) has finite colimits (Proposition 3.5), we can construct a symmetric monoidal
+double category of Z-structured cospans [BC20, Theorem 3.9]. As we have argued before, it will be
+isomorphic to Open(Para(Dynam)).
+With this definition, we can construct double functors between open graphs and open param-
+eterized dynamical systems, but the vertex parameters under Lotka-Volterra dynamics cause a
+twist in the story compared to Baez and Pollard’s compositionality result for mass-action kinetics
+[BP17, Theorem 18]. When composing open dynamical systems in the image of the Lotka-Volterra
+functor, one takes a coproduct of the parameter variables, i.e., a direct sum of the parameter spaces,
+belonging to identified vertices. However, if one composes the open graphs first, then the identified
+vertices receive a single copy of the parameters from the Lotka-Volterra functor. Thus this functor
+does not preserve composition of open systems, not even up to isomorphism. Nevertheless, there is
+a natural (noninvertible) comparison between them: given a pair of parameters in the direct sum,
+we can reduce them to a single parameter simply by summing them. In mathematical terms, we get
+a lax double functor: a double functor that strictly preserves vertical composition, as usual, but
+preserves horizontal composition only up to specified comparison maps.5
+Theorem 3.10 (Open Lotka-Volterra models). There is a symmetric monoidal lax double functor
+LV : Open(FinGraph) → Open(Para(Dynam))
+that acts
+• on objects and vertical arrows, as the identity;
+5The precise definition of a lax double functor can be found in the textbook [Gra19, §3.5], among other sources.
+24
+
+• on horizontal arrows and cells, by the functor LV : FinGraph → Para(Dynam) on graphs and
+graph homomorphisms and as the identity on the associated cospans and cospan morphisms:
+�
+X, A0
+ℓ0
+−→ X(V )
+ℓ1
+←− A1
+�
+�→
+�
+LV(X), A0
+ℓ0
+−→ X(V )
+ℓ1
+←− A1
+�
+.
+The comparison cells are defined using the morphisms of linearly parameterized dynamical systems
+αS : Z(S) → LV(Disc S), where
+αS := (0S, 1S) : (∅, S, 0) → (S, S, ⃗
+LV(Disc S)),
+S ∈ FinSet.
+• Given composable open graphs (X, A → X(V ) ← B) and (Y, B → Y (V ) ← C), the comparison
+cell for horizontal composition is given by the morphism of systems
+LV(X) +Z(B) LV(Y )
+id +αB id
+−−−−−−→ LV(X) +LV(Disc B) LV(Y )
+∼
+=
+−→ LV(X +Disc B Y ).
+• Given a finite set A, the comparison cell for the horizontal unit is given by the morphism of
+systems αA : Z(A) → LV(Disc A).
+Similarly, there is a symmetric monoidal lax double functor
+LV : Open(FinSgnGraph) → Open(Para(Dynam+)).
+Proof. To construct the lax double functor, we use a lax version of [BC20, Theorem 4.3]. The
+family of morphisms αS : Z(S) → LV(Disc S), S ∈ FinSet, in the theorem statement assemble into
+a natural transformation
+FinSet
+FinGraph
+FinSet
+Para(Dynam)
+L=Disc
+LV
+L′=Z
+α
+.
+The functors involved in this cell all preserve finite colimits: the top and bottom ones because they
+are left adjoints and the right one by Theorem 3.7. The hypotheses of [BC20, Theorem 4.3] are
+therefore satisfied, except that α is not a natural isomorphism but merely a natural transformation.
+By inspection of the proof, the result still holds except that the resulting double functor is lax rather
+than pseudo. We obtain a lax double functor
+Open(FinGraph) ∼= LCsp(FinGraph) → L′Csp(Para(Dynam)) ∼= Open(Para(Dynam)).
+To show that this double functor is the same one in the theorem statement, we once again
+use the adjunctions to pass between L-structured and R-decorated cospans (recalling terminology
+introduced in the proof of Proposition 2.4). Notice that the natural transformation α has as its
+mate [CGR14, §1] the identity transformation ¯α = 1evV :
+FinSet
+FinGraph
+FinSet
+Para(Dynam)
+R=evV
+LV
+R′=πS
+¯α
+.
+25
+
+Thus the action of the double functor F := LV on L-structured cospans simplifies to the identity
+when translated to R-decorated cospans.
+L(A0)
+X
+L(A1)
+L′(A0)
+F(L(A0))
+F(X)
+F(L(A1))
+L′(A1)
+↭
+A0
+R(X)
+A1
+A0
+R(X)
+R′(F(X))
+R(X)
+A1
+ℓ0
+ℓ1
+αA0
+F(ℓ0)
+F(ℓ1)
+αA1
+¯ℓ0
+¯ℓ1
+¯ℓ0
+¯αX
+¯αX
+¯ℓ1
+A similar statement holds for the action of the double functor on morphisms of L-structured and
+R-decorated cospans.
+4
+Conclusion
+Summary.
+Regulatory networks are a minimalistic but widely used tool to describe the interactions
+between molecules in biochemical systems. We have made the first functorial study of regulatory
+networks, formalized as signed graphs, and their connections with other mathematical models in
+biochemistry. Among the latter, we have studied reaction networks, formalized as Petri nets with
+signed links, and parameterized dynamical systems, focusing on Lotka-Volterra dynamics. This
+project fits into a broader program by applied category theorists and other scientists aiming to
+systematize, in a completely precise way, the language and methods of describing, composing, and
+transforming scientific models.
+The major categories of this paper, and the functors between them, are summarized in the
+following diagram, where “LV” is the Lotka-Volterra dynamics functor (§3.2).
+SgnCat (§2.2)
+SgnGraph (§2.1)
+SgnPetri (§2.3)
+FinSgnGraph
+Para(Dynam+) (§3.1)
+Path
+U
+LV
+Net
+⊣
+Most of the main results extend from closed systems to open systems, which compose by gluing
+along their boundaries. Of the diagram above, we have extended the following parts to double
+categories of open systems and double functors between them.
+Open(SgnCat)
+Open(SgnGraph)
+Open(FinSgnGraph)
+Open(Para(Dynam+)) (§3.3)
+LV
+Path
+26
+
+Future work.
+Of many possible directions for future work, we mention a few. As noted in the
+introduction, Lotka-Volterra dynamics are only one of numerous dynamics that could be considered
+as a canonical model for regulatory networks, and they are not even among the most commonly
+studied in the biochemistry literature [TLK19]. It would be desirable to have dynamics functors for
+regulatory networks that draw on more flexible or more biologically plausible classes of dynamical
+systems. In another direction, the two halves of this paper—qualitative and quantitative—are not as
+tightly as integrated as one might hope. How does the presence of a motif in a regulatory network,
+such as an incoherent feedforward loop perhaps even of a specific type, manifest in the continuous
+dynamics of that network? Put in category-theoretic terms, the Lotka-Volterra dynamics functor is
+defined on signed graphs, so how does it relate to the freely generated signed categories in which
+motifs are expressed? These intriguing questions are suggestive of “feedback loop analysis” in the
+field of system dynamics [Ric95], to which stronger connections should be made.
+Acknowledgments.
+The authors thank the American Mathematical Society (AMS) for hosting
+the 2022 Mathematical Research Community (MRC) on Applied Category Theory, where this
+research project began. The AMS MRC was supported by NSF grant 1916439. We thank John Baez,
+our group’s mentor at the MRC, for suggesting this project and for much helpful advice along the
+way. Authors Fairbanks, Patterson, and Shapiro acknowledge subsequent support from the DARPA
+ASKEM and Young Faculty Award programs through grants HR00112220038 and W911NF2110323.
+Author Ocal acknowledges subsequent support from an AMS-Simons Travel Grant and from the
+Hausdorff Research Institute for Mathematics funded by the German Research Foundation (DFG)
+under Germany’s Excellence Strategy - EXC-2047/1 - 390685813.
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+Uhlmann, John J. Tyson, and Béla Novák. “Cell cycle regulation by feed-forward
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+Alfred J. Lotka. Elements of physical biology. Williams & Wilkins, 1925. url: https:
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+Saunders Mac Lane and Ieke Moerdijk. Sheaves in geometry and logic: A first introduction
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+polynomial dynamic models with biological applications. Academic Press, 2018. doi:
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+S. Stanhope, Jonathan E. Rubin, and David Swigon. “Identifiability of linear and linear-
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+John D. Sterman. Business dynamics: systems thinking and modeling for a complex
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+Ross Street. “The petit topos of globular sets”. Journal of Pure and Applied Algebra
+154.1-3 (2000), pp. 299–315. doi: 10.1016/S0022-4049(99)00183-8.
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+behavior of biochemical regulatory networks”. Journal of Theoretical Biology 462 (2019),
+pp. 514–527. doi: 10.1016/j.jtbi.2018.11.034.
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+Annual Review of Physical Chemistry 61 (2010), pp. 219–240. doi: 10.1146/annurev.
+physchem.012809.103457.
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+Eberhard O. Voit. Computational analysis of biochemical systems: A practical guide for
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+29
+
diff --git a/19AzT4oBgHgl3EQfe_xt/content/tmp_files/load_file.txt b/19AzT4oBgHgl3EQfe_xt/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf,len=892
+page_content='A compositional account of motifs, mechanisms, and dynamics in  biochemical regulatory networks  Rebekah Aduddell  James Fairbanks  Amit Kumar  Pablo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Ocal  Evan Patterson  Brandon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Shapiro  Abstract  Regulatory networks depict promoting or inhibiting interactions between molecules in a  biochemical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We introduce a category-theoretic formalism for regulatory networks, using  signed graphs to model the networks and signed functors to describe occurrences of one network  in another, especially occurrences of network motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' With this foundation, we establish functorial  mappings between regulatory networks and other mathematical models in biochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We  construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory  networks, enabling us to precisely define when a reaction network could be a physical mechanism  underlying a regulatory network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Turning to quantitative models, we associate a regulatory  network with a Lotka-Volterra system of differential equations, defining a functor from the  category of signed graphs to a category of parameterized dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We extend this  result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not  only inclusions and collapsings of regulatory networks, but also the process of building up complex  regulatory networks by gluing together simpler pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Formally, we use the theory of structured  cospans to produce a lax double functor from the double category of open signed graphs to that  of open parameterized dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Throughout the paper, we ground the categorical  formalism in examples inspired by systems biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  1  Introduction  The genes, proteins, and RNA molecules that comprise living cells interact in complex, varied ways  to sustain the cell throughout its lifecycle and respond to changes in its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Intensive  experimental study of these interactions is distilled in an idealized form as regulatory networks,  a kind of directed graph in which vertices represent molecules and edges represent interactions  between molecules (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The edges are labeled with a positive or negative sign according to  whether the interaction is activating or inhibiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Regulatory networks are the subject of a large  body of experimental and theoretical work, notably reviewed by Alon [Alo07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Alo19] and Tyson  et al [TN10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' TLK19] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Particular attention has been paid to network motifs [Alo07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  TN10], the simple but functionally meaningful patterns that recur frequently in regulatory networks,  and to various quantitative dynamics [TLK19] that can be assigned to the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Although regulatory networks are simple enough to define mathematically—we shall define them  to be directed graphs, possibly with multiple edges and loops, whose edges are assigned a positive  or negative sign—important scientific concepts involving them, such as occurrences of motifs in  networks and biochemical mechanisms generating networks, are often treated imprecisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Likewise  for relationships between regulatory networks and other mathematical models in biochemistry,  particularly dynamical models based on ordinary or stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Hence a first  aim of this paper is to put certain concepts and relations concerning regulatory networks on a firm  mathematical footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To do so, we will use methods from category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='01445v1  [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='MN]  4 Jan 2023  Ash1  Cdk1/ClbS  Sld2  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1: A small biochemical regulatory network: regulation of Sld2 by Cdk1 or ClbS with Ash1 as a  predicted transcription factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Adapted from Csikász-Nagy et al [Csi+09, Figure 3C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Category theory, in both the small and the large, is a natural tool for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In saying that  a motif occurs in a network, one should allow for the possibility that the occurrence is indirect,  involving a sequence of appropriately signed interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For example, positive autoregulation  can occur directly but also indirectly through a double-negative feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Since a small  category is exactly a graph in which consecutive edges can be composed, subject to certain laws,  regulatory networks should be viewed not only as signed graphs (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1) but also as signed  categories freely generated by those (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Sign-preserving functors, unlike sign-preserving  graph homomorphisms, can express indirect occurrences and are in this sense a better notion of  morphism for regulatory networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Here we are doing category theory in the small, using categories  as algebraic structures comparable to familiar ones like graphs, groups, and monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Having laid these foundations for regulatory networks, we turn to category theory in the large, a  mathematical theory of structure well suited to describe the passages between regulatory networks  and other mathematical models of biochemical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Formally speaking, these passages are  functors into or out of the category of regulatory networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Making a functor is significantly stronger  than making an objects-only mapping, as is typically done in the literature, since if morphisms  of signed graphs formalize relationships between different regulatory networks, then functorality  requires that these relationships be transported to or from other models of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By contrast,  an objects-only mapping is, abstractly speaking, entirely unconstrained and so is capable of acting  highly irregularly across different models of a given class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Functorality thus serves as a kind of  safeguard for model transformation: it does not, on its own, ensure that a transformation makes  good scientific sense but it does impose nontrivial logical constraints and coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A first illustration of this principle is the connection between regulatory networks and biochemical  reaction networks (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' When modeling the complex biochemical systems that constitute a  living cell, it is often practically necessary to abstract away certain details of the underlying chemical  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Regulatory networks generally do not capture all the species or reactions involved in  a given system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' nor can they capture multispecies reactions faithfully because they describe only  pairwise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Given that regulatory networks are, to some degree, phenomenological models,  it is natural to ask whether a given network could arise as a summary of a specific chemical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The latter are described by biochemical reaction networks, graph-like structures allowing reactions  or transitions with multiple inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Inspired by graphical syntax from systems biology  [Voi00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Voi13], we formalize reaction networks as “Petri nets with links,” and we construct a functor  from the category of Petri nets with signed links to the category of signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This functor  enables us to propose a formal definition for when a reaction network could be a mechanism for a  regulatory network, a concept that is rarely if ever treated in a precise way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  This concludes the content of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In Section 3, we turn from qualitative to quantitative  analysis, seeking a functorial assignment of continuous dynamics to regulatory networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Although  rarely made explicitly functorial, systematic ways to formulate a model belonging to a mathematically  homogeneous class of models are ubiquitous in science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Voit calls these “canonical representations”  2  or “canonical models,”1 and identifies Lotka-Volterra models and BST models/S-systems as two  prominent examples in biology [Voi13, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Reflecting their phenomenological status, regulatory  networks do not admit a single, obvious dynamical interpretation, and so a wide variety of dynamical  models have been considered, spanning the discrete and continuous, deterministic and stochastic  [TLK19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We consider the Lotka-Volterra systems of ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' While not  necessarily the most biologically plausible, it is among the simplest continuous models and hence a  natural place to begin a functorial study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A Lotka-Volterra system of equations has the form  ˙xi = ρi xi +  n  �  j=1  βi,j xi xj,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  or equivalently, has logarithmic derivatives that are affine functions of the state variables:  d  dt[log xi(t)] = ρi +  n  �  j=1  βi,j xj(t),  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The coefficients ρi specify baseline rates of growth or decay, according to their sign, and the  coefficients βi,j rates of activation or inhibition, according to their sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We construct a functor  that sends a signed graph (regulatory network) to a Lotka-Volterra model suitably constraining the  signs of the rate coefficients (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' As a prerequisite, we define a category of parameterized  dynamical systems (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1), a construction of intrinsic interest that is by no means confined to  Lotka-Volterra dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In order to comprehend complex biological systems, we must decompose them into small, readily  understandable pieces and then compose them back together to reproduce the behavior of the  original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This is the mantra of systems biology, which stresses that compositionality is no  less important than reductionism in biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' With this motivation, a secondary aim of this paper  is to extend the above constructions from closed systems to open ones, which can be composed  together by gluing them along their interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Mathematically, we pass from categories to double  categories [Gra19], two-dimensional categorical structures in which the usual morphisms of systems  compose along one direction (by convention, the “vertical” one) and open systems compose along  the other direction (the “horizontal” one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Among other results, we show that the Lotka-Volterra  dynamics functor extends to a lax double functor from the double category of open signed graphs  to a double category of open parameterized dynamical systems (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The mathematics developed here is motivated by biochemistry but need not be restricted to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Famously, Lotka-Volterra systems originated in ecology to model predator-prey dynamics [Lot25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Regulatory networks and Lotka-Volterra systems can be used as generic models of entities that  “regulate” each other in some manner, be it at the scale of individual cells or animal ecosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Regulatory networks are highly reminiscent of the causal loop diagrams in system dynamics [Ste00,  Chapter 5], where the latter explicitly label feedback loops and their polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The language of category theory is indispensable to this work but the level of knowledge assumed  by the reader is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We assume throughout that the reader is familiar with the basic  notions of category theory, such as categories, functors, and natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Our main  reference for facts about category theory is Riehl’s text [Rie16], although there are many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In the definitions and theorems, we have tried to minimize the technical level and explicate the  statements in concrete terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In the proofs, we have aimed for efficiency and freely use concepts  1This usage of “canonical” should not be confused with the unrelated, in fact incompatible, meaning of “canonical”  in pure mathematics, especially category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  3  and results from the literature that do not appear in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The reader can omit the proofs  without disrupting the continuity of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  2  Qualitative analysis: motifs and mechanisms  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1  Regulatory networks as signed graphs  To begin, we clarify the notion of graph to be used throughout in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The following definition  is standard among category theorists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In other fields, it might be called a “directed multigraph,”  but we will call it simply a “graph.”  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1 (Graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The schema for graphs is the category Sch(Graph) freely generated  by two parallel morphisms:  V  E  src  tgt  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A graph is a functor X : Sch(Graph) → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A graph homomorphism from a graph X to another  graph Y is a natural transformation φ : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Graphs and graph homomorphisms form the  category Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To restate the definition in explicit terms, a graph X consists of  a set X(V ) of vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  a set X(E) of edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' and  functions X(src), X(tgt) : X(E) → X(V ), assigning to each edge its source and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A graph homomorphism φ : X → Y consists of a function φV : X(V ) → Y (V ), the vertex map,  and another function φE : X(E) → Y (E), the edge map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' These maps must preserve sources and  targets, meaning that the following squares commute:  X(E)  X(V )  Y (E)  Y (V )  X(src)  φE  Y (src)  φV  X(E)  X(V )  Y (E)  Y (V )  X(tgt)  φE  Y (tgt)  φV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We now turn to the main notion of this section, signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Write Sgn for the set of (nonzero)  signs, whose two elements may be denoted {1, −1} or {+, −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The set of signs is an abelian group,  isomorphic to the cyclic group Z2, under the usual multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2 (Signed graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of signed graphs is the slice category  SgnGraph := Graph/Sgn,  where, by abuse of notation, Sgn is regarded as a graph with one vertex and two loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Unpacking the definition, a signed graph is seen to be a graph X equipped with a function  X(sgn) : X(E) → Sgn that assigns a sign to each edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Given signed graphs X and Y , a morphism  of signed graphs from X to Y is a graph homomorphism φ that preserves signs, meaning that  the following triangle commutes:  X(E)  Y (E)  Sgn  X(sgn)  Y (sgn)  φE  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  4  Signed graphs are a mathematical description of the regulatory networks studied in systems  biology [Alo07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' TN10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  For the purposes of this paper, we will simply define a regulatory  network to be a signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The vertices of the graph represent the components of the network,  which could be proteins, genes, or RNA molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Signed edges represent interactions between  components, where the source has the effect of either activating/promoting the target (positive sign)  or inhibiting/repressing it (negative sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' As is customary, we denote activation interactions by  arrows with pointed heads (−→) and inhibition interactions by arrows with flat heads (−−⊣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For  instance, the two drawings  x  y  +  −  ↭  x  y  represent the same network, a negative feedback loop in which x activates y, which in turn inhibits  x [TN10, Scheme 1, Motif B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In the literature [TN10], regulatory networks are often modeled as sign-valued matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This  approach is a special case of ours in that an n-by-n matrix valued in {+1, −1, 0} can be interpreted  as a simple signed graph on n vertices, with signed edges defined by the nonzero matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Unlike the matricial formalism, our formalism allows multiple edges between the same pair of edges,  which can model multiple interactions based on different mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Allowing multiple edges and  self-loops also ensures that graphs and signed graphs form well behaved categories, as the following  proposition shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of signed graphs is complete (has all limits) and cocomplete (has  all colimits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Because Graph is a copresheaf category, it is complete and cocomplete [Rie16, Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The slice category SgnGraph = Graph/Sgn is hence also complete and cocomplete [Rie16,  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' alternatively, this follows because slices of copresheaf categories are again  (equivalent to) copresheaf categories [Str00, Remark p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' 303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Colimits of signed graphs can be used to construct a category, or rather a double category, of  open signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Composition of open signed graphs formalizes the process of building large  regulatory networks from smaller pieces, including network motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4 (Open signed graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a symmetric monoidal double category of open  signed graphs, Open(SgnGraph), having  as objects, sets A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as vertical arrows, functions f : A → B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as horizontal arrows, open signed graphs, which consist of a signed graph X together with  a cospan of sets A0  ℓ0  −→ X(V )  ℓ1  ←− A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as cells, morphisms of open signed graphs (X, ℓ0, ℓ1) → (Y, m0, m1), which consist of a  map of signed graphs φ : X → Y along with functions fi : Ai → Bi, i = 0, 1, making the  following diagram commute:  A0  X(V )  A1  B0  Y (V )  B1  ℓ0  f0  ℓ1  φV  m0  f1  m1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  5  Vertical composition is by composition in Set and in SgnGraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Horizontal composition and monoidal  products are by pushouts and coproducts in SgnGraph, respectively, viewing the sets in the feet of the  cospans as discrete signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To construct this symmetric monoidal double category, we use the method of structured  cospans [FS07] in its double-categorical form [BC20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The categories of sets and of signed graphs  are related by an adjoint pair of functors  Set  SgnGraph  Disc  evV  ⊣  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Here evV : SgnGraph → Set is the evaluation at V functor, sending a signed graph X to its set of  vertices X(V ) and a morphism of signed graphs φ to its vertex map φV , and Disc : Set → SgnGraph  is the discrete signed graph functor, sending a set A to the signed graph with vertex set A and no  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We obtain a symmetric monoidal double category of open signed graphs as the L-structured  cospans for the functor L := Disc : Set → SgnGraph [BC20, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To show that this symmetric monoidal double category is the same one in the proposition  statement, suppose that L ⊣ R : A → X is an adjoint pair of functors, where in our application  L = Disc and R = evV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By the defining bijection of an adjunction, L-structured cospans, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=',  objects A0 and A1 in A together with a cospan L(A0) → X ← L(A1) in X, correspond exactly  to “R-decorated cospans,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=', an object X in X together with a cospan A0 → R(X) ← A1 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Furthermore, by the naturality of this bijection [Rie16, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3], morphisms of L-structured  and R-decorated cospans  L(A0)  X  L(A1)  L(B0)  Y  L(B1)  L(f0)  φ  L(f1)  ↭  A0  R(X)  A1  B0  R(Y )  B1  f0  R(φ)  f1  related by the adjunction are equivalent in that one diagram commutes if and only if the other does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We will tacitly reuse this reasoning in future constructions, such as Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A morphism of signed graphs can do two things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Most obviously, it can pick out a signed graph  as a subobject of another one, via a sign-preserving subgraph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A signed graph morphism  can also collapse multiple vertices onto a single vertex, and multiple edges onto a single edge with  the same sign, in the fairly restrictive sense permitted by a graph homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To illustrate,  consider the following morphism inspired by Alon’s review [Alo07, Figure 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  argR  argCBH  argD  argE  argF  argI  −→  argR  arg∗  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1)  The network in the domain is a “single-input module” in the arginine biosynthesis system, in which  the regulator argR represses five different enzymes (argCHB, argD, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=') involved in producing  arginine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The morphism above forgets the distinction between these enzymes, collapsing them  into a catch-all entity labeled “arg∗”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' These two functions—embedding and collapsing—are all  that a signed graph morphism can do, because any such morphism factors essentially uniquely as  6  an epimorphism (morphism with surjective vertex and edge maps) followed by a monomorphism  (morphism with injective vertex and edge maps), using the epi-mono factorization available in any  copresheaf category, or more generally in any topos [MM94, §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2  Refining regulatory networks using signed categories and functors  While morphisms of signed graph have their uses, they do not capture the important idea of refining  regulatory networks, in which an interaction in one network is realized as a composite of several  interactions in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To express refinement, we must generalize our notion of morphism from  graph homomorphisms to functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This, in turn, requires the concept of a signed category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5 (Signed categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of signed categories is the slice category  SgnCat := Cat/Sgn,  where Cat is the category of small categories and the group of signs, Sgn, is regarded as a category  with one object and two morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Unpacking the definition, a signed category is a category C in which every morphism f is  assigned a sign sgn(f) ∈ {1, −1} in a functorial way, meaning that  sgn(x0  f1  −→ x1  f2  −→ · · ·  fn  −→ xn) =  n  �  i=1  sgn(fi)  for every n ≥ 0 and every sequence of composable morphisms f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' , fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In particular (n = 0), the  identity morphisms have positive sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A morphism of signed categories, or signed functor,  is a functor F : C → D between signed categories that preserves the signs, meaning that  sgnD(F(f)) = sgnC(f)  for every morphism f in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Since our aim is to have a more flexible notion of morphism between signed graphs, we will  mostly restrict ourselves to those signed categories that are freely generated by a signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The  free signed category or signed path category functor  Path : SgnGraph → SgnCat  sends a signed graph X to the signed category Path(X) having  as objects, the vertices of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as morphisms from x to y, the paths in X from x to y, whose sign is defined to be the product  of the signs of the edges comprising the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Composition of paths is by concatenation, which clearly preserves the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The identity morphism  at x is the empty path at x, which has positive sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By convention, if X and Y are signed graphs,  we say that a signed functor from X to Y is a signed functor F : Path(X) → Path(Y ) between  the corresponding signed path categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Since the morphisms of Path(X) are freely generated  by the edges in X, a signed functor from X to Y is uniquely determined by a morphism of signed  graphs from X to the underlying signed graph of Path(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This means that each edge in X is  sent to an appropriately signed path of edges in Y , which can be regarded as a refinement of the  relationship that the edge represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  7  Motif  Generic instance  Positive autoregulation  L+ :=  � �  Negative autoregulation  L− :=  � �  Coherent feedforward loop  I++ :=  � �  Incoherent feedforward loop  I± :=  � �  Positive feedback loop  L++ :=  � �  Negative feedback loop  L± :=  � �  Double-negative feedback loop  L−− :=  �• �  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1: Common motifs in biochemical regulation networks [Alo07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' TN10]  We now have a precise language with which to classify network motifs and their occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  As a first example, Alon identifies four types of incoherent feedforward loop (FFL) involving three  components,  x  x  x  x  y  y  y  y  z  z  z  z  ,  those of type 1, 2, 3, and 4, respectively [Alo07, Figure 2a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Besides having three components, what  these motifs have in common is that there exists a signed functor into each of them from the signed  graph I± :=  � �  having two parallel arrows of opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The network I± is thus the  “generic” incoherent feedforward loop, in the sense that signed functors out of it refine the pattern  in specific ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A similar situation holds for other common network motifs (Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1), which  motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6 (Motif instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Given a signed graph A, regarded as a motif, an instance or  occurrence of the motif A in a network X is a monic signed functor A ↣ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Note that a signed functor is a monomorphism exactly when the functor is an embedding of  categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=', an injective-on-objects, faithful functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Requiring the functor in the definition to  be monic excludes “degenerate instances” of motifs where vertices or edges are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Now, should the incoherent FFL be regarded as a network motif, or is it the more specific types,  such as the incoherent FFL of type 1, that are motifs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' From our point of view, they are all equally  motifs but they have different degrees of specificity, and the functorial language clarifies how motifs  are iteratively refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Specifically, an instance of an incoherent FFL of type 1 in a network X  also gives an instance of an incoherent FFL in X (of unspecified type), simply by composing the  monomorphisms involved:  I± ∼=  �  x  z  �  ↣  �  x  y  z  �  ↣  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  8  Similarly, in the notation of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1, any instance of double-negative feedback (L−−) also gives an  instance of positive autoregulation (L+) [CP09], via the monomorphism L+ ↣ L−− that sends the  positive loop to the double-negative 2-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  For any choice of motif A, the mapping that sends a regulatory network X to the set of  occurrences of A in X is a functor  HomSgnCatm(Path(A), Path(−)) : SgnGraphm → Set,  where SgnGraphm and SgnCatm denote the wide subcategories of monomorphisms in SgnGraph and  SgnCat, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This functor is almost, but not quite, representable, due to the distinction  between signed graphs and signed categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' More importantly, the existence of this functor means  that a monomorphism between regulatory networks induces a map between instances of A, for any  motif A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We now extend the construction of open signed graphs to open signed categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of signed categories is complete and cocomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Because the category Cat is complete and cocomplete [Rie16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6], its slice  SgnCat = Cat/Sgn is also [Rie16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='8 (Open signed categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a symmetric monoidal double category of open  signed categories, Open(SgnCat), having  as objects, sets A, B, C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as vertical arrows, functions f : A → B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as horizontal arrows, open signed categories, which consist of a signed category C together  with a cospan of sets A0  ℓ0  −→ Ob(C)  ℓ1  ←− A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as cells, morphisms of open signed categories (C, ℓ0, ℓ1) → (D, m0, m1), which consist of  a signed functor F : C → D along with functions fi : Ai → Bi, i = 0, 1, making the diagram  commute:  A0  Ob(C)  A1  B0  Ob(D)  B1  ℓ0  f0  ℓ1  Ob(F)  m0  f1  m1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Vertical composition is by composition in Set and in SgnCat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Horizontal composition and monoidal  products are by pushouts and coproducts in SgnCat, respectively, viewing the sets in the feet of  cospans as discrete signed categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Moreover, the signed path category functor extends to a symmetric monoidal double functor  Path : Open(SgnGraph) → Open(SgnCat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We take Open(SgnCat) to be the symmetric monoidal double category of L′-structured  cospans for the functor L′ := Disc : Set → SgnCat involved the composite adjunction  Set  SgnCat  =  Set  SgnGraph  SgnCat  Disc  Ob  Disc  evV  U  Path  ⊣  ⊣  ⊣  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  On the right hand side, the first adjunction was already used in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4, and  the second adjunction is the free-forgetful adjunction between signed graphs and signed categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  9  To prove the last statement, we notice that all functors involved in the commutative square  Set  SgnGraph  Set  SgnCat  L=Disc  L′=Disc  Path  are left adjoints, hence preserve colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We can therefore appeal to [BC20, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3] to obtain  a symmetric monoidal double functor  Open(SgnGraph) ∼= LCsp(SgnGraph) → L′Csp(SgnCat) ∼= Open(SgnCat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3  Mechanistic models as Petri nets with links  However challenging they may be to identify through experiments and data analysis, regulatory  networks still only summarize how the components of a complex biochemical system interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Regulatory networks typically include only a subset of the system’s components, and they do not  model individual reactions and processes, only pairwise promoting or inhibiting interactions between  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In this sense, regulatory networks are not fully mechanistic models, even if they have  a stronger causal interpretation than, say, a correlation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  By contrast, mechanistic models in biochemistry model individual reactions, which requires a  different formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Pictures like the following, adapted from Voit’s review [Voi13, Figure 4], are  common in systems biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A  B  D  C  −  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2)  This diagram possesses two distinctive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' First, directed hyperedges represent reactions  having a number of inputs or outputs different than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There are, for example, hyperedges from  B and C to D, from nothing to A (an inflow), and from D to nothing (an outflow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' If, in lieu of  hyperedges, we introduce a second type of vertex, we obtain a structure similar to a Petri net  A  B  C  D  −  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3)  but with the second distinctive feature of having signed links from the first type of vertices (species)  to the second type (transitions), whose signs indicate promotion or inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In this section, we explain how a Petri net with signed links can provide a mechanism for a  regulatory network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This involves constructing a functor from Petri nets with signed links to signed  graphs, approximating the former as the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' As a prerequisite, we need a rigorous definition of a  Petri net with links, which seems to be absent from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  10  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='9 (Petri net with links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The schema for Petri nets with links is the category  Sch(LPetri) freely generated by these objects and morphisms:  I  S  O  T  L  srcL  tgtL  srcI  tgtI  tgtO  srcO  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A Petri nets with links is a functor P : Sch(LPetri) → Set, and a morphism of these is a natural  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A morphism φ : P → Q preserves arities if the naturality squares associated  with the morphisms I → T and O → T are also pullback squares:  P(I)  P(T)  Q(I)  Q(T)  P(tgtI)  φI  φT  P(tgtI)  ⌟  P(O)  P(T)  Q(O)  Q(T)  P(srcO)  φO  φT  P(srcO)  ⌟  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Petri nets with links and their morphisms form the category LPetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To explicate the definition, a Petri net with links P consists of  a set P(S) of species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  a set P(T) of transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  a set P(I) of input arcs going from species to transitions, via maps P(srcI) : P(I) → P(S)  and P(tgtI) : P(I) → P(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  a set P(O) of output arcs going from transitions to species, via maps P(srcO) : P(O) → P(T)  and P(tgtO) : P(O) → P(S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' and finally  a set P(L) of links going from species to transitions, via maps P(srcL) : P(L) → P(S) and  P(tgtL) : P(L) → P(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The property of preserving arities, called “etale” by Kock [Koc22, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4], means that a morphism  φ : P → Q of Petri nets with links preserves the input and output arities of all transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Namely,  for each transition t in the net P the map φI : P(I) → Q(I) restricts to a bijection between the  input arcs to t and to φT (t), and similarly the map φO : P(O) → Q(O) restricts to a bijection  between the output arcs from t and from φT (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This property seems appropriate for many purposes,  including in biochemistry, but for mathematical convenience we will not always assume it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='10 (Related literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Our definition of a Petri net with links, while apparently novel,  is the obvious joint generalization of two existing concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Kock has described Petri nets as  copresheaves on a category with objects S, T, I, O [Koc22], calling them whole-grain Petri nets to  distinguish them from classical Petri nets, whose semantics are subtly different [Bae+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Meanwhile,  the concept of a link is essential to stock and flow diagrams, originating in the field of system  dynamics [For61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Ste00] and recently given a rigorous categorical account [Bae+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We also note  that the Petri nets with catalysts proposed by Baez, Foley, and Moeller [BFM19] differ significantly  from Petri nets with links: the former fix a subset of the species to be catalysts throughout the net,  whereas the latter uses links to make catalyzation specific to individual reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  11  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='11 (Petri nets as typed graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Like bare Petri nets, Petri nets with links can be described  as graphs with two types of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To see this, take the graph  TLPetri :=  �  �  �  �  �  �  �  S  T  I  L  O  �  �  �  �  �  �  �  with vertices S and T and edges I, O, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of Petri nets with links and natural  transformations is isomorphic to the slice category Graph/TLPetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Moreover, the schema category  Sch(LPetri) is isomorphic to the category of elements of the functor TLPetri : Sch(Graph) → Set,  exemplifying a general fact about slices of copresheaf categories [Str00, Remark p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' 303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Petri nets with signed links are defined analogously to signed graphs (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='12 (Petri nets with signed links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of Petri nets with signed links  is the slice category  SgnPetri := LPetri/PSgn,  where PSgn is the Petri net with links having one species, one transition, one input arc, one output  arc, and two links, namely the elements of Sgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Incidentally, the morphism P → PSgn defining a Petri net with signed links does not preserve  arities unless every transition in P has exactly one input and one output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We now turn to the main construction of this section, a functor that “approximates” a Petri net  with signed links as a signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' On the example in Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3), this functor gives  the signed graph  A  B  D  C  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4)  In general, the resulting signed graph has, as vertices, the Petri net’s species and has signed edges  for each of the four cases:  (a) for every input-output pair to a transition, a positive edge from input to output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  (b) for every input to a transition, a negative self loop, representing consumption by the reaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  (c) for every signed link, an edge of opposite sign going from the linked species to each input to  the linked transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  (d) for every signed link, an edge of equal sign going from the linked species to each output from  the linked transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  All four cases are visible in the example of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Set-theoretically, each of these cases is  the result of a conjunctive query, or equivalently of a representable functor  HomLPetri(P, −) : LPetri → Set  associated with a particular Petri net with links P, the generic instance for that query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The four  generic instances we need are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Their sum is a disjoint union of conjunctive  queries, or duc-query for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  12  (a) Input-output pair to tran-  sition  (b) Input to transition  (c) Input to transition with  incident link  (d) Output from transition  with incident link  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1: Four different Petri nets with links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For each of these instances P, evaluating the representable  functor HomLPetri(P, −) : LPetri → Set gives the edges for one case in the case analysis that defines the functor  from Petri nets with signed links to signed graphs (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To make the construction just sketched on objects fully precise and functorial, we use Spivak’s  theory of functorial data migration based on duc-queries [Spi21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In order to apply it, we fully  schematize the definitions of signed graphs and Petri nets with signed links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The schema for signed graphs is the category Sch(SgnGraph) freely generated by these objects  and morphisms:  V  E  A  neg  src  tgt  sgn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A signed graph as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2 is equivalent to a functor X : Sch(SgnGraph) → Set such that  X(A) = Sgn, the set of signs, and X(neg) : Sgn → Sgn is negation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=', multiplication by −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Note  that negation is not needed to define the data of a signed graph but is relevant to the data migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A morphism of signed graphs X → Y is a natural transformation φ : X → Y whose component at  A is the identity function, φA = 1Sgn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We thus obtain a category isomorphic to SgnGraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Similarly, the schema for Petri nets with signed links is the category Sch(SgnPetri) freely  generated by:  S  I  O  L  A  T  neg  srcL  tgtL  srcI  tgtI  tgtO  srcO  sgn  one  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A Petri net with signed links, as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='12, is equivalent to a functor P : Sch(SgnPetri) → Set  such that P(A) = Sgn, the map P(neg) : Sgn → Sgn is negation, and P(one) : P(T) → Sgn is the  constant map at +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Again, these maps are needed for data migration, not for the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A  morphism of Petri nets with signed links is a natural transformation φ : P → Q such φA = 1Sgn,  yielding a category isomorphic to SgnPetri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='13 (Regulatory net induced by Petri net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A functor  Net : SgnPetri → SgnGraph  13  is specified by the following functor from Sch(SgnGraph) to the category of duc-queries on Sch(SgnPetri):  Sch(SgnGraph) → ⨿  ��  SetSch(SgnPetri)�op�  V �→ S  E �→ I ×T O + I + I ×T L + O ×T L  A �→ A  src �→ [srcI ◦πI, srcI, srcL ◦πL, srcL ◦πL]  tgt �→ [tgtO ◦πO, srcI, srcI ◦πI, tgtO ◦πO]  sgn �→ [one ◦πT , neg ◦ one ◦ tgtI, neg ◦ sgn ◦πL, sgn ◦πL]  neg �→ neg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We will define the functor Net : SgnPetri → SgnGraph as the restriction of a functor  SetD → SetC between the categories of copresheaves on D := Sch(SgnPetri) and C := Sch(SgnGraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In fact, the functor SetD → SetC will be of the special kind known as a parametric right adjoint  [Str00, Definition p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' 311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  According to the theory of data migration [Spi21, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6], giving a parametric right  adjoint SetD → SetC is equivalent to giving a functor from C to ⨿((SetD)op), the free coproduct  completion of the free limit completion of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Our functor C → ⨿((SetD)op) is defined by Equa-  tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The assignment of E ∈ C can also be described as the sum of the four representables  associated with the Petri nets with links in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Finally, the assignments A �→ A and neg �→ neg ensure that if P is a copresheaf on D with  P(A) = Sgn and P(neg) is negation, then applying this parametric right adjoint functor to P yields  a copresheaf X on C where again X(A) = Sgn and X(neg) is negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Thus, this functor between  copresheaf categories restricts to a functor SgnPetri → SgnGraph as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  With this construction, we can give a formal account of what it means to have a mechanistic  model for a regulatory network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='14 (Mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A mechanistic model for a regulatory network X is a Petri  net with signed links P together with an occurrence of X in Net(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=', a monic signed functor  X ↣ Net(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  For example, the Petri net with signed links in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3) is a mechanistic model for a  regulatory network in which A and D participate in a positive feedback loop:  A  D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  3  Quantitative analysis: parameters and dynamics  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1  Parameterized dynamical systems  Pioneering the idea of functorial semantics for scientific models, Baez and Pollard extended the  mass-action model of reaction networks to a functor from the category of Petri nets with rates  into a category of dynamical systems [BP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In this picture, the reaction rate coefficients are  known constants associated with the reaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In practice, however, rate coefficients are  often unknown and must be extracted from existing literature or estimated from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We therefore change our perspective slightly and consider dynamical systems not in isolation but  as parameterized families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This shift also turns out to have formal advantages: the category of  14  parameterized dynamical systems is better behaved than the category of dynamical systems, which  has too few morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To begin, we recall the dynamics functor, nearly identical to Baez-Pollard’s [BP17, Lemma 15]:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1 (Dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a functor Dynam : FinSet → VectR that sends  a finite set S to the vector space of algebraic vector fields v : RS → RS, where algebraic  means that the components of the vector field are polynomials in the state variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  a function f : S → S′ between finite sets to the linear transformation  (v : RS → RS)  �→  (f∗ ◦ v ◦ f∗ : RS′ → RS′),  where the linear map f∗ : RS′ → RS is the pullback along f  f∗(x′)(i) := x′(f(i)),  x′ ∈ RS′, i ∈ S,  and the linear map f∗ : RS → RS′ is the pushforward along f  f∗(x)(i′) :=  �  i∈f−1(i′)  x(i),  x ∈ RS, i′ ∈ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The functor Dynam : FinSet → VectR can be constructed as the composite  FinSet  ⟨D,F⟩  −−−−→ Vectop  R × VectR  Poly(−,−)  −−−−−−→ VectR,  where F : FinSet → VectR is the free vector space functor (restricted to finite sets);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' D : FinSetop →  VectR is the dual vector space functor (restricted to F), whose underlying set-valued functor is  VectR(F(−), R) ∼= Set(−, R) : FinSetop → Set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  and Poly(−, −) is the VectR-enriched hom-functor that sends a pair of vector spaces to the vector  space of polynomial maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2  The dynamics functor is the same one studied by Baez and Pollard except that we take the  vector space, rather than merely the set, of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' That is because we are interested in  linearly parameterized dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In calling the functor “dynamics,” we implictly identify a  vector field with the differentiable dynamical system that it generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This common practice is not  entirely innocent since even when a system of differential equations depends linearly on parameters,  its solutions rarely do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We also note that the restriction to algebraic vector fields, as opposed to  smooth or even just continuous ones, is inessential but suffices for us and agrees with Baez-Pollard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The dynamics functor is the main building block in constructing a category of parameterized  dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2 (Linear parameterizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of linearly parameterized dynami-  cal systems is the comma category  Para(Dynam) := F/ Dynam,  where F : FinSet → VectR, X �→ RX is the free vector space functor restricted to finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  2For a coordinate-free description of polynomial maps between vector spaces, see [Car71, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  15  So, by definition, a linearly parameterized dynamical system consists of a finite set P, the  parameter variables, and a finite set S, the state variables, together with a linear map  v : RP → Dynam(S)  sending each choice of parameters θ ∈ RP to an algebraic vector field v(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In more conventional  notation, we can write v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' θ) := v(θ)(x) for x ∈ RS and θ ∈ RP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A morphism (P, S, v) → (P ′, S′, v′)  of linearly parameterized dynamical systems is a pair of functions q : P → P ′ and f : S → S′  making the square  RP  Dynam(S)  RP ′  Dynam(S′)  v  f∗◦(−)◦f∗  v′  q∗  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1)  commute, or equivalently making the equation  f∗(v(f∗(x′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' θ)) = v′(x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' q∗(θ))  hold for all x′ ∈ RS′ and θ ∈ RP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  While certainly not all dynamical models depend linearly on their parameters, a great many  of them do, including several important canonical models in biology and chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The law of  mass action defines dynamical systems that depend linearly on the rate coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The generalized  Lotka-Volterra equations, studied in the next section, are linear in the rate and affinity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Of course, the mass-action and Lotka-Volterra equations are nonlinear ODEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' linearity of a vector  field in state or in parameters are separate matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Nevertheless, even for nonlinear models such  as Lotka-Volterra, linearity in parameters is a useful assumption that aides in the identifiability  analysis of the model [SRS14, §5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To express important physical constraints and to define a semantics for signed graphs, we will  restrict the dynamical system and its parameters to be nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This is straightforward enough  but requires a bit of additional formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Write R+ := {x ∈ R : x ≥ 0} for the semiring of nonnegative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A module over R+  is called a conical space, and the category of conical spaces and conic-linear (R+-linear) maps is  denoted Con := ModR+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A conical space is a structure in which one can take linear combinations  with nonnegative real coefficients, just as a real vector space (R-module) is a structure in which  one can take linear combinations with arbitrary real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Any convex cone in a real vector  space is a conical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Our main example is the nonnegative orthant of RS for some set S: the  function space RS  + := {x : S → R+}, with conical combinations taken pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A real vector space  can itself be regarded as a conical space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' more precisely, the inclusion of semirings R+ �→ R induces  a forgetful functor VectR → Con by restriction of scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Recall that a dynamical system is nonnegative if whenever the initial condition is in the  nonnegative orthant, its trajectory always remains in the nonnegative orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' A dynamical system  of form ˙x = v(x) is nonnegative if and only if vi(x) ≥ 0 whenever x ≥ 0 componentwise and  xi = 0 [HCH10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1], in which case the vector field v is called essentially nonnegative  [HCH10, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Using this criterion, it is easy to see that a reaction network with mass-  action kinetics is nonnegative assuming the rate constants are nonnegative, as is a Lotka-Volterra  system for any choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Hence both systems satisfy the obvious physical constraint  that no species should have negative concentration or population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3 (Nonnegative dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a functor Dynam+ : FinSet → Con that sends a  finite set S to the conical space of essentially nonnegative, algebraic vector fields v : RS → RS and  sends a function f : S → S′ to the transformation v �→ f∗ ◦ v ◦ f∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The proof is similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' It is clear that the essentially nonnegative functions  are stable under pointwise conical combinations, hence form a conical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' (They are, of course, not  stable under arbitrary linear combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=') We just need to check that if v : RS → RS is essentially  nonnegative, then so is the transformed vector field f∗ ◦ v ◦ f∗ : RS′ → RS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Fix x′ ∈ RS′  + and i′ ∈ S′,  and suppose that x′(i′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For every i ∈ f−1(i′), we have f∗(x′)(i) = x′(f(i)) = x′(i′) = 0 and so  v(f∗(x′))(i) ≥ 0, whence the inequality of essential nonnegativity follows:  (f∗ ◦ v ◦ f∗)(x′)(i′) =  �  i∈f−1(i′)  v(f∗(x′))(i) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We now define the conical analogue of linearly parameterized dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4 (Conical parameterizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category of conically parameterized non-  negative dynamical systems is the comma category  Para(Dynam+) := F+/ Dynam+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  where F+ : FinSet → Con, X �→ RX  + is the free conical space functor restricted to finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  So, a conically parameterized nonnegative dynamical system consists of finite sets P and S  together with a conic-linear map  v : RP  + → Dynam+(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5 (Colimits of parameterized dynamical systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The categories of linearly and  conically parameterized dynamical systems are finitely cocomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Moreover, these finite colimits  are computed by colimits in FinSet of the parameter variables and of the state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The category FinSet has finite colimits and the functors F : FinSet → VectR and F+ :  FinSet → Con preserve finite colimits, since they are composites of the inclusion FinSet �→ Set with  the left adjoints  Set  VectR  F  U  ⊣  and  Set  Con  F+  U+  ⊣  to the underlying set functors on vector spaces and conical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6 below, the comma  categories Para(Dynam) = F/ Dynam and Para(Dynam+) = F+/ Dynam+ have finite colimits,  which are preserved by the projection functors onto FinSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To illustrate, we describe the initial object and binary coproducts in Para(Dynam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The initial  linearly parameterized dynamical system has no parameter variables, no state variables, and the  unique (trivial) vector field on the zero vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The coproduct of two linearly parameterized  dynamical systems (P1, S1, v1) and (P2, S2, v2) has parameter variables P1 + P2, state variables  S1 + S2, and parameterized vector field  RP1+P2 ∼= RP1 ⊕ RP2  v1⊕v2  −−−−→ Dynam(S1) ⊕ Dynam(S2)  [Dynam(ι1),Dynam(ι2))]  −−−−−−−−−−−−−−−→ Dynam(S1 + S2),  where ι1 : S1 → S1 +S2 and ι2 : S2 → S1 +S2 are the canonical inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In conventional notation,  the coproduct system has parameterized vector field  v  ��  x1  x2  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  �  θ1  θ2  ��  =  �  v1(x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' θ1)  v2(x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' θ2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5, as well as of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='8 below, depends on the following  technical lemma about comma categories, which the reader can omit without loss of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  17  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6 (Colimits in comma categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Let C0  F0  −→ C  F1  ←− C1 be a cospan of categories such  that C0 and C1 have colimits of shape J and F0 preserves J-shaped colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Then the comma category  F0/F1 also has J-shaped colimits, and the projection functors πi : F0/F1 → Ci, i = 0, 1, preserve  those colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Furthermore, a functor G : X → F0/F1 into the comma category preserves J-shaped colimits  whenever the associated functors Gi := πi ◦ G : X → Ci, i = 0, 1, do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Colimits in the comma category F0/F1 are constructed in [RB88, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To make the rest of  the proof self-contained, we recall the construction here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  By the universal property of the comma category, a diagram D : J → F0/F1 is equivalent to  diagrams Di := πi ◦ D : J → Ci, i = 0, 1, along with a natural transformation ⃗D : F0 ◦ D0 ⇒ F1 ◦ D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Let (ci, λi) be a colimit cocone for the diagram Di in Ci, having legs Di(j)  λi  j  −→ ci for each j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The family of morphisms  F0(D0(j))  ⃗Dj  −−→ F1(D1(j))  F1(λ1  j)  −−−−→ F1(c1),  j ∈ J,  is then a cocone under F0 ◦ D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Since F0 preserves J-shaped limits, (F0(c0), F0 ∗ λ0) is a colimit  cocone for F0 ◦ D0, so by its universal property, there exists a unique morphism f : F0(c0) → F1(c1)  making the squares commute:  F0(D0(j))  F1(D1(j))  F0(c0)  F1(c1)  ⃗Dj  f  F0(λ0  j)  F1(λ1  j) ,  j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Setting λ := (λ0  j, λ1  j)j∈J, the cocone ((c0, c1, f), λ) can be shown to be a colimit of the diagram D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To prove the last statement about colimit preservation, let D : J → X be a diagram with colimit  cocone (x, λ), having legs Dj  λj  −→ y for j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We must show that its image cocone (G(x), G ∗ λ) is  a colimit of the diagram G ◦ D in F0/F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By the universal property of the comma category, the  functor G : X → F0/F1 is equivalent to the functors Gi : X → Ci, i = 0, 1, along with a natural  transformation ⃗G : F0 ◦ G0 ⇒ F1 ◦ G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The image cocone (G(x), G ∗ λ) then amounts to cocones  (G0(x), G0 ∗ λ) and (G1(x), G1 ∗ λ), which by hypothesis are colimits of the diagrams G0 ◦ D and  G1 ◦ D in C0 and C1, together with a family of commutative squares in C:  F0(G0(Dj))  F1(G1(Dj))  F0(G0(x))  F1(G1(x))  ⃗GDj  F0(G0(λj))  ⃗Gx  F1(G1(λj)) ,  j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  But a morphism ⃗Gx making all these squares commute is already uniquely determined by the  universal property of the colimit cocone (F0(G0(x)), F0 ∗ G0 ∗ λ) for the diagram F0 ◦ G0 ◦ D, using  the hypothesis that F0 preserves J-shaped colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Indeed, this is precisely how one constructs the  colimit of the diagram G ◦ D in F0/F1, as shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' It follows that (G(x), G ∗ λ) is a colimit  cocone for G ◦ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2  The Lotka-Volterra dynamical model  A Lotka-Volterra system with n species has, using matrix notation, the vector field  v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) := x ⊙ (ρ + βx) = diag(x)(ρ + βx)  with state vector x ∈ Rn and arbitrary real-valued parameters ρ ∈ Rn and β ∈ Rn×n [SMH18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In coordinates, the vector field is  vi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = xi  �  �ρi +  n  �  j=1  βi,jxj  �  � = ρixi +  n  �  j=1  βi,jxixj,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The parameter ρi sets the baseline rate of growth (when positive) or decay (when negative) for  species i, whereas βi,j defines a promoting (when positive) or inhibiting (when negative) effect of  species j on species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In typical applications the signs of the parameters are fixed and known in  advance of any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For example, in the famous predator-prey Lotka-Volterra system  ˙x = ax − bxy  ˙y = dxy − cy,  with prey x and predators y, the parameters ρ =  �  a  −c  �  and β =  �  0  −b  d  0  �  are specified by nonnegative  real numbers a, b, c, d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In this section, we define quantitative semantics for graphs and signed graphs using the Lotka-  Volterra dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To illustrate the main ideas, we first construct a functor from finite  graphs (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1) to linearly parameterized dynamical systems (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2), giving a  semantics for unlabeled graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' It is more useful to have a semantics for regulatory networks, which  we have defined to be signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We therefore construct a second functor from finite signed  graphs (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2) to conically parameterized nonnegative dynamical systems (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Recall that a graph is finite if its vertex and edge sets are both finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Let FinGraph denote the  full subcategory of Graph spanned by finite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='7 (Lotka-Volterra model for finite graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a functor  LV : FinGraph → Para(Dynam)  that sends  a finite graph X to the linearly parameterized dynamical system with parameter variables  P := X(V ) + X(E), state variables S := X(V ), and algebraic vector field3  v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β)(i) := ρ(i) x(i) +  �  (e:i′→i)∈X  β(e) x(i′) x(i),  x ∈ RX(V ), i ∈ X(V ),  parameterized by vectors ρ ∈ RX(V ) and β ∈ RX(E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  a graph homomorphism φ : X → Y to a morphism of systems with parameter variable map  φV + φE : X(V ) + X(E) → Y (V ) + Y (E) and state variable map φV : X(V ) → Y (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Moreover, the functor LV preserves finite colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  3For a fixed graph X and vertex i ∈ X(V ), the notation (e : i′ → i) ∈ X means any edge e ∈ X(tgt)−1(i) incoming  to i, whose source i′ = X(src)(e) varies with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By the universal property of the comma category Para(Dynam) = F/ Dynam, to give a  functor LV : FinGraph → Para(Dynam) is to give a pair of functors LV0, LV1 : FinGraph → FinSet  along with a natural transformation  ⃗  LV : (F ◦ LV0) ⇒ (Dynam ◦ LV1) : FinGraph → VectR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We set LV0(X) := X(V ) + X(E) and LV1(X) := X(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Using the universal property of the  coproduct in VectR, the components  ⃗  LVX : RX(V ) ⊕ RX(E) ∼= RX(V )+X(E) → Dynam(X(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  of the transformation ⃗  LV themselves decompose into two parts, call them  v0  X := ⃗  LV  0  X : RX(V ) → Dynam(X(V ))  and  v1  X := ⃗  LV  1  X : RX(E) → Dynam(X(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  We define these to be  v0  X(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ)(i) := ρ(i) x(i)  and  v1  X(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' β)(i) :=  �  e∈X(tgt)−1(i)  β(e) x(X(src)(e)) x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Putting the pieces back together reproduces the first statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We just need to  check that the transformation ⃗  LV is, in fact, natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Given a graph homomorphism φ : X → Y , the naturality square for the transformation ⃗  LV is  RX(V )+X(E)  Dynam(X(V ))  RY (V )+Y (E)  Dynam(Y (V ))  ⃗  LVX  (φV +φE)∗  (φV )∗◦(−)◦(φV )∗  ⃗  LVY  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2)  which decomposes into two squares,  RX(V )  Dynam(X(V ))  RY (V )  Dynam(Y (V ))  v0  X  (φV )∗  (φV )∗◦(−)◦(φV )∗  v0  Y  and  RX(E)  Dynam(X(V ))  RY (E)  Dynam(Y (V ))  v1  X  (φE)∗  (φV )∗◦(−)◦(φV )∗  v1  Y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Let us check that both squares commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For the first, we have  (φV )∗(v0  X(φ∗  V (y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ))(j) =  �  i∈φ−1  V (j)  v0  X(y ◦ φV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ)(i) =  �  i∈φ−1  V (j)  ρ(i) y(φV (i))  =  �  �  �  �  i∈φ−1  V (j)  ρ(i)  �  �  � y(j) = v0  Y (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' (φV )∗(ρ))(j)  20  for all y ∈ RY (V ), ρ ∈ RX(V ), and j ∈ Y (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' For the second, we have  (φV )∗(v1  X(φ∗  V (y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' β))(j) =  �  i∈φ−1  V (j)  v1  X(y ◦ φV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' β)(i)  =  �  i∈φ−1  V (j)  �  e∈X(tgt)−1(i)  β(e) y(φV (X(src)(e))) y(j)  =  �  f∈Y (tgt)−1(j)  �  e∈φ−1  E (f)  β(e) y(Y (src)(φE(e))) y(j)  =  �  f∈Y (tgt)−1(j)  �  �  �  �  e∈φ−1  E (f)  β(e)  �  �  � y(Y (src)(f))y(j)  = v1  Y (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' (φE)∗(β))(j),  for all y ∈ RY (V ), β ∈ RX(E), and j ∈ Y (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' When exchanging the order of the summations we  have used the facts that the graph homomorphism φ : X → Y preserves sources and targets, the  latter in its contravariant form  X(E)  X(V )  Y (E)  Y (V )  X(tgt)  φE  Y (tgt)  φV  ⇝  P(Y (V ))  P(X(V ))  P(Y (E))  P(X(E))  X(tgt)−1  φ−1  E  Y (tgt)−1  φ−1  V  ,  where P(S) denotes the power set of a set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Finally, we must verify that the functor LV preserves finite colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='6, that  happens provided both functors LV0, LV1 : FinGraph → FinSet preserve finite colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The functor  LV1 = evV is an evaluation functor on a copresheaf category, hence preserves colimits [Rie16,  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Since coproducts commute with colimits, the pointwise coproduct of two  evaluation functors  LV0 =  �FinGraph  ⟨evV ,evE⟩  −−−−−−→ FinSet × FinSet +  −→ FinSet  �  also preserves colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  A quantitative semantics for signed graphs can be defined similarly, subject to a caveat about  the vertex parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Our notion of signed graph, designed to capture regulatory networks as  studied in the biochemistry literature, attaches signs only to edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We are thus led to assume that,  in the Lotka-Volterra dynamical model, all species have baseline rates of decay rather than growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  This assumption is generally valid for protein regulatory networks, but not for gene regulatory  networks in which mediating proteins are ignored [TN10], nor for predator-prey models in ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  More flexible approaches are certainly possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' It would be straightforward to attach signs to  vertices as well as edges and use them in the quantitative semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Alternatively, at the expense  of a more cumbersome formalism, one could define dynamical systems with mixed linear-conical  parameterizations, allowing the vertex parameters to be arbitrary reals while the edge parameters  are constrained to be nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4 For simplicity and uniformity of presentation, we do not describe  these extensions further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Let FinSgnGraph denote the full subcategory of SgnGraph spanned by finite signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  4Similar mixed parameterizations are a practical necessity for parametric statistical models, studied in detail in  one author’s PhD thesis [Pat20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  21  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='8 (Lotka-Volterra model for finite signed graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a functor  LV : FinSgnGraph → Para(Dynam+)  that sends a finite signed graph X to the conically parameterized nonnegative dynamical system with  parameter variables P := X(V ) + X(E), state variables S := X(V ), and essentially nonnegative,  algebraic vector field  v(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β)(i) := −ρ(i) x(i) +  �  (e:i′→i)∈X  X(sgn)(e) β(e) x(i′) x(i),  x ∈ RX(V ), i ∈ X(V ),  parameterized by ρ ∈ RX(V )  +  and β ∈ RX(E)  +  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Moreover, the functor LV preserves finite colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Similarly to the previous proof, the functor LV : FinSgnGraph → Para(Dynam+) is defined  by functors LV0, LV1 : FinSgnGraph → FinSet along with a natural transformation  ⃗  LV : F+ ◦ LV0 ⇒ Dynam+ ◦ LV1 : FinSgnGraph → Con,  now having components ⃗  LVX given by the copairing of  v0  X := ⃗  LV  0  X : RX(V )  +  → Dynam+(X(V ))  and  v1  X := ⃗  LV  1  X : RX(E)  +  → Dynam+(X(V )),  where we define  v0  X(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ)(i) := −ρ(i) x(i)  and  v1  X(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' β)(i) :=  �  e∈X(tgt)−1(i)  X(sgn)(e) β(e) x(X(src)(e)) x(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The proof of naturality is essentially the same as before, using the crucial additional fact that  morphisms of signed graphs preserve signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The proof that the functor LV preserves finite colimits  is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To exemplify the theorem, let us see how the Lotka-Volterra dynamics functor acts on a  monomorphism and on an epimorphism of signed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  In order to compare the dynamics of two species A and B involved in a negative feedback loop  versus A and B in isolation, we take the inclusion of signed graphs  A  B  A  B  ι  Labeling the edges in the feedback loop as AB and BA, the morphism LV(ι) sends the conically  parameterized dynamical system  �  vA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ) = −ρA xA  vB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ) = −ρB xB  ,  ρ ∈ R{A,B}  +  ,  to the parameterized dynamical system  �  vA(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρA xA − βBA xB xA  vB(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρB xB + βAB xA xB  ,  ρ ∈ R{A,B}  +  , β ∈ R{AB,BA}  +  ,  by setting the latter’s interaction coefficients to zero: βAB = βBA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  This formalizes the  commonsense fact that the first system is a degenerate case of the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  22  For a more interesting example, we return to the projection map between regulatory networks  given by Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1) of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1, inspired by the arginine biosynthesis system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Call this  projection map p, and abbreviate the regulator molecule as R and the enzymes as S := {C, D, E, F, I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The morphism LV(p) sends the parameterized dynamical system  �  �  �  �  �  �  �  vR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρR xR − βR x2  R  vC(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρC xC − βC xR xC  vD(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρD xD − βD xR xD  vE(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρE xE − βE xR xE  vF (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρF xF − βF xR xF  vI(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρI xI − βI xR xI  with state variables {R} + S and parameters ρ, β ∈ R{R}+S  +  to the parameterized dynamical system  �  vR(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρR xR − βR x2  R  v∗(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ρ, β) = −ρ∗ x∗ − β∗ xR x∗  with state variables {R, ∗} and parameters ρ, β ∈ R{R,∗}  +  , in two different but equivalent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The  first way sets the latter system’s coefficients equal to sums of the former’s coefficients, namely  ρ∗ =  �  i∈S  ρi  and  β∗ =  �  i∈S  βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The second way substitutes x∗ for each xi, i ∈ S, in the first system and then takes the vector field  v∗ to be the sum of the vi’s, i ∈ S, with these substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The equivalence of these operations  is precisely the condition for LV(p) to be a morphism of parameterized dynamical systems, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3  Composing Lotka-Volterra models  To complete this part of the story, we extend the Lotka-Volterra dynamics functors between  graphs and parameterized dynamical systems, constructed in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='8, to double  functors between open graphs and open parameterized dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We begin by making  parameterized dynamical systems into open systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='9 (Open parameterized dynamical systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a symmetric monoidal double  category of open linearly parameterized dynamical systems, Open(Para(Dynam)), having  as objects, finite sets A, A′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as vertical arrows, functions f : A → A′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as horizontal arrows, open linearly parameterized dynamical systems, which consist of  a linearly parameterized dynamical system (P, S, v : RP → Dynam(S)) along with a cospan  A0  ℓ0  −→ S  ℓ1  ←− A1 whose apex is the set S of state variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  as cells, morphisms of such open systems (P, S, v, ℓ0, ℓ1) → (P ′, S′, v′, ℓ′  0, ℓ′  1), which  consist of a morphism (q, f) : (P, S, v) → (P ′, S′, v′) between linearly parameterized dynamical  systems along with functions f0 : A0 → A′  0 and f1 : A1 → A′  1 making the diagram commute:  A0  S  A1  A′  0  S′  A′  1  ℓ0  ℓ1  f0  f  ℓ′  0  f1  ℓ′  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  23  Vertical composition is by composition in FinSet and in Para(Dynam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Horizontal composition and  monoidal products are by pushouts and coproducts in Para(Dynam), respectively, interpreting the  finite sets in the feet of the cospans as linearly parameterized dynamical systems with no parameter  variables and identically zero vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Similarly, there is a symmetric monoidal double category Open(Para(Dynam+)) of open conically  parameterized nonnegative dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We perform the construction for linearly parameterized dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The construction  for conically parameterized nonnegative dynamical systems is perfectly analogous, replacing R with  R+ and vector spaces with conical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The projection functor πS : Para(Dynam) → FinSet, (P, S, v) �→ S that sends a linearly parame-  terized dynamical systems to its set of state variables has a left adjoint Z : FinSet → Para(Dynam)  that sends a finite set S to the system (∅, S, 0) with empty set of parameter variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' By linearity, its  parameterized vector field 0 ∼= R∅ → Dynam(S) is necessarily the zero vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This indeed gives  an adjunction Z ⊣ πS, because to any function f : S → S′ and linearly parameterized dynamical  system (P ′, S′, v′) there corresponds a unique morphism (0P ′, f) : Z(S) → (P ′, S′, v′), where the  required square  0  Dynam(S)  RP ′  Dynam(S′)  f∗◦(−)◦f∗  v′  commutes trivially, since the zero vector space is initial in VectR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Since Para(Dynam) has finite colimits (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5), we can construct a symmetric monoidal  double category of Z-structured cospans [BC20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' As we have argued before, it will be  isomorphic to Open(Para(Dynam)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  With this definition, we can construct double functors between open graphs and open param-  eterized dynamical systems, but the vertex parameters under Lotka-Volterra dynamics cause a  twist in the story compared to Baez and Pollard’s compositionality result for mass-action kinetics  [BP17, Theorem 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' When composing open dynamical systems in the image of the Lotka-Volterra  functor, one takes a coproduct of the parameter variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=', a direct sum of the parameter spaces,  belonging to identified vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' However, if one composes the open graphs first, then the identified  vertices receive a single copy of the parameters from the Lotka-Volterra functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Thus this functor  does not preserve composition of open systems, not even up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Nevertheless, there is  a natural (noninvertible) comparison between them: given a pair of parameters in the direct sum,  we can reduce them to a single parameter simply by summing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In mathematical terms, we get  a lax double functor: a double functor that strictly preserves vertical composition, as usual, but  preserves horizontal composition only up to specified comparison maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='10 (Open Lotka-Volterra models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' There is a symmetric monoidal lax double functor  LV : Open(FinGraph) → Open(Para(Dynam))  that acts  on objects and vertical arrows, as the identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  5The precise definition of a lax double functor can be found in the textbook [Gra19, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='5], among other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  24  on horizontal arrows and cells, by the functor LV : FinGraph → Para(Dynam) on graphs and  graph homomorphisms and as the identity on the associated cospans and cospan morphisms:  �  X, A0  ℓ0  −→ X(V )  ℓ1  ←− A1  �  �→  �  LV(X), A0  ℓ0  −→ X(V )  ℓ1  ←− A1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The comparison cells are defined using the morphisms of linearly parameterized dynamical systems  αS : Z(S) → LV(Disc S), where  αS := (0S, 1S) : (∅, S, 0) → (S, S, ⃗  LV(Disc S)),  S ∈ FinSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Given composable open graphs (X, A → X(V ) ← B) and (Y, B → Y (V ) ← C), the comparison  cell for horizontal composition is given by the morphism of systems  LV(X) +Z(B) LV(Y )  id +αB id  −−−−−−→ LV(X) +LV(Disc B) LV(Y )  ∼  =  −→ LV(X +Disc B Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Given a finite set A, the comparison cell for the horizontal unit is given by the morphism of  systems αA : Z(A) → LV(Disc A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Similarly, there is a symmetric monoidal lax double functor  LV : Open(FinSgnGraph) → Open(Para(Dynam+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' To construct the lax double functor, we use a lax version of [BC20, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The  family of morphisms αS : Z(S) → LV(Disc S), S ∈ FinSet, in the theorem statement assemble into  a natural transformation  FinSet  FinGraph  FinSet  Para(Dynam)  L=Disc  LV  L′=Z  α  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The functors involved in this cell all preserve finite colimits: the top and bottom ones because they  are left adjoints and the right one by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The hypotheses of [BC20, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3] are  therefore satisfied, except that α is not a natural isomorphism but merely a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  By inspection of the proof, the result still holds except that the resulting double functor is lax rather  than pseudo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We obtain a lax double functor  Open(FinGraph) ∼= LCsp(FinGraph) → L′Csp(Para(Dynam)) ∼= Open(Para(Dynam)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  To show that this double functor is the same one in the theorem statement, we once again  use the adjunctions to pass between L-structured and R-decorated cospans (recalling terminology  introduced in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Notice that the natural transformation α has as its  mate [CGR14, §1] the identity transformation ¯α = 1evV :  FinSet  FinGraph  FinSet  Para(Dynam)  R=evV  LV  R′=πS  ¯α  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  25  Thus the action of the double functor F := LV on L-structured cospans simplifies to the identity  when translated to R-decorated cospans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  L(A0)  X  L(A1)  L′(A0)  F(L(A0))  F(X)  F(L(A1))  L′(A1)  ↭  A0  R(X)  A1  A0  R(X)  R′(F(X))  R(X)  A1  ℓ0  ℓ1  αA0  F(ℓ0)  F(ℓ1)  αA1  ¯ℓ0  ¯ℓ1  ¯ℓ0  ¯αX  ¯αX  ¯ℓ1  A similar statement holds for the action of the double functor on morphisms of L-structured and  R-decorated cospans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  4  Conclusion  Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Regulatory networks are a minimalistic but widely used tool to describe the interactions  between molecules in biochemical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We have made the first functorial study of regulatory  networks, formalized as signed graphs, and their connections with other mathematical models in  biochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Among the latter, we have studied reaction networks, formalized as Petri nets with  signed links, and parameterized dynamical systems, focusing on Lotka-Volterra dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' This  project fits into a broader program by applied category theorists and other scientists aiming to  systematize, in a completely precise way, the language and methods of describing, composing, and  transforming scientific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The major categories of this paper, and the functors between them, are summarized in the  following diagram, where “LV” is the Lotka-Volterra dynamics functor (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  SgnCat (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2)  SgnGraph (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1)  SgnPetri (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3)  FinSgnGraph  Para(Dynam+) (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1)  Path  U  LV  Net  ⊣  Most of the main results extend from closed systems to open systems, which compose by gluing  along their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Of the diagram above, we have extended the following parts to double  categories of open systems and double functors between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Open(SgnCat)  Open(SgnGraph)  Open(FinSgnGraph)  Open(Para(Dynam+)) (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='3)  LV  Path  26  Future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Of many possible directions for future work, we mention a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' As noted in the  introduction, Lotka-Volterra dynamics are only one of numerous dynamics that could be considered  as a canonical model for regulatory networks, and they are not even among the most commonly  studied in the biochemistry literature [TLK19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' It would be desirable to have dynamics functors for  regulatory networks that draw on more flexible or more biologically plausible classes of dynamical  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' In another direction, the two halves of this paper—qualitative and quantitative—are not as  tightly as integrated as one might hope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' How does the presence of a motif in a regulatory network,  such as an incoherent feedforward loop perhaps even of a specific type, manifest in the continuous  dynamics of that network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Put in category-theoretic terms, the Lotka-Volterra dynamics functor is  defined on signed graphs, so how does it relate to the freely generated signed categories in which  motifs are expressed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' These intriguing questions are suggestive of “feedback loop analysis” in the  field of system dynamics [Ric95], to which stronger connections should be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  The authors thank the American Mathematical Society (AMS) for hosting  the 2022 Mathematical Research Community (MRC) on Applied Category Theory, where this  research project began.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' The AMS MRC was supported by NSF grant 1916439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' We thank John Baez,  our group’s mentor at the MRC, for suggesting this project and for much helpful advice along the  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Authors Fairbanks, Patterson, and Shapiro acknowledge subsequent support from the DARPA  ASKEM and Young Faculty Award programs through grants HR00112220038 and W911NF2110323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Author Ocal acknowledges subsequent support from an AMS-Simons Travel Grant and from the  Hausdorff Research Institute for Mathematics funded by the German Research Foundation (DFG)  under Germany’s Excellence Strategy - EXC-2047/1 - 390685813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='9 (2017), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Kershaw Publishing Company, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  27  [CGR14]  Eugenia Cheng, Nick Gurski, and Emily Riehl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='2 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' arXiv: 1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [CP09]  Stephen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Crews and Joseph C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Pearson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Transcriptional autoregulation in develop-  ment”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Current Biology 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='cub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='  [Csi+09]  Attila Csikász-Nagy, Orsolya Kapuy, Attila Tóth, Csaba Pál, Lars J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Jensen, Frank  Uhlmann, John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Tyson, and Béla Novák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Cell cycle regulation by feed-forward  loops coupling transcription and phosphorylation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Molecular Systems Biology 5 (2009),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1038/msb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Forrester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Industrial dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' MIT Press, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [FS07]  José Luiz Fiadeiro and Vincent Schmitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Structured co-spans: an algebra of interaction  protocols”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' International Conference on Algebra and Coalgebra in Computer Science  (CALCO 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1007/978-3-540-73859-6_14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' World  Scientific, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Nonnegative and compart-  mental dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Princeton University Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [Koc22]  Joachim Kock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='1145/3559103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='org/details/elementsofphysic017171mbp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Sheaves in geometry and logic: A first introduction  to topos theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='1007/978-1-4612-0927-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' “Loop polarity, loop dominance, and the concept of dominant po-  larity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' System Dynamics Review 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='1002/sdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='  [Rie16]  Emily Riehl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Category theory in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Courier Dover Publications, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' url: http:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Academic Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1016/C2015-0-06035-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Rubin, and David Swigon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Identifiability of linear and linear-  in-parameters dynamical systems from a single trajectory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' SIAM Journal on Applied  Dynamical Systems 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='4 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1137/130937913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [Ste00]  John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' Business dynamics: systems thinking and modeling for a complex  world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' McGraw-Hill, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [Str00]  Ross Street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “The petit topos of globular sets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Journal of Pure and Applied Algebra  154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content=' 299–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1016/S0022-4049(99)00183-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  28  [TLK19]  John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Tyson, Teeraphan Laomettachit, and Pavel Kraikivski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Modeling the dynamic  behavior of biochemical regulatory networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Journal of Theoretical Biology 462 (2019),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' 514–527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='jtbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+page_content='034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [TN10]  John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Tyson and Béla Novák.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Functional motifs in biochemical reaction networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  Annual Review of Physical Chemistry 61 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' 219–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  physchem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='012809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='103457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [Voi00]  Eberhard O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Voit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Computational analysis of biochemical systems: A practical guide for  biochemists and molecular biologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Cambridge University Press, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  [Voi13]  Eberhard O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' Voit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' “Biochemical systems theory: a review”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' International Scholarly  Research Notices 2013 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='1155/2013/897658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
+page_content='  29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfe_xt/content/2301.01445v1.pdf'}
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+arXiv:2301.11392v1  [cond-mat.mes-hall]  26 Jan 2023
+Noise and Thermal Depinning of Wigner Crystals
+C. Reichhardt and C. J. O. Reichhardt
+Theoretical Division and Center for Nonlinear Studies, Los Alamos National
+Laboratory, Los Alamos, New Mexico 87545, USA
+30 January 2023
+Abstract.
+We examine changes in the depinning threshold and conduction noise
+fluctuations for driven Wigner crystals in the presence of quenched disorder. At low
+temperatures there is a well defined depinning threshold and a strong peak in the noise
+power with 1/f noise characteristics. At higher temperatures, the depinning threshold
+shifts to lower drives and the noise, which is also reduced in power, becomes more white
+in character. At lower temperatures, a washboard frequency appears when the system
+depins elastically or forms a moving smectic state; however, this washboard signal is
+strongly reduced for higher temperatures and completely disappears above the melting
+temperature of a system without quenched disorder. Our results are in good agreement
+with recent transport and noise studies for systems where electron crystal depinning is
+believed to arise, and also show how noise can be used to distinguish between crystal,
+glass, and liquid phases.
+1. Introduction
+When a collection of interacting particles is driven over quenched disorder, the system
+can exhibit a pinned phase, a depinning threshold, and a sliding phase [1, 2].
+The
+existence of these phases can be deduced from changes in transport measures such
+as the velocity-force and differential resistance curves [1, 2, 3, 4, 5, 6, 7, 8].
+If the
+particles maintain the same neighbors during the depinning and sliding process, the
+depinning is considered elastic and is associated with specific scaling features in the
+velocity-force curves [1, 2, 9, 10], while if there is tearing or mixing of the particles,
+the behavior is plastic and can produce multiple steps or jumps in the transport curves
+[1, 2, 9, 11].
+In the sliding phase, there can also be dynamical transitions between
+different types of plastic flow or fluidlike flow, as well as dynamical ordering transitions
+where the driven particles move so rapidly over the substrate that the effectiveness of
+the pinning is reduced and a disordered system can organize into a moving crystal or
+smectic state [2, 12, 13, 14, 15, 16, 17, 18, 19, 20]. When thermal effects are included,
+additional behaviors can occur both during depinning and in the sliding states.
+In
+general, sharp depinning thresholds become rounded due to thermal creep; however, a
+peak in the differential velocity can still arise near the T = 0 depinning threshold due
+to a transition from creep to sliding dynamics [1, 2, 13]. If the temperature is higher
+
+Noise and Thermal Depinning of Wigner Crystals
+2
+than the melting temperature of the quenched disorder-free system, a system containing
+quenched disorder will always be in a disordered state, and can form a glass phase with
+thermal creep or a fluctuating liquid state at high drives [2, 13, 21, 22].
+Another method to examine the driven dynamics is to measure changes in the noise
+for systems in which time series of the velocity or density fluctuations can be obtained
+as a function of drive [2, 4, 17, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].
+One of
+the most common characterization techniques is to determine the power spectrum of
+the fluctuations and to measure the noise power over some frequency range. For elastic
+depinning or ordered moving states where the particles maintain a fixed set of neighbors
+and travel at speed v, there is typically a narrow band noise signature containing peaks
+at specific frequencies ω = 2πv/a that are associated with the average spacing a between
+the particles [2, 4]. Additional peaks appear at higher harmonics of these characteristic
+frequencies. Narrow band noise signatures have been observed for sliding charge density
+waves [4], superconducting vortex lattices [17, 26, 27, 28, 34], moving charge crystals [33],
+and skyrmion systems [30, 35]. In some cases it is possible to observe multiple frequencies
+when the system is broken up into several large sections that locally behave elastically
+but globally provide multiple degrees of freedom, permitting switching behavior to occur
+[33, 36]. If the depinning is strongly plastic, the narrow band noise signal is lost and is
+typically replaced by 1/f α noise with 0.75 < α < 2.0 [2, 17, 25, 27, 28], while for a fluid
+the noise power is often low and the fluctuations have white noise characteristics with
+α = 0. In other cases, the fluctuations are Lorentizan and the noise has a 1/f shape at
+low frequencies but becomes white above a characteristic frequency ωc that is associated
+with the average time between collisions of particles with pinning sites [2]. Broad band
+1/f noise has been observed near the depinning transition for superconducting vortices
+[2, 17, 25, 28], magnetic skyrmions [30, 32], and charge crystals [33, 37]. The noise
+measurements can also be used to identify a transition between different dynamical
+states such as plastic flow to dynamically ordered flow. In this case, 1/f noise occurs in
+the plastic flow phase just above depinning, but at higher drives there is a crossover to
+narrow band noise as dynamical ordering emerges [2, 17, 28, 30]. These different noise
+features can be modified significantly when thermal effects become important [2].
+Another example of an assembly of particle-like objects that can be coupled to
+quenched disorder and driven is electron crystals or Wigner crystals [38, 39, 40, 41, 42],
+where transport measures provide evidence for a conduction threshold that is consistent
+with the existence of a depinning transition [39, 40, 43, 44, 45, 46, 36, 47, 48]. Recently, a
+growing number of materials have been identified that can support Wigner crystals, such
+as moir`e superlattices [49, 50], transition metal dichalcogenide monolayers [51, 52, 53],
+and systems where Wigner crystals are stable at zero magnetic field [54].
+It would
+be interesting to examine conduction and noise measures as a function of drive and
+temperature in these new systems. Previous experiments that showed evidence of a
+conduction threshold also revealed a large increase in the conduction noise just above
+depinning [46], and previous numerical studies of driven Wigner crystals also showed
+both a conduction threshold and 1/f noise features near depinning followed by a
+
+Noise and Thermal Depinning of Wigner Crystals
+3
+crossover to narrow band noise at higher drives [55]. Brussarski et al. [56] examined the
+transport and noise of Wigner crystals near depinning as function of temperature, and
+found that at low temperature, there is a sharp depinning threshold that is correlated
+with a large peak in the noise power. Additionally, the noise near depinning is of 1/f 0.75
+form. As the temperature is increased, the depinning threshold shifts to lower values
+and the peak noise power is also reduced. This suggests that at higher temperature, the
+system forms a Wigner liquid in which the correlated motion associated with glassy or
+plastic flow phases and large noise power is lost. Noise studies have also been performed
+near the metal-insulator transition, which could be associated with a change from a
+Wigner glass to a Wigner liquid, and a drop in the noise power is observed at higher
+temperatures where a fluid phase may be present [57, 58]. Particle-based simulations
+across a Wigner glass to Wigner fluid crossover show high power 1/f α noise in the
+Wigner glass state and lower noise power with a white spectrum at higher temperatures
+in the fluid state [59]. Thermal effects and thermal melting in Wigner crystals have
+also been extensively studied [60, 61, 62, 63, 64], so it should be feasible to perform
+experimental noise and transport measures across a thermal melting transition while
+the system is being driven.
+In this work, we consider thermally induced transport and noise measurements for
+a two-dimensional (2D) electron system driven over quenched disorder. Previous work
+on this system focused on the T = 0 case, and showed that for plastic depinning, there is
+strong 1/f noise with a peak in the noise power near the depinning transition, followed
+by a drop in the noise power and a transition to white or narrow band noise at high
+driving where a moving smectic or moving crystal phase emerges [55, 65]. Here we find
+that as we increase the temperature, the depinning threshold decreases and the noise
+power drops, in agreement with experiments. Additionally, we find that the narrow
+band noise visible for T = 0 at high drives is strongly reduced at higher temperatures
+and vanishes above the temperature Tm at which the system melts in the absence of
+quenched disorder. This suggests that narrow band noise signals may only be accessible
+at temperatures well below melting.
+We map out the dynamic phase diagram as a
+function of drive versus temperature and show that at Tm there is a divergence in the
+drive at which a transition to ordered or partially ordered flow occurs, similar to the
+dynamic phase diagram proposed by Koshelev and Vinokur for driven superconducting
+vortex systems [13, 22]. For the case of elastic depinning, we find a thermally induced
+creep regime in which the lattice moves by one lattice constant at a time, and show
+that a narrow band signal can still arise even in the creep regime. The spectral peaks
+become sharper and shift to higher frequencies with increasing drive, but the narrow
+band signature is lost with increasing temperature even before the system reaches the
+clean melting temperature Tm.
+
+Noise and Thermal Depinning of Wigner Crystals
+4
+2. Simulation and System
+We model a 2D classical Wigner crystal with charge density n = Ne/L2, where Ne is the
+number of electrons and L is the system size. We employ periodic boundary conditions
+in the x and y directions, and the sample contains Np randomly placed pinning sites
+modeled as short range attractive wells with a density of np = Np/L2. Throughout
+this work we fix n = 0.208 and np = 0.25. At T = 0 and in the absence of quenched
+disorder, the charges form a triangular lattice that has a well defined melting transition
+temperature Tm [66]. Additionally, when T = 0 there is a well defined quenched disorder
+strength above which the system disorders [66].
+We represent the charges using a
+previously studied model [55, 67, 65, 66, 68, 69, 70, 71], where the equation of motion
+for charge i is
+αdvi =
+N
+�
+j
+∇U(rij) + Fp + FD + FT
+i .
+(1)
+Here αd is a damping term and Ui = q2/r is the long range Coulomb repulsion between
+charges of magnitude q. As in previous work [55, 71], we employ a Lekner summation to
+evaluate the long range interactions. The second term on the right hand side represents
+pinning sites modeled as finite range parabolic traps that impart a maximum pinning
+force of Fp at radius rp. The thermal fluctuations are applied with the term FT , which
+has the following properties: ⟨F T⟩ = 0 and ⟨F T(ti)F T(t′
+j)⟩ = 2kBTδijδ(t − t′). The
+initial positions of the charges are obtained through simulated annealing at zero drive.
+Once the system has been initialized, we apply a driving force FD = FDˆx representing
+an applied voltage. The drive can be set to a constant value, in which case we wait
+for the system to reach a steady state before measuring the average velocity per charge
+⟨V ⟩ =
+�Ne
+i
+vi · ˆx or obtaining a time series of the velocity to examine the temporal
+fluctuations. By considering a range of drives and measuring the average velocity at
+each drive, we can create a current-voltage curve. If there is a magnetic field present,
+the changes experience an additional force qB × vi that can create a Hall angle for the
+electron motion. This effect is generally small and we neglect it in the present work,
+but we have studied it in detail elsewhere [71].
+3. Elastic and Plastic Regimes
+In Fig. 1 we plot the fraction P6 of six-fold coordinated charges versus temperature T/Tm
+for a system with no quenched disorder. The melting temperature Tm is defined to be the
+temperature at which a proliferation of topological defects or non-sixfold coordinated
+charges occurs. For T/Tm < 1.0, P6 is close to 1.0, as expected for a triangular lattice,
+while for T/Tm > 1.0, a large number of fivefold and sevenfold coordinated charges
+appear, causing P6 to drop.
+Once we have defined Tm by measuring a clean system, we introduce quenched
+disorder in order to study the conduction noise and transport response above and below
+Tm for varied disorder strengths Fp. We apply a constant drive with FD = 0.01 to
+
+Noise and Thermal Depinning of Wigner Crystals
+5
+0
+0.5
+1
+1.5
+T/Tm
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P6
+Figure 1.
+The average fraction of sixfold-coordinated charges P6 versus temperature
+T/Tm in a system with no quenched disorder. Tm is defined to be the temperature at
+which a proliferation of non-sixfold coordinated charges occurs in a clean system.
+samples with different Fp and measure the time average velocity per charge ⟨V ⟩ over
+4 × 106 simulation time steps. When Fp = 0, the charge velocity V0 is identical to the
+driving force, V0 = FD = 0.01, so a measurement of ⟨V ⟩/V0 = 1 indicates that the
+flow of the charges has reached the pin-free limit. In Fig. 2(a), where we plot ⟨V/V0⟩
+versus Fp, at T/Tm = 0 there is a large drop in ⟨V ⟩/V0 near Fp = 0.75. Figure 2(b)
+shows the corresponding values of P6 versus Fp, where for T/Tm = 0 there is a well
+defined transition from an ordered Wigner crystal to a disordered Wigner glass, and the
+proliferation of defects correlates with the velocity drop in Fig. 2(a). At T/Tm = 0.3,
+the overall velocity is higher than for the T/Tm = 0 sample due to the lowering of
+the effectiveness of the pinning by the thermal fluctuations. Additionally, the pinning
+strength required to disorder the system is shifted upward to a value close to Fp = 0.1,
+which is again due to the partial reduction of the pinning effectiveness by the thermal
+fluctuations. A similar trend occurs for T/Tm = 0.6, where the velocity is higher. For
+T/Tm = 1.03, the system is disordered for all values of Fp and the velocity is even higher
+but has a gradual drop with increasing Fp, and the same trend occurs for T/Tm = 1.36. A
+more detailed study of the general phase diagram for the disordered and ordered phases
+as a function of pinning strength versus temperature appears in Ref. [66]. The results in
+
+Noise and Thermal Depinning of Wigner Crystals
+6
+0
+0.2
+0.4
+0.6
+0.8
+1
+<V>/V0
+0
+0.05
+0.1
+0.15
+Fp
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+P6
+Figure 2. (a) The average charge velocity ⟨V ⟩/V0 vs pinning strength Fp at FD = 0.01,
+where V0 = FD = 0.01 is the average velocity in a disorder-free system. T/Tm = 0
+(dark blue), 0.3 (light blue), 0.6 (green), 1.03 (yellow) and 1.38 (red).
+(b) The
+corresponding P6 vs Fp showing that for T/Tm < 1.0, there is a well defined pinning-
+induced order to disorder transition.
+Fig. 1 and Fig. 2 indicate that the system exhibits three distinct regimes. These are an
+ordered or crystal regime containing sixfold-coordinated charges, which occurs at low
+temperatures or low pinning strengths; a disordered or plastic regime where the system
+has low mobility and is strongly affected by the pinning; and a high temperature fluid
+phase where the effectiveness of the pinning is reduced and the system is in a strongly
+fluctuating state. In terms of transport, in the presence of pinning the ordered state
+exhibits elastic depinning in which the charges maintain their same neighbors. The glass
+state undergoes plastic depinning, and the fluid state does not have a pinned phase but
+can still have a regime in which the charges are trapped for a time before thermally
+
+Noise and Thermal Depinning of Wigner Crystals
+7
+hopping out of the pinning sites.
+4. Transport and Noise in the Plastic Regime
+We next examine the noise and transport in the three regimes identified above. We
+consider samples with Fp = 0.5, a pinning strength at which the charges are disordered
+for T/Tm = 0, so the system is in a strongly disordered glass phase. The T/Tm = 0
+plastic depinning that occurs in this regime was studied in detail in [66], where a pinned
+phase, a filamentary flow phase, a disordered flow phase, and a dynamically ordered
+moving smectic phase appear in sequence as a function of increasing drive.
+In Fig. 3(a) we show a snapshot of the charge locations, pinning site locations, and
+trajectories in the plastic flow regime for FD = 0.15 at T/Tm = 0.15, where a portion
+of the charges are moving in a series of well defined channels, with occasional jumps
+between the channels when certain channels open or close again. In general, for strong
+pinning, at low temperature the system exhibits channel flow just above depinning,
+similar to that studied in other systems at zero temperature. Figure 3(b) shows the
+same system at T/Tm = 0.606 where there is a combination of channel flow and random
+thermal hopping, indicating that as the temperature increases, there is a transition
+from one-dimensional (1D) channels to two-dimensional (2D) flow.
+In Fig. 3(c), at
+T/Tm = 1.03 the motion is 2D and fluidlike. For higher temperatures, the images look
+similar to what is shown in Fig. 3(c).
+In Fig. 4(a) we plot ⟨V ⟩ versus FD for the system in Fig. 3 at T/Tm = 0, 0.606,
+0.9, 1.21, 1.81, and 2.42. For the lower temperatures, there is a well-defined depinning
+threshold followed by a nonlinear regime, while when T/Tm > 1.0, the threshold is
+replaced by a creep regime and the nonlinear regime at higher drives persists. At the
+highest drives, all of the curves approach the pin-free limit. In Fig. 4(b) we show the
+corresponding d⟨V ⟩/dFD versus FD curves, where for T/Tm ≥ 1.21 there is a peak
+in d⟨V ⟩/dFD due to the S shape of the velocity-force curves.
+Similar peaks in the
+differential conductivity were observed for driven superconducting vortices in the plastic
+flow regime [12, 17, 21, 22]. For T/Tm > 1.21, the peaks are lost and a creep regime
+appears. The dashed line is the differential conductivity for the pin free system, and
+all of the curves approach this value at high drives. We note that for T/Tm = 0, at
+low drives there are a number of jumps in the conduction as well as a few regimes of
+negative differential conduction. This arises due to a filamentary flow channel effect
+that is described in more detail in [65]. For T/Tm > 0.5, the jumps associated with
+the filamentary flow phase are lost and a single large peak in d⟨V ⟩/dFD appears in the
+plastic flow regime where there is a combination of moving and pinned charges.
+In Fig. 5 we plot P6 versus FD for the system in Fig. 4 for T/Tm = 0, 0.303, 0.606,
+0.757, 0.91, 1.21, and 2.42. For T/Tm < 1.0 there is an initial dip in P6 at the onset of
+plastic flow, and at high drives where d⟨V ⟩/dFD starts to flatten, P6 approaches values
+of 0.9 or higher as the system forms a moving smectic phase. In the moving smectic
+state, the charges move in well defined channels and a small number of dislocations are
+
+Noise and Thermal Depinning of Wigner Crystals
+8
+x
+(a)
+y
+x
+(b)
+y
+x
+(c)
+y
+Figure 3.
+Charge locations (red circles), trajectories (blue lines), and pinning
+site locations (black circles) for the system in Fig. 2 at FD = 0.15 and Fp = 0.5. (a)
+Filamentary flow at T/Tm = 0.15. (b) Disordered flow with channels at T/Tm = 0.606.
+(c) T/Tm = 1.03.
+present that have their Burgers vectors aligned with the driving direction [2, 55, 65].
+As T/Tm increases further, the drive at which the smectic state emerges shifts to higher
+values of FD, and for T/Tm > 1.0, the system no longer forms a moving smectic but
+instead becomes a moving fluid.
+From the features in the transport curves and P6 plotted in Figs. 4 and 5, we
+construct a dynamic phase diagram of the evolution of the different phases as a function
+of FD versus T/Tm in Fig. 6. At low drives we find a pinned or creep regime denoted
+C, where d⟨V ⟩/dFD < 0.5. The dynamically ordered moving smectic phase MS appears
+
+Noise and Thermal Depinning of Wigner Crystals
+9
+0
+0.2
+0.4
+0.6
+0.8
+1
+<V>
+0
+0.25
+0.5
+0.75
+1
+FD
+0
+0.5
+1
+1.5
+d<V>/dFD
+(a)
+(b)
+Figure 4.
+(a) Velocity ⟨V ⟩ vs drive FD for the system in Fig. 3 with Fp = 0.5 at
+T/Tm = 0.0 (purple), 0.606 (blue), 0.91 (dark green), 1.21 (light green), 1.81 (orange),
+and 2.42 (red). (b) The corresponding d⟨V ⟩/dFD vs FD curves. The dashed lines
+indicate the pin-free limit.
+when P6 > 0.9. The disordered regime is where the system is structurally disordered but
+moving, and it can be either a moving glass MG for T/Tm < 1.0, or a moving liquid ML
+for T/Tm > 1.0. The overall features of the phase diagram are similar to those observed
+in driven superconducting vortex systems with quenched disorder, as first proposed by
+Koshelev and Vinokur [13], where the transition between the MG and MS states shifts to
+higher drives as T/Tm is approached. In Ref. [13], the transition line from the disordered
+to moving ordered phase was argued to be proportional to A/(Tm −T), where A is some
+prefactor and the moving ordered phase can be described in terms of having an effective
+temperature that is decreasing toward zero. This picture assumes the formation of a
+
+Noise and Thermal Depinning of Wigner Crystals
+10
+0
+0.5
+1
+1.5
+2
+FD
+0.4
+0.6
+0.8
+1
+P6
+Figure 5.
+P6 vs FD for the system in Fig. 4 with Fp = 0.5 at T/Tm = 0 (purple),
+0.303 (dark blue), 0.606 (light blue), 0.757 (green), 0.91 (yellow), 1.21 (orange), and
+2.42 (red). The system reaches an ordered state at high drives for T/Tm < 1.0.
+moving crystal at high drive, and is somewhat modified in our system since the moving
+state we observe is a smectic in which the dynamic fluctuations are anisotropic [16]. We
+find that a better fit to the transition line in our case is (Tm − T)−0.7, which is likely
+due to the anisotropic nature of the moving smectic.
+Now that we have established the dynamic phase diagram as a function of drive
+versus temperature, we can ask how the velocity fluctuation power spectra change as a
+function of FD and T. The power spectrum as a function of ω = 2πf can be calculated
+using the time series v(t) of the velocity fluctuations,
+S(ω) =
+����
+�
+v(t)e−iωt
+����
+2
+(2)
+At T = 0 the noise has a 1/f α signature with α ≈ 0.8, in agreement with recent
+experiments [56]. The noise power is reduced at high drives and shows a crossover to
+a narrow band signature when the system forms a moving smectic phase; however, the
+experiments do not detect a narrow band noise signature at higher drives, suggesting
+that thermal effects could be coming into play.
+In Fig. 7 we plot power spectra of the velocity time series for the system in Fig. 6
+at a drive of FD = 0.15 for T/Tm = 0, 0.303, 0.606, and 1.03. For T/Tm = 0, the
+
+Noise and Thermal Depinning of Wigner Crystals
+11
+0
+0.5
+1
+1.5
+T/Tm
+0
+1
+2
+3
+4
+FD
+MS
+ML
+MG
+C
+Figure 6.
+Dynamic phase diagram as a function of FD vs T/Tm for the system in
+Figs. 4 and 5 with Fp = 0.5. There is a pinned or creep phase C (red), a disordered
+moving phase (blue) that is a moving glass, MG, at lower temperatures and a moving
+liquid, ML, at higher temperatures, and a moving smectic MS (green).
+low frequency noise has a 1/f α form, where the dashed line is a fit with α = −0.8,
+while at higher frequencies the noise tail has α = −2.0. At T/Tm = 0.5, the lower
+frequency noise power is reduced and α drops closer to α = 0, the value expected for
+white noise; however, the high frequency noise still has a 1/f 2 form. For higher T/Tm,
+the low frequency noise power is further reduced while the higher frequency noise power
+is enhanced, and the spectrum becomes much whiter overall.
+To better characterize the system, we measure the noise power S0, which is the
+value of the spectral power integrated in a small window around a specific frequency
+ω = 20. In Fig. 8 we show S0 versus FD for T/Tm = 0, 0.303, 0.606, and 1.03 on a log-
+linear plot. For T = 0 there is a large peak in S0 over the range 0.01 < FD < 0.5, which
+corresponds to the appearance of 1/f 0.85 noise. The noise is white for 0.5 < FD < 0.9,
+and for FD > 0.9 a narrow band noise signal appears. For T/Tm = 0.303 and 0.606,
+there is still a peak in the noise near FD = 0.2, but as the temperature increases, the
+peak power diminishes and the peak location shifts to lower drives. This is correlated
+with a whitening of the low frequency noise, as shown in Fig. 7. For T/Tm = 1.03, the
+
+Noise and Thermal Depinning of Wigner Crystals
+12
+10
+100
+1000
+ω
+10
+-9
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+S(ω)
+Figure 7.
+Power spectra S(ω) vs ω for the system in Fig. 6 with FD = 0.15 for
+T/Tm = 0 (dark blue), 0.303 (light blue), 0.606 (yellow), and 1.03 (red). The spectral
+signature changes from 1/f to white at low frequencies as the temperature increases,
+while the amount of noise power at higher frequencies increases with increasing T .
+sharp noise power peak is lost. At large FD, we find that the noise power increases with
+increasing temperature due to the transition from flow through narrow 1D channels
+in the smectic state to a 2D Brownian like motion in the liquid state.
+The overall
+behavior of the noise power that we find is in agreement with experimental observations
+[56], where there is a large peak in the noise power near the depinning threshold at
+low temperatures, while for higher temperatures the noise power peak is reduced and
+shifts to lower drives before disappearing at sufficiently high temperature.
+Another
+feature that is also observed in the experiments is that the noise power increases with
+temperature at large drives.
+We next consider thermal effects in the high drive limit where the system forms a
+moving smectic at T = 0. In Fig. 9(a) we show S(ω) vs ω for the system from Fig. 6 at
+FD = 1.5 and T/Tm = 0, where there are a series of peaks associated with a narrow band
+noise signature. At T/Tm = 0.303 in Fig. 9(b), there are still strong peaks associated
+with the narrow band noise but the higher harmonic peaks are strongly reduced in
+power. In Fig. 9(c) at T/Tm = 0.606, the level of background noise has increased and
+the narrow band peaks are diminished in size, while at T/Tm = 1.03 in Fig. 9(d), the
+
+Noise and Thermal Depinning of Wigner Crystals
+13
+0
+0.5
+1
+1.5
+2
+FD
+10
+-10
+10
+-9
+10
+-8
+10
+-7
+10
+-6
+10
+-5
+S0
+Figure 8.
+The noise power S0 at fixed ω = 20 vs FD for the system in Fig. 7 at
+FD = 0.15 for T/Tm = 0 (dark blue), 0.303 (light blue), 0.606 (yellow), and 1.03 (red).
+moving smectic phase is lost and the narrow band peaks disappear into the background
+noise. To better characterize the change in the narrow band noise signature, in Fig. 10
+we plot the noise power S0 at ω = 323, which is the location of the most pronounced
+narrow band noise peak in Fig. 9(a). For T/Tm < 0.5 there is a strong narrow band
+noise signal; however, at T/Tm = 0.75 the narrow band noise level is close to the value
+of the background noise. This suggests that thermal effects can strongly reduce the
+narrow band noise signal even at temperatures well below T/Tm = 1.0, which could
+explain why the narrow band noise signals are difficult to see in experiment. To better
+understand the origins of the changes in the noise signals, in Fig. 11(a) we plot the
+trajectories of the charges at T/Tm = 0.606 where narrow band noise is present. The
+system is still in a moving smectic state but the channels have been broadened by the
+thermal fluctuations, and there are several regions in which the channel structures are
+starting to break down. Figure 11(b) shows the trajectories for T/Tm = 1.07, where
+the 1D channel structure is lost, there is a significant amount of transverse diffusion,
+and the narrow band noise peaks disappear. This result indicates that the narrow band
+noise occurs only when the motion of the charges is mostly 1D in character.
+
+Noise and Thermal Depinning of Wigner Crystals
+14
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+2.0×10
+-5
+2.5×10
+-5
+S(ω)
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+2.0×10
+-5
+2.5×10
+-5
+S(ω)
+0
+1000
+2000
+3000
+ω
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+2.0×10
+-5
+2.5×10
+-5
+S(ω)
+0
+1000 2000 3000 4000 5000
+ω
+0.0
+5.0×10
+-6
+1.0×10
+-5
+1.5×10
+-5
+S(ω)
+(a)
+(b)
+(c)
+(d)
+Figure 9.
+S(ω) vs ω for the system in Fig. 6 at FD = 1.5 where the system is in the
+moving smectic phase. T/Tm = (a) 0, (b) 0.303, (c) 0.606, and (d) 1.03.
+5. Thermal Depinning Noise in the Elastic Regime
+We next consider the thermal depinning and noise in the elastic regime where the charges
+maintain their same neighbors. From Fig. 2 we select a value of Fp = 0.05, well below
+the T/Tm = 0 disordering threshold of Fp = 0.075. In Fig. 12(a) we show ⟨V ⟩ versus
+FD at Fp = 0.05 for T/Tm = 0, 0.0378, 0.0756, 0.17, 0.303, 0.606, and 1.03, and we plot
+the corresponding d⟨V ⟩/dFD curves in Fig. 12(b). As T/Tm increases, the depinning
+threshold shifts to lower FD, and in Fig. 12(b), the peak in d⟨V ⟩/dFD that appears for
+T = 0 is lost for T/Tm > 0.303. We note that the system remains in an ordered state
+up to T/Tm = 1.0 for all values of FD. The d⟨V ⟩/dFD curves also show a multiple peak
+feature at high temperatures, with one peak at the finite temperature threshold and a
+second peak near the T = 0 depinning threshold. In between these two peaks, the flow
+is creep-like in nature.
+In Fig. 13 we plot S(ω) versus ω for the system in Fig. 12 at T/Tm = 0.1515 for
+different values of FD = 0.025, 0.03, 0.04, 0.046, 0.06, and 0.01. At FD = 0.025, the
+motion occurs mostly in the form of avalanches, and no clear narrow band signatures
+are present but the low frequency noise has high power. For FD = 0.03, the system
+starts to develop a narrow band noise signature that sharpens with increasing drive,
+and for FD ≥ 0.06, which is above the zero temperature depinning threshold, the low
+frequency noise is strongly suppressed and the narrow band noise peaks become much
+
+Noise and Thermal Depinning of Wigner Crystals
+15
+0
+0.5
+1
+T/Tm
+1×10
+-5
+S0
+Figure 10.
+The value of the noise power S0 at ω = 323, the frequency of the largest
+narrow band noise peak in Fig. 9, vs T/Tm for the system in Fig. 8 with FD = 0.15.
+Here the narrow band noise peaks are lost near T/Tm = 0.75.
+x
+(a)
+y
+x
+(b)
+y
+Figure 11.
+Charge locations (red circles), trajectories (blue lines), and pinning site
+locations (black circles) for the system in Figs. 9 and 10 at FD = 1.5 for T/Tm = (a)
+0.606 and (b) 1.03.
+
+Noise and Thermal Depinning of Wigner Crystals
+16
+0
+0.01
+0.02
+<V>
+0
+0.01
+0.02
+FD
+0
+2
+4
+6
+d<V>/dFD
+(a)
+(b)
+Figure 12. (a) ⟨V ⟩ vs FD for a system that exhibits elastic depinning at T/Tm = 0,
+where Fp = 0.05. The different curves are for temperatures of T/Tm = 0, 0.0378,
+0.0756, 0.17, 0.303, 0.606, and 1.03, from right to left. The dashed line is the expected
+curve in the pin free limit. (b) The corresponding d⟨V ⟩/dFD vs FD curves.
+sharper. This result shows that in the elastic flow regime, the narrow band noise signal
+is more robust than in the plastic phase, and it appears once the system has depinned.
+In Fig. 14 we plot S(ω) vs ω for the system in Fig. 13 at T/Tm = 0 and T/Tm = 0.303
+at a drive of FD = 0.02. At T/Tm = 0, there is a strong narrow band noise feature.
+Interestingly, at T/Tm = 0.303, although the level of background noise has increased,
+the primary narrow band noise peak is enhanced in power. The increase in the narrow
+band peak occurs when thermal effects weaken the effectiveness of the pinning and allow
+the charges to become better ordered. This effect is diminished in the case of strong
+pinning.
+
+Noise and Thermal Depinning of Wigner Crystals
+17
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+-10
+10
+-9
+10
+-8
+10
+-7
+S(ω)
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+ω
+10
+-11
+10
+-10
+10
+-9
+10
+-8
+S(ω)
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+ω
+10
+-11
+10
+-10
+10
+-9
+10
+-8
+10
+-7
+S(ω)
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 13.
+S(ω) vs ω for the system in Fig. 12 with Fp = 0.05 at T/Tm = 0.1515
+for FD = (a) 0.025, (b) 0.03, (c) 0.04, (d) 0.046, (e) 0.06, and (f) 0.01.
+To better characterize the narrow band noise behavior for the system in Figs. 12 and
+13, in Fig. 15 we plot the noise power S0 versus T/Tm for FD = 0.02 where the system is
+always in a moving state at the narrow band peak of ω = 80 and the background noise
+signal at ω = 300, along with the difference between these two noise powers. Unlike the
+case for strong pinning, the power of the narrow band noise signal generally increases
+with increasing T/Tm; however, the background noise power also increases, and the
+amount of power in the two signals becomes equal near T/Tm = 1.0. The narrow band
+noise peak has the greatest amount of additional power compared to the background
+noise near T/Tm = 0.3. This is again due to thermal effects washing out any additional
+avalanche-like motion and permitting the charge lattice to become better ordered.
+6. Discussion
+Narrow band noise has been observed experimentally in superconducting vortex [26, 28],
+magnetic skyrmion [32], and charge density wave [4] systems, but has not been seen for
+Wigner crystals. There have been reports of periodic noise in charge ordering systems
+such as stripe or bubble forming states [33, 37]; however, this noise generally appears at
+low frequencies and is probably not associated with the lattice-scale narrow band noise,
+
+Noise and Thermal Depinning of Wigner Crystals
+18
+0
+200
+400
+600
+ω
+0
+2×10
+-8
+4×10
+-8
+6×10
+-8
+S(ω)
+Figure 14.
+The power spectra S(ω) vs ω for the system in Fig. 13 with Fp = 0.05
+at FD = 0.02 in the moving phase for T/Tm = 0 (blue) and T/Tm = 0.303 (green).
+At T/Tm = 0.303, although the overall background noise power is higher, there is an
+enhancement of the narrow band noise signal.
+but instead arises due to the motion of some other periodically moving macroscopic scale
+structure. In the experiments of Brussarski et al. [56], the peak noise power decreased
+with increasing temperature, similar to what we observe, but no narrow band noise signal
+was observed. This could be the result of several possible factors. If the drive applied to
+the system is not uniform, there could still be strong plastic flow at low drives; however,
+at high drives the system may not form a uniformly ordered moving state but could
+instead break into several locally ordered regions that are moving at different speeds.
+Related to this, if the quenched disorder has a wide range of strength so that some of
+the charges are moving while a small number remain pinned, a disordered flow regime
+would emerge in which narrow band noise is absent. A narrow band noise signal could
+also be masked by strong background noise. In this case, the signal could be boosted
+by applying an additional ac drive. If the frequency of this ac drive is swept, phase
+locking or Shapiro steps would appear when the frequency comes into resonance with
+the narrow band signal [26]. Another possible issue is that the narrow band frequency
+could be too high to detect with the available experimental setup; however, for a system
+in the elastic depinning limit, fairly low frequency periodic signals could be generated in
+the creep regime. The lack of experimentally observed narrow band noise may suggest
+
+Noise and Thermal Depinning of Wigner Crystals
+19
+0
+0.5
+1
+T/Tm
+0
+2×10
+-8
+4×10
+-8
+6×10
+-8
+8×10
+-8
+S0
+Figure 15.
+The noise power S0 vs T/Tm for the system in Figs. 12 and 13 with
+Fp = 0.05 at FD = 0.02 for the narrow band frequency of ω = 80 (green circles), the
+background noise at ω = 300 (red squares), and the difference (blue triangles).
+that elastic depinning of the Wigner crystal is not occurring and that the systems are
+generally in the disordered or plastic flow regimes where the only available narrow band
+noise signals are of the moving smectic type. In principle, we think that the best place
+to look for a narrow band noise signature is in a sample with relatively weak pinning just
+above the depinning threshold. In our work, we focused on samples that were entirely
+within the elastic or plastic regimes; however, close to the transition between the elastic
+and plastic regimes, the plastic flow noise may be enhanced.
+7. Summary
+We have investigated the thermally induced depinning and noise fluctuations for driven
+Wigner crystal systems with quenched disorder. We identify an elastic regime in which
+the charges maintain the same neighbors at depinning as well as a plastic regime in which
+the system is broken up into moving and non-moving regions. In the plastic depinning
+regime, the velocity noise has a 1/f shape and there is a peak in the noise power above
+the depinning threshold at lower temperatures, while for large temperatures, the noise
+power peak is reduced and the spectrum becomes white, in agreement with experiments.
+For high drives at low temperatures in the plastic regime, the system forms a moving
+
+Noise and Thermal Depinning of Wigner Crystals
+20
+smectic with a narrow band noise signal. We find that this narrow band signal persists
+up to T/Tm = 0.75, where Tm is the temperature at which the charge lattice melts in the
+absence of quenched disorder. In the elastic regime, the system remains ordered up to
+temperatures approaching T/Tm = 1.0, although thermal effects reduce the depinning
+threshold. In the elastic regime, 1/f noise appears only in the creep regime where there
+are avalanches or jumps of motion, while in the sliding regime, pronounced narrow band
+noise appears that reaches its lowest power at the disorder-free melting temperature.
+Our results show that measurements of the velocity noise spectra and noise power can
+be used in connection with transport curves to distinguish different phases of driven
+Wigner crystals.
+Acknowledgments
+We gratefully acknowledge the support of the U.S. Department of Energy through
+the LANL/LDRD program for this work.
+This work was supported by the US
+Department of Energy through the Los Alamos National Laboratory.
+Los Alamos
+National Laboratory is operated by Triad National Security, LLC, for the National
+Nuclear Security Administration of the U. S. Department of Energy (Contract No.
+892333218NCA000001).
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+
diff --git a/3tFIT4oBgHgl3EQf6Cuj/content/tmp_files/load_file.txt b/3tFIT4oBgHgl3EQf6Cuj/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf,len=1385
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='11392v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='mes-hall]  26 Jan 2023  Noise and Thermal Depinning of Wigner Crystals  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Reichhardt and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Reichhardt  Theoretical Division and Center for Nonlinear Studies, Los Alamos National  Laboratory, Los Alamos, New Mexico 87545, USA  30 January 2023  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  We examine changes in the depinning threshold and conduction noise  fluctuations for driven Wigner crystals in the presence of quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At low  temperatures there is a well defined depinning threshold and a strong peak in the noise  power with 1/f noise characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At higher temperatures, the depinning threshold  shifts to lower drives and the noise, which is also reduced in power, becomes more white  in character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At lower temperatures, a washboard frequency appears when the system  depins elastically or forms a moving smectic state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, this washboard signal is  strongly reduced for higher temperatures and completely disappears above the melting  temperature of a system without quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Our results are in good agreement  with recent transport and noise studies for systems where electron crystal depinning is  believed to arise, and also show how noise can be used to distinguish between crystal,  glass, and liquid phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Introduction  When a collection of interacting particles is driven over quenched disorder, the system  can exhibit a pinned phase, a depinning threshold, and a sliding phase [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The  existence of these phases can be deduced from changes in transport measures such  as the velocity-force and differential resistance curves [1, 2, 3, 4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  If the  particles maintain the same neighbors during the depinning and sliding process, the  depinning is considered elastic and is associated with specific scaling features in the  velocity-force curves [1, 2, 9, 10], while if there is tearing or mixing of the particles,  the behavior is plastic and can produce multiple steps or jumps in the transport curves  [1, 2, 9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In the sliding phase, there can also be dynamical transitions between  different types of plastic flow or fluidlike flow, as well as dynamical ordering transitions  where the driven particles move so rapidly over the substrate that the effectiveness of  the pinning is reduced and a disordered system can organize into a moving crystal or  smectic state [2, 12, 13, 14, 15, 16, 17, 18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' When thermal effects are included,  additional behaviors can occur both during depinning and in the sliding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In  general, sharp depinning thresholds become rounded due to thermal creep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, a  peak in the differential velocity can still arise near the T = 0 depinning threshold due  to a transition from creep to sliding dynamics [1, 2, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' If the temperature is higher  Noise and Thermal Depinning of Wigner Crystals  2  than the melting temperature of the quenched disorder-free system, a system containing  quenched disorder will always be in a disordered state, and can form a glass phase with  thermal creep or a fluctuating liquid state at high drives [2, 13, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Another method to examine the driven dynamics is to measure changes in the noise  for systems in which time series of the velocity or density fluctuations can be obtained  as a function of drive [2, 4, 17, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  One of  the most common characterization techniques is to determine the power spectrum of  the fluctuations and to measure the noise power over some frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For elastic  depinning or ordered moving states where the particles maintain a fixed set of neighbors  and travel at speed v, there is typically a narrow band noise signature containing peaks  at specific frequencies ω = 2πv/a that are associated with the average spacing a between  the particles [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Additional peaks appear at higher harmonics of these characteristic  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Narrow band noise signatures have been observed for sliding charge density  waves [4], superconducting vortex lattices [17, 26, 27, 28, 34], moving charge crystals [33],  and skyrmion systems [30, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In some cases it is possible to observe multiple frequencies  when the system is broken up into several large sections that locally behave elastically  but globally provide multiple degrees of freedom, permitting switching behavior to occur  [33, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' If the depinning is strongly plastic, the narrow band noise signal is lost and is  typically replaced by 1/f α noise with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75 < α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0 [2, 17, 25, 27, 28], while for a fluid  the noise power is often low and the fluctuations have white noise characteristics with  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In other cases, the fluctuations are Lorentizan and the noise has a 1/f shape at  low frequencies but becomes white above a characteristic frequency ωc that is associated  with the average time between collisions of particles with pinning sites [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Broad band  1/f noise has been observed near the depinning transition for superconducting vortices  [2, 17, 25, 28], magnetic skyrmions [30, 32], and charge crystals [33, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The noise  measurements can also be used to identify a transition between different dynamical  states such as plastic flow to dynamically ordered flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In this case, 1/f noise occurs in  the plastic flow phase just above depinning, but at higher drives there is a crossover to  narrow band noise as dynamical ordering emerges [2, 17, 28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' These different noise  features can be modified significantly when thermal effects become important [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Another example of an assembly of particle-like objects that can be coupled to  quenched disorder and driven is electron crystals or Wigner crystals [38, 39, 40, 41, 42],  where transport measures provide evidence for a conduction threshold that is consistent  with the existence of a depinning transition [39, 40, 43, 44, 45, 46, 36, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Recently, a  growing number of materials have been identified that can support Wigner crystals, such  as moir`e superlattices [49, 50], transition metal dichalcogenide monolayers [51, 52, 53],  and systems where Wigner crystals are stable at zero magnetic field [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  It would  be interesting to examine conduction and noise measures as a function of drive and  temperature in these new systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Previous experiments that showed evidence of a  conduction threshold also revealed a large increase in the conduction noise just above  depinning [46], and previous numerical studies of driven Wigner crystals also showed  both a conduction threshold and 1/f noise features near depinning followed by a  Noise and Thermal Depinning of Wigner Crystals  3  crossover to narrow band noise at higher drives [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Brussarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' [56] examined the  transport and noise of Wigner crystals near depinning as function of temperature, and  found that at low temperature, there is a sharp depinning threshold that is correlated  with a large peak in the noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Additionally, the noise near depinning is of 1/f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' As the temperature is increased, the depinning threshold shifts to lower values  and the peak noise power is also reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This suggests that at higher temperature, the  system forms a Wigner liquid in which the correlated motion associated with glassy or  plastic flow phases and large noise power is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Noise studies have also been performed  near the metal-insulator transition, which could be associated with a change from a  Wigner glass to a Wigner liquid, and a drop in the noise power is observed at higher  temperatures where a fluid phase may be present [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Particle-based simulations  across a Wigner glass to Wigner fluid crossover show high power 1/f α noise in the  Wigner glass state and lower noise power with a white spectrum at higher temperatures  in the fluid state [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Thermal effects and thermal melting in Wigner crystals have  also been extensively studied [60, 61, 62, 63, 64], so it should be feasible to perform  experimental noise and transport measures across a thermal melting transition while  the system is being driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In this work, we consider thermally induced transport and noise measurements for  a two-dimensional (2D) electron system driven over quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Previous work  on this system focused on the T = 0 case, and showed that for plastic depinning, there is  strong 1/f noise with a peak in the noise power near the depinning transition, followed  by a drop in the noise power and a transition to white or narrow band noise at high  driving where a moving smectic or moving crystal phase emerges [55, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Here we find  that as we increase the temperature, the depinning threshold decreases and the noise  power drops, in agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Additionally, we find that the narrow  band noise visible for T = 0 at high drives is strongly reduced at higher temperatures  and vanishes above the temperature Tm at which the system melts in the absence of  quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This suggests that narrow band noise signals may only be accessible  at temperatures well below melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  We map out the dynamic phase diagram as a  function of drive versus temperature and show that at Tm there is a divergence in the  drive at which a transition to ordered or partially ordered flow occurs, similar to the  dynamic phase diagram proposed by Koshelev and Vinokur for driven superconducting  vortex systems [13, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For the case of elastic depinning, we find a thermally induced  creep regime in which the lattice moves by one lattice constant at a time, and show  that a narrow band signal can still arise even in the creep regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The spectral peaks  become sharper and shift to higher frequencies with increasing drive, but the narrow  band signature is lost with increasing temperature even before the system reaches the  clean melting temperature Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Noise and Thermal Depinning of Wigner Crystals  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Simulation and System  We model a 2D classical Wigner crystal with charge density n = Ne/L2, where Ne is the  number of electrons and L is the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We employ periodic boundary conditions  in the x and y directions, and the sample contains Np randomly placed pinning sites  modeled as short range attractive wells with a density of np = Np/L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Throughout  this work we fix n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='208 and np = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At T = 0 and in the absence of quenched  disorder, the charges form a triangular lattice that has a well defined melting transition  temperature Tm [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Additionally, when T = 0 there is a well defined quenched disorder  strength above which the system disorders [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  We represent the charges using a  previously studied model [55, 67, 65, 66, 68, 69, 70, 71], where the equation of motion  for charge i is  αdvi =  N  �  j  ∇U(rij) + Fp + FD + FT  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  (1)  Here αd is a damping term and Ui = q2/r is the long range Coulomb repulsion between  charges of magnitude q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' As in previous work [55, 71], we employ a Lekner summation to  evaluate the long range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The second term on the right hand side represents  pinning sites modeled as finite range parabolic traps that impart a maximum pinning  force of Fp at radius rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The thermal fluctuations are applied with the term FT , which  has the following properties: ⟨F T⟩ = 0 and ⟨F T(ti)F T(t′  j)⟩ = 2kBTδijδ(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The  initial positions of the charges are obtained through simulated annealing at zero drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Once the system has been initialized, we apply a driving force FD = FDˆx representing  an applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The drive can be set to a constant value, in which case we wait  for the system to reach a steady state before measuring the average velocity per charge  ⟨V ⟩ =  �Ne  i  vi · ˆx or obtaining a time series of the velocity to examine the temporal  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' By considering a range of drives and measuring the average velocity at  each drive, we can create a current-voltage curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' If there is a magnetic field present,  the changes experience an additional force qB × vi that can create a Hall angle for the  electron motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This effect is generally small and we neglect it in the present work,  but we have studied it in detail elsewhere [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Elastic and Plastic Regimes  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 1 we plot the fraction P6 of six-fold coordinated charges versus temperature T/Tm  for a system with no quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The melting temperature Tm is defined to be the  temperature at which a proliferation of topological defects or non-sixfold coordinated  charges occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, P6 is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, as expected for a triangular lattice,  while for T/Tm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, a large number of fivefold and sevenfold coordinated charges  appear, causing P6 to drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Once we have defined Tm by measuring a clean system, we introduce quenched  disorder in order to study the conduction noise and transport response above and below  Tm for varied disorder strengths Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We apply a constant drive with FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01 to  Noise and Thermal Depinning of Wigner Crystals  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  T/Tm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9  1  P6  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The average fraction of sixfold-coordinated charges P6 versus temperature  T/Tm in a system with no quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Tm is defined to be the temperature at  which a proliferation of non-sixfold coordinated charges occurs in a clean system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  samples with different Fp and measure the time average velocity per charge ⟨V ⟩ over  4 × 106 simulation time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' When Fp = 0, the charge velocity V0 is identical to the  driving force, V0 = FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01, so a measurement of ⟨V ⟩/V0 = 1 indicates that the  flow of the charges has reached the pin-free limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 2(a), where we plot ⟨V/V0⟩  versus Fp, at T/Tm = 0 there is a large drop in ⟨V ⟩/V0 near Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Figure 2(b)  shows the corresponding values of P6 versus Fp, where for T/Tm = 0 there is a well  defined transition from an ordered Wigner crystal to a disordered Wigner glass, and the  proliferation of defects correlates with the velocity drop in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='3,  the overall velocity is higher than for the T/Tm = 0 sample due to the lowering of  the effectiveness of the pinning by the thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Additionally, the pinning  strength required to disorder the system is shifted upward to a value close to Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='1,  which is again due to the partial reduction of the pinning effectiveness by the thermal  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' A similar trend occurs for T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6, where the velocity is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For  T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, the system is disordered for all values of Fp and the velocity is even higher  but has a gradual drop with increasing Fp, and the same trend occurs for T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' A  more detailed study of the general phase diagram for the disordered and ordered phases  as a function of pinning strength versus temperature appears in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The results in  Noise and Thermal Depinning of Wigner Crystals  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8  1  <V>/V0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15  Fp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9  1  P6  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' (a) The average charge velocity ⟨V ⟩/V0 vs pinning strength Fp at FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01,  where V0 = FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01 is the average velocity in a disorder-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' T/Tm = 0  (dark blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='3 (light blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6 (green), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03 (yellow) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='38 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  (b) The  corresponding P6 vs Fp showing that for T/Tm < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, there is a well defined pinning-  induced order to disorder transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 2 indicate that the system exhibits three distinct regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' These are an  ordered or crystal regime containing sixfold-coordinated charges, which occurs at low  temperatures or low pinning strengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' a disordered or plastic regime where the system  has low mobility and is strongly affected by the pinning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' and a high temperature fluid  phase where the effectiveness of the pinning is reduced and the system is in a strongly  fluctuating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In terms of transport, in the presence of pinning the ordered state  exhibits elastic depinning in which the charges maintain their same neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The glass  state undergoes plastic depinning, and the fluid state does not have a pinned phase but  can still have a regime in which the charges are trapped for a time before thermally  Noise and Thermal Depinning of Wigner Crystals  7  hopping out of the pinning sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Transport and Noise in the Plastic Regime  We next examine the noise and transport in the three regimes identified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We  consider samples with Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5, a pinning strength at which the charges are disordered  for T/Tm = 0, so the system is in a strongly disordered glass phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The T/Tm = 0  plastic depinning that occurs in this regime was studied in detail in [66], where a pinned  phase, a filamentary flow phase, a disordered flow phase, and a dynamically ordered  moving smectic phase appear in sequence as a function of increasing drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 3(a) we show a snapshot of the charge locations, pinning site locations, and  trajectories in the plastic flow regime for FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15 at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15, where a portion  of the charges are moving in a series of well defined channels, with occasional jumps  between the channels when certain channels open or close again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In general, for strong  pinning, at low temperature the system exhibits channel flow just above depinning,  similar to that studied in other systems at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Figure 3(b) shows the  same system at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 where there is a combination of channel flow and random  thermal hopping, indicating that as the temperature increases, there is a transition  from one-dimensional (1D) channels to two-dimensional (2D) flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 3(c), at  T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03 the motion is 2D and fluidlike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For higher temperatures, the images look  similar to what is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 4(a) we plot ⟨V ⟩ versus FD for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 3 at T/Tm = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='21, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='81, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For the lower temperatures, there is a well-defined depinning  threshold followed by a nonlinear regime, while when T/Tm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, the threshold is  replaced by a creep regime and the nonlinear regime at higher drives persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At the  highest drives, all of the curves approach the pin-free limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 4(b) we show the  corresponding d⟨V ⟩/dFD versus FD curves, where for T/Tm ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='21 there is a peak  in d⟨V ⟩/dFD due to the S shape of the velocity-force curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Similar peaks in the  differential conductivity were observed for driven superconducting vortices in the plastic  flow regime [12, 17, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='21, the peaks are lost and a creep regime  appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The dashed line is the differential conductivity for the pin free system, and  all of the curves approach this value at high drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We note that for T/Tm = 0, at  low drives there are a number of jumps in the conduction as well as a few regimes of  negative differential conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This arises due to a filamentary flow channel effect  that is described in more detail in [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5, the jumps associated with  the filamentary flow phase are lost and a single large peak in d⟨V ⟩/dFD appears in the  plastic flow regime where there is a combination of moving and pinned charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 5 we plot P6 versus FD for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 4 for T/Tm = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='757, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='91, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='21, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0 there is an initial dip in P6 at the onset of  plastic flow, and at high drives where d⟨V ⟩/dFD starts to flatten, P6 approaches values  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9 or higher as the system forms a moving smectic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In the moving smectic  state, the charges move in well defined channels and a small number of dislocations are  Noise and Thermal Depinning of Wigner Crystals  8  x  (a)  y  x  (b)  y  x  (c)  y  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Charge locations (red circles), trajectories (blue lines), and pinning  site locations (black circles) for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 2 at FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15 and Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' (a)  Filamentary flow at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' (b) Disordered flow with channels at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  (c) T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  present that have their Burgers vectors aligned with the driving direction [2, 55, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  As T/Tm increases further, the drive at which the smectic state emerges shifts to higher  values of FD, and for T/Tm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, the system no longer forms a moving smectic but  instead becomes a moving fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  From the features in the transport curves and P6 plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 4 and 5, we  construct a dynamic phase diagram of the evolution of the different phases as a function  of FD versus T/Tm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At low drives we find a pinned or creep regime denoted  C, where d⟨V ⟩/dFD < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The dynamically ordered moving smectic phase MS appears  Noise and Thermal Depinning of Wigner Crystals  9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8  1  <V>  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75  1  FD  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  d<V>/dFD  (a)  (b)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  (a) Velocity ⟨V ⟩ vs drive FD for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 3 with Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 at  T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0 (purple), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='91 (dark green), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='21 (light green), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='81 (orange),  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='42 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' (b) The corresponding d⟨V ⟩/dFD vs FD curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The dashed lines  indicate the pin-free limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  when P6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The disordered regime is where the system is structurally disordered but  moving, and it can be either a moving glass MG for T/Tm < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, or a moving liquid ML  for T/Tm > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The overall features of the phase diagram are similar to those observed  in driven superconducting vortex systems with quenched disorder, as first proposed by  Koshelev and Vinokur [13], where the transition between the MG and MS states shifts to  higher drives as T/Tm is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' [13], the transition line from the disordered  to moving ordered phase was argued to be proportional to A/(Tm −T), where A is some  prefactor and the moving ordered phase can be described in terms of having an effective  temperature that is decreasing toward zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This picture assumes the formation of a  Noise and Thermal Depinning of Wigner Crystals  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  2  FD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8  1  P6  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  P6 vs FD for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 4 with Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 at T/Tm = 0 (purple),  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303 (dark blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 (light blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='757 (green), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='91 (yellow), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='21 (orange), and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='42 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The system reaches an ordered state at high drives for T/Tm < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  moving crystal at high drive, and is somewhat modified in our system since the moving  state we observe is a smectic in which the dynamic fluctuations are anisotropic [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We  find that a better fit to the transition line in our case is (Tm − T)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='7, which is likely  due to the anisotropic nature of the moving smectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Now that we have established the dynamic phase diagram as a function of drive  versus temperature, we can ask how the velocity fluctuation power spectra change as a  function of FD and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The power spectrum as a function of ω = 2πf can be calculated  using the time series v(t) of the velocity fluctuations,  S(ω) =  ����  �  v(t)e−iωt  ����  2  (2)  At T = 0 the noise has a 1/f α signature with α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8, in agreement with recent  experiments [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The noise power is reduced at high drives and shows a crossover to  a narrow band signature when the system forms a moving smectic phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, the  experiments do not detect a narrow band noise signature at higher drives, suggesting  that thermal effects could be coming into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 7 we plot power spectra of the velocity time series for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 6  at a drive of FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15 for T/Tm = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm = 0, the  Noise and Thermal Depinning of Wigner Crystals  11  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  T/Tm  0  1  2  3  4  FD  MS  ML  MG  C  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Dynamic phase diagram as a function of FD vs T/Tm for the system in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 4 and 5 with Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' There is a pinned or creep phase C (red), a disordered  moving phase (blue) that is a moving glass, MG, at lower temperatures and a moving  liquid, ML, at higher temperatures, and a moving smectic MS (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  low frequency noise has a 1/f α form, where the dashed line is a fit with α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='8,  while at higher frequencies the noise tail has α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5, the lower  frequency noise power is reduced and α drops closer to α = 0, the value expected for  white noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, the high frequency noise still has a 1/f 2 form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For higher T/Tm,  the low frequency noise power is further reduced while the higher frequency noise power  is enhanced, and the spectrum becomes much whiter overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  To better characterize the system, we measure the noise power S0, which is the  value of the spectral power integrated in a small window around a specific frequency  ω = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 8 we show S0 versus FD for T/Tm = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03 on a log-  linear plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T = 0 there is a large peak in S0 over the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01 < FD < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5, which  corresponds to the appearance of 1/f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='85 noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The noise is white for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 < FD < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9,  and for FD > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='9 a narrow band noise signal appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606,  there is still a peak in the noise near FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='2, but as the temperature increases, the  peak power diminishes and the peak location shifts to lower drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This is correlated  with a whitening of the low frequency noise, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, the  Noise and Thermal Depinning of Wigner Crystals  12  10  100  1000  ω  10  9  10  8  10  7  10  6  10  5  S(ω)  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Power spectra S(ω) vs ω for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 6 with FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15 for  T/Tm = 0 (dark blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303 (light blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 (yellow), and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The spectral  signature changes from 1/f to white at low frequencies as the temperature increases,  while the amount of noise power at higher frequencies increases with increasing T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  sharp noise power peak is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At large FD, we find that the noise power increases with  increasing temperature due to the transition from flow through narrow 1D channels  in the smectic state to a 2D Brownian like motion in the liquid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The overall  behavior of the noise power that we find is in agreement with experimental observations  [56], where there is a large peak in the noise power near the depinning threshold at  low temperatures, while for higher temperatures the noise power peak is reduced and  shifts to lower drives before disappearing at sufficiently high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Another  feature that is also observed in the experiments is that the noise power increases with  temperature at large drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  We next consider thermal effects in the high drive limit where the system forms a  moving smectic at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9(a) we show S(ω) vs ω for the system from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 6 at  FD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 and T/Tm = 0, where there are a series of peaks associated with a narrow band  noise signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9(b), there are still strong peaks associated  with the narrow band noise but the higher harmonic peaks are strongly reduced in  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9(c) at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606, the level of background noise has increased and  the narrow band peaks are diminished in size, while at T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9(d), the  Noise and Thermal Depinning of Wigner Crystals  13  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  2  FD  10  10  10  9  10  8  10  7  10  6  10  5  S0  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The noise power S0 at fixed ω = 20 vs FD for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 7 at  FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15 for T/Tm = 0 (dark blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303 (light blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 (yellow), and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  moving smectic phase is lost and the narrow band peaks disappear into the background  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' To better characterize the change in the narrow band noise signature, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 10  we plot the noise power S0 at ω = 323, which is the location of the most pronounced  narrow band noise peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For T/Tm < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 there is a strong narrow band  noise signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75 the narrow band noise level is close to the value  of the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This suggests that thermal effects can strongly reduce the  narrow band noise signal even at temperatures well below T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, which could  explain why the narrow band noise signals are difficult to see in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' To better  understand the origins of the changes in the noise signals, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 11(a) we plot the  trajectories of the charges at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 where narrow band noise is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The  system is still in a moving smectic state but the channels have been broadened by the  thermal fluctuations, and there are several regions in which the channel structures are  starting to break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Figure 11(b) shows the trajectories for T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='07, where  the 1D channel structure is lost, there is a significant amount of transverse diffusion,  and the narrow band noise peaks disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This result indicates that the narrow band  noise occurs only when the motion of the charges is mostly 1D in character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Noise and Thermal Depinning of Wigner Crystals  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  S(ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  S(ω)  0  1000  2000  3000  ω  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  S(ω)  0  1000 2000 3000 4000 5000  ω  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0×10  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5×10  5  S(ω)  (a)  (b)  (c)  (d)  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  S(ω) vs ω for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 6 at FD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 where the system is in the  moving smectic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' T/Tm = (a) 0, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606, and (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Thermal Depinning Noise in the Elastic Regime  We next consider the thermal depinning and noise in the elastic regime where the charges  maintain their same neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 2 we select a value of Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05, well below  the T/Tm = 0 disordering threshold of Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12(a) we show ⟨V ⟩ versus  FD at Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05 for T/Tm = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0378, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0756, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, and we plot  the corresponding d⟨V ⟩/dFD curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' As T/Tm increases, the depinning  threshold shifts to lower FD, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12(b), the peak in d⟨V ⟩/dFD that appears for  T = 0 is lost for T/Tm > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We note that the system remains in an ordered state  up to T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0 for all values of FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The d⟨V ⟩/dFD curves also show a multiple peak  feature at high temperatures, with one peak at the finite temperature threshold and a  second peak near the T = 0 depinning threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In between these two peaks, the flow  is creep-like in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 13 we plot S(ω) versus ω for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12 at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='1515 for  different values of FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='046, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='06, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='025, the  motion occurs mostly in the form of avalanches, and no clear narrow band signatures  are present but the low frequency noise has high power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' For FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, the system  starts to develop a narrow band noise signature that sharpens with increasing drive,  and for FD ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='06, which is above the zero temperature depinning threshold, the low  frequency noise is strongly suppressed and the narrow band noise peaks become much  Noise and Thermal Depinning of Wigner Crystals  15  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  T/Tm  1×10  5  S0  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The value of the noise power S0 at ω = 323, the frequency of the largest  narrow band noise peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9, vs T/Tm for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 8 with FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Here the narrow band noise peaks are lost near T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  x  (a)  y  x  (b)  y  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Charge locations (red circles), trajectories (blue lines), and pinning site  locations (black circles) for the system in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 9 and 10 at FD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5 for T/Tm = (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606 and (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Noise and Thermal Depinning of Wigner Crystals  16  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='02  <V>  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='02  FD  0  2  4  6  d<V>/dFD  (a)  (b)  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' (a) ⟨V ⟩ vs FD for a system that exhibits elastic depinning at T/Tm = 0,  where Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The different curves are for temperatures of T/Tm = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0378,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0756, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='606, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The dashed line is the expected  curve in the pin free limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' (b) The corresponding d⟨V ⟩/dFD vs FD curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This result shows that in the elastic flow regime, the narrow band noise signal  is more robust than in the plastic phase, and it appears once the system has depinned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 14 we plot S(ω) vs ω for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 13 at T/Tm = 0 and T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303  at a drive of FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' At T/Tm = 0, there is a strong narrow band noise feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Interestingly, at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, although the level of background noise has increased,  the primary narrow band noise peak is enhanced in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The increase in the narrow  band peak occurs when thermal effects weaken the effectiveness of the pinning and allow  the charges to become better ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This effect is diminished in the case of strong  pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Noise and Thermal Depinning of Wigner Crystals  17  10  10  10  9  10  8  S(ω)  10  10  10  9  10  8  S(ω)  10  10  10  9  10  8  S(ω)  10  10  10  9  10  8  10  7  S(ω)  10  0  10  1  10  2  10  3  10  4  ω  10  11  10  10  10  9  10  8  S(ω)  10  0  10  1  10  2  10  3  10  4  ω  10  11  10  10  10  9  10  8  10  7  S(ω)  (a)  (b)  (c)  (d)  (e)  (f)  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  S(ω) vs ω for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12 with Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05 at T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='1515  for FD = (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='025, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='03, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='04, (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='046, (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='06, and (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  To better characterize the narrow band noise behavior for the system in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12 and  13, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 15 we plot the noise power S0 versus T/Tm for FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='02 where the system is  always in a moving state at the narrow band peak of ω = 80 and the background noise  signal at ω = 300, along with the difference between these two noise powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Unlike the  case for strong pinning, the power of the narrow band noise signal generally increases  with increasing T/Tm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, the background noise power also increases, and the  amount of power in the two signals becomes equal near T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The narrow band  noise peak has the greatest amount of additional power compared to the background  noise near T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This is again due to thermal effects washing out any additional  avalanche-like motion and permitting the charge lattice to become better ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Discussion  Narrow band noise has been observed experimentally in superconducting vortex [26, 28],  magnetic skyrmion [32], and charge density wave [4] systems, but has not been seen for  Wigner crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' There have been reports of periodic noise in charge ordering systems  such as stripe or bubble forming states [33, 37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, this noise generally appears at  low frequencies and is probably not associated with the lattice-scale narrow band noise,  Noise and Thermal Depinning of Wigner Crystals  18  0  200  400  600  ω  0  2×10  8  4×10  8  6×10  8  S(ω)  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The power spectra S(ω) vs ω for the system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 13 with Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05  at FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='02 in the moving phase for T/Tm = 0 (blue) and T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  At T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='303, although the overall background noise power is higher, there is an  enhancement of the narrow band noise signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  but instead arises due to the motion of some other periodically moving macroscopic scale  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In the experiments of Brussarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' [56], the peak noise power decreased  with increasing temperature, similar to what we observe, but no narrow band noise signal  was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' This could be the result of several possible factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' If the drive applied to  the system is not uniform, there could still be strong plastic flow at low drives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however,  at high drives the system may not form a uniformly ordered moving state but could  instead break into several locally ordered regions that are moving at different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Related to this, if the quenched disorder has a wide range of strength so that some of  the charges are moving while a small number remain pinned, a disordered flow regime  would emerge in which narrow band noise is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' A narrow band noise signal could  also be masked by strong background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In this case, the signal could be boosted  by applying an additional ac drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' If the frequency of this ac drive is swept, phase  locking or Shapiro steps would appear when the frequency comes into resonance with  the narrow band signal [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Another possible issue is that the narrow band frequency  could be too high to detect with the available experimental setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, for a system  in the elastic depinning limit, fairly low frequency periodic signals could be generated in  the creep regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' The lack of experimentally observed narrow band noise may suggest  Noise and Thermal Depinning of Wigner Crystals  19  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='5  1  T/Tm  0  2×10  8  4×10  8  6×10  8  8×10  8  S0  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  The noise power S0 vs T/Tm for the system in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' 12 and 13 with  Fp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='05 at FD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='02 for the narrow band frequency of ω = 80 (green circles), the  background noise at ω = 300 (red squares), and the difference (blue triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  that elastic depinning of the Wigner crystal is not occurring and that the systems are  generally in the disordered or plastic flow regimes where the only available narrow band  noise signals are of the moving smectic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In principle, we think that the best place  to look for a narrow band noise signature is in a sample with relatively weak pinning just  above the depinning threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In our work, we focused on samples that were entirely  within the elastic or plastic regimes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' however, close to the transition between the elastic  and plastic regimes, the plastic flow noise may be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Summary  We have investigated the thermally induced depinning and noise fluctuations for driven  Wigner crystal systems with quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We identify an elastic regime in which  the charges maintain the same neighbors at depinning as well as a plastic regime in which  the system is broken up into moving and non-moving regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In the plastic depinning  regime, the velocity noise has a 1/f shape and there is a peak in the noise power above  the depinning threshold at lower temperatures, while for large temperatures, the noise  power peak is reduced and the spectrum becomes white, in agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  For high drives at low temperatures in the plastic regime, the system forms a moving  Noise and Thermal Depinning of Wigner Crystals  20  smectic with a narrow band noise signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' We find that this narrow band signal persists  up to T/Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='75, where Tm is the temperature at which the charge lattice melts in the  absence of quenched disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In the elastic regime, the system remains ordered up to  temperatures approaching T/Tm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='0, although thermal effects reduce the depinning  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' In the elastic regime, 1/f noise appears only in the creep regime where there  are avalanches or jumps of motion, while in the sliding regime, pronounced narrow band  noise appears that reaches its lowest power at the disorder-free melting temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Our results show that measurements of the velocity noise spectra and noise power can  be used in connection with transport curves to distinguish different phases of driven  Wigner crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Acknowledgments  We gratefully acknowledge the support of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Department of Energy through  the LANL/LDRD program for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  This work was supported by the US  Department of Energy through the Los Alamos National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  Los Alamos  National Laboratory is operated by Triad National Security, LLC, for the National  Nuclear Security Administration of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Department of Energy (Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  892333218NCA000001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content='  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
+page_content=' Fisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFIT4oBgHgl3EQf6Cuj/content/2301.11392v1.pdf'}
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+Information Flow Tracking Methods for Protecting
+Cyber-Physical Systems against Hardware Trojans
+- a Survey -
+Sofia Maragkou
+Institute of Computer Technology, TU Wien
+Vienna University of Technology
+Gusshausstr. 27–29 / 384, 1040 Wien, Austria
+sofia.maragkou@tuwien.ac.at
+Axel Jantsch
+Institute of Computer Technology, TU Wien
+Vienna University of Technology
+Gusshausstr. 27–29 / 384, 1040 Wien, Austria
+axel.jantsch@tuwien.ac.at
+Abstract—Cyber-physical systems (CPS) provide profitable
+surfaces for hardware attacks such as hardware Trojans. Hard-
+ware Trojans can implement stealthy attacks such as leaking
+critical information, taking control of devices or harm humans.
+In this article we review information flow tracking (IFT) methods
+for protecting CPS against hardware Trojans, and discuss their
+current limitations. IFT methods are a promising approach for
+the detection of hardware Trojans in complex systems because the
+detection mechanism does not necessarily rely on potential Trojan
+behavior. However, in order to maximize the benefits research
+should focus more on black-box design models and consider real-
+world attack scenarios.
+Index Terms—hardware Trojans, detection, hardware security,
+real hardware attacks, information flow tracking, cyber-physical
+production systems, cyber-physical systems
+I. INTRODUCTION
+Hardware security began facing desultory challenges much
+later than software [1]. In 1996 a timing attack was published
+[2] based on which sensitive information could be leaked from
+a cryptographic component. After this point, hardware security
+research became more systematic. From 2005 on [1, 3] the
+field of hardware security has gained ground in the academic
+and the industrial world because it breaks the chain of trust
+known so far.
+This chain of trust, from the hardware security perspective,
+begins at the integrated circuit (IC) supply chain, where
+security vulnerabilities are formed by the needs of the market
+for fast and cheap technology. The involvement of external en-
+tities in the design process and the internationally outsourced
+fabrication can create security breaches that can be even
+relevant for national security. Design houses, in order to stay
+competitive, purchase third-party intellectual property (3PIP)
+cores from vendors and outsource the fabrication process
+without always verifying the returned product with respect
+to hardware security breaches. The reason for that is that the
+verification of the purchased cores is an expensive process that
+requires resources and time. Those intellectual property (IP)
+cores or chips are integrated and distributed to the customers.
+Consequently, hardware security has to deal with attacks like
+IP piracy, reverse engineering, counterfeit chips and hardware
+Trojans.
+In a real world scenario, when an IP core is being purchased,
+the design house requests some design specification and the
+3PIP core vendor replies with the IP core and the specifications
+of the IP core. Throughout this information exchange, the only
+trusted part is the specification requested by the design house.
+The core in return, is considered untrusted and it is treated
+as black box. Information flow tracking (IFT) methods are
+a promising research direction for the detection of hardware
+Trojans because the verification can be based on the security
+specification of the application and not only on potentially
+malicious designs. Thus, the verification methods can be
+adapted based on the application. In addition, those methods
+can be flexible regarding new attacks, and can be expandable
+in case of the alteration of the security specifications.
+A. Known Real World Attacks
+The real world hardware attacks are much more complicated
+than the attacks developed by the research community, since
+real world attacks interact with different layers of the comput-
+ing system and communicate with external systems over long
+distance. Compared to software, real world hardware attacks
+are less frequent. The information that is publicly available
+about real world attacks is limited and specific details are
+rarely known to the public.
+The real world attack that received most attention is the
+2007 attack on a Syrian military radar [4, 5]. Even though
+the details were not officially revealed, all the indications
+suggest that the radar at a nuclear installation in Syria has
+been tampered. The attack took place in September 2007 and
+the nuclear installation was completely destroyed by Israeli
+bombing jets. The Israeli jets, took off from southern Israel,
+crossed the Mediterranean Sea and the Syrian-Turkish borders
+and returned four hours later. The state of the art radars did not
+detect the jets, which raised suspicions for malicious alteration
+of their functionality. Adee [4] suspects a kill-switch or a
+backdoor in the off-the-shelf microprocessor that could block
+a bombing radar by an apparently remote command (trigger)
+arXiv:2301.02620v1  [cs.CR]  27 Nov 2022
+
+without shutting down the whole system. The difference of
+a kill switch and a backdoor is that the kill switch will shut
+off a specific chip when triggered, but a backdoor requires
+an intruder to implement the same effect. The hypothesis of
+the kill switch is more likely and, in order to be implemented,
+requires the injection of extra logic. The HW and SW overhead
+for such an attack is very small which makes it hard to
+detect during testing, and the threat models discussed are
+the malicious designer and the malicious manufacturer. The
+microprocessor used remains unknown. This is not the only
+occasion where microprocessors including a kill switch have
+supposedly been used. According to anonymous sources from
+U.S defense department, it is known that a European chip
+maker is building microprocessors with a kill switch, and the
+French defense uses this technology for military applications.
+Undocumented microchips were found in the servers assem-
+bled by Supermicro [6, 7], that implemented a doorway to the
+network of the original system, which incorporated memory,
+networking capacity and processing power. The attack aimed
+at leaking sensitive information over a long term.
+Stuxnet attack provides an example of the real world attack
+capabilities in the industrial environment [8]. Stuxnet is a
+worm that was introduced in the Microsoft Windows operating
+systems and it was targeting specific industrial control systems
+of Siemens which were used in Iran to run centrifuges. Until
+the target was found the worm was updating itself. The worm
+was compromising the targeted system by exploiting ’zero-
+day’ vulnerabilities. After monitoring the operation, the worm
+was taking the control of the control system and it ran the
+centrifuges to the point of failure, returning false feedback to
+cover the failure until the damage was irreversible.
+Hybrid attacks are very common in real world scenar-
+ios. The hybrid attacks can include hardware, software and
+firmware parts. Such an attack can be malicious software that
+exploits vulnerabilities of the hardware, damaging physical
+resources such as Stuxnet [8].
+B. Cyber-Physical Production Systems
+Cyber-physical systems (CPS) are sophisticated systems that
+combine physical and cyber units. They are used in many
+different applications and they are the fundamental units of
+the internet of things (IoT) . Their functionality is based on
+the information exchange and the interaction with each other.
+According to the [9], the nature of the CPS makes them
+particularly sensitive to attacks, due to their heterogeneous
+nature, their reliance on data and their large scale.
+When those systems are integrated in the production envi-
+ronment then we refer to them as cyber-physical production
+systems (CPPS). Often, CPPS expose a profitable surface to
+adversaries for hardware Trojan introduction, because they are
+complex, sophisticated structures that manage sensitive infor-
+mation with extend communication among them, which facili-
+tates malicious functionality to stay hidden. Consequently, we
+consider securing the CPPS an emerging, critical issue.
+According to [10], the pyramid of the automation hierar-
+chy known until recently, is decentralized in the concept of
+Industry 4.0. The information processing has been distributed
+in many control units which exchanging information with the
+goal to optimize the production process. The control units have
+moved closer to the technical processes for efficiency, creating
+an interactive communication net among heterogeneous sys-
+tems. This creates the challenge to secure those components.
+Assume that a hardware Trojan is included in one of
+the control units. In Industry 4.0 machines use machine
+to machine (M2M) communication for sensitive information
+exchange. That means that the authentication keys are stored
+and processed in the machines. If the hardware Trojan leaks
+an authentication key to the adversary, she can take the control
+of the unit and possibly the control of the factory.
+In such a demanding environment the CPPS should stay
+consistent to the security requirements. Availability, integrity
+and confidentiality are only the basic guidelines of the prop-
+erties that should be taken into consideration. The proof that
+the units of those systems comply to those properties and to
+more detailed ones can be achieved with IFT methods as we
+discuss in the next sections.
+C. Scope
+The scope of this report is to survey how IFT methodologies
+can secure CPS against hardware Trojan attacks and how those
+methods need to be further developed in order to be applicable
+in real world scenarios.
+The remainder of this survey is organised as follows:
+Section II provides basic information about hardware Trojans.
+Section III refers to basic information for IFT methods and
+presents state of the art methodologies against information
+leakage. Finally, in section V we compare the IFT methods
+and we discuss future steps for research.
+II. HARDWARE TROJANS
+Hardware Trojans are circuits with hidden, unspecified,
+malicious functionality that can be included in any phase
+of the IC supply chain. In the environment of Industry 4.0,
+stealthy attacks like hardware Trojans can implement any kind
+of effect, including information leakage. In this report we are
+interested in this kind of malicious activity.
+Figure 1 shows a time bomb hardware Trojan from [11].
+This hardware Trojan is activated when the counter reaches
+the value 2k − 1. When the trigger is activated, the output
+value at ER* becomes different from the initial signal ER.
+The circuitry with the counter is the trigger and the circuitry
+that changes the value of the signal ER is the payload. This
+is a simplified example. More sophisticated mechanisms have
+been proposed from the research community like the Trojans
+mentioned above.
+According to the taxonomy of R. Karri, J. Rajendran, K.
+Rosenfeld, M. Tehranipoor [12], a hardware Trojan can be
+described by the insertion phase, the abstraction level, the
+activation mechanism (trigger), the effects (payload) and the
+location in the design.
+
+…
+0
+1
+K-1
+CLK
+ER
+ER*
+Fig. 1. Time bomb hardware Trojan based on [11]
+1) Insertion phase: The earlier a hardware Trojan is in-
+troduced in the design the broader the range of its impact is
+and the lower the cost of the attack is. For instance, assume
+that a third party vendor infects an IP core with a hardware
+Trojan. This IP core can be integrated in more than one
+design, increasing the number of infected systems. On the
+other hand, the scenario of the malicious manufacturer is
+design-specific. The attacker, in order to introduce a Trojan,
+should be aware of the design details which can be acquired by
+reverse engineering, a technique that needs special knowledge
+and is expensive in time and resources. Consequently, the
+phase of the hardware Trojan introduction, in combination
+with the value of the protected assets should be taken into
+consideration, during the development of countermeasures.
+2) Abstraction level: Depending on the abstraction level of
+the design, a hardware Trojan can be injected at system level,
+at the development environment, at register-transfer level as
+soft IP core, at gate level as firm IP core, at transistor level as
+hard IP core or at the physical level.
+3) Triggers: There are hardware Trojans exploiting don’t
+care conditions for their trigger mechanisms [13], or data
+patterns in specific memory addresses [14], or even dedicated
+input images [15]. Some attacks have even more sophisticated
+triggers which are activated during the design flow, leaving no
+trigger signal to the possible detection algorithm [16, 17].
+4) Payload: The most common attacks realized by hard-
+ware Trojans are sensitive information leakage and denial
+of service (DoS) attacks. Other attacks can be functional
+alteration, downgrade performance, data corruption, circuit
+aging, chip destruction, etc.
+5) Attack targets: The most common targets for hardware
+Trojan attacks are memory elements [18–21] and crypto-
+graphic components [13, 22, 23]. However, there are many
+proposals for attacking cores such as UARTs [24] or AXI4-
+bus interconnects [25], FPGA LUTs [16], CPUs [26–28], etc.
+6) Resources required: For the majority of the Trojans we
+study, the attacker needs knowledge of the design and access
+to it (e.g. bitstream [29], netlists [30] or access to the design
+tools [16, 17, 31]).
+III. INFORMATION FLOW TRACKING
+The basic idea behind IFT methods is that they track the
+influence of information of a system during computation. In
+order to achieve that, they assign tags (usually binary values)
+for each of the data element of the design and they update
+the value of the tag based on the applied method and the
+applied security properties. The verification is achieved by the
+observation of the value of the tags.
+IFT methods can be used with different verification tech-
+niques as it is described in the taxonomy in [32]. More
+specifically they can verify security properties through static
+methods like simulation, formal verification, emulation, and
+virtual prototyping or through dynamic methods like runtime
+monitoring techniques.
+There are many IFT methods used with different verification
+techniques and at different abstraction levels and tackling dif-
+ferent problems, since not all those methods address hardware
+Trojans.
+Here in this paper we chose to present different IFT ap-
+proaches and discuss their limitations and requirements. We
+present IFT static methods that tackle information leakage.
+Information leakage is the most common hardware Trojan
+effect and in the case of CPS it can cause economic loss or
+even set a human life in danger.
+As we discussed earlier, the runtime monitoring methods
+can be expensive in resources, and the recovery from those
+attacks can be costly too. Based on that, we chose to focus
+on the static IFT verification methods. Static IFT methods are
+applied in design-time, identifying the malicious behavior soon
+enough to minimize the recovery cost. Moreover, they do not
+add overhead in the original designs resources.
+IV. IFT METHODS AGAINST HARDWARE TROJANS
+Many methodologies are using theorem proving to verify
+the information flow in the designs [33–36]. In those methods
+the security properties are expressed as theorems and theorem
+proving tools such as Coq are used to verify them. In the
+proof-carrying hardware IP (PCHIP) framework [33] the IP
+vendors are required to deliver the HDL code of the design
+with formal proofs that the code is according to some security
+properties predefined among the two parties. For instance, such
+a property could describe that an instruction is allowed to
+access memory locations, which are defined in its op-code.
+With the provided security tags to the signals PCHIP can
+track the information flow in the design. The disadvantage
+of theorem proving methods is the manual conversion of
+the HDL core to the theorem proving language and proof
+checking environment (e.g. Coq and CoqIDE). Even though
+a conversion from HDL to Coq has been proposed [33, 34],
+theorem proving is far from an automated technique.
+The approach proposed in [37] addresses black box models.
+It is based on information flow security (IFS) verification
+which detects violations of security properties. An asset is
+modeled as stuck-at-0 and stuck-at-1 faults and, by leverag-
+ing the automatic test pattern generation (ATPG), faults are
+searched for. When a fault is detected, it means that there
+is an information flow from the asset to observation points.
+Finally the trigger mechanisms is extracted. This methodology
+is based on the fact that the trigger mechanism is injected in
+the original circuit.
+
+The tool Register Transfer Level Information Flow Tracking
+(RTLIFT)[38], can be applied directly to HDL code. Secu-
+rity tags (or labels) are assigned to every signal. Register
+transfer level information flow tracking (RTLIFT) uses IFT
+logic to securely propagate the tags throughout the design.
+The functionality of the additional IFT logic depends on the
+precision required. For instance, the output of an operation can
+be tainted when any of the inputs is tainted. If an untainted
+input influences the output to be untainted even though the
+other input is tainted, a false positive may occur. To avoid
+inaccuracies, the modules implementing the flow tracking
+logic take such cases into consideration. Based on the required
+trade off between complexity and precision, different precision
+levels can be achieved. Given the Verilog code, the control
+and the data flow precision flags (which define the required
+precision level), the tool generates a functionally equivalent
+Verilog code including IFT logic (IFT-Verilog code). The IFT-
+Verilog code is tested against the security properties requested
+for the design through simulation or formal verification. If the
+design passes this process, the extra logic is removed and the
+design is sent for fabrication. If it fails, the design has to be
+altered and to go through this process again.
+The methodology described in [39], gate-level information-
+flow tracking (GLIFT), can detect hardware Trojans injected
+by malicious third-party vendors, that alter the functionality
+of the original circuit or leak sensitive information. According
+to GLIFT, each data bit is assigned to a security label. This
+is implemented with additional tracking logic. It is up to the
+designers to define the security properties and use the GLIFT
+to verify the cores. For example, assume that the goal is
+to track the flow of a cryptographic key in order to ensure
+that it does not leak. The security labels of the keys will
+take the value ’confidential’ and the security property that
+verifies that there is no leakage should ensure that no bit with
+’confidential’ label ends up in an output or memory with the
+label ’untrusted’. Thus, this technique can identify violations
+of confidentiality and integrity and, hence, expose a hardware
+Trojan.
+Both methods discussed above [38, 39] face the problem of
+false positives results, which have to be resolved manually.
+The method proposed by Wang et al. [40], called HLIFT,
+detects hardware Trojans based on the trigger behavior at
+register transfer level (RTL) with the use of control and
+data flow graphs (CDFG). The method can identify hardware
+Trojans that leak information through specific outputs pins
+or side channel, without functional modification and through
+unspecified output pins. This approach is based on a feature
+matching methodology that captures specific Trojan features.
+The features are based on three kind of Trojan triggers: always-
+on, immediate-on, sequential-on. This methodology can be
+divided in the predefinition flow and the application flow.
+During the predefinition flow, statement CDFGs are build
+based on already known infected RTL designs. Statement
+CDFGs are abstract, high-level and compact RTL netlists. That
+way unnecessary information is removed which decreases the
+complexity. IFT is applied on the CDFGs and a list of Trojan
+IFT features is created. At the application flow, the statement-
+level CDFG is extracted from the unknown RTL design, and
+it is compared for matches with the list of the extracted Trojan
+features.
+The methodology proposed in [41] uses virtual prototyping
+(SystemC TLM 2.0) to identify information leakage or un-
+trusted access. At the behavioral level there is a lack of design
+details. Thus, the security properties applied are very strict.
+This can lead to false positives. This approach identifies the
+vulnerable paths and reports them to the user for inspection.
+Consequently, the inspection process is done manually, adding
+time overhead.
+The approach in [42], creates IFT models and optimizes
+them according to specific security properties. The security
+properties are compiled to security constraints and assertions,
+which are combined with the trimmed IFT model. Finally, the
+combination of the IFT model with the security constraints and
+assertions is verified through simulation, emulation or formal
+verification.
+In contrast to the methods presented above, the method in
+[43] does not use any of the mentioned verification methods.
+The HDL code is converted to an abstract syntax tree (AST)
+to identify, track and localize anomaly behavior. The AST is
+converted to directed data-flow graph (DFG). This process
+automatically recognizes interaction between IP cores. By
+identifying the sink and the source signals, the tool detects
+vulnerabilities and finally locates the threats.
+V. DISCUSSION AND CONCLUSIONS
+The development of hardware Trojans is flourishing as they
+attract interest from the academia and industry. As counter-
+measures, IFT methodologies are very promising, because
+they can be flexible, adaptable and expandable based on the
+application.
+However, the IFT verification methodologies proposed so
+far, cannot be applied in real world scenarios. To the best
+of our knowledge, usually the purchased IP cores are not in
+a white box form (usually the cores are purchased locked
+in order to avoid IP piracy), or the specifications of the
+cores provided are considered untrusted. Thus, the IP cores
+purchased are treated as black boxes. That means that the
+internals of the purchased modules are unknown and can
+be leveraged from other layers of the systems (firmware or
+software) for potential attacks.
+Thus, there is a need to explore more IFT methods for black
+box designs without the usage of known hardware Trojan
+behaviors. The reason we suggest, that the known Trojan
+behaviors should not be taken into consideration is because the
+attackers want their Trojans to stay hidden, pushing the limits
+of the current known Trojan behaviors, in order to make them
+more stealthy. A case in point is the development of trigger
+mechanisms. In recent years there is the tendency to include
+the trigger mechanisms in the design flow, so that the detection
+methods searching for trigger behaviors cannot detect them.
+On the other hand, methods that are based on security
+properties to identify unwanted or unspecified behavior in the
+
+TABLE I
+STATIC IFT METHODS - WB=WHITE BOX, BB=BLACK BOX,
+TP=THEOREM PROVING, MC=MODEL CHECKING, GL= GATE LEVEL,
+SL=SEQUENCIAL LOGIC
+Method
+Abstraction
+BB/
+Verification
+Limitations
+level
+WB
+method
+[33]
+RTL
+WB
+TP
+based on
+conservative
+rules [44]
+[35]
+GL
+WB
+TP
+manual proof
+or RTL
+construction
+[36]
+GL
+WB
+TP
+proof of genuine
+benchmark ,
+does not
+support SL
+[34]
+GL
+WB
+TP and MC
+high complexity,
+false positives
+[37]
+GL
+BB
+partial scan ATPG
+based on
+analysis
+trigger condition
+[38]
+RTL
+WB
+simulation or
+challenged in
+SAT solving
+complex
+structures
+[39]
+GL
+WB
+simulation
+creates
+false positives
+[40]
+RTL
+WB
+feature matching
+based on HT
+features
+[41]
+behavioral
+WB
+virtual prototypes
+lack of design
+details,
+manual inspection
+[42]
+RTL
+WB
+assertion based
+false positives
+or GL
+simulation
+emulation
+[43]
+RTL
+WB
+If-tracker
+false positives
+designs seem more flexible with respect to unknown attacks.
+However, the completeness of the security properties is an
+open problem. Another issue is the definition of the security
+properties by the engineers. Manual processes can result in
+vulnerabilities of the systems which can be leveraged by
+adversaries.
+Identifying a hardware Trojan in a real world example can
+be very challenging, especially since the trigger mechanism is
+not necessarily part of the original design. In some concepts
+a fault, a vulnerability, or a backdoor may be no different
+from a well covered Trojan. From the real world attacks we
+can conclude that the attack scenarios implemented are much
+more complete than the ones provided by academia. In the
+real world examples mentioned above we identify mechanisms
+that can communicate at great distance and can affect state of
+the art systems. The attacks were sophisticated enough with
+complicated mechanisms with more than negligible overhead.
+It will be useful for the research community to explore more
+complicated attacks, across the levels of a computing system
+in order to facilitate corresponding countermeasures.
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+
diff --git a/5dE0T4oBgHgl3EQfvgHM/content/tmp_files/load_file.txt b/5dE0T4oBgHgl3EQfvgHM/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf,len=631
+page_content='Information Flow Tracking Methods for Protecting  Cyber-Physical Systems against Hardware Trojans  a Survey -  Sofia Maragkou  Institute of Computer Technology, TU Wien  Vienna University of Technology  Gusshausstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' 27–29 / 384, 1040 Wien, Austria  sofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='maragkou@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='at  Axel Jantsch  Institute of Computer Technology, TU Wien  Vienna University of Technology  Gusshausstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' 27–29 / 384, 1040 Wien, Austria  axel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='jantsch@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='at  Abstract—Cyber-physical systems (CPS) provide profitable  surfaces for hardware attacks such as hardware Trojans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Hard-  ware Trojans can implement stealthy attacks such as leaking  critical information, taking control of devices or harm humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  In this article we review information flow tracking (IFT) methods  for protecting CPS against hardware Trojans, and discuss their  current limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' IFT methods are a promising approach for  the detection of hardware Trojans in complex systems because the  detection mechanism does not necessarily rely on potential Trojan  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' However, in order to maximize the benefits research  should focus more on black-box design models and consider real-  world attack scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Index Terms—hardware Trojans, detection, hardware security,  real hardware attacks, information flow tracking, cyber-physical  production systems, cyber-physical systems  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' INTRODUCTION  Hardware security began facing desultory challenges much  later than software [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In 1996 a timing attack was published  [2] based on which sensitive information could be leaked from  a cryptographic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' After this point, hardware security  research became more systematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' From 2005 on [1, 3] the  field of hardware security has gained ground in the academic  and the industrial world because it breaks the chain of trust  known so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  This chain of trust, from the hardware security perspective,  begins at the integrated circuit (IC) supply chain, where  security vulnerabilities are formed by the needs of the market  for fast and cheap technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The involvement of external en-  tities in the design process and the internationally outsourced  fabrication can create security breaches that can be even  relevant for national security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Design houses, in order to stay  competitive, purchase third-party intellectual property (3PIP)  cores from vendors and outsource the fabrication process  without always verifying the returned product with respect  to hardware security breaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The reason for that is that the  verification of the purchased cores is an expensive process that  requires resources and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Those intellectual property (IP)  cores or chips are integrated and distributed to the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Consequently, hardware security has to deal with attacks like  IP piracy, reverse engineering, counterfeit chips and hardware  Trojans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  In a real world scenario, when an IP core is being purchased,  the design house requests some design specification and the  3PIP core vendor replies with the IP core and the specifications  of the IP core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Throughout this information exchange, the only  trusted part is the specification requested by the design house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The core in return, is considered untrusted and it is treated  as black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Information flow tracking (IFT) methods are  a promising research direction for the detection of hardware  Trojans because the verification can be based on the security  specification of the application and not only on potentially  malicious designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Thus, the verification methods can be  adapted based on the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In addition, those methods  can be flexible regarding new attacks, and can be expandable  in case of the alteration of the security specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Known Real World Attacks  The real world hardware attacks are much more complicated  than the attacks developed by the research community, since  real world attacks interact with different layers of the comput-  ing system and communicate with external systems over long  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Compared to software, real world hardware attacks  are less frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The information that is publicly available  about real world attacks is limited and specific details are  rarely known to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The real world attack that received most attention is the  2007 attack on a Syrian military radar [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Even though  the details were not officially revealed, all the indications  suggest that the radar at a nuclear installation in Syria has  been tampered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The attack took place in September 2007 and  the nuclear installation was completely destroyed by Israeli  bombing jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The Israeli jets, took off from southern Israel,  crossed the Mediterranean Sea and the Syrian-Turkish borders  and returned four hours later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The state of the art radars did not  detect the jets, which raised suspicions for malicious alteration  of their functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Adee [4] suspects a kill-switch or a  backdoor in the off-the-shelf microprocessor that could block  a bombing radar by an apparently remote command (trigger)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='02620v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='CR]  27 Nov 2022  without shutting down the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The difference of  a kill switch and a backdoor is that the kill switch will shut  off a specific chip when triggered, but a backdoor requires  an intruder to implement the same effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The hypothesis of  the kill switch is more likely and, in order to be implemented,  requires the injection of extra logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The HW and SW overhead  for such an attack is very small which makes it hard to  detect during testing, and the threat models discussed are  the malicious designer and the malicious manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The  microprocessor used remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This is not the only  occasion where microprocessors including a kill switch have  supposedly been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' According to anonymous sources from  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='S defense department, it is known that a European chip  maker is building microprocessors with a kill switch, and the  French defense uses this technology for military applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Undocumented microchips were found in the servers assem-  bled by Supermicro [6, 7], that implemented a doorway to the  network of the original system, which incorporated memory,  networking capacity and processing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The attack aimed  at leaking sensitive information over a long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Stuxnet attack provides an example of the real world attack  capabilities in the industrial environment [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Stuxnet is a  worm that was introduced in the Microsoft Windows operating  systems and it was targeting specific industrial control systems  of Siemens which were used in Iran to run centrifuges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Until  the target was found the worm was updating itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The worm  was compromising the targeted system by exploiting ’zero-  day’ vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' After monitoring the operation, the worm  was taking the control of the control system and it ran the  centrifuges to the point of failure, returning false feedback to  cover the failure until the damage was irreversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Hybrid attacks are very common in real world scenar-  ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The hybrid attacks can include hardware, software and  firmware parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Such an attack can be malicious software that  exploits vulnerabilities of the hardware, damaging physical  resources such as Stuxnet [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Cyber-Physical Production Systems  Cyber-physical systems (CPS) are sophisticated systems that  combine physical and cyber units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' They are used in many  different applications and they are the fundamental units of  the internet of things (IoT) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Their functionality is based on  the information exchange and the interaction with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  According to the [9], the nature of the CPS makes them  particularly sensitive to attacks, due to their heterogeneous  nature, their reliance on data and their large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  When those systems are integrated in the production envi-  ronment then we refer to them as cyber-physical production  systems (CPPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Often, CPPS expose a profitable surface to  adversaries for hardware Trojan introduction, because they are  complex, sophisticated structures that manage sensitive infor-  mation with extend communication among them, which facili-  tates malicious functionality to stay hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Consequently, we  consider securing the CPPS an emerging, critical issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  According to [10], the pyramid of the automation hierar-  chy known until recently, is decentralized in the concept of  Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The information processing has been distributed  in many control units which exchanging information with the  goal to optimize the production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The control units have  moved closer to the technical processes for efficiency, creating  an interactive communication net among heterogeneous sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This creates the challenge to secure those components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Assume that a hardware Trojan is included in one of  the control units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='0 machines use machine  to machine (M2M) communication for sensitive information  exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' That means that the authentication keys are stored  and processed in the machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' If the hardware Trojan leaks  an authentication key to the adversary, she can take the control  of the unit and possibly the control of the factory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  In such a demanding environment the CPPS should stay  consistent to the security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Availability, integrity  and confidentiality are only the basic guidelines of the prop-  erties that should be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The proof that  the units of those systems comply to those properties and to  more detailed ones can be achieved with IFT methods as we  discuss in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Scope  The scope of this report is to survey how IFT methodologies  can secure CPS against hardware Trojan attacks and how those  methods need to be further developed in order to be applicable  in real world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The remainder of this survey is organised as follows:  Section II provides basic information about hardware Trojans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Section III refers to basic information for IFT methods and  presents state of the art methodologies against information  leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Finally, in section V we compare the IFT methods  and we discuss future steps for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' HARDWARE TROJANS  Hardware Trojans are circuits with hidden, unspecified,  malicious functionality that can be included in any phase  of the IC supply chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In the environment of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='0,  stealthy attacks like hardware Trojans can implement any kind  of effect, including information leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In this report we are  interested in this kind of malicious activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Figure 1 shows a time bomb hardware Trojan from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  This hardware Trojan is activated when the counter reaches  the value 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' When the trigger is activated, the output  value at ER* becomes different from the initial signal ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The circuitry with the counter is the trigger and the circuitry  that changes the value of the signal ER is the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This  is a simplified example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' More sophisticated mechanisms have  been proposed from the research community like the Trojans  mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  According to the taxonomy of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Karri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Rajendran, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Rosenfeld, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Tehranipoor [12], a hardware Trojan can be  described by the insertion phase, the abstraction level, the  activation mechanism (trigger), the effects (payload) and the  location in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  …  0  1  K-1  CLK  ER  ER*  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Time bomb hardware Trojan based on [11]  1) Insertion phase: The earlier a hardware Trojan is in-  troduced in the design the broader the range of its impact is  and the lower the cost of the attack is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' For instance, assume  that a third party vendor infects an IP core with a hardware  Trojan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This IP core can be integrated in more than one  design, increasing the number of infected systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' On the  other hand, the scenario of the malicious manufacturer is  design-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The attacker, in order to introduce a Trojan,  should be aware of the design details which can be acquired by  reverse engineering, a technique that needs special knowledge  and is expensive in time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Consequently, the  phase of the hardware Trojan introduction, in combination  with the value of the protected assets should be taken into  consideration, during the development of countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  2) Abstraction level: Depending on the abstraction level of  the design, a hardware Trojan can be injected at system level,  at the development environment, at register-transfer level as  soft IP core, at gate level as firm IP core, at transistor level as  hard IP core or at the physical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  3) Triggers: There are hardware Trojans exploiting don’t  care conditions for their trigger mechanisms [13], or data  patterns in specific memory addresses [14], or even dedicated  input images [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Some attacks have even more sophisticated  triggers which are activated during the design flow, leaving no  trigger signal to the possible detection algorithm [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  4) Payload: The most common attacks realized by hard-  ware Trojans are sensitive information leakage and denial  of service (DoS) attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Other attacks can be functional  alteration, downgrade performance, data corruption, circuit  aging, chip destruction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  5) Attack targets: The most common targets for hardware  Trojan attacks are memory elements [18–21] and crypto-  graphic components [13, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' However, there are many  proposals for attacking cores such as UARTs [24] or AXI4-  bus interconnects [25], FPGA LUTs [16], CPUs [26–28], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  6) Resources required: For the majority of the Trojans we  study, the attacker needs knowledge of the design and access  to it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' bitstream [29], netlists [30] or access to the design  tools [16, 17, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' INFORMATION FLOW TRACKING  The basic idea behind IFT methods is that they track the  influence of information of a system during computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In  order to achieve that, they assign tags (usually binary values)  for each of the data element of the design and they update  the value of the tag based on the applied method and the  applied security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The verification is achieved by the  observation of the value of the tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  IFT methods can be used with different verification tech-  niques as it is described in the taxonomy in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' More  specifically they can verify security properties through static  methods like simulation, formal verification, emulation, and  virtual prototyping or through dynamic methods like runtime  monitoring techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  There are many IFT methods used with different verification  techniques and at different abstraction levels and tackling dif-  ferent problems, since not all those methods address hardware  Trojans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Here in this paper we chose to present different IFT ap-  proaches and discuss their limitations and requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' We  present IFT static methods that tackle information leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Information leakage is the most common hardware Trojan  effect and in the case of CPS it can cause economic loss or  even set a human life in danger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  As we discussed earlier, the runtime monitoring methods  can be expensive in resources, and the recovery from those  attacks can be costly too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Based on that, we chose to focus  on the static IFT verification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Static IFT methods are  applied in design-time, identifying the malicious behavior soon  enough to minimize the recovery cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Moreover, they do not  add overhead in the original designs resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' IFT METHODS AGAINST HARDWARE TROJANS  Many methodologies are using theorem proving to verify  the information flow in the designs [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In those methods  the security properties are expressed as theorems and theorem  proving tools such as Coq are used to verify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In the  proof-carrying hardware IP (PCHIP) framework [33] the IP  vendors are required to deliver the HDL code of the design  with formal proofs that the code is according to some security  properties predefined among the two parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' For instance, such  a property could describe that an instruction is allowed to  access memory locations, which are defined in its op-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  With the provided security tags to the signals PCHIP can  track the information flow in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The disadvantage  of theorem proving methods is the manual conversion of  the HDL core to the theorem proving language and proof  checking environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Coq and CoqIDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Even though  a conversion from HDL to Coq has been proposed [33, 34],  theorem proving is far from an automated technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The approach proposed in [37] addresses black box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  It is based on information flow security (IFS) verification  which detects violations of security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' An asset is  modeled as stuck-at-0 and stuck-at-1 faults and, by leverag-  ing the automatic test pattern generation (ATPG), faults are  searched for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' When a fault is detected, it means that there  is an information flow from the asset to observation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Finally the trigger mechanisms is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This methodology  is based on the fact that the trigger mechanism is injected in  the original circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The tool Register Transfer Level Information Flow Tracking  (RTLIFT)[38], can be applied directly to HDL code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Secu-  rity tags (or labels) are assigned to every signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Register  transfer level information flow tracking (RTLIFT) uses IFT  logic to securely propagate the tags throughout the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The functionality of the additional IFT logic depends on the  precision required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' For instance, the output of an operation can  be tainted when any of the inputs is tainted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' If an untainted  input influences the output to be untainted even though the  other input is tainted, a false positive may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' To avoid  inaccuracies, the modules implementing the flow tracking  logic take such cases into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Based on the required  trade off between complexity and precision, different precision  levels can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Given the Verilog code, the control  and the data flow precision flags (which define the required  precision level), the tool generates a functionally equivalent  Verilog code including IFT logic (IFT-Verilog code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The IFT-  Verilog code is tested against the security properties requested  for the design through simulation or formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' If the  design passes this process, the extra logic is removed and the  design is sent for fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' If it fails, the design has to be  altered and to go through this process again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The methodology described in [39], gate-level information-  flow tracking (GLIFT), can detect hardware Trojans injected  by malicious third-party vendors, that alter the functionality  of the original circuit or leak sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' According  to GLIFT, each data bit is assigned to a security label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This  is implemented with additional tracking logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' It is up to the  designers to define the security properties and use the GLIFT  to verify the cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' For example, assume that the goal is  to track the flow of a cryptographic key in order to ensure  that it does not leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The security labels of the keys will  take the value ’confidential’ and the security property that  verifies that there is no leakage should ensure that no bit with  ’confidential’ label ends up in an output or memory with the  label ’untrusted’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Thus, this technique can identify violations  of confidentiality and integrity and, hence, expose a hardware  Trojan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Both methods discussed above [38, 39] face the problem of  false positives results, which have to be resolved manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The method proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' [40], called HLIFT,  detects hardware Trojans based on the trigger behavior at  register transfer level (RTL) with the use of control and  data flow graphs (CDFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The method can identify hardware  Trojans that leak information through specific outputs pins  or side channel, without functional modification and through  unspecified output pins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This approach is based on a feature  matching methodology that captures specific Trojan features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The features are based on three kind of Trojan triggers: always-  on, immediate-on, sequential-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This methodology can be  divided in the predefinition flow and the application flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  During the predefinition flow, statement CDFGs are build  based on already known infected RTL designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Statement  CDFGs are abstract, high-level and compact RTL netlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' That  way unnecessary information is removed which decreases the  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' IFT is applied on the CDFGs and a list of Trojan  IFT features is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' At the application flow, the statement-  level CDFG is extracted from the unknown RTL design, and  it is compared for matches with the list of the extracted Trojan  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The methodology proposed in [41] uses virtual prototyping  (SystemC TLM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='0) to identify information leakage or un-  trusted access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' At the behavioral level there is a lack of design  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Thus, the security properties applied are very strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  This can lead to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This approach identifies the  vulnerable paths and reports them to the user for inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Consequently, the inspection process is done manually, adding  time overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The approach in [42], creates IFT models and optimizes  them according to specific security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The security  properties are compiled to security constraints and assertions,  which are combined with the trimmed IFT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Finally, the  combination of the IFT model with the security constraints and  assertions is verified through simulation, emulation or formal  verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  In contrast to the methods presented above, the method in  [43] does not use any of the mentioned verification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  The HDL code is converted to an abstract syntax tree (AST)  to identify, track and localize anomaly behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The AST is  converted to directed data-flow graph (DFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' This process  automatically recognizes interaction between IP cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' By  identifying the sink and the source signals, the tool detects  vulnerabilities and finally locates the threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSIONS  The development of hardware Trojans is flourishing as they  attract interest from the academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' As counter-  measures, IFT methodologies are very promising, because  they can be flexible, adaptable and expandable based on the  application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  However, the IFT verification methodologies proposed so  far, cannot be applied in real world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' To the best  of our knowledge, usually the purchased IP cores are not in  a white box form (usually the cores are purchased locked  in order to avoid IP piracy), or the specifications of the  cores provided are considered untrusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Thus, the IP cores  purchased are treated as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' That means that the  internals of the purchased modules are unknown and can  be leveraged from other layers of the systems (firmware or  software) for potential attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Thus, there is a need to explore more IFT methods for black  box designs without the usage of known hardware Trojan  behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The reason we suggest, that the known Trojan  behaviors should not be taken into consideration is because the  attackers want their Trojans to stay hidden, pushing the limits  of the current known Trojan behaviors, in order to make them  more stealthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' A case in point is the development of trigger  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In recent years there is the tendency to include  the trigger mechanisms in the design flow, so that the detection  methods searching for trigger behaviors cannot detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' methods that are based on security  properties to identify unwanted or unspecified behavior in the  TABLE I  STATIC IFT METHODS - WB=WHITE BOX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' BB=BLACK BOX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  TP=THEOREM PROVING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' MC=MODEL CHECKING,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' GL= GATE LEVEL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  SL=SEQUENCIAL LOGIC  Method  Abstraction  BB/  Verification  Limitations  level  WB  method  [33]  RTL  WB  TP  based on  conservative  rules [44]  [35]  GL  WB  TP  manual proof  or RTL  construction  [36]  GL  WB  TP  proof of genuine  benchmark ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  does not  support SL  [34]  GL  WB  TP and MC  high complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  false positives  [37]  GL  BB  partial scan ATPG  based on  analysis  trigger condition  [38]  RTL  WB  simulation or  challenged in  SAT solving  complex  structures  [39]  GL  WB  simulation  creates  false positives  [40]  RTL  WB  feature matching  based on HT  features  [41]  behavioral  WB  virtual prototypes  lack of design  details,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  manual inspection  [42]  RTL  WB  assertion based  false positives  or GL  simulation  emulation  [43]  RTL  WB  If-tracker  false positives  designs seem more flexible with respect to unknown attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  However, the completeness of the security properties is an  open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Another issue is the definition of the security  properties by the engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' Manual processes can result in  vulnerabilities of the systems which can be leveraged by  adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  Identifying a hardware Trojan in a real world example can  be very challenging, especially since the trigger mechanism is  not necessarily part of the original design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In some concepts  a fault, a vulnerability, or a backdoor may be no different  from a well covered Trojan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' From the real world attacks we  can conclude that the attack scenarios implemented are much  more complete than the ones provided by academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' In the  real world examples mentioned above we identify mechanisms  that can communicate at great distance and can affect state of  the art systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content=' The attacks were sophisticated enough with  complicated mechanisms with more than negligible overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
+page_content='  It will be useful for the research community to explore more  complicated attacks, across the levels of a computing system  in order to facilitate corresponding countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
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+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
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+page_content=' Iskander, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE0T4oBgHgl3EQfvgHM/content/2301.02620v1.pdf'}
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+Deep Breath: A Machine Learning Browser
+Extension to Tackle Online Misinformation
+Marc Kydd
+School of Design and Informatics
+Division of Cyber Security
+Abertay University
+Dundee, United Kingdom
+m.kydd1800@abertay.ac.uk
+Lynsay A. Shepherd
+School of Design and Informatics
+Division of Cyber Security
+Abertay University
+Dundee, United Kingdom
+lynsay.shepherd@abertay.ac.uk
+Abstract—Over the past decade, the media landscape has seen
+a radical shift. As more of the public stay informed of current
+events via online sources, competition has grown as outlets vie
+for attention. This competition has prompted some online outlets
+to publish sensationalist and alarmist content to grab readers’
+attention. Such practices may threaten democracy by distorting
+the truth and misleading readers about the nature of events.
+This paper proposes a novel system for detecting, processing,
+and warning users about misleading content online to combat the
+threats posed by misinformation. By training a machine learning
+model on an existing dataset of 32,000 clickbait news article
+headlines, the model predicts how sensationalist a headline is and
+then interfaces with a web browser extension which constructs
+a unique content warning notification based on existing design
+principles and incorporates the models’ prediction. This research
+makes a novel contribution to machine learning and human-
+centred security with promising findings for future research. By
+warning users when they may be viewing misinformation, it is
+possible to prevent spontaneous reactions, helping users to take
+a deep breath and approach online media with a clear mind.
+Index Terms—Misinformation, Machine Learning, Human-
+Centred Security, Cyber Security, Web Technologies.
+I. INTRODUCTION AND BACKGROUND
+The Internet has rapidly become the dominant means for
+users to stay connected, even more so in the wake of the
+COVID-19 pandemic; however, this has led to several prob-
+lems [1]. From connecting with friends and family, comment-
+ing on recent events, or just being entertained, the Internet
+plays an integral role in how we inform and stay informed
+[2]. However, in a landscape that rewards attention rather than
+quality, some have turned to more dubious practices such as
+sensationalism, misinformation, and shocking imagery.
+A. The Incentivisation of Misinformation & Clickbait
+As more and more content is published every second, focus
+shifts from producing quality content to hijacking attention.
+When users’ interest is constantly pulled from one article,
+video, or image to the next, publishers need to find new
+ways to garner clicks. Consequently, this can easily lead
+to sensationalism, clickbait, and in the worst case, outright
+misinformation without the user being aware.
+The constant news cycle of the Information Age presents
+difficulties for news outlets to write up each emerging story.
+Instead, some outlets adopt the practice of aggregating content,
+either lightly editorialising a piece of existing content or
+directing users to another source to read the information there.
+The method of news aggregation is controversial, with some
+feeling that it is blatant theft of content, whilst others see
+it as the only viable solution to surviving in a fast-paced
+information economy [3] [4].
+B. The Impact of Sharing Misinformation and Clickbait
+Regardless of the ethics of clickbait and similar practices,
+publishing misinformation can be harmful once it reaches a
+broader audience. Many internet users utilise social network-
+ing, which compounds misinformation’s effects by allowing
+for rapid dissemination of falsehoods and half-truths. Often
+this creates a chain-reaction scenario where one user shares
+a story with another user, who in turn shares it with another,
+and so on. The impact of this phenomenon does not require a
+user to have a vast following on social media platforms. Cˆot´e
+and Darling [5] found that once a Twitter account surpassed
+approximately 1000 followers, it was much more likely to
+attract other users from a wide variety of backgrounds, in-
+cluding members of the general public, outreach groups, and
+even influential decision makers. This dramatically increases
+the potential audience that a piece of content can reach - and be
+shared by. Such a situation can result in users being presented
+with the same information multiple times, albeit being posted
+by different users.
+Fazio, Rand and Pennycook analysed the effects of
+widespread sharing of information [6]. The authors exam-
+ined how repeated exposure to information (both true and
+false) affects individuals’ perception of the data presented.
+Participants were asked to rate a series of statements on a
+percentage scale based on their believability. Statements were
+presented in two ways: some were shown only once, while
+others were repeatedly shown and were interspersed with one-
+off statements. Participants’ perception of truth across the
+accuracy scale rose without any influencing factor, other than
+being repeatedly shown the same information.
+When rated by participants, information regarded as the
+truth was rated as such. Similarly, information strongly con-
+sidered false did not influence participants sufficiently to
+arXiv:2301.03301v1  [cs.HC]  9 Jan 2023
+
+cross over into the “considered as true” category. This was
+noticeable in the 1%-30% bin, where repeated exposure did not
+generate a pronounced change in participant opinion. One of
+the paper’s key findings relates to information considered to be
+ambiguous. Information initially considered unclear gradually
+became regarded as accurate and authentic the more the user
+was exposed to the statement. This was particularly evident
+in the 41-70% bin which saw, on average, an increase of 6
+points compared to their original rating.
+Existing research [3]–[6] suggests that, in the current hyper-
+social age, users risk having their perceptions warped by
+misleading statements, hyperbole, and repeated exposure to
+information. This repetition does not have to stem purely from
+repeatedly seeing the same news article displayed but also
+from the conversation surrounding the topic. Increasingly frag-
+mented and confrontational viewpoints presented online com-
+plicates how users determine what is true or false. However,
+simply stating that a user is ‘misinformed’ or ‘wrong’ does
+not make them more inclined to reassess their understanding.
+Instead, a more thoughtful approach is required, giving users
+a starting point from which to come to their own conclusions.
+As Bronstein et al. [7] suggests, interventions catering towards
+the promotion of ‘open-minded and analytical thinking’ could
+be of benefit in curbing the impact of misinformation.
+C. Informing Users of Risk Online
+As
+misinformation
+has
+become
+more
+apparent
+and
+widespread in recent years, research into warning users when
+they are being misled has received increased interest, both in
+academia and industry. Many social media sites have taken
+steps to curb the impact of misinformation on their respective
+platforms, typically by presenting users with a warning label.
+This approach offers a simple means of quickly informing
+users that the content they are viewing may be misleading;
+proving popular with various social media platforms adopting
+similar practices including Twitter [8] and Meta’s Facebook
+[9].
+Ross et al. [10] analysed the effectiveness of warning labels
+adopted by major social platforms. Focusing primarily on the
+methods used to dissuade users from sharing misinformation,
+the authors tested two different label styles, one replicating
+those used by Meta and another informed by contemporary
+research. Participants (N = 151) were shown content consisting
+of six true and six false stories. Group one were presented
+all stories without any warning label. Group two were shown
+half the stories with a warning and the other half without. The
+participants in each group were then asked to determine which
+stories were manipulated or fake and which were unaltered.
+The research found that neither of the warning messages
+changed user behaviour. Users did not appear to be more
+suspicious of labelled content and were just as likely to interact
+with the content as that which was unlabelled.
+D. Designing Effective Warning Labels
+Ross et al. [10] indicate that providing additional context to
+the user can be beneficial in curbing the impact of misinfor-
+mation. Careful consideration in the design process on what
+information is communicated to the user and how could be
+instrumental in limiting misinformation’s reach.
+Shepherd and Renaud [11] conducted a literature review
+on designing effective security warning labels in browsers.
+Assessing existing work in this area, the authors found that
+time and resources are not adequately allocated to designing
+warning labels leading to user frustration and confusion. This
+sentiment appeared to be validated by Ross et al. [10]. The au-
+thors found that the effectiveness of current solutions differed.
+Some warnings wrongly assumed that users had background
+knowledge on a topic which decreased their effectiveness,
+whilst others were too vague in their language that users did
+not understand the ramifications of their choices.
+To combat the aforementioned issues, Shepherd and Renaud
+[11] concluded with the proposal of a set of design guidelines
+for browser warnings. The authors note that warnings designed
+for privacy and those designed for security differed, suggesting
+that different priorities must be considered depending on the
+intended use- case. The proposed guidelines recommend using
+simple and concise language to alert users to a potential
+issue and using neutral colours to avoid an undue emotional
+response. Furthermore, the guidelines propose linking to ad-
+ditional resources should the initial description not prove
+sufficient. Although the research in question is primarily
+targeted toward warning labels for security purposes, there
+is still value in applying these recommendations to tackling
+misinformation.
+E. Designing Effective Browser Warnings and Labels
+Much work has been done previously in the field of usable
+security concerning the design of warning labels. Early at-
+tempts at warning systems typically used contextual measures
+such as a small on-page popup informing the user of potential
+risk. However, work by Wu, Miller and Garfinkel [12] illus-
+trated these popups are often ignored, misunderstood, or users
+do not even recognise they are there.
+Further research in this area took on a different approach,
+utilising interstitial warnings. This approach required users to
+interact with the warning label before proceeding, the under-
+lying theory of this approach being that making the warning
+the central focus of the users’ attention would increase the
+likelihood of users reading and making an informed decision
+on the contents of the warning.
+Until recently, most research on effective warning design
+has been limited to web-security topics such as expired cer-
+tificates or phishing links. Kaiser et al. [13] examined warning
+label design to inform users of potential disinformation online.
+Evaluating several different warning design styles, ranging
+from information-dense with minimal colouring, to warnings
+with a strong visual impact but minimal detail, the authors
+assessed how users responded to the designs in a realistic
+environment.
+In a survey conducted with 238 participants, the authors
+found that participants responded most favourably to designs
+which featured a reference to the perceived risk (“This page
+
+contains misinformation”) and the recommended next step
+(“Consider finding alternative sources.”). The authors note
+that none of the designs evaluated showed any significant
+difference in how likely users were to consult a second
+or alternative source afterwards. The authors propose that
+changes in behaviour were more likely to stem from the
+friction caused by having to manually click through a warning
+rather than the content of the warning itself.
+Multiple factors play a role in shaping how users respond
+to warning labels, including the language used within them.
+Findings from a research study conducted by Mozilla [14] to
+understand how to design better warning labels highlighted
+that employing opinionated design was more important than
+providing objective information. This means it is more im-
+portant to convey the idea of a threat rather than what the
+threat is - prior research suggested overly technical warnings
+lead to confusion among users. Mozilla implemented this
+by simplifying the warning heading to feature abstract but
+understandable language. Additionally, for scenarios where
+users want to know details of the underlying issue, the warning
+provides an accessible description of the risks associated with
+the security fault.
+The issue of labelling misleading content is a challenging
+one. What counts as misinformation must be determined,
+and designing warning systems that promote critical thinking
+rather than knee-jerk reactions is still an ongoing area of re-
+search. Although the means of warning users adopted by major
+social platforms may have limited efficacy [10], Shepherd and
+Renaud [11] indicate that warning labels can cater to users’
+assumed knowledge and understanding without provoking
+undue alarm or concern. Similarly, Kaiser et al. [13] suggest
+that such research can be used to combat misinformation.
+F. Detecting Clickbait with Machine Learning
+The vast array of content posted online every second makes
+it impossible for human moderators to assess and review all
+dubious uploads. The use of an automated system is merited,
+one capable of rapidly and reliably analysing content for
+potentially misleading information which can integrate inter-
+vention measures. Machine learning’s inherent capabilities for
+finding and predicting patterns in information are well-suited
+to tackling misinformation. Furthermore, machine learning
+has seen renewed interest over the past decade as computing
+power and data storage have matured to enable real-world
+applications across a host of use cases.
+Chen, Conroy and Rubin [15] explored if clickbait, and by
+extension, misinformation, could be detected using machine
+learning methods. Conducting a holistic view of research in
+the field, the authors note four unique means of detecting
+clickbait. Initially focusing on the textual content of clickbait
+articles, the authors found clickbait often displays lexical and
+semantic features unique to its form. The authors cited work
+by Lex, Juffinger and Granitzer [16] which analysed clickbait
+based upon factors such as word length, word choice, and
+terminology, and found that a machine learning model could
+be trained to detect clickbait with 77% accuracy regardless of
+the topic discussed.
+Appealing to users’ innate curiosity by using unresolved
+pronouns or alluding to content within the article was also
+consistent with clickbait. As a subsequent paper by Rubin et al.
+[17] noted, automated fact-checking and verification systems
+could help detect language patterns in text and warn users
+that the content they are about to read may be misleading.
+The authors also note that such a tool could prove helpful
+for journalists too, alerting them when they may be conflating
+claims or accidentally misleading.
+Clickbait is not strictly limited to the text of the article
+in question; the authors also found that surrounding factors
+such as imagery and how the user interacts with the article
+play a key role. Regarding the former, the authors [17] cite
+Ecker et al. [18] who found that clickbait articles were likely
+to feature images which were incongruent with the headline.
+In such articles, an image can be used to grab the would-be
+readers’ attention with an impactful but unrelated image or
+shape opinion before the article was read.
+The authors [17] also noted that previous research had found
+clickbait outlets typically aimed to attract user attention before
+funnelling them towards sponsored content or advertising.
+Additionally, the time difference between “time spent reading
+the article” and “time spent sharing and commenting about
+the article” could also be a signifier of clickbait. In this
+aspect, clickbait articles tend to use alarmist or sensationalist
+headlines to provoke knee-jerk responses (whether that be
+commenting or sharing) before the user has actually read the
+content within.
+G. Problem Space
+The rapid rise of the information age has led some to
+adopt unethical practices to drive engagement. Whether these
+practices are deployed purposefully or not, they pose a serious
+risk to society. Although existing work has explored the use of
+warning labels, depending on how these are designed, these
+may be ineffective. Given the amount of content published
+every second, it would be impossible to label the accuracy
+of content manually. Instead, machine learning offers a com-
+pelling alternative. Misinformation poses a severe threat; there-
+fore combining advances in machine learning and warning
+design means an effective solution can be proposed to keep
+users safe.
+II. METHODOLOGY
+The proposed method consists of two-components: the
+machine learning model, for analysing and classifying content,
+and the web extension for communicating potential risk to the
+user. A simplified pipeline can been seen in Figure 1.
+Using TensorFlow [19] and adopting the same dataset as
+used by Chakraborty et al. [20], a Sequential Model was
+trained on 32,000 news article headlines, labelled as either
+‘clickbait’ or ‘non-clickbait’. The model, consisting of four
+layers (excluding the input layer), tokenises input text into a
+64-dimensional dense vector before running it through a global
+
+average pooling filter. Output is then fed through a layer of
+ReLU nodes, followed by a final layer of Sigmoid nodes to
+arrive at a real number between zero (indicating neutral) and
+one (indicating strongly misleading).
+The model was then connected to a browser extension via a
+Native Manifest, which allowed the browser to send portions
+of an article (e.g., the headline) to the model to analyse and
+generate a rating. The rating is then returned to the browser,
+after which a relevant warning can be presented to the user.
+Warnings were designed to be informative and actionable for
+the user, presenting clear detailing about the perceived risk
+and recommended next steps.
+A. Developing the machine learning model
+1) Dataset: The same dataset used by Chakraborty et al.
+[20] was adopted for this project and consisted of 32,000
+news article headlines labelled as either ‘clickbait’ or ‘non-
+clickbait’. The dataset offered a robust and relevant base upon
+which to build. In particular, clickbait and misinformation rely
+on emotional language to provoke a response, suggesting the
+dataset would help develop a model well suited to detecting
+such language.
+2) Model: In practice, the backend of this project cen-
+tres around binary classification: Is this piece of text click-
+bait/misinformation or not? As such, using a Sequential model
+was deemed the most suitable due to its singular input-output
+structure, a structure well suited to classification tasks such as
+this.
+Fig. 1. Example simplified data pipeline.
+3) Preprocessing Data : The dataset was split into 26,666
+training samples and 5,334 testing samples which equates to
+83% of the dataset for training and the remaining 17% for test-
+ing. Before training, the dataset had to be formatted such that
+the model could determine distinctions between clickbait and
+non-clickbait. This was achieved by converting, the headlines
+in the dataset into a vocabulary of word embeddings (Figure
+2). To ensure uniformity across the dataset, all sequences
+were padded to 24 tokens long which was considered a safe
+maximum length for a headline. Headlines longer than 24
+words long were automatically truncated to the maximum
+length.
+Fig. 2. Example word embedding.
+4) Building the Model: The model is visualised in Figure
+3, and consists of the input layer, two hidden layers, and
+the output layer. The first layer takes the input (a tokenised
+sentence) and transforms it into a 64-dimensional dense vector.
+The usage of dense vectors allows for the semantic meaning
+Fig. 3. Visualisation of the Model.
+of the sentence to be compressed, ensuring better general-
+isation. Despite their ability to derive underlying meanings
+and connections for a given sentence, the aforementioned
+dense vectors can result in overfitting if they become too
+detailed. To address the issue, the second layer consists of
+a Global Average Pool. This layer takes the 64-dimensional
+vector and determines the mean of each input channel (the 24-
+dimensional token sequences) which allows the model to learn
+approximations of embeddings rather than their exact values
+(Figure 4).
+Fig. 4. Visualisation of 1-Dimensional global average pooling.
+At this stage, the input data is now formatted and approx-
+imated to limit overfitting, and layers can be constructed,
+which will inform the output of the model. The first activation
+function of the model uses a rectified linear activation function
+(or ‘ReLU’). ReLU ensures that the next layer of the network
+receives a positive value as ReLU outputs 0 for input values
+equal to or less than 0 or the original value for those greater
+than 0. Finally, the data is passed through a Sigmoid activation
+layer, ensuring the resultant output falls between 0 and 1, i.e.,
+“Is this piece of text clickbait or not?”.
+The model measures its performance based on the ac-
+curacy of its predictions. The task involves classification;
+
+9 makeup tips you won't believe!
+Model
+ Clickbait: 1If Disney Princesses Were From Florida
+[10122 
+752
+6586 4 1 737
+0 
+0 0 (
+0 (
+0 (
+000000000000(
+01input:
+[(None, 24)]
+InputLayer
+output:
+[(None, 24)]
+input:
+(None,24)
+Embedding
+output:
+(None, 24, 64)
+input:
+(None, 24, 64)
+GlobalAveragePooling1D
+output:
+(None, 64)
+input:
+(None, 64)
+Dense
+output:
+(None, 2)
+input:
+(None, 2)
+Dense
+output:
+(None, 1)4
+6
+4
+2
+4Fig. 5. Accuracy graph during model training (Higher is better).
+thus, a binary cross-entropy loss function is used. The func-
+tion calculates how far the models’ predictions stray from
+the dataset’s labels. A gradient descent with a momentum
+optimiser (also known as Stochastic Gradient Descent or
+SGD) further minimises the loss function. The optimiser helps
+improve the model’s training rate by minimising loss across
+training iterations. By doing so, it is intended that the model
+predictions will gradually trend towards the expected output.
+Fig. 6. Loss graph during model training (Lower is better).
+The model was trained for 80 epochs with a batch size
+of 128. Although batch sizes larger than 32 can lead to
+underfitting, it was intended that the larger epoch size would
+gradually result in greater accuracy and convergence, which
+can be seen in Figures 5 and 6. After the model was compiled,
+trained, and evaluated, it was exported for use by the web
+extension.
+B. Web extension and warning messages
+1) Creating the web extension: A web browser extension
+was developed to ensure the model could be deployed in a real-
+world context. The extension means the model can analyse
+news articles as the user views them, providing a warning if
+misleading content is found.
+Native Manifests [21] were used to allow a web browser to
+interface with a native application, passing data back and forth
+between the two. These manifests allowed the TensorFlow
+model to perform predictions locally on the machine and then
+send the resultant prediction to the browser for further analysis
+and output.
+Within the browser, the analysis begins when the headline
+of the page the user has visited is fetched. Initially, white
+space and control characters are trimmed. The headline is
+processed, and a value is returned indicating how sensationalist
+the headline is.
+On the user’s device, the program transfers over to the native
+application, which handles parsing the headline into a format
+suitable for the model before computing a rating which is
+returned to the browser. Data from the browser is JSON-
+encapsulated and is sent via standard input (stdin), which
+the program reads from. Following this, the script begins
+importing libraries for loading the model and formatting the
+incoming data accordingly. The model is loaded, and the
+tokeniser is instantiated to convert the incoming headline. At
+this stage, the initial setup is complete and the model is ready
+for use.
+The core of the script features a loop which waits for a
+message from the browser to be received, at which point the
+decoding process can take place. Now that the headline of the
+article is available, the script tokenises it and provides padding
+to ensure compatibility with the model. From here, a standard
+model prediction call can be made, encoded, and returned to
+the browser for display to the user.
+2) Presenting warnings to the user: When the native ap-
+plication produces a result, it is returned to the browser. The
+browser extension then generates a message sent to the news
+article’s page, with the native applications result stored in a
+variable. This message is received by the content script, which
+is injected into each page by the extension.
+The content script dynamically generates a warning label
+for the user. This is done by waiting to receive a message
+(the result) from the background script. This value is then
+multiplied by ten (to accommodate any floating-point issues
+that may arise (i.e. converting 0.8 to 8)). To prevent repeatedly
+warning the user about innocuous content, only headlines that
+score above five out of ten have a warning generated. Scrolling
+is disabled whilst the warning is on-screen, ensuring the user
+has to acknowledge it.
+With regards to this project, the existing literature points
+towards interstitial warnings being the most likely to promote
+change in user behaviour. Additionally, even if most users do
+not appear to actually read the content of a warning label, they
+do show a preference for such information being present.
+Informed by the papers discussed in Section I-D, the web
+extension was designed to ensure strong visual clarity to
+effectively convey risks associated with a piece of misleading
+content.
+The warning adopts a paywall-style design, mimicking an
+approach that users will likely already be familiar with from
+other news sites. This helps to ensure that the warning is
+not overlooked, which can happen with contextual warnings.
+To further ensure that the warning is brought to the user’s
+attention, an overlay is used to darken the article and scrolling
+
+model accuracy
+tain
+0.85
+val
+0.80
+0.75
+accuracy
+0.70
+0.65
+0.60
+0.55
+0.50
+10
+70
+0
+20
+30
+40
+50
+60
+80
+epochmodel loss
+0.70
+tain
+val
+0.65
+0.60
+los5
+0.55
+0.50
+0.45
+0.40
+10
+20
+30
+40
+50
+0
+70
+60
+80
+epochis prevented while the warning is on screen. The warning
+design comes in 5 variants - ranging from most minor to most
+severe, depending on the article’s rating. Exemplar designs can
+be seen in Figure 7
+Fig. 7. Sample of warning designs.
+A vital problem with previous warning designs is the poor
+communication of risk, whereby warnings may be obscured
+by jargon [13]. The design of the warnings seeks to minimise
+existing issues by conveying as much relevant information
+as possible in an easy-to-read format. Prominent symbols
+represent increasing risk levels based on the article’s rating.
+Articles lower on the risk scale are given a more general
+’magnifying glass’ symbol, promoting the notion of thinking
+more critically about the article’s merit. If an article includes
+more severe levels of misinformation, increasingly prominent
+’alert’-oriented symbols are deployed, such as warning signs,
+stop signs and symbols of authority such as police figures.
+Additionally, an oscillating gradient is placed behind the
+warning. Depending on the severity of the warning, the colour
+used will shift from yellow to orange to red. The subtle
+movement of the gradient is intended to draw the user’s eye
+to the warning, with the unique colour of each warning also
+helping the user understand the associated level of risk.
+Ultimately, the extension seeks to change user behaviour
+and provide education on meaningful steps users can take to
+protect themselves from misinformation in the future. As such,
+each warning label features unique phrasing that informs the
+user of not just what the perceived risk is but also advice on
+actionable next steps.
+Two buttons are presented to the user at the bottom of the
+warning, allowing them to dismiss the warning and continue,
+or navigate away from the page. To indicate the intended
+behaviour, the option to navigate away is displayed in promi-
+nent green with a ’Recommended’ label included in brackets.
+Conversely, the option to dismiss the warning is presented in
+red and is slightly faded out to deliberately be obscured against
+the background until the user hovers over the button.
+III. RESULTS AND DISCUSSION
+Fig. 8. Accuracy comparison between training and evaluation.
+During training, the model achieved an accuracy of approx-
+imately 85% and a loss of 0.39, and when pitted against
+the evaluation dataset, the model achieved an accuracy of
+approximately 45% with a loss of 1.15 (Figure 8, Figure 9).
+This decline in accuracy likely stems from inconsistencies in
+the existing evaluation dataset, e.g., improperly formatted data,
+such as some of the labels assigned to the headlines do not
+appear to be correct. This could be resolved via an additional
+data cleaning. Another point of note is that the evaluation
+dataset used only binary labels (Is this headline ‘clickbait’ or
+‘news’?), which may have also contributed to the discrepancy
+in accuracy as the model was producing a result between 0
+and 1 instead of a pure binary output.
+
+ThisContent PutsYouatRisk
+Misleadingarticlescanwarpyourperceptionofevents
+Think twice before continuing
+Deep Breathhas detectedthatthis article is misleading
+and should not be viewed.
+Backto safety (Recommended)
+Misleading Content Alert
+Sensationalistcontentcantrickyouinto consumingfalse
+information.Consultmultiplesources
+Websites can use sensationalist languageto mislead and
+deceive.Deep Breaths detection algorithmbelieves that
+this article is misleading.
+Dismiss and continue
+Back to safety (Recommended)
+SensationalistContentWarning
+Thispagemaycontainsensationalistormisleading
+content.Considerconsulting additionalsources.
+Websites can use sensationalist languagetograb your
+attention.DeepBreathsdetectionalgorithmbelievesthat
+this article may be misleading.
+Dismiss and continue
+Backto safety (Recommended)100
+90
+85%
+80
+70
+Accuracy
+60
+50
+45%
+40
+30
+20
+10
+0
+Training
+Evaluation
+(Higher is better.)Fig. 9. Loss comparison between training and evaluation.
+With regards to machine learning, the project confirms the
+findings of Lex, Juffinger and Granitzer (2010) that clickbait
+and misinformation can be detected based upon lexical seman-
+tics, namely word choice, word length, and word commonality,
+i.e., Word x appears frequently alongside word y.
+IV. CONCLUSION AND FUTURE WORK
+The model demonstrated in this paper has shown a re-
+liable degree of performance, however, it could be refined
+further to derive even better results. The models’ accuracy
+and loss were still increasing and decreasing, respectively,
+suggesting better performance could be obtained before the
+curves flattened out. Furthermore, the capability of the model
+could be extended further. The model has been trained only on
+clickbait-styled headlines, which was effective. However, more
+robust results may be achieved by training on the contents of
+clickbait articles which would allow the model to develop a
+deeper understanding of the article and make a more nuanced
+prediction. The model used in this paper is a Sequential model
+designed to take a single output and produce a single result.
+Although this is effective at classifying a single headline as
+used in this paper, greater functionality could be achieved
+by allowing multiple inputs and outputs. This could include
+assessing the article’s headline but also a selection of sentences
+from the article. The rating assigned to content is dynamic;
+however, the underlying warning remains static. By expanding
+the models’ capabilities, it may also be possible to provide
+personalised warnings relevant to the content. In practice, this
+could mean warning the user about specific aspects of the
+article, such as sensationalist authors, misleading sentences,
+and miscaptioned images. Although every effort was taken to
+ensure the model made balanced and accurate predictions, no
+system is infallible. Conducting user testing and introducing
+the option for users to report when the model makes a
+perceived miscalculation could help adjust for missteps.
+The work presented in this paper makes promising ad-
+vances toward tackling the issue of misinformation online
+by combining machine learning, human-computer interaction
+research, and web technologies. Findings validate and build
+upon prior research, and incorporating machine learning with
+usable security is still a relatively under-explored area of study.
+REFERENCES
+[1] H. S. Lallie, L. A. Shepherd, J. R. Nurse, A. Erola, G. Epiphaniou,
+C. Maple, and X. Bellekens, “Cyber security in the age of covid-19:
+A timeline and analysis of cyber-crime and cyber-attacks during the
+pandemic,” Computers & Security, vol. 105, p. 102248, 2021.
+[2] A.
+Mitchell,
+E.
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+and
+G.
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+[Online]. Available: https://www.pewresearch.org/journalism/2021/11/
+15/news-on-twitter-consumed-by-most-users-and-trusted-by-many/
+[3] H. I. Chyi, S. C. Lewis, and N. Zheng, “Parasite or partner? coverage
+of google news in an era of news aggregation,” Journalism & Mass
+Communication Quarterly, vol. 93, no. 4, pp. 789–815, 2016.
+[4] M. Coddington, “Gathering evidence of evidence: News aggregation as
+an epistemological practice,” Journalism, vol. 21, no. 3, pp. 365–380,
+2020.
+[5] I. M. Cˆot´e and E. S. Darling, “Scientists on twitter: Preaching to the
+choir or singing from the rooftops?” Facets, vol. 3, no. 1, pp. 682–694,
+2018.
+[6] L. K. Fazio, D. G. Rand, and G. Pennycook, “Repetition increases
+perceived truth equally for plausible and implausible statements,” Psy-
+chonomic bulletin & review, vol. 26, no. 5, pp. 1705–1710, 2019.
+[7] M. V. Bronstein, G. Pennycook, A. Bear, D. G. Rand, and T. D. Can-
+non, “Belief in fake news is associated with delusionality, dogmatism,
+religious fundamentalism, and reduced analytic thinking,” Journal of
+applied research in memory and cognition, vol. 8, no. 1, pp. 108–117,
+2019.
+[8] Y.
+Roth
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+Pickles,
+“Updating
+our
+ap-
+proach
+to
+misleading
+information,”
+2020,
+https://blog.twitter.com/en us/topics/product/2020/updating-our-
+approach-to-misleading-information (Accessed 8 February 2022).
+[9] Facebook, “Taking Action Against People Who Repeatedly Share
+Misinformation,”
+2021,
+https://about.fb.com/news/2021/05/taking-
+action-against-people-who-repeatedly-share-misinformation/ (Accessed
+8 February 2022).
+[10] B. Ross, A. Jung, J. Heisel, and S. Stieglitz, “Fake news on social media:
+The (in) effectiveness of warning messages,” in CIS 2018 Proceedings.
+16.
+Association for Information Systems, 2018.
+[11] L. A. Shepherd and K. Renaud, “How to design browser security
+and privacy alerts,” in 2018 AISB Convention: Symposium on Digital
+Behaviour Intervention for Cyber Security.
+Society for the Study of
+Artificial Intelligence and Simulation for . . . , 2018, pp. 21–28.
+[12] M. Wu, R. C. Miller, and S. L. Garfinkel, “Do security toolbars actually
+prevent phishing attacks?” in Proceedings of the SIGCHI conference on
+Human Factors in computing systems, 2006, pp. 601–610.
+[13] B. Kaiser, J. Wei, E. Lucherini, K. Lee, J. N. Matias, and J. Mayer,
+“Adapting security warnings to counter online disinformation,” in 30th
+USENIX Security Symposium (USENIX Security 21), 2021, pp. 1163–
+1180.
+[14] M.
+Walkington,
+“Designing
+Better
+Security
+Warnings,”
+2019,
+https://blog.mozilla.org/ux/2019/03/designing-better-security-warnings/
+(Accessed 8 February 2022).
+[15] Y. Chen, N. J. Conroy, and V. L. Rubin, “Misleading online content:
+recognizing clickbait as” false news”,” in Proceedings of the 2015 ACM
+on workshop on multimodal deception detection, 2015, pp. 15–19.
+[16] E. Lex, A. Juffinger, and M. Granitzer, “Objectivity classification in
+online media,” in Proceedings of the 21st ACM conference on Hypertext
+and hypermedia, 2010, pp. 293–294.
+[17] V. L. Rubin, N. Conroy, Y. Chen, and S. Cornwell, “Fake news or
+truth? using satirical cues to detect potentially misleading news,” in
+Proceedings of the second workshop on computational approaches to
+deception detection, 2016, pp. 7–17.
+[18] U. K. Ecker, S. Lewandowsky, E. P. Chang, and R. Pillai, “The effects
+of subtle misinformation in news headlines.” Journal of experimental
+psychology: applied, vol. 20, no. 4, p. 323, 2014.
+[19] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S.
+Corrado, A. Davis, J. Dean, M. Devin et al., “Tensorflow: Large-scale
+machine learning on heterogeneous distributed systems,” arXiv preprint
+arXiv:1603.04467, 2016.
+
+1.2
+1.15
+1.1
+1
+0.9
+0.8
+.Oss
+0.7
+0.6
+L
+0.5
+0.39
+0.4
+0.3
+0.2
+0.1
+0
+Training
+Evaluation
+(Lower is better.)[20] A. Chakraborty, B. Paranjape, S. Kakarla, and N. Ganguly, “Stop
+clickbait: Detecting and preventing clickbaits in online news media,”
+in Advances in Social Networks Analysis and Mining (ASONAM), 2016
+IEEE/ACM International Conference on.
+IEEE, 2016, pp. 9–16.
+[21] Mozilla, “Native Manifests,” 2022, https://developer.mozilla.org/en-
+US/docs/Mozilla/Add-ons/WebExtensions/Native manifests
+(Accessed
+5 March 2022).
+
diff --git a/6NE1T4oBgHgl3EQfmwRz/content/tmp_files/load_file.txt b/6NE1T4oBgHgl3EQfmwRz/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf,len=492
+page_content='Deep Breath: A Machine Learning Browser  Extension to Tackle Online Misinformation  Marc Kydd  School of Design and Informatics  Division of Cyber Security  Abertay University  Dundee, United Kingdom  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='kydd1800@abertay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='uk  Lynsay A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Shepherd  School of Design and Informatics  Division of Cyber Security  Abertay University  Dundee, United Kingdom  lynsay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='shepherd@abertay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='uk  Abstract—Over the past decade, the media landscape has seen  a radical shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' As more of the public stay informed of current  events via online sources, competition has grown as outlets vie  for attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This competition has prompted some online outlets  to publish sensationalist and alarmist content to grab readers’  attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Such practices may threaten democracy by distorting  the truth and misleading readers about the nature of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  This paper proposes a novel system for detecting, processing,  and warning users about misleading content online to combat the  threats posed by misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' By training a machine learning  model on an existing dataset of 32,000 clickbait news article  headlines, the model predicts how sensationalist a headline is and  then interfaces with a web browser extension which constructs  a unique content warning notification based on existing design  principles and incorporates the models’ prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This research  makes a novel contribution to machine learning and human-  centred security with promising findings for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' By  warning users when they may be viewing misinformation, it is  possible to prevent spontaneous reactions, helping users to take  a deep breath and approach online media with a clear mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Index Terms—Misinformation, Machine Learning, Human-  Centred Security, Cyber Security, Web Technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' INTRODUCTION AND BACKGROUND  The Internet has rapidly become the dominant means for  users to stay connected, even more so in the wake of the  COVID-19 pandemic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' however, this has led to several prob-  lems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' From connecting with friends and family, comment-  ing on recent events, or just being entertained, the Internet  plays an integral role in how we inform and stay informed  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' However, in a landscape that rewards attention rather than  quality, some have turned to more dubious practices such as  sensationalism, misinformation, and shocking imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The Incentivisation of Misinformation & Clickbait  As more and more content is published every second, focus  shifts from producing quality content to hijacking attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  When users’ interest is constantly pulled from one article,  video, or image to the next, publishers need to find new  ways to garner clicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Consequently, this can easily lead  to sensationalism, clickbait, and in the worst case, outright  misinformation without the user being aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The constant news cycle of the Information Age presents  difficulties for news outlets to write up each emerging story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Instead, some outlets adopt the practice of aggregating content,  either lightly editorialising a piece of existing content or  directing users to another source to read the information there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The method of news aggregation is controversial, with some  feeling that it is blatant theft of content, whilst others see  it as the only viable solution to surviving in a fast-paced  information economy [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The Impact of Sharing Misinformation and Clickbait  Regardless of the ethics of clickbait and similar practices,  publishing misinformation can be harmful once it reaches a  broader audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Many internet users utilise social network-  ing, which compounds misinformation’s effects by allowing  for rapid dissemination of falsehoods and half-truths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Often  this creates a chain-reaction scenario where one user shares  a story with another user, who in turn shares it with another,  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The impact of this phenomenon does not require a  user to have a vast following on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Cˆot´e  and Darling [5] found that once a Twitter account surpassed  approximately 1000 followers, it was much more likely to  attract other users from a wide variety of backgrounds, in-  cluding members of the general public, outreach groups, and  even influential decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This dramatically increases  the potential audience that a piece of content can reach - and be  shared by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Such a situation can result in users being presented  with the same information multiple times, albeit being posted  by different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Fazio, Rand and Pennycook analysed the effects of  widespread sharing of information [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The authors exam-  ined how repeated exposure to information (both true and  false) affects individuals’ perception of the data presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Participants were asked to rate a series of statements on a  percentage scale based on their believability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Statements were  presented in two ways: some were shown only once, while  others were repeatedly shown and were interspersed with one-  off statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Participants’ perception of truth across the  accuracy scale rose without any influencing factor, other than  being repeatedly shown the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  When rated by participants, information regarded as the  truth was rated as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Similarly, information strongly con-  sidered false did not influence participants sufficiently to  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='03301v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='HC]  9 Jan 2023  cross over into the “considered as true” category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This was  noticeable in the 1%-30% bin, where repeated exposure did not  generate a pronounced change in participant opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' One of  the paper’s key findings relates to information considered to be  ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Information initially considered unclear gradually  became regarded as accurate and authentic the more the user  was exposed to the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This was particularly evident  in the 41-70% bin which saw, on average, an increase of 6  points compared to their original rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Existing research [3]–[6] suggests that, in the current hyper-  social age, users risk having their perceptions warped by  misleading statements, hyperbole, and repeated exposure to  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This repetition does not have to stem purely from  repeatedly seeing the same news article displayed but also  from the conversation surrounding the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Increasingly frag-  mented and confrontational viewpoints presented online com-  plicates how users determine what is true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' However,  simply stating that a user is ‘misinformed’ or ‘wrong’ does  not make them more inclined to reassess their understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Instead, a more thoughtful approach is required, giving users  a starting point from which to come to their own conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  As Bronstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [7] suggests, interventions catering towards  the promotion of ‘open-minded and analytical thinking’ could  be of benefit in curbing the impact of misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Informing Users of Risk Online  As  misinformation  has  become  more  apparent  and  widespread in recent years, research into warning users when  they are being misled has received increased interest, both in  academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Many social media sites have taken  steps to curb the impact of misinformation on their respective  platforms, typically by presenting users with a warning label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  This approach offers a simple means of quickly informing  users that the content they are viewing may be misleading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  proving popular with various social media platforms adopting  similar practices including Twitter [8] and Meta’s Facebook  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [10] analysed the effectiveness of warning labels  adopted by major social platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Focusing primarily on the  methods used to dissuade users from sharing misinformation,  the authors tested two different label styles, one replicating  those used by Meta and another informed by contemporary  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Participants (N = 151) were shown content consisting  of six true and six false stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Group one were presented  all stories without any warning label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Group two were shown  half the stories with a warning and the other half without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The  participants in each group were then asked to determine which  stories were manipulated or fake and which were unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The research found that neither of the warning messages  changed user behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Users did not appear to be more  suspicious of labelled content and were just as likely to interact  with the content as that which was unlabelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Designing Effective Warning Labels  Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [10] indicate that providing additional context to  the user can be beneficial in curbing the impact of misinfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Careful consideration in the design process on what  information is communicated to the user and how could be  instrumental in limiting misinformation’s reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Shepherd and Renaud [11] conducted a literature review  on designing effective security warning labels in browsers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Assessing existing work in this area, the authors found that  time and resources are not adequately allocated to designing  warning labels leading to user frustration and confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This  sentiment appeared to be validated by Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The au-  thors found that the effectiveness of current solutions differed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Some warnings wrongly assumed that users had background  knowledge on a topic which decreased their effectiveness,  whilst others were too vague in their language that users did  not understand the ramifications of their choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  To combat the aforementioned issues, Shepherd and Renaud  [11] concluded with the proposal of a set of design guidelines  for browser warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The authors note that warnings designed  for privacy and those designed for security differed, suggesting  that different priorities must be considered depending on the  intended use- case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The proposed guidelines recommend using  simple and concise language to alert users to a potential  issue and using neutral colours to avoid an undue emotional  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Furthermore, the guidelines propose linking to ad-  ditional resources should the initial description not prove  sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Although the research in question is primarily  targeted toward warning labels for security purposes, there  is still value in applying these recommendations to tackling  misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Designing Effective Browser Warnings and Labels  Much work has been done previously in the field of usable  security concerning the design of warning labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Early at-  tempts at warning systems typically used contextual measures  such as a small on-page popup informing the user of potential  risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' However, work by Wu, Miller and Garfinkel [12] illus-  trated these popups are often ignored, misunderstood, or users  do not even recognise they are there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Further research in this area took on a different approach,  utilising interstitial warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This approach required users to  interact with the warning label before proceeding, the under-  lying theory of this approach being that making the warning  the central focus of the users’ attention would increase the  likelihood of users reading and making an informed decision  on the contents of the warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Until recently, most research on effective warning design  has been limited to web-security topics such as expired cer-  tificates or phishing links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [13] examined warning  label design to inform users of potential disinformation online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Evaluating several different warning design styles, ranging  from information-dense with minimal colouring, to warnings  with a strong visual impact but minimal detail, the authors  assessed how users responded to the designs in a realistic  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  In a survey conducted with 238 participants, the authors  found that participants responded most favourably to designs  which featured a reference to the perceived risk (“This page  contains misinformation”) and the recommended next step  (“Consider finding alternative sources.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The authors note  that none of the designs evaluated showed any significant  difference in how likely users were to consult a second  or alternative source afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The authors propose that  changes in behaviour were more likely to stem from the  friction caused by having to manually click through a warning  rather than the content of the warning itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Multiple factors play a role in shaping how users respond  to warning labels, including the language used within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Findings from a research study conducted by Mozilla [14] to  understand how to design better warning labels highlighted  that employing opinionated design was more important than  providing objective information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This means it is more im-  portant to convey the idea of a threat rather than what the  threat is - prior research suggested overly technical warnings  lead to confusion among users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Mozilla implemented this  by simplifying the warning heading to feature abstract but  understandable language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Additionally, for scenarios where  users want to know details of the underlying issue, the warning  provides an accessible description of the risks associated with  the security fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The issue of labelling misleading content is a challenging  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' What counts as misinformation must be determined,  and designing warning systems that promote critical thinking  rather than knee-jerk reactions is still an ongoing area of re-  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Although the means of warning users adopted by major  social platforms may have limited efficacy [10], Shepherd and  Renaud [11] indicate that warning labels can cater to users’  assumed knowledge and understanding without provoking  undue alarm or concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Similarly, Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [13] suggest  that such research can be used to combat misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Detecting Clickbait with Machine Learning  The vast array of content posted online every second makes  it impossible for human moderators to assess and review all  dubious uploads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The use of an automated system is merited,  one capable of rapidly and reliably analysing content for  potentially misleading information which can integrate inter-  vention measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Machine learning’s inherent capabilities for  finding and predicting patterns in information are well-suited  to tackling misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Furthermore, machine learning  has seen renewed interest over the past decade as computing  power and data storage have matured to enable real-world  applications across a host of use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Chen, Conroy and Rubin [15] explored if clickbait, and by  extension, misinformation, could be detected using machine  learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Conducting a holistic view of research in  the field, the authors note four unique means of detecting  clickbait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Initially focusing on the textual content of clickbait  articles, the authors found clickbait often displays lexical and  semantic features unique to its form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The authors cited work  by Lex, Juffinger and Granitzer [16] which analysed clickbait  based upon factors such as word length, word choice, and  terminology, and found that a machine learning model could  be trained to detect clickbait with 77% accuracy regardless of  the topic discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Appealing to users’ innate curiosity by using unresolved  pronouns or alluding to content within the article was also  consistent with clickbait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' As a subsequent paper by Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  [17] noted, automated fact-checking and verification systems  could help detect language patterns in text and warn users  that the content they are about to read may be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The authors also note that such a tool could prove helpful  for journalists too, alerting them when they may be conflating  claims or accidentally misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Clickbait is not strictly limited to the text of the article  in question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' the authors also found that surrounding factors  such as imagery and how the user interacts with the article  play a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Regarding the former, the authors [17] cite  Ecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [18] who found that clickbait articles were likely  to feature images which were incongruent with the headline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  In such articles, an image can be used to grab the would-be  readers’ attention with an impactful but unrelated image or  shape opinion before the article was read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The authors [17] also noted that previous research had found  clickbait outlets typically aimed to attract user attention before  funnelling them towards sponsored content or advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Additionally, the time difference between “time spent reading  the article” and “time spent sharing and commenting about  the article” could also be a signifier of clickbait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' In this  aspect, clickbait articles tend to use alarmist or sensationalist  headlines to provoke knee-jerk responses (whether that be  commenting or sharing) before the user has actually read the  content within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Problem Space  The rapid rise of the information age has led some to  adopt unethical practices to drive engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Whether these  practices are deployed purposefully or not, they pose a serious  risk to society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Although existing work has explored the use of  warning labels, depending on how these are designed, these  may be ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Given the amount of content published  every second, it would be impossible to label the accuracy  of content manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Instead, machine learning offers a com-  pelling alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Misinformation poses a severe threat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' there-  fore combining advances in machine learning and warning  design means an effective solution can be proposed to keep  users safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' METHODOLOGY  The proposed method consists of two-components: the  machine learning model, for analysing and classifying content,  and the web extension for communicating potential risk to the  user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' A simplified pipeline can been seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Using TensorFlow [19] and adopting the same dataset as  used by Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' [20], a Sequential Model was  trained on 32,000 news article headlines, labelled as either  ‘clickbait’ or ‘non-clickbait’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The model, consisting of four  layers (excluding the input layer), tokenises input text into a  64-dimensional dense vector before running it through a global  average pooling filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Output is then fed through a layer of  ReLU nodes, followed by a final layer of Sigmoid nodes to  arrive at a real number between zero (indicating neutral) and  one (indicating strongly misleading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The model was then connected to a browser extension via a  Native Manifest, which allowed the browser to send portions  of an article (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=', the headline) to the model to analyse and  generate a rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The rating is then returned to the browser,  after which a relevant warning can be presented to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Warnings were designed to be informative and actionable for  the user, presenting clear detailing about the perceived risk  and recommended next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Developing the machine learning model  1) Dataset: The same dataset used by Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  [20] was adopted for this project and consisted of 32,000  news article headlines labelled as either ‘clickbait’ or ‘non-  clickbait’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The dataset offered a robust and relevant base upon  which to build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' In particular, clickbait and misinformation rely  on emotional language to provoke a response, suggesting the  dataset would help develop a model well suited to detecting  such language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  2) Model: In practice, the backend of this project cen-  tres around binary classification: Is this piece of text click-  bait/misinformation or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' As such, using a Sequential model  was deemed the most suitable due to its singular input-output  structure, a structure well suited to classification tasks such as  this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Example simplified data pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  3) Preprocessing Data : The dataset was split into 26,666  training samples and 5,334 testing samples which equates to  83% of the dataset for training and the remaining 17% for test-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Before training, the dataset had to be formatted such that  the model could determine distinctions between clickbait and  non-clickbait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This was achieved by converting, the headlines  in the dataset into a vocabulary of word embeddings (Figure  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' To ensure uniformity across the dataset, all sequences  were padded to 24 tokens long which was considered a safe  maximum length for a headline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Headlines longer than 24  words long were automatically truncated to the maximum  length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Example word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  4) Building the Model: The model is visualised in Figure  3, and consists of the input layer, two hidden layers, and  the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The first layer takes the input (a tokenised  sentence) and transforms it into a 64-dimensional dense vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The usage of dense vectors allows for the semantic meaning  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Visualisation of the Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  of the sentence to be compressed, ensuring better general-  isation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Despite their ability to derive underlying meanings  and connections for a given sentence, the aforementioned  dense vectors can result in overfitting if they become too  detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' To address the issue, the second layer consists of  a Global Average Pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This layer takes the 64-dimensional  vector and determines the mean of each input channel (the 24-  dimensional token sequences) which allows the model to learn  approximations of embeddings rather than their exact values  (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Visualisation of 1-Dimensional global average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  At this stage, the input data is now formatted and approx-  imated to limit overfitting, and layers can be constructed,  which will inform the output of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The first activation  function of the model uses a rectified linear activation function  (or ‘ReLU’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' ReLU ensures that the next layer of the network  receives a positive value as ReLU outputs 0 for input values  equal to or less than 0 or the original value for those greater  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Finally, the data is passed through a Sigmoid activation  layer, ensuring the resultant output falls between 0 and 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=',  “Is this piece of text clickbait or not?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The model measures its performance based on the ac-  curacy of its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The task involves classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content="  9 makeup tips you won't believe!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Model  Clickbait: 1If Disney Princesses Were From Florida  [10122  752  6586 4 1 737  0  0 0 (  0 (  0 (  000000000000(  01input:  [(None, 24)]  InputLayer  output:  [(None, 24)]  input:  (None,24)  Embedding  output:  (None, 24, 64)  input:  (None, 24, 64)  GlobalAveragePooling1D  output:  (None, 64)  input:  (None, 64)  Dense  output:  (None, 2)  input:  (None, 2)  Dense  output:  (None, 1)4  6  4  2  4Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Accuracy graph during model training (Higher is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  thus, a binary cross-entropy loss function is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The func-  tion calculates how far the models’ predictions stray from  the dataset’s labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' A gradient descent with a momentum  optimiser (also known as Stochastic Gradient Descent or  SGD) further minimises the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The optimiser helps  improve the model’s training rate by minimising loss across  training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' By doing so, it is intended that the model  predictions will gradually trend towards the expected output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Loss graph during model training (Lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The model was trained for 80 epochs with a batch size  of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Although batch sizes larger than 32 can lead to  underfitting, it was intended that the larger epoch size would  gradually result in greater accuracy and convergence, which  can be seen in Figures 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' After the model was compiled,  trained, and evaluated, it was exported for use by the web  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Web extension and warning messages  1) Creating the web extension: A web browser extension  was developed to ensure the model could be deployed in a real-  world context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The extension means the model can analyse  news articles as the user views them, providing a warning if  misleading content is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Native Manifests [21] were used to allow a web browser to  interface with a native application, passing data back and forth  between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' These manifests allowed the TensorFlow  model to perform predictions locally on the machine and then  send the resultant prediction to the browser for further analysis  and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Within the browser, the analysis begins when the headline  of the page the user has visited is fetched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Initially, white  space and control characters are trimmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The headline is  processed, and a value is returned indicating how sensationalist  the headline is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  On the user’s device, the program transfers over to the native  application, which handles parsing the headline into a format  suitable for the model before computing a rating which is  returned to the browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Data from the browser is JSON-  encapsulated and is sent via standard input (stdin), which  the program reads from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Following this, the script begins  importing libraries for loading the model and formatting the  incoming data accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The model is loaded, and the  tokeniser is instantiated to convert the incoming headline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' At  this stage, the initial setup is complete and the model is ready  for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The core of the script features a loop which waits for a  message from the browser to be received, at which point the  decoding process can take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Now that the headline of the  article is available, the script tokenises it and provides padding  to ensure compatibility with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' From here, a standard  model prediction call can be made, encoded, and returned to  the browser for display to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  2) Presenting warnings to the user: When the native ap-  plication produces a result, it is returned to the browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The  browser extension then generates a message sent to the news  article’s page, with the native applications result stored in a  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This message is received by the content script, which  is injected into each page by the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The content script dynamically generates a warning label  for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This is done by waiting to receive a message  (the result) from the background script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This value is then  multiplied by ten (to accommodate any floating-point issues  that may arise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' converting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='8 to 8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' To prevent repeatedly  warning the user about innocuous content, only headlines that  score above five out of ten have a warning generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Scrolling  is disabled whilst the warning is on-screen, ensuring the user  has to acknowledge it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  With regards to this project, the existing literature points  towards interstitial warnings being the most likely to promote  change in user behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Additionally, even if most users do  not appear to actually read the content of a warning label, they  do show a preference for such information being present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Informed by the papers discussed in Section I-D, the web  extension was designed to ensure strong visual clarity to  effectively convey risks associated with a piece of misleading  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The warning adopts a paywall-style design, mimicking an  approach that users will likely already be familiar with from  other news sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This helps to ensure that the warning is  not overlooked, which can happen with contextual warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  To further ensure that the warning is brought to the user’s  attention, an overlay is used to darken the article and scrolling  model accuracy  tain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='85  val  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='75  accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='50  10  70  0  20  30  40  50  60  80  epochmodel loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='70  tain  val  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='60  los5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='40  10  20  30  40  50  0  70  60  80  epochis prevented while the warning is on screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The warning  design comes in 5 variants - ranging from most minor to most  severe, depending on the article’s rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Exemplar designs can  be seen in Figure 7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Sample of warning designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  A vital problem with previous warning designs is the poor  communication of risk, whereby warnings may be obscured  by jargon [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The design of the warnings seeks to minimise  existing issues by conveying as much relevant information  as possible in an easy-to-read format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Prominent symbols  represent increasing risk levels based on the article’s rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Articles lower on the risk scale are given a more general  ’magnifying glass’ symbol, promoting the notion of thinking  more critically about the article’s merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' If an article includes  more severe levels of misinformation, increasingly prominent  ’alert’-oriented symbols are deployed, such as warning signs,  stop signs and symbols of authority such as police figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Additionally, an oscillating gradient is placed behind the  warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Depending on the severity of the warning, the colour  used will shift from yellow to orange to red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The subtle  movement of the gradient is intended to draw the user’s eye  to the warning, with the unique colour of each warning also  helping the user understand the associated level of risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Ultimately, the extension seeks to change user behaviour  and provide education on meaningful steps users can take to  protect themselves from misinformation in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' As such,  each warning label features unique phrasing that informs the  user of not just what the perceived risk is but also advice on  actionable next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Two buttons are presented to the user at the bottom of the  warning, allowing them to dismiss the warning and continue,  or navigate away from the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' To indicate the intended  behaviour, the option to navigate away is displayed in promi-  nent green with a ’Recommended’ label included in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Conversely, the option to dismiss the warning is presented in  red and is slightly faded out to deliberately be obscured against  the background until the user hovers over the button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Accuracy comparison between training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  During training, the model achieved an accuracy of approx-  imately 85% and a loss of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='39, and when pitted against  the evaluation dataset, the model achieved an accuracy of  approximately 45% with a loss of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='15 (Figure 8, Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  This decline in accuracy likely stems from inconsistencies in  the existing evaluation dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=', improperly formatted data,  such as some of the labels assigned to the headlines do not  appear to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This could be resolved via an additional  data cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Another point of note is that the evaluation  dataset used only binary labels (Is this headline ‘clickbait’ or  ‘news’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' ), which may have also contributed to the discrepancy  in accuracy as the model was producing a result between 0  and 1 instead of a pure binary output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  ThisContent PutsYouatRisk  Misleadingarticlescanwarpyourperceptionofevents  Think twice before continuing  Deep Breathhas detectedthatthis article is misleading  and should not be viewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Backto safety (Recommended)  Misleading Content Alert  Sensationalistcontentcantrickyouinto consumingfalse  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='Consultmultiplesources  Websites can use sensationalist languageto mislead and  deceive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='Deep Breaths detection algorithmbelieves that  this article is misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Dismiss and continue  Back to safety (Recommended)  SensationalistContentWarning  Thispagemaycontainsensationalistormisleading  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='Considerconsulting additionalsources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Websites can use sensationalist languagetograb your  attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='DeepBreathsdetectionalgorithmbelievesthat  this article may be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Dismiss and continue  Backto safety (Recommended)100  90  85%  80  70  Accuracy  60  50  45%  40  30  20  10  0  Training  Evaluation  (Higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=')Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Loss comparison between training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  With regards to machine learning, the project confirms the  findings of Lex, Juffinger and Granitzer (2010) that clickbait  and misinformation can be detected based upon lexical seman-  tics, namely word choice, word length, and word commonality,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=', Word x appears frequently alongside word y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  The model demonstrated in this paper has shown a re-  liable degree of performance, however, it could be refined  further to derive even better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The models’ accuracy  and loss were still increasing and decreasing, respectively,  suggesting better performance could be obtained before the  curves flattened out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Furthermore, the capability of the model  could be extended further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The model has been trained only on  clickbait-styled headlines, which was effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' However, more  robust results may be achieved by training on the contents of  clickbait articles which would allow the model to develop a  deeper understanding of the article and make a more nuanced  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The model used in this paper is a Sequential model  designed to take a single output and produce a single result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  Although this is effective at classifying a single headline as  used in this paper, greater functionality could be achieved  by allowing multiple inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' This could include  assessing the article’s headline but also a selection of sentences  from the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' The rating assigned to content is dynamic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  however, the underlying warning remains static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' By expanding  the models’ capabilities, it may also be possible to provide  personalised warnings relevant to the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' In practice, this  could mean warning the user about specific aspects of the  article, such as sensationalist authors, misleading sentences,  and miscaptioned images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Although every effort was taken to  ensure the model made balanced and accurate predictions, no  system is infallible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Conducting user testing and introducing  the option for users to report when the model makes a  perceived miscalculation could help adjust for missteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content='  The work presented in this paper makes promising ad-  vances toward tackling the issue of misinformation online  by combining machine learning, human-computer interaction  research, and web technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
+page_content=' Findings validate and build  upon prior research, and incorporating machine learning with  usable security is still a relatively under-explored area of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
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+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
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+page_content='1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
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+page_content='  [21] Mozilla, “Native Manifests,” 2022, https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfmwRz/content/2301.03301v1.pdf'}
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+arXiv:2301.04861v1  [cs.IT]  12 Jan 2023
+Grant-Free Random Access of IoT devices in
+Massive MIMO with Partial CSI
+Gilles Callebaut1, Franc¸ois Rottenberg1, Liesbet Van der Perre1, and Erik G. Larsson2
+1Department of Electrical Engineering (ESAT-DRAMCO), KU Leuven, 9000 Ghent, Belgium
+2Department of Electrical Engineering (ISY), Link¨oping University, Link¨oping, Sweden
+Abstract—The number of wireless devices is drastically in-
+creasing, resulting in many devices contending for radio re-
+sources. In this work, we present an algorithm to detect ac-
+tive devices for unsourced random access, i.e., the devices are
+uncoordinated. The devices use a unique, but non-orthogonal
+preamble, known to the network, prior to sending the payload
+data. They do not employ any carrier sensing technique and
+blindly transmit the preamble and data. To detect the active
+users, we exploit partial channel state information (CSI), which
+could have been obtained through a previous channel estimate.
+For static devices, e.g., Internet of Things nodes, it is shown that
+CSI is less time-variant than assumed in many theoretical works.
+The presented iterative algorithm uses a maximum likelihood
+approach to estimate both the activity and a potential phase
+offset of each known device. The convergence of the proposed
+algorithm is evaluated. The performance in terms of probability
+of miss detection and false alarm is assessed for different qualities
+of partial CSI and different signal-to-noise ratio.
+Index Terms—activity detection, grant-free, massive MIMO,
+maximum likelihood, random access.
+I. INTRODUCTION
+Massive machine-typed communication (mMTC) is envi-
+sioned to enable low-power connectivity to a very large
+number of devices and open up new applications. While it
+has been put forward as a key building block in 5G and
+beyond, it has so far received less attention than enhanced
+Mobile Broadband (eMBB) and Ultra-Reliable Low Latency
+Communications (URLLC). The challenge in mMTC, and
+in general Internet of Things (IoT) systems, resides in the
+low-power operation, sporadic nature of the traffic and a
+large amount of uncoordinated devices. As these devices are
+often battery-powered, they are constrained in the signalling
+overhead they can handle [2]. Furthermore, they are often
+deployed in remote areas, where network coverage is low or
+non-existent [2, 3]. To address this, novel protocols need to
+be designed tailored to the low-power, sporadic and massive
+nature of mMTC traffic. One promising direction, and the
+focus of this work, is the use of massive MIMO, where a
+large number of antennas is present at the base station.
+Two multiple access approaches can be taken, grant-based
+and grant-free random access. The former requires the devices
+to obtain a grant from the network, where after it can use a
+The research reported herein was partly funded by the European Union’s
+Horizon 2020 research and innovation programme under grant agreement No
+101013425.
+This paper is presented at IEEE WCNC 2023. G. Callebaut, F. Rottenberg,
+L. Van der Perre, and E. G. Larsson, “Grant-Free random access of IoT devices
+in massive MIMO with partial CSI,” in 2023 IEEE Wireless Communications
+and Networking Conference (WCNC) (IEEE WCNC 2023), Glasgow, United
+Kingdom (Great Britain), Mar. 2023
+collision-free radio resource to transmit its data. The disadvan-
+tage of this approach for IoT and mMTC is that devices need
+to compete for grants, requiring, e.g., dedicated preambles or a
+lot of signalling for collision resolution. Due to the number of
+competing devices such schemes are not practical and scalable.
+Therefore, grant-free approaches are advocated over grant-
+based solutions [4–6]. In grant-free random access, devices
+do not contend for a grant, but just access the network when
+required. Often these devices are unaware of the other devices
+in the system and operate in an uncoordinated fashion, which
+is called unsourced grant-free random access [6]. Different
+approaches have been studied to detect the active devices in
+massive MIMO systems. Detecting active devices is facilitated
+when each device uses a unique and orthogonal preamble
+sequence. This entails that each preamble should have the
+same length as the number of devices to have no collisions,
+which is unpractical. Therefore a number of studies have
+considered a pool of orthogonal pilots [7] or the use of non-
+orthogonal pilots. Both of these strategies have been elaborated
+in [4], where they formulate the device-activity detection
+problem as a compressed sensing problem, where the sparsity
+of the active devices at each time slot is exploited. However,
+compressed sensing requires the preamble length to be larger
+than the active devices, increasing the energy expenditure of
+data transfer. To combat this, in [6, 8] a covariance-based
+method is suggested using two estimators for device activity
+recovery, i.e., maximum-likelihood (ML) and non-negative
+least squares (NNLS). This work is generalized by Ganesan,
+Bjornson, and Larsson [9, 10] to the cell-free case where mul-
+tiple access points (APs) are geographically distributed. They
+considered different large-scale fading coefficients per APs.
+They employed a simplification of their proposed algorithm
+by including only the AP with the strongest contribution. They
+concluded that co-located massive MIMO is highly sensitive to
+low signal-to-noise ratios (SNRs), while cell-free deployment
+is better suited against shadowing fading effects.
+Contributions. Inspired by these works, we propose a
+new algorithm exploiting the static nature of IoT devices. As
+demonstrated in [11] and elaborated in Secion II, the channel
+state information (CSI) can remain almost invariant over large
+period of times (hours) for scenarios with low mobility. This
+was not yet leveraged by practical algorithms in the literature
+to the best of our knowledge. Given that the base station
+knows a part of the CSI of each device the performance of
+the device activity scheme can be improved. To do so, we
+formulate the maximum-likelihood activity detection problem
+using partial CSI. Furthermore, a phase offset can be estimated
+which occurs due to carrier frequency offset (CFO) (when
+
+the CFO is static over the preamble duration). We validate
+the convergence of the iterative algorithm, study the impact
+of different initialization vectors for the device activity, the
+impact of the quality of the partial CSI and the SNR on the
+performance of the algorithm.
+The
+used
+mathematical
+notations
+are
+described
+in
+https://github.com/wavecore-research/math-notations.
+II. MOTIVATION – RECURRENCE IN CSI
+IoT technologies are often put in the field with a deploy-
+and-forget strategy, where the devices remain immobile after-
+wards. As such, we can expect that the channel conditions
+are less time-variant than typically assumed in literature [12,
+13]. An experimental campaign to investigate the long-term
+behavior of the channel is presented in [11]. The long-
+term behavior is measured by taking the channel correlation
+δi,j = |¯hH
+i ·¯hj|/∥¯hi∥∥¯hj∥ of the first channel estimate and the
+other channel N measurements, i.e., δ1,j for j ∈ {1, . . ., N}.
+The observed channel correlations are depicted in Fig. 1.
+It illustrates that most of the time the correlation coefficient
+is close to 1, indicating that the channel is highly correlated
+with the first estimate and thus can be considered static. It also
+shows that, while in some occasions the correlation drops,
+the channel quickly becomes again highly correlated with
+the first channel instance. More than 90% of the measured
+channels1 have a correlation coefficient higher than 0.9. This
+demonstrates the potential of re-using channel estimates in IoT
+contexts.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+0.6
+0.8
+1
+Sample index
+Channel Correlation δ1,j
+Node 1
+Node 2
+Figure 1: Channel correlation over a full day (9h24-17h48) with over 10 000
+channel instances, with an average correlation of 0.935/0.968 and a standard
+deviation of 0.06/0.03 for Node 1/Node 2.
+III. SYSTEM MODEL
+There is a total of K single-antenna devices and a total of M
+co-located base station (BS) antennas. The set of active users
+trying to access the network is denoted by Ka, with |Ka| ≤ K.
+To access the network, each active device k ∈ Ka sends a
+unique, non-orthogonal preamble of length T , known to the
+network. The pilot symbol of the preamble sent by device k at
+pilot symbol t is denoted by sk,t. The channel vector between
+the receive antenna m and user k is denoted by hk,m ∈ C, and
+1More specifically, 91% and 95% for Node 1 and Node 2, respectively.
+2We use here the term “re-using” to indicate that we no longer operate in
+a block fading model with independent channel realizations. Typically, these
+blocks are considered in the order of 50 ms.
+is considered fixed over the preamble duration. The received
+symbol at the m-th BS antenna at time t is
+yt,m =
+K−1
+�
+k=0
+hk,msk,tγk + wt,m,
+(1)
+where γk is an unknown complex scalar and wt,m ∈ C is
+independently and identically distributed (i.i.d.) zero mean
+circularly symmetric complex Gaussian (ZMCSCG) noise with
+variance σ2. The unknown complex scalar γk can be developed
+as
+γk = √ρkakejφk,
+(2)
+where ρk ∈ R+ is the transmit power of device k, ak ∈ {0, 1}
+is the device activity and φk models a potential phase offset.
+This offset φk can account for a CFO, where the phase is
+considered constant over the preamble duration. This occurs
+when no frequency drift has occurred during the preamble
+duration, which is a feasible assumption as the CFO is
+typically low, yielding negligible phase rotations during the
+preamble interval. By assuming that all M antennas are
+perfectly synchronized, this offset is only dependent on the
+device. In case the device is inactive, γk will be zero. We will
+introduce the term activity indicator to denote γk.
+Let us consider that the BS knows a part of the CSI, i.e.,
+gk,m in
+hk,m = gk,m + λkǫk,m,
+(3)
+where ǫk,m are i.i.d. ZMCSCG variables with unit variance,
+and λk ∈ R+ models the unknown part of the CSI. The large-
+scale fading coefficient of user k is βk = E(∥hk∥2 /M) =
+∥gk∥2 /M +λ2
+k. The factor λk models the quality of the known
+CSI. Hence, it quantifies the correlation of the actual channel
+hk,m to the known partial CSI gk,m. In the extreme case with
+λk = 0, the CSI is perfectly known and there is no uncertainty
+left, as was studied in [11]. This could be the case in a
+fully static environment and if the CSI estimates are noiseless.
+However, for a realistic IoT scenario, even for static devices,
+CSI is not perfect due to i) environment dynamics and ii) noisy
+estimates. The parameter λk then quantifies this imperfection.
+It is here assumed to be known3. Another extreme case,
+as considered in [8, 10], is obtained when gk,m = 0, ∀m,
+implying that only the large scale fading coefficient of user k
+is known, i.e., λk = √βk.
+IV. DEVICE ACTIVITY DETECTION
+This section describes the proposed activity detection al-
+gorithm. First, the log-likelihood of the received preamble is
+derived. Then, given its non-convex expression, an iterative
+approach is proposed to estimate the parameters γk ∀k. Finally,
+activity detection is performed.
+A. Log-Likelihood of the Received Symbols
+Combining (1) and (3), the symbol, received at BS antenna
+m and for pilot symbol t, is given by
+yt,m =
+K−1
+�
+k=0
+(gk,m + ǫk,mλk)st,kγk + wt,m.
+3It could be set to a certain value depending on the user activity profile
+and/or tracked for each user over time.
+
+Stacking the observations at antenna m gives
+ym =
+K−1
+�
+k=0
+gk,mskγk +
+K−1
+�
+k=0
+ǫk,mλkskγk + wm,
+where
+ym =
+
+
+
+y0,m
+...
+yT −1,m
+
+
+ , sk =
+
+
+
+s0,k
+...
+sT −1,k
+
+
+ , wm =
+
+
+
+wm,0
+...
+wT −1,m
+
+
+ .
+For a given value of γk, ym|γk has a circularly symmetric
+Gaussian distribution with mean �K−1
+k=0 gk,mskγk. After defin-
+ing the vector θm = �K−1
+k=0 ǫk,mλkskγk+wm, the covariance
+matrix is
+C = E
+�
+θmθH
+m
+�
+=
+K−1
+�
+k=0
+λ2
+k|γk|2sksH
+k + σ2IT ,
+(4)
+where we used the fact that ǫk,m were assumed to be i.i.d. and
+the additive noise is white. Note that this covariance matrix
+does not depend on the antenna index m and is thus valid for
+all ym. Defining the vector γ = (γ0, ..., γK−1)T , and Θm =
+ym − �K−1
+k=0 gk,mskγk the log-likelihood of the observation
+vector ym is
+log p(ym|γ) = − ln (|C|) − T ln(π) − ΘH
+mC−1Θm.
+Given the conditional independence of ǫk,m and wt,m over the
+antennas, the different ym are independent as well. Hence, the
+log-likelihood of the aggregated observations at all antennas
+y = (yT
+0 , ..., yT
+M−1)T becomes
+log p(y|γ) = −M ln (|C|) − MT ln(π) −
+M−1
+�
+m=0
+ΘH
+mC−1Θm.
+(5)
+B. Iterative Algorithm for Maximizing Likelihood
+The maximum likelihood estimator of γ is obtained by
+maximizing
+ˆγML = arg max
+γ
+log p(y|γ).
+This problem is not trivial to solve given the nonlinear and
+non-convex dependence of the log-likelihood, more specifi-
+cally the covariance matrix C, in γ. An idea to maximize
+the likelihood is to use an iterative approach, similarly as [8,
+10]: at each iteration, all γk are kept fixed but one, which is
+optimized and updated. This way, they get updated one by
+one until convergence is attained, i.e., a maximum number of
+iterations or a certain tolerance is reached. A block diagram
+of the algorithm is given in Figure 2 and the pseudocode is
+summarized in Algorithm 1.
+Let us consider that the complex-valued γk′ needs to be
+updated. Using the definition introduced in (2), we can rewrite
+γk′ with a phase-amplitude decomposition: γk′ = rk′eφk′,
+with rk′ = |γk′| = √ρk′ak′. The optimization with respect
+to γk′ is done in the following in several steps: i) optimizing
+the phase φk′ for a fixed value of rk′, ii) re-inserting this
+expression in the objective function to remove the dependence
+in φk′ and iii) optimizing the amplitude rk′.
+1) Phase optimization: To highlight the dependence of the
+objective function in γk′ for constant values of other γk, k ̸=
+k′, let us define the vector
+yk′,m = ym −
+K−1
+�
+k=0,k̸=k′
+gk,mskγk,
+(8)
+which can be seen as a cancellation of device interference to
+isolate the contribution from device k′. Hence, the objective
+function to maximize can be written as in (7)4.
+where we explicitly express the dependence in (rk′, φk′)
+while the other (rk, φk), k ̸= k′ do not appear since they are
+considered constant. Note that the matrix C, defined in (4),
+does not depend on φk′ but only rk′. In the extreme case of no
+prior CSI, i.e., gk′,m = 0 ∀m, the dependence of f(rk′, φk′)
+in φk′ disappears and there is an underdetermination and no
+estimate of the phase offset can be obtained. In other cases,
+we can find that, after some manipulations,
+ˆφk′ = ∠sH
+k′C−1
+M−1
+�
+m=0
+g∗
+k′,myk′,m.
+(9)
+This result has an intuitive understanding: the optimal phase
+φ′
+k tends to align the partial CSI with the observations due to
+device k′.
+2) Removing the phase dependence: Inserting this optimal
+value in the objective function f(rk′, φk′) makes the depen-
+dence in φk′ vanish and gives (6).
+3) Amplitude optimization: To alleviate the dependence on
+rk′ in (6), let us define
+C−k′ = C − λ2
+k′|γk′|2sk′sH
+k′
+(10)
+=
+�
+k\k′
+λ2
+k|γk|2sksH
+k + σ2IT ,
+which does not depend on rk′ and is full rank, thus invertible.
+Applying the Sherman-Morrison formulas [14] to C−1 and
+C−1sk′ gives
+C−1 = C−1
+−k′ − C−1
+−k′sk′sH
+k′C−1
+−k′r2
+k′λ2
+k′
+1 + sH
+k′C−1
+−k′sk′r2
+k′λ2
+k′
+(11)
+We insert these expressions in (6) and we omit terms that
+do not depend on rk′, which will vanish after taking the
+derivative. This gives
+˜f(rk′) = −M ln (|C|) + αr2
+k′ + βrk′
+1 + δr2
+k′
+,
+(12)
+where we defined the constants (independent of rk′) α, β and
+δ, as
+α =
+M−1
+�
+m=0
+|yH
+k′,mC−1
+−k′sk′|2λ2
+k′ − sH
+k′C−1
+−k′sk′
+M−1
+�
+m=0
+|gk′,m|2
+β = 2
+�����
+M−1
+�
+m=0
+yH
+k′,mC−1
+−k′sk′gk′,m
+�����
+δ = sH
+k′C−1
+−k′sk′λ2
+k′.
+(13)
+4For clarity, we omit in the following the constant term MT ln(π) which
+does not affect optimization as it does not depend on γ and vanishes after
+differentiation.
+
+f(rk′) = −M ln (|C|) −
+M−1
+�
+m=0
+yH
+k′,mC−1yk′,m − r2
+k′sH
+k′C−1sk′
+M−1
+�
+m=0
+|gk′,m|2 + 2
+�����
+M−1
+�
+m=0
+yH
+k′,mC−1sk′gk′,m
+����� rk′
+(6)
+f(rk′, φk′) = −M ln (|C|) −
+M−1
+�
+m=0
+�
+yk′,m − gk′,msk′rk′eφk′�H C−1 �
+yk′,m − gk′,msk′rk′eφk′�
+(7)
+Inputs
+σ2, λk, gm,k,
+ym ∀m, k
+Initialization
+k′ ← 0, ˆγ ← ˆγinit
+Amplitude Optimization
+(14), (16) or (17)
+ˆrk′ ← arg max ˜f(rk′)
+Phase Optimization (9)
+ˆφk′ ← ∠sH
+k′C−1 �M−1
+m=0 g∗
+k′,myk′,m
+Converged?
+Iterative maximum likelihood estimator
+Tolerance
+Max iteration
+Activity Detection
+ˆγk ≤ γth,k ∀k
+No
+Yes
+k′ ← k′ + 1 mod K
+Updated ˆγk′ ← ˆrk′eφk′
+ˆγ
+Figure 2: Block diagram of the iterative maximum likelihood estimator and activity detection.
+One can note that ˜f(rk′) in (12) can now be differentiated
+with respect to rk′, using the differential rule ∂(log |A|) =
+tr[A−1∂A]. Setting the derivative to zero gives, noting that
+the denominator is always strictly positive,
+0 = −r3
+k′2Mδ2 − r2
+k′βδ + rk′(−2Mδ + 2α) + β,
+(14)
+which is a polynomial of degree 3 in rk′. There are closed-
+form solutions for the roots of such polynomials. Following
+Descarte’s rule of signs, (14) has only one real and positive
+root, as required, in case the terms are non-zero. Below we
+discuss the special cases when the terms are not non-zero, i.e.,
+when there is no or complete CSI knowledge.
+The algorithm is summarized in the pseudocode Algo-
+rithm 1. At each iteration, the constants α, β and δ can
+be easily re-evaluated based on (13). They require the matrix
+inversion C−1
+−k′. To avoid re-computing a full inverse at each
+iteration, one can rely on the Sherman-Morrison formula and
+on the current knowledge of C−1, which is updated at the end
+of each iteration by inserting the obtained value of rk′ in (11).
+Using (10), we find
+C−1
+−k′ = C−1 + λ2
+k′|γk′|2C−1sk′sH
+k′C−1
+1 − λ2
+k′|γk′|2sH
+k′C−1sk′ .
+(15)
+Moreover, computations can be optimized as several quantities
+appear multiple times and can be computed only once. The
+complexity of the proposed algorithm is O(IMT 2), where I
+is the number of iterations.5
+We now investigate two particular cases, to gain further
+insights.
+Algorithm 1 Iterative maximum likelihood device activity
+detector
+Require: σ2, λk, ym, gk,m, ˆγinit ∀k, m
+k′ ← 0
+ˆγ ← ˆγinit
+C−1 ←
+��K−1
+k=0 λ2
+k|ˆγk|2sksH
+k + σ2IT
+�−1
+while Not converged do
+Compute yk′,m, C−1
+−k′, α, β and δ based on (8), (15) and
+(13)
+ˆrk′ ← arg max ˜f(rk′)
+⊲ Update amplitude based on (14),
+(16) or (17)
+ˆφk′ ← ∠sH
+k′C−1 �M−1
+m=0 g∗
+k′,myk′,m
+⊲ Update phase
+ˆγk′ ← ˆrk′e ˆφk′
+C−1 ← C−1
+−k′ −
+C−1
+−k′ sk′ sH
+k′ C−1
+−k′ r2
+k′ λ2
+k′
+1+sH
+k′ C−1
+−k′ sk′ r2
+k′ λ2
+k′
+k′ ← k′ + 1 mod K
+end while
+a) Particular case: device with no CSI: Now consider
+that, for a given k′, gk′,m = 0 ∀m. This could be because
+this device is new or moving a lot, such that its CSI is
+outdated. Only, its parameter λk′ is known, which is equal
+5Note that I will in practice depend on K as we will iterate N times over
+all users K, but this is not a requirement.
+
+to the large-scale fading coefficient √βk′. At iteration of user
+k′, evaluating (13) for gk′,m = 0 ∀m implies that β = 0.
+Hence, (14) simplifies to
+0 = 2rk′(−r2
+k′Mδ2 − Mδ + α),
+which has a trivial solution in rk′ = 0. One of the other roots
+is always negative. Keeping only the positive one, we find the
+amplitude update
+ˆrk′ =
+�
+α − Mδ
+Mδ2
+.
+(16)
+If this root is imaginary, we set ˆrk′ to zero. As discussed
+before introducing the phase update equation (9), in the case
+of no prior CSI, the phase ambiguity cannot be resolved.
+This particular case gives an update relatively similar to the
+maximum likelihood estimator derived in [8, (23)], where
+their ML expression estimates the error, while ours estimates
+directly the coefficient ˆrk′, given the same estimate of γk′ at
+each iteration.
+b) Particular case: device with complete prior CSI: Now,
+consider that, for a given k′, λk′ = 0, so that the CSI is
+perfectly known. Only the phase shift and the transmit power
+are unknown. At iteration of user k′, evaluating (13) for λk′ =
+0 implies that δ = 0. Hence, (14) simplifies to a linear equation
+0 = rk′2α + β, which gives the following amplitude update
+ˆrk′ = −β
+2α =
+|sH
+k′C−1 �M−1
+m=0 g∗
+k′,myk′,m|
+sH
+k′C−1sk′ �
+m′ |gk′,m′|2
+,
+(17)
+while the phase is updated according to (9).
+4) Initialization: To start the iterative algorithm, we con-
+sider different choices to initialize ˆγinit. A simple choice is
+to initialize to zero, i.e., ˆγ0
+init = 0. Another choice is to
+initialize solely based on the available prior CSI, considering
+that λk ≈ 0, ∀k. The estimator is similar to [11], except that,
+here, prior CSI is used instead of complete CSI.
+If λk ≈ 0, ∀k, the covariance matrix C, defined in (4),
+simplifies to C = σ−2IT , which is independent of γk. Hence,
+many terms of the log-likelihood in (5) become independent
+of γ. Maximizing (5) becomes equivalent to the following
+minimization
+max
+γ
+log p(y|γ) = min
+γ
+M−1
+�
+m=0
+�����ym −
+K−1
+�
+k=0
+gk,mskγk
+�����
+2
+= min
+γ ∥y − Γγ∥2 ,
+where we defined the vector and matrix notations
+y =
+�y0 . . . yM−1
+�T , Γ =
+�Γ0 . . . ΓM−1
+�T ,
+Γm =
+�s0 . . . sK−1
+�
+diag(g0,m . . . gK−1,m).
+This minimization problem is a quadratic function of γ, which
+is a least squares problem. The estimate has the following
+closed-form expression
+ˆγZF
+init =
+�
+ΓHΓ
+�−1
+ΓHy.
+(18)
+This last estimator can be seen as a zero-forcing (ZF) es-
+timator, which requires a matrix inversion. To avoid ill-
+conditioning, a first necessary condition is that K ≤ MT .
+This condition is not sufficient as the channel and preamble
+of two devices could be correlated, especially when K is on
+the order of MT . Moreover, if no prior information is available
+for a given user k′, i.e., gk′,m = 0, ∀m, the inverse will also
+be ill-conditioned. This implies that the k′-th column of Γ
+becomes null and thus Γ is rank deficient. Moreover, the prior
+CSI might be noisy, leading to unstable results.
+To make initialization more robust, we can use an least
+minimum mean square error (LMMSE) criterion. To do this,
+some prior knowledge must be assumed on the statistics of
+γ, more specifically, its first and second order moments.
+We here make the following assumptions: i) the activity of
+each device is independent of one another, ii) the average
+activity and average transmit power of each device is known
+and iii) no prior information is known on the phase offset
+so that φk is considered uniformly distributed between 0
+and 2π. Under these assumptions, we have E(γ) = 0 and
+D = E(γγH) = diag (E(a0)E(ρ0), ..., E(aK−1)E(ρK−1)).
+Hence, for the linear observation model y = Γγ + w, still
+considering that λk ≈ 0, ∀k, the LMMSE estimator of γ is
+then given by [15]
+ˆγLMMSE
+init
+=
+�
+ΓHΓ + σ2D−1�−1
+ΓHy.
+(19)
+Note that the matrix to be inverted is always well-conditioned.
+Finally, a matched filter (MF) estimator could be used to avoid
+the need for matrix inversion.
+ˆγMF
+init =
+�
+diag(ΓHΓ) + σ2D−1�−1
+ΓHy.
+(20)
+C. Activity Detection
+A non-negative activity threshold γth,k is applied for each
+device k. A device is considered active if |ˆγk| ≥ γth,k. The
+real-valued threshold is defined as,
+γth,k = v
+�
+SNRk
+−1,
+(21)
+where v is chosen to have a desired probability of false alarm
+and miss detection performance and with SNRk = Mβk/σ2 =
+E(∥hk∥2)/σ2 = (∥gk∥2 + Mλ2
+k)/σ2.
+Miss detection happens when a device was undetected,
+while it was actually transmitting. Equivalently, a false alarm
+occurs if a device is considered active by the algorithm but
+was not. We define the probability of miss detection as the
+average ratio of undetected devices to the number of active
+devices Pmd = 1 −
+����Ka ∩ ˆKa
+��� / |Ka|
+�
+, where Ka is the set
+of active devices and ˆKa = {k|ˆak = 1, ∀ ∈ [1, K]} denotes
+the estimated set of active devices. Note that on average
+|Ka| = Ka. Similarly, the probability of false alarm is the ratio
+of inactive devices considered active to the number of inactive
+devices and is given by Pfa =
+���� ˆKa \ Ka
+��� /(K − |Ka|)
+�
+.
+A trade-off can be made between the two probabilities
+by varying v in (21). A lower v yields a lower activity
+threshold, resulting in more devices considered active. This in
+turn lowers the probability of miss detection, while increasing
+the probability of generating a false alarm. In the simulations,
+the parameter v is swept across the range [−40, 40]dB.
+
+Table I: Simulation parameter set with default values.
+Parameter
+Symbol
+Default value
+Number of devices
+K
+500
+Number of total BS antennas
+M
+64
+Signal-to-noise ratio
+SNR
+20 dB
+Device activity probability
+ǫa
+0.1
+Pilot sequence
+sk
+∼ CN (0, 1)
+Pilot length
+τp
+10 symbols
+Phase offset
+φk
+∼ U[0,2π]
+Number of simulations
+Nsim
+>10 000
+Number of algorithm iterations
+Niter
+K · 4
+Initialization vector
+ˆ
+γinit
+ˆγLMMSE
+init
+(19)
+Unknown part of the CSI
+λ
+0.3
+V. NUMERICAL ASSESSMENT
+The default simulation configurations are summarized in
+Table I. The device activity profile is generated randomly and
+independently for each device with a probability ǫa = 0.1,
+meaning that on average ǫaK = 50 devices are active simul-
+taneously. Or equivalently, the devices have an average duty
+cycle of 10%, which is high for typical IoT applications [2].
+The channel between the BS and device k is modelled as
+in (3). The pilot sequence is randomly generated from a
+complex Gaussian distribution sk ∼ CN (0, 1), and is assumed
+to be known by the BS. Each device uses a pilot sequence
+of 10 symbols. A random phase offset φk
+∼ U[0,2π] is
+generated to simulate a carrier frequency offset (considered
+time-invariant over the preamble duration). The source code
+for all simulations can be accessed online6.
+A. Convergence of different initialization vectors
+The convergence of different initializations is evaluated with
+respect to the genie-aided approach. In the genie-aided case,
+the algorithm is initialized with the real activity indicators, i.e.,
+γ. The convergence is assessed via the likelihood (5) and the
+mean square error (MSE). The former should monotonically
+increase with each iteration, while the MSE can vary as it can
+not directly be minimized. The performance of the different
+initialization vectors for ˆγinit are depicted in Figure 3. The
+bottom figures zoom in on a smaller region to distinguish
+the performance of the initialization vectors when converging
+closer to the genie-aided case. While all initialization methods
+approximate the genie-aided case, the initialization vector has
+a non-negligible impact on the performance of the algorithm.
+An intuitive approach is to initialize with 0 because the activity
+probability is low and hence, on average, 90 % of the devices
+are expected to be inactive. As illustrated in Figure 3, ˆγinit = 0
+requires considerably more iterations to approach the other
+initialization methods.
+B. Impact of the quality of prior CSI
+Figure 4 illustrates the performance of the detector algo-
+rithms for different correlations between the actual channel
+and the known CSI, i.e., λ. With increased λ, and thus
+decreased channel knowledge, both the LMMSE estimator
+and the proposed algorithm have an increased probability of
+miss detection. The figure also demonstrates the gain of the
+6https://github.com/wavecore-research/grant-free-random-access-partial-csi
+0
+5
+10
+15
+20
+−8,000
+−6,000
+−4,000
+−2,000
+log p(y|γ) (5)
+LMMSE (19)
+ZF (18)
+MF (20)
+zeros
+genie (γ)
+0
+5
+10
+15
+20
+−8,000
+−6,000
+−4,000
+−2,000
+log p(y|γ) (5)
+LMMSE (19)
+ZF (18)
+MF (20)
+zeros
+genie (γ)
+0
+5
+10
+15
+20
+10−2
+10−1
+100
+101
+Number of iterations (Niter/K)
+MSE(ˆγ)
+0
+5
+10
+15
+20
+10−2
+10−1
+100
+101
+Number of iterations (Niter/K)
+MSE(ˆγ)
+Figure 3: The log-likelihood (5) and MSE of the estimated activity indicators
+for different initialization vectors. While with all initialization vectors the
+genie-aided case is approximated, different number of iterations are required.
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+10−4
+10−3
+10−2
+10−1
+λ
+Pmd
+Pfa
+0.1
+0.01
+0.001
+Figure 4: Performance of the proposed algorithm (
+) versus the LMMSE
+estimator (
+) for different values of channel knowledge. The probability
+of miss detection is shown for different values of Pfa (10 %, 1 %, 0.1 %).
+proposed algorithm with respect to the LMMSE estimator. The
+algorithm outperforms the LMMSE estimator for all λ and
+is most effective when the prior CSI has a strong correlation
+with the actual channel, and diminishes with decreased channel
+knowledge.
+C. Impact of the signal-to-noise ratio
+Figure 5 shows the false alarm and miss detection prob-
+ability of the LMMSE estimator and the iterative maximum
+likelihood device activity detector for different device SNRs.
+The full CSI case is included as a baseline for comparison,
+
+10−5
+10−4
+10−3
+10−2
+10−5
+10−4
+10−3
+10−2
+21x
+Pfa
+Pmd
+−6.67 dB
+6.67 dB
+20 dB
+Figure 5: The probability of false alarm and miss detection for different device
+SNRs for the LMMSE estimator when having full CSI (
+) and partial CSI
+(
+), and the proposed iterative algorithm with partial CSI (
+). No
+miss detection or false alarm occurred in the full CSI case for SNR values of
+−6.67 dB and 20 dB.
+where the full CSI is known instead of only a portion (λ).
+Fig. 5 demonstrates the large performance gain of the proposed
+algorithm with respect to the LMMSE estimator. The graph
+demonstrates that the iterative algorithm lowers the probability
+of miss detection by a factor of 21 for the same probability
+of false alarm.7 The performance is only marginally increased
+for very low SNRs (below zero).
+VI. CONCLUSION
+We formulated an iterative maximum-likelihood (ML) al-
+gorithm to detect active devices using prior channel state
+information (CSI) when performing grant-free random access.
+Previous experimental work has demonstrated that, in many
+massive machine-typed communication (mMTC) applications,
+the CSI is less time-variant than assumed in theoretical models.
+Given the static nature of Internet of Things (IoT) devices, we
+have exploited this feature in the activity detection estimator.
+During grant-free access, the devices transmit a unique, but
+non-orthogonal preamble, which is used for activity detection.
+Next to this, the algorithm is also able to detect a device-
+specific phase offset, which could be caused by carrier fre-
+quency offset (CFO). The algorithm is numerically evaluated
+and compared to the conventional least minimum mean square
+error (LMMSE) estimator with full channel knowledge and
+partial CSI. The presented results indicate that the iterative al-
+gorithm converges and outperforms the conventional LMMSE
+estimator. For a signal-to-noise ratio (SNR) of 6.67 dB, the
+probability of not detecting an active device is 21 times lower
+for the proposed iterative ML estimator than the LMMSE
+estimator for the same probability of wrongly considering an
+inactive device as an active device.
+This work can be extended to the cell-free or distributed case
+with geographically distributed access points. In that case, the
+partial CSI becomes access point-dependent.
+7Notably, the LMMSE estimator does not employ an iterative approach.
+Therefore, our proposed algorithm will be compared in future work with
+other iterative approaches.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf,len=574
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='04861v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='IT]  12 Jan 2023  Grant-Free Random Access of IoT devices in  Massive MIMO with Partial CSI  Gilles Callebaut1, Franc¸ois Rottenberg1, Liesbet Van der Perre1, and Erik G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Larsson2  1Department of Electrical Engineering (ESAT-DRAMCO), KU Leuven, 9000 Ghent, Belgium  2Department of Electrical Engineering (ISY), Link¨oping University, Link¨oping, Sweden  Abstract—The number of wireless devices is drastically in-  creasing, resulting in many devices contending for radio re-  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In this work, we present an algorithm to detect ac-  tive devices for unsourced random access, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', the devices are  uncoordinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The devices use a unique, but non-orthogonal  preamble, known to the network, prior to sending the payload  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' They do not employ any carrier sensing technique and  blindly transmit the preamble and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To detect the active  users, we exploit partial channel state information (CSI), which  could have been obtained through a previous channel estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  For static devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', Internet of Things nodes, it is shown that  CSI is less time-variant than assumed in many theoretical works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  The presented iterative algorithm uses a maximum likelihood  approach to estimate both the activity and a potential phase  offset of each known device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The convergence of the proposed  algorithm is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The performance in terms of probability  of miss detection and false alarm is assessed for different qualities  of partial CSI and different signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Index Terms—activity detection, grant-free, massive MIMO,  maximum likelihood, random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' INTRODUCTION  Massive machine-typed communication (mMTC) is envi-  sioned to enable low-power connectivity to a very large  number of devices and open up new applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' While it  has been put forward as a key building block in 5G and  beyond, it has so far received less attention than enhanced  Mobile Broadband (eMBB) and Ultra-Reliable Low Latency  Communications (URLLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The challenge in mMTC, and  in general Internet of Things (IoT) systems, resides in the  low-power operation, sporadic nature of the traffic and a  large amount of uncoordinated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' As these devices are  often battery-powered, they are constrained in the signalling  overhead they can handle [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Furthermore, they are often  deployed in remote areas, where network coverage is low or  non-existent [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To address this, novel protocols need to  be designed tailored to the low-power, sporadic and massive  nature of mMTC traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' One promising direction, and the  focus of this work, is the use of massive MIMO, where a  large number of antennas is present at the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Two multiple access approaches can be taken, grant-based  and grant-free random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The former requires the devices  to obtain a grant from the network, where after it can use a  The research reported herein was partly funded by the European Union’s  Horizon 2020 research and innovation programme under grant agreement No  101013425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This paper is presented at IEEE WCNC 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Callebaut, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Rottenberg,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Van der Perre, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Larsson, “Grant-Free random access of IoT devices  in massive MIMO with partial CSI,” in 2023 IEEE Wireless Communications  and Networking Conference (WCNC) (IEEE WCNC 2023), Glasgow, United  Kingdom (Great Britain), Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' 2023  collision-free radio resource to transmit its data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The disadvan-  tage of this approach for IoT and mMTC is that devices need  to compete for grants, requiring, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', dedicated preambles or a  lot of signalling for collision resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Due to the number of  competing devices such schemes are not practical and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Therefore, grant-free approaches are advocated over grant-  based solutions [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In grant-free random access, devices  do not contend for a grant, but just access the network when  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Often these devices are unaware of the other devices  in the system and operate in an uncoordinated fashion, which  is called unsourced grant-free random access [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Different  approaches have been studied to detect the active devices in  massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Detecting active devices is facilitated  when each device uses a unique and orthogonal preamble  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This entails that each preamble should have the  same length as the number of devices to have no collisions,  which is unpractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Therefore a number of studies have  considered a pool of orthogonal pilots [7] or the use of non-  orthogonal pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Both of these strategies have been elaborated  in [4], where they formulate the device-activity detection  problem as a compressed sensing problem, where the sparsity  of the active devices at each time slot is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' However,  compressed sensing requires the preamble length to be larger  than the active devices, increasing the energy expenditure of  data transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To combat this, in [6, 8] a covariance-based  method is suggested using two estimators for device activity  recovery, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', maximum-likelihood (ML) and non-negative  least squares (NNLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This work is generalized by Ganesan,  Bjornson, and Larsson [9, 10] to the cell-free case where mul-  tiple access points (APs) are geographically distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' They  considered different large-scale fading coefficients per APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  They employed a simplification of their proposed algorithm  by including only the AP with the strongest contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' They  concluded that co-located massive MIMO is highly sensitive to  low signal-to-noise ratios (SNRs), while cell-free deployment  is better suited against shadowing fading effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Inspired by these works, we propose a  new algorithm exploiting the static nature of IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' As  demonstrated in [11] and elaborated in Secion II, the channel  state information (CSI) can remain almost invariant over large  period of times (hours) for scenarios with low mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This  was not yet leveraged by practical algorithms in the literature  to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Given that the base station  knows a part of the CSI of each device the performance of  the device activity scheme can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To do so, we  formulate the maximum-likelihood activity detection problem  using partial CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Furthermore, a phase offset can be estimated  which occurs due to carrier frequency offset (CFO) (when  the CFO is static over the preamble duration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' We validate  the convergence of the iterative algorithm, study the impact  of different initialization vectors for the device activity, the  impact of the quality of the partial CSI and the SNR on the  performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  The  used  mathematical  notations  are  described  in  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='com/wavecore-research/math-notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' MOTIVATION – RECURRENCE IN CSI  IoT technologies are often put in the field with a deploy-  and-forget strategy, where the devices remain immobile after-  wards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' As such, we can expect that the channel conditions  are less time-variant than typically assumed in literature [12,  13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' An experimental campaign to investigate the long-term  behavior of the channel is presented in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The long-  term behavior is measured by taking the channel correlation  δi,j = |¯hH  i ·¯hj|/∥¯hi∥∥¯hj∥ of the first channel estimate and the  other channel N measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', δ1,j for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  The observed channel correlations are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  It illustrates that most of the time the correlation coefficient  is close to 1, indicating that the channel is highly correlated  with the first estimate and thus can be considered static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' It also  shows that, while in some occasions the correlation drops,  the channel quickly becomes again highly correlated with  the first channel instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' More than 90% of the measured  channels1 have a correlation coefficient higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This  demonstrates the potential of re-using channel estimates in IoT  contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='8  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='8  1  Sample index  Channel Correlation δ1,j  Node 1  Node 2  Figure 1: Channel correlation over a full day (9h24-17h48) with over 10 000  channel instances, with an average correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='935/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='968 and a standard  deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='03 for Node 1/Node 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' SYSTEM MODEL  There is a total of K single-antenna devices and a total of M  co-located base station (BS) antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The set of active users  trying to access the network is denoted by Ka, with |Ka| ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  To access the network, each active device k ∈ Ka sends a  unique, non-orthogonal preamble of length T , known to the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The pilot symbol of the preamble sent by device k at  pilot symbol t is denoted by sk,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The channel vector between  the receive antenna m and user k is denoted by hk,m ∈ C, and  1More specifically, 91% and 95% for Node 1 and Node 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  2We use here the term “re-using” to indicate that we no longer operate in  a block fading model with independent channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Typically, these  blocks are considered in the order of 50 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  is considered fixed over the preamble duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The received  symbol at the m-th BS antenna at time t is  yt,m =  K−1  �  k=0  hk,msk,tγk + wt,m,  (1)  where γk is an unknown complex scalar and wt,m ∈ C is  independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=') zero mean  circularly symmetric complex Gaussian (ZMCSCG) noise with  variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The unknown complex scalar γk can be developed  as  γk = √ρkakejφk,  (2)  where ρk ∈ R+ is the transmit power of device k, ak ∈ {0, 1}  is the device activity and φk models a potential phase offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This offset φk can account for a CFO, where the phase is  considered constant over the preamble duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This occurs  when no frequency drift has occurred during the preamble  duration, which is a feasible assumption as the CFO is  typically low, yielding negligible phase rotations during the  preamble interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' By assuming that all M antennas are  perfectly synchronized, this offset is only dependent on the  device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In case the device is inactive, γk will be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' We will  introduce the term activity indicator to denote γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Let us consider that the BS knows a part of the CSI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=',  gk,m in  hk,m = gk,m + λkǫk,m,  (3)  where ǫk,m are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' ZMCSCG variables with unit variance,  and λk ∈ R+ models the unknown part of the CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The large-  scale fading coefficient of user k is βk = E(∥hk∥2 /M) =  ∥gk∥2 /M +λ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The factor λk models the quality of the known  CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Hence, it quantifies the correlation of the actual channel  hk,m to the known partial CSI gk,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In the extreme case with  λk = 0, the CSI is perfectly known and there is no uncertainty  left, as was studied in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This could be the case in a  fully static environment and if the CSI estimates are noiseless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  However, for a realistic IoT scenario, even for static devices,  CSI is not perfect due to i) environment dynamics and ii) noisy  estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The parameter λk then quantifies this imperfection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  It is here assumed to be known3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Another extreme case,  as considered in [8, 10], is obtained when gk,m = 0, ∀m,  implying that only the large scale fading coefficient of user k  is known, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', λk = √βk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' DEVICE ACTIVITY DETECTION  This section describes the proposed activity detection al-  gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' First, the log-likelihood of the received preamble is  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Then, given its non-convex expression, an iterative  approach is proposed to estimate the parameters γk ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Finally,  activity detection is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Log-Likelihood of the Received Symbols  Combining (1) and (3), the symbol, received at BS antenna  m and for pilot symbol t, is given by  yt,m =  K−1  �  k=0  (gk,m + ǫk,mλk)st,kγk + wt,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  3It could be set to a certain value depending on the user activity profile  and/or tracked for each user over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Stacking the observations at antenna m gives  ym =  K−1  �  k=0  gk,mskγk +  K−1  �  k=0  ǫk,mλkskγk + wm,  where  ym =  \uf8eb  \uf8ec  \uf8ed  y0,m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  yT −1,m  \uf8f6  \uf8f7  \uf8f8 , sk =  \uf8eb  \uf8ec  \uf8ed  s0,k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  sT −1,k  \uf8f6  \uf8f7  \uf8f8 , wm =  \uf8eb  \uf8ec  \uf8ed  wm,0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  wT −1,m  \uf8f6  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  For a given value of γk, ym|γk has a circularly symmetric  Gaussian distribution with mean �K−1  k=0 gk,mskγk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' After defin-  ing the vector θm = �K−1  k=0 ǫk,mλkskγk+wm, the covariance  matrix is  C = E  �  θmθH  m  �  =  K−1  �  k=0  λ2  k|γk|2sksH  k + σ2IT ,  (4)  where we used the fact that ǫk,m were assumed to be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' and  the additive noise is white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Note that this covariance matrix  does not depend on the antenna index m and is thus valid for  all ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Defining the vector γ = (γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', γK−1)T , and Θm =  ym − �K−1  k=0 gk,mskγk the log-likelihood of the observation  vector ym is  log p(ym|γ) = − ln (|C|) − T ln(π) − ΘH  mC−1Θm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Given the conditional independence of ǫk,m and wt,m over the  antennas, the different ym are independent as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Hence, the  log-likelihood of the aggregated observations at all antennas  y = (yT  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', yT  M−1)T becomes  log p(y|γ) = −M ln (|C|) − MT ln(π) −  M−1  �  m=0  ΘH  mC−1Θm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (5)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Iterative Algorithm for Maximizing Likelihood  The maximum likelihood estimator of γ is obtained by  maximizing  ˆγML = arg max  γ  log p(y|γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This problem is not trivial to solve given the nonlinear and  non-convex dependence of the log-likelihood, more specifi-  cally the covariance matrix C, in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' An idea to maximize  the likelihood is to use an iterative approach, similarly as [8,  10]: at each iteration, all γk are kept fixed but one, which is  optimized and updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This way, they get updated one by  one until convergence is attained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', a maximum number of  iterations or a certain tolerance is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' A block diagram  of the algorithm is given in Figure 2 and the pseudocode is  summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Let us consider that the complex-valued γk′ needs to be  updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Using the definition introduced in (2), we can rewrite  γk′ with a phase-amplitude decomposition: γk′ = rk′e\uf6beφk′,  with rk′ = |γk′| = √ρk′ak′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The optimization with respect  to γk′ is done in the following in several steps: i) optimizing  the phase φk′ for a fixed value of rk′, ii) re-inserting this  expression in the objective function to remove the dependence  in φk′ and iii) optimizing the amplitude rk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  1) Phase optimization: To highlight the dependence of the  objective function in γk′ for constant values of other γk, k ̸=  k′, let us define the vector  yk′,m = ym −  K−1  �  k=0,k̸=k′  gk,mskγk,  (8)  which can be seen as a cancellation of device interference to  isolate the contribution from device k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Hence, the objective  function to maximize can be written as in (7)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  where we explicitly express the dependence in (rk′, φk′)  while the other (rk, φk), k ̸= k′ do not appear since they are  considered constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Note that the matrix C, defined in (4),  does not depend on φk′ but only rk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In the extreme case of no  prior CSI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', gk′,m = 0 ∀m, the dependence of f(rk′, φk′)  in φk′ disappears and there is an underdetermination and no  estimate of the phase offset can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In other cases,  we can find that, after some manipulations,  ˆφk′ = ∠sH  k′C−1  M−1  �  m=0  g∗  k′,myk′,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (9)  This result has an intuitive understanding: the optimal phase  φ′  k tends to align the partial CSI with the observations due to  device k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  2) Removing the phase dependence: Inserting this optimal  value in the objective function f(rk′, φk′) makes the depen-  dence in φk′ vanish and gives (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  3) Amplitude optimization: To alleviate the dependence on  rk′ in (6), let us define  C−k′ = C − λ2  k′|γk′|2sk′sH  k′  (10)  =  �  k\\k′  λ2  k|γk|2sksH  k + σ2IT ,  which does not depend on rk′ and is full rank, thus invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Applying the Sherman-Morrison formulas [14] to C−1 and  C−1sk′ gives  C−1 = C−1  −k′ − C−1  −k′sk′sH  k′C−1  −k′r2  k′λ2  k′  1 + sH  k′C−1  −k′sk′r2  k′λ2  k′  (11)  We insert these expressions in (6) and we omit terms that  do not depend on rk′, which will vanish after taking the  derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This gives  ˜f(rk′) = −M ln (|C|) + αr2  k′ + βrk′  1 + δr2  k′  ,  (12)  where we defined the constants (independent of rk′) α, β and  δ, as  α =  M−1  �  m=0  |yH  k′,mC−1  −k′sk′|2λ2  k′ − sH  k′C−1  −k′sk′  M−1  �  m=0  |gk′,m|2  β = 2  �����  M−1  �  m=0  yH  k′,mC−1  −k′sk′gk′,m  �����  δ = sH  k′C−1  −k′sk′λ2  k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (13)  4For clarity, we omit in the following the constant term MT ln(π) which  does not affect optimization as it does not depend on γ and vanishes after  differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  f(rk′) = −M ln (|C|) −  M−1  �  m=0  yH  k′,mC−1yk′,m − r2  k′sH  k′C−1sk′  M−1  �  m=0  |gk′,m|2 + 2  �����  M−1  �  m=0  yH  k′,mC−1sk′gk′,m  ����� rk′  (6)  f(rk′, φk′) = −M ln (|C|) −  M−1  �  m=0  �  yk′,m − gk′,msk′rk′e\uf6beφk′�H C−1 �  yk′,m − gk′,msk′rk′e\uf6beφk′�  (7)  Inputs  σ2, λk, gm,k,  ym ∀m, k  Initialization  k′ ← 0, ˆγ ← ˆγinit  Amplitude Optimization  (14), (16) or (17)  ˆrk′ ← arg max ˜f(rk′)  Phase Optimization (9)  ˆφk′ ← ∠sH  k′C−1 �M−1  m=0 g∗  k′,myk′,m  Converged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Iterative maximum likelihood estimator  Tolerance  Max iteration  Activity Detection  ˆγk ≤ γth,k ∀k  No  Yes  k′ ← k′ + 1 mod K  Updated ˆγk′ ← ˆrk′e\uf6beφk′  ˆγ  Figure 2: Block diagram of the iterative maximum likelihood estimator and activity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  One can note that ˜f(rk′) in (12) can now be differentiated  with respect to rk′, using the differential rule ∂(log |A|) =  tr[A−1∂A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Setting the derivative to zero gives, noting that  the denominator is always strictly positive,  0 = −r3  k′2Mδ2 − r2  k′βδ + rk′(−2Mδ + 2α) + β,  (14)  which is a polynomial of degree 3 in rk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' There are closed-  form solutions for the roots of such polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Following  Descarte’s rule of signs, (14) has only one real and positive  root, as required, in case the terms are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Below we  discuss the special cases when the terms are not non-zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=',  when there is no or complete CSI knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  The algorithm is summarized in the pseudocode Algo-  rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' At each iteration, the constants α, β and δ can  be easily re-evaluated based on (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' They require the matrix  inversion C−1  −k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To avoid re-computing a full inverse at each  iteration, one can rely on the Sherman-Morrison formula and  on the current knowledge of C−1, which is updated at the end  of each iteration by inserting the obtained value of rk′ in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Using (10), we find  C−1  −k′ = C−1 + λ2  k′|γk′|2C−1sk′sH  k′C−1  1 − λ2  k′|γk′|2sH  k′C−1sk′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (15)  Moreover, computations can be optimized as several quantities  appear multiple times and can be computed only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The  complexity of the proposed algorithm is O(IMT 2), where I  is the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='5  We now investigate two particular cases, to gain further  insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Algorithm 1 Iterative maximum likelihood device activity  detector  Require: σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' ym,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' ˆγinit ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' m  k′ ← 0  ˆγ ← ˆγinit  C−1 ←  ��K−1  k=0 λ2  k|ˆγk|2sksH  k + σ2IT  �−1  while Not converged do  Compute yk′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' C−1  −k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' β and δ based on (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' (15) and  (13)  ˆrk′ ← arg max ˜f(rk′)  ⊲ Update amplitude based on (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (16) or (17)  ˆφk′ ← ∠sH  k′C−1 �M−1  m=0 g∗  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='myk′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='m  ⊲ Update phase  ˆγk′ ← ˆrk′e\uf6be ˆφk′  C−1 ← C−1  −k′ −  C−1  −k′ sk′ sH  k′ C−1  −k′ r2  k′ λ2  k′  1+sH  k′ C−1  −k′ sk′ r2  k′ λ2  k′  k′ ← k′ + 1 mod K  end while  a) Particular case: device with no CSI: Now consider  that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' for a given k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' gk′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='m = 0 ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This could be because  this device is new or moving a lot, such that its CSI is  outdated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Only, its parameter λk′ is known, which is equal  5Note that I will in practice depend on K as we will iterate N times over  all users K, but this is not a requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  to the large-scale fading coefficient √βk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' At iteration of user  k′, evaluating (13) for gk′,m = 0 ∀m implies that β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Hence, (14) simplifies to  0 = 2rk′(−r2  k′Mδ2 − Mδ + α),  which has a trivial solution in rk′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' One of the other roots  is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Keeping only the positive one, we find the  amplitude update  ˆrk′ =  �  α − Mδ  Mδ2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (16)  If this root is imaginary, we set ˆrk′ to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' As discussed  before introducing the phase update equation (9), in the case  of no prior CSI, the phase ambiguity cannot be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This particular case gives an update relatively similar to the  maximum likelihood estimator derived in [8, (23)], where  their ML expression estimates the error, while ours estimates  directly the coefficient ˆrk′, given the same estimate of γk′ at  each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  b) Particular case: device with complete prior CSI: Now,  consider that, for a given k′, λk′ = 0, so that the CSI is  perfectly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Only the phase shift and the transmit power  are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' At iteration of user k′, evaluating (13) for λk′ =  0 implies that δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Hence, (14) simplifies to a linear equation  0 = rk′2α + β, which gives the following amplitude update  ˆrk′ = −β  2α =  |sH  k′C−1 �M−1  m=0 g∗  k′,myk′,m|  sH  k′C−1sk′ �  m′ |gk′,m′|2  ,  (17)  while the phase is updated according to (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  4) Initialization: To start the iterative algorithm, we con-  sider different choices to initialize ˆγinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' A simple choice is  to initialize to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', ˆγ0  init = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Another choice is to  initialize solely based on the available prior CSI, considering  that λk ≈ 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The estimator is similar to [11], except that,  here, prior CSI is used instead of complete CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  If λk ≈ 0, ∀k, the covariance matrix C, defined in (4),  simplifies to C = σ−2IT , which is independent of γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Hence,  many terms of the log-likelihood in (5) become independent  of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Maximizing (5) becomes equivalent to the following  minimization  max  γ  log p(y|γ) = min  γ  M−1  �  m=0  �����ym −  K−1  �  k=0  gk,mskγk  �����  2  = min  γ ∥y − Γγ∥2 ,  where we defined the vector and matrix notations  y =  �y0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' yM−1  �T , Γ =  �Γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' ΓM−1  �T ,  Γm =  �s0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' sK−1  �  diag(g0,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' gK−1,m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This minimization problem is a quadratic function of γ, which  is a least squares problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The estimate has the following  closed-form expression  ˆγZF  init =  �  ΓHΓ  �−1  ΓHy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (18)  This last estimator can be seen as a zero-forcing (ZF) es-  timator, which requires a matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To avoid ill-  conditioning, a first necessary condition is that K ≤ MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This condition is not sufficient as the channel and preamble  of two devices could be correlated, especially when K is on  the order of MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Moreover, if no prior information is available  for a given user k′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', gk′,m = 0, ∀m, the inverse will also  be ill-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This implies that the k′-th column of Γ  becomes null and thus Γ is rank deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Moreover, the prior  CSI might be noisy, leading to unstable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  To make initialization more robust, we can use an least  minimum mean square error (LMMSE) criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' To do this,  some prior knowledge must be assumed on the statistics of  γ, more specifically, its first and second order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  We here make the following assumptions: i) the activity of  each device is independent of one another, ii) the average  activity and average transmit power of each device is known  and iii) no prior information is known on the phase offset  so that φk is considered uniformly distributed between 0  and 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Under these assumptions, we have E(γ) = 0 and  D = E(γγH) = diag (E(a0)E(ρ0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', E(aK−1)E(ρK−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Hence, for the linear observation model y = Γγ + w, still  considering that λk ≈ 0, ∀k, the LMMSE estimator of γ is  then given by [15]  ˆγLMMSE  init  =  �  ΓHΓ + σ2D−1�−1  ΓHy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (19)  Note that the matrix to be inverted is always well-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Finally, a matched filter (MF) estimator could be used to avoid  the need for matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  ˆγMF  init =  �  diag(ΓHΓ) + σ2D−1�−1  ΓHy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  (20)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Activity Detection  A non-negative activity threshold γth,k is applied for each  device k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' A device is considered active if |ˆγk| ≥ γth,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The  real-valued threshold is defined as,  γth,k = v  �  SNRk  −1,  (21)  where v is chosen to have a desired probability of false alarm  and miss detection performance and with SNRk = Mβk/σ2 =  E(∥hk∥2)/σ2 = (∥gk∥2 + Mλ2  k)/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Miss detection happens when a device was undetected,  while it was actually transmitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Equivalently, a false alarm  occurs if a device is considered active by the algorithm but  was not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' We define the probability of miss detection as the  average ratio of undetected devices to the number of active  devices Pmd = 1 −  ����Ka ∩ ˆKa  ��� / |Ka|  �  , where Ka is the set  of active devices and ˆKa = {k|ˆak = 1, ∀ ∈ [1, K]} denotes  the estimated set of active devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Note that on average  |Ka| = Ka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Similarly, the probability of false alarm is the ratio  of inactive devices considered active to the number of inactive  devices and is given by Pfa =  ���� ˆKa \\ Ka  ��� /(K − |Ka|)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  A trade-off can be made between the two probabilities  by varying v in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' A lower v yields a lower activity  threshold, resulting in more devices considered active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' This in  turn lowers the probability of miss detection, while increasing  the probability of generating a false alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In the simulations,  the parameter v is swept across the range [−40, 40]dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Table I: Simulation parameter set with default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Parameter  Symbol  Default value  Number of devices  K  500  Number of total BS antennas  M  64  Signal-to-noise ratio  SNR  20 dB  Device activity probability  ǫa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='1  Pilot sequence  sk  ∼ CN (0, 1)  Pilot length  τp  10 symbols  Phase offset  φk  ∼ U[0,2π]  Number of simulations  Nsim  >10 000  Number of algorithm iterations  Niter  K · 4  Initialization vector  ˆ  γinit  ˆγLMMSE  init  (19)  Unknown part of the CSI  λ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='3  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' NUMERICAL ASSESSMENT  The default simulation configurations are summarized in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The device activity profile is generated randomly and  independently for each device with a probability ǫa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='1,  meaning that on average ǫaK = 50 devices are active simul-  taneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Or equivalently, the devices have an average duty  cycle of 10%, which is high for typical IoT applications [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  The channel between the BS and device k is modelled as  in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The pilot sequence is randomly generated from a  complex Gaussian distribution sk ∼ CN (0, 1), and is assumed  to be known by the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Each device uses a pilot sequence  of 10 symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' A random phase offset φk  ∼ U[0,2π] is  generated to simulate a carrier frequency offset (considered  time-invariant over the preamble duration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The source code  for all simulations can be accessed online6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Convergence of different initialization vectors  The convergence of different initializations is evaluated with  respect to the genie-aided approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In the genie-aided case,  the algorithm is initialized with the real activity indicators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=',  γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The convergence is assessed via the likelihood (5) and the  mean square error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The former should monotonically  increase with each iteration, while the MSE can vary as it can  not directly be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The performance of the different  initialization vectors for ˆγinit are depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The  bottom figures zoom in on a smaller region to distinguish  the performance of the initialization vectors when converging  closer to the genie-aided case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' While all initialization methods  approximate the genie-aided case, the initialization vector has  a non-negligible impact on the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  An intuitive approach is to initialize with 0 because the activity  probability is low and hence, on average, 90 % of the devices  are expected to be inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' As illustrated in Figure 3, ˆγinit = 0  requires considerably more iterations to approach the other  initialization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Impact of the quality of prior CSI  Figure 4 illustrates the performance of the detector algo-  rithms for different correlations between the actual channel  and the known CSI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=', λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' With increased λ, and thus  decreased channel knowledge, both the LMMSE estimator  and the proposed algorithm have an increased probability of  miss detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The figure also demonstrates the gain of the  6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='com/wavecore-research/grant-free-random-access-partial-csi  0  5  10  15  20  −8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  −6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  log p(y|γ) (5)  LMMSE (19)  ZF (18)  MF (20)  zeros  genie (γ)  0  5  10  15  20  −8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  −6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  −2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='000  log p(y|γ) (5)  LMMSE (19)  ZF (18)  MF (20)  zeros  genie (γ)  0  5  10  15  20  10−2  10−1  100  101  Number of iterations (Niter/K)  MSE(ˆγ)  0  5  10  15  20  10−2  10−1  100  101  Number of iterations (Niter/K)  MSE(ˆγ)  Figure 3: The log-likelihood (5) and MSE of the estimated activity indicators  for different initialization vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' While with all initialization vectors the  genie-aided case is approximated, different number of iterations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='9  10−4  10−3  10−2  10−1  λ  Pmd  Pfa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='001  Figure 4: Performance of the proposed algorithm (  ) versus the LMMSE  estimator (  ) for different values of channel knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The probability  of miss detection is shown for different values of Pfa (10 %, 1 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='1 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  proposed algorithm with respect to the LMMSE estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The  algorithm outperforms the LMMSE estimator for all λ and  is most effective when the prior CSI has a strong correlation  with the actual channel, and diminishes with decreased channel  knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' Impact of the signal-to-noise ratio  Figure 5 shows the false alarm and miss detection prob-  ability of the LMMSE estimator and the iterative maximum  likelihood device activity detector for different device SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  The full CSI case is included as a baseline for comparison,  10−5  10−4  10−3  10−2  10−5  10−4  10−3  10−2  21x  Pfa  Pmd  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='67 dB  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='67 dB  20 dB  Figure 5: The probability of false alarm and miss detection for different device  SNRs for the LMMSE estimator when having full CSI (  ) and partial CSI  (  ), and the proposed iterative algorithm with partial CSI (  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' No  miss detection or false alarm occurred in the full CSI case for SNR values of  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='67 dB and 20 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  where the full CSI is known instead of only a portion (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' 5 demonstrates the large performance gain of the proposed  algorithm with respect to the LMMSE estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The graph  demonstrates that the iterative algorithm lowers the probability  of miss detection by a factor of 21 for the same probability  of false alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='7 The performance is only marginally increased  for very low SNRs (below zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' CONCLUSION  We formulated an iterative maximum-likelihood (ML) al-  gorithm to detect active devices using prior channel state  information (CSI) when performing grant-free random access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Previous experimental work has demonstrated that, in many  massive machine-typed communication (mMTC) applications,  the CSI is less time-variant than assumed in theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Given the static nature of Internet of Things (IoT) devices, we  have exploited this feature in the activity detection estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  During grant-free access, the devices transmit a unique, but  non-orthogonal preamble, which is used for activity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Next to this, the algorithm is also able to detect a device-  specific phase offset, which could be caused by carrier fre-  quency offset (CFO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The algorithm is numerically evaluated  and compared to the conventional least minimum mean square  error (LMMSE) estimator with full channel knowledge and  partial CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' The presented results indicate that the iterative al-  gorithm converges and outperforms the conventional LMMSE  estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' For a signal-to-noise ratio (SNR) of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='67 dB, the  probability of not detecting an active device is 21 times lower  for the proposed iterative ML estimator than the LMMSE  estimator for the same probability of wrongly considering an  inactive device as an active device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  This work can be extended to the cell-free or distributed case  with geographically distributed access points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content=' In that case, the  partial CSI becomes access point-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  7Notably, the LMMSE estimator does not employ an iterative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
+page_content='  Therefore, our proposed algorithm will be compared in future work with  other iterative approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfCgub/content/2301.04861v1.pdf'}
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diff --git a/89FQT4oBgHgl3EQfIjX8/content/tmp_files/2301.13253v1.pdf.txt b/89FQT4oBgHgl3EQfIjX8/content/tmp_files/2301.13253v1.pdf.txt
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+Dense Nuclear Matter Equation of State from Heavy-Ion Collisions
+Agnieszka Sorensen
+Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA
+Kshitij Agarwal
+Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany
+Kyle W. Brown
+Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA and
+Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA
+Zbigniew Chajecki
+Department of Physics, Western Michigan University, Kalamazoo, MI, 49008, USA
+Pawe�l Danielewicz, William G. Lynch, Scott Pratt, and ManYee Betty Tsang
+Department of Physics and Astronomy and Facility for Rare Isotope
+Beams Michigan State University, East Lansing, MI 48824 USA
+Christian Drischler
+Department of Physics and Astronomy and Institute of Nuclear
+and Particle Physics, Ohio University, Athens, OH 45701, USA
+Stefano Gandolfi and Ingo Tews
+Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
+Jeremy W. Holt and Che-Ming Ko
+Department of Physics and Astronomy and Cyclotron Institute,
+Texas A&M University, College Station, TX 77843, USA
+Matthias Kaminski
+Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA
+Rohit Kumar
+Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA
+Bao-An Li and William Newton
+Texas A& M University-Commerce, Commerce, TX 75429, USA
+Alan B. McIntosh
+Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA
+Oleh Savchuk
+Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA and
+Bogolyubov Institute for Theoretical Physics, 03680 Kyiv, Ukraine
+Maria Stefaniak
+GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany
+arXiv:2301.13253v1  [nucl-th]  30 Jan 2023
+
+2
+Ramona Vogt
+Nuclear and Chemical Sciences Division, Lawrence Livermore
+National Laboratory, Livermore, CA 94551, USA and
+Department of Physics and Astronomy, University of California, Davis, CA 95616, USA
+Hermann Wolter
+Faculty of Physics, University of Munich, D-85748 Garching, Germany
+Hanna Zbroszczyk
+Faculty of Physics, Warsaw University of Technology, Koszykowa 75, Warsaw, Poland
+Endorsing authors:
+Anton Andronic
+Westf¨alische Wilhelms-Universit¨at M¨unster, Institut f¨ur Kernphysik, 48149 M¨unster, Germany
+Steffen A. Bass
+Department of Physics, Duke University, Durham NC 27708
+Abdelouahad Chbihi
+GANIL, CEA/DRF-CNRS/IN2P3, Boulevard Henri Becquerel, F-14076 Caen Cedex, France
+Maria Colonna
+INFN-LNS, Laboratori Nazionali del Sud, 95123 Catania, Italy
+Mircea Dan Cozma
+IFIN-HH, Reactorului 30, 077125 Mˇagurele-Bucharest, Romania
+Veronica Dexheimer
+Department of Physics, Kent State University, Kent OH 44242 USA
+Xin Dong, Jørgen Randrup, and Nu Xu
+Lawrence Berkeley National Laboratory, Berkeley, CA 94720
+Travis Dore
+Fakult¨at f¨ur Physik, Universit¨at at Bielefeld, D-33615 Bielefeld, Germany
+Lipei Du
+Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada
+Steven P. Harris, Larry McLerran, and Sanjay Reddy
+Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA
+Huan Zhong Huang
+University of California, Los Angeles, CA 90095
+
+3
+Jos´e C. Jim´enez
+Instituto de F´ısica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1371, 05508–090 S˜ao Paulo-SP, Brazil
+Joseph Kapusta
+School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 USA
+Arnaud Le F`evre, Christian Sturm, and Wolfgang Trautmann
+GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany
+Jacquelyn Noronha-Hostler
+University of Illinois at Urbana-Champaign, Urbana, IL 61801
+Christopher Plumberg
+Natural Science Division, Pepperdine University, Malibu, CA 90263, USA
+Hans-Rudolf Schmidt
+Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany and
+GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany
+Peter Senger
+Facility for Antiproton and Ion Research, Planckstr. 1, Darmstadt, Germany
+Richard Seto
+University of California-Riverside, Riverside, California 92521, USA
+Chun Shen
+Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA and
+RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
+Jan Steinheimer
+Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany
+Joachim Stroth
+Institut f¨ur Kernphysik, Goethe-Universit¨at, 60438 Frankfurt, Germany and
+GSI Helmholtz Centre for Heavy-ion Research, Planckstr. 1, 64291 Darmstadt, Germany
+Kai-Jia Sun
+Institute of Modern Physics, Fudan University, 200438,Shanghai,China
+Giuseppe Verde
+INFN Sezione di Catania, 64 Via Santa Sofia, I-95123 Catania, Italy
+Volodymyr Vovchenko
+Physics Department, University of Houston, Box 351550, Houston, TX 77204, USA
+(Dated: February 1, 2023)
+
+4
+Executive Summary
+The nuclear equation of state (EOS) is at the center of numerous theoretical and experimental
+efforts in nuclear physics, motivated by its crucial role in our understanding of the properties of
+nuclear matter found on Earth, in neutron stars, and in neutron-star mergers. With advances in
+microscopic theories for nuclear interactions, the availability of experiments probing nuclear matter
+under conditions not reached before, and the advent of multi-messenger astronomy, the next decade
+will bring new opportunities for determining the nuclear matter EOS.
+• Profound questions challenging our understanding of strong interactions remain
+unanswered: It is still unknown whether the transition between a hadronic gas and a
+quark-gluon plasma, which at zero baryon density is known to be consistent with a crossover
+transition predicted by Lattice QCD, becomes of first order in the finite-density region of
+the QCD phase diagram accessible in terrestrial experiments. The isospin-dependence of
+the EOS, crucial to our understanding of both the structure of neutron-rich nuclei and the
+properties of neutron stars, is poorly known above nuclear saturation density. Moreover,
+recent observations of very heavy compact stars indicate that the EOS in neutron-rich mat-
+ter becomes very stiff at densities of the order of a few times saturation density, leading to
+values of the speed of sound exceeding 1/
+√
+3 of the speed of light (breaking the conformal
+limit). Not only is the mechanism behind this striking behavior not known, but it is also
+unknown whether a similar stiffening occurs in symmetric or nearly-symmetric nuclear mat-
+ter. Resolving these and other questions about the properties of dense nuclear
+matter is possible by taking advantage of the unique opportunities for studying
+the nuclear matter EOS in heavy-ion collision experiments.
+• Among controlled terrestrial experiments, collisions of heavy nuclei at interme-
+diate beam energies (from a few tens of MeV/nucleon to about 25 GeV/nucleon
+in the fixed-target frame) probe the widest ranges of baryon density and tem-
+perature, enabling studies of nuclear matter from a few tenths to about 5 times the nuclear
+saturation density and for temperatures from a few to well above a hundred MeV, respec-
+tively. In the next decade, numerous efforts worldwide will be devoted to uncovering the
+dense nuclear matter EOS through heavy-ion collisions, including studies at FRIB where
+the isospin-dependence of the EOS can be probed in energetic collisions of rare isotopes.
+Modern detectors and refined analysis techniques will yield measurements that
+will elucidate the dependence of the EOS on density, temperature, and isospin
+asymmetry.
+• Hadronic transport simulations are currently the only means of interpreting
+observables measured in heavy-ion collision experiments at intermediate beam
+energies. This means that capitalizing on the enormous scientific effort aimed at uncovering
+the dense nuclear matter EOS, both at RHIC and at FRIB, depends on the continued
+development of state-of-the-art hadronic transport simulations. Support for the hadronic
+transport community, and in particular for viable career pathways for early
+career researchers, is imperative to maintain the health of and diversify the U.S.
+hadronic transport community, and to fully realize the potential of U.S. efforts
+leading the exploration of the dense nuclear matter EOS.
+
+5
+CONTENTS
+I. Introduction
+7
+A. Constraining the nuclear matter EOS using heavy-ion collisions
+8
+B. Connections to fundamental questions in nuclear physics
+9
+C. Upcoming opportunities
+11
+D. Needs
+12
+II. The equation of state from 0 to 5n0
+13
+A. Transport model simulations of heavy-ion collisions
+13
+1. Transport theory
+14
+2. Selected constraints on the EOS obtained from heavy-ion collisions
+17
+3. Challenges and opportunities
+19
+B. Microscopic calculations of the EOS
+25
+1. Status
+25
+2. Challenges and opportunities
+27
+C. Neutron star theory
+28
+1. Status
+28
+2. Challenges and opportunities
+31
+III. Heavy-ion collision experiments
+33
+A. Experiments to extract the EOS of symmetric nuclear matter
+35
+1. Measurements sensitive to the EOS
+35
+2. Experiments probing densities between 1–2.5n0
+36
+3. Experiments probing densities above 2.5n0
+38
+4. Challenges and opportunities
+39
+B. Experiments to extract the symmetry energy
+42
+1. Experiments that probe low densities
+42
+2. Measurements to extract symmetry energy up to 1.5n0
+42
+3. Selected constraints on the symmetry energy around 1.5n0
+44
+4. Challenges and opportunities
+46
+IV. The equation of state from combined constraints
+50
+A. Constraints
+51
+B. EOS obtained by combining various constraint sets
+53
+V. Connections to other areas of nuclear physics
+54
+A. Applications of hadronic transport
+54
+1. Detector design
+55
+2. Space exploration, radiation therapy, and nuclear data
+55
+B. Hydrodynamics
+57
+1. Status
+57
+2. Range of applicability
+58
+3. Challenges and opportunities
+60
+VI. Exploratory directions
+60
+A. Dense nuclear matter EOS meeting extreme gravity and dark matter in supermassive
+neutron stars
+60
+B. Nuclear EOS with reduced spatial dimensions
+61
+
+6
+C. Interplay between nucleonic and partonic degrees of freedom: SRC effects on nuclear
+EOS, heavy-ion reactions, and neutron stars
+62
+D. High-density symmetry energy above 2n0
+63
+E. Density-dependence of neutron-proton effective mass splitting in neutron-rich matter
+66
+Acknowledgments
+68
+References
+68
+
+7
+I.
+INTRODUCTION
+The equation of state (EOS) is a fundamental property of nuclear matter, describing its emergent
+macroscopic properties originating from the underlying strong interactions. Around the saturation
+density of nuclear matter, the EOS controls the structure of nuclei through the binding energy and
+the incompressibility. The EOS also determines, among other things, the neutron-skin thickness
+in neutron-rich nuclei as well as the properties of nuclear matter at extreme densities and/or tem-
+peratures, corresponding to conditions produced in experiments colliding heavy nuclei or observed
+in neutron stars and neutron star mergers. Far beyond describing the properties of matter com-
+posed of only protons and neutrons, the EOS can also reflect the appearance of new degrees of
+freedom, e.g., strange particles in the cores of neutron stars or quarks and gluons in ultrarelativistic
+heavy-ion collisions, or the emergence of new states of matter, e.g., chirally-restored matter, meson
+condensates, or quarkyonic matter.
+In heavy-ion collision experiments, the EOS is studied by detecting particles emerging from the
+collision zone and measuring observables sensitive to the properties of nuclear matter. Crucially,
+any interpretation of these observables, including quantitative constraints on the EOS, requires
+comparisons of experimentally measured observables to results obtained in dynamic simulations.
+This white paper highlights the essential role of hadronic transport simulations of
+heavy-ion collisions in advancing our understanding of the EOS. It also elucidates the
+many connections between inferences of the EOS from heavy-ion collision data and
+other efforts aiming to describe and understand the properties of nuclear matter.
+FIG. 1. Schematic depiction of the ranges of density and temperature probed in experiments and astronom-
+ical observations sensitive to the EOS of nuclear matter (counterclockwise from bottom left): neutron star
+crust physics, including nuclear pasta structures; properties of nuclei; structure of neutron stars; dynamics of
+neutron star mergers; and outcomes of heavy-ion collisions which can probe both symmetric and asymmetric
+matter. Figures adapted from [1–5].
+
+100
+HIC (sym)
+temperature [MeV]
+10
+HIC (asym)
+S
+ROOKHAVEN
+NS mergers
+.NS crust
+nuclear
+properties
+0.1
+Z
+neutron stars (NS)
+0
+1
+2
+3
+4
+5
+density np/no8
+A.
+Constraining the nuclear matter EOS using heavy-ion collisions
+FIG. 2.
+Constraints on the zeroth (Sv) and
+first (L) coefficient of the symmetry energy ex-
+pansion.
+Experimental constraints are derived
+from heavy-ion collisions (HIC) [6], neutron-skin
+thicknesses of Sn isotopes [7], giant dipole res-
+onances (GDR) [8], the dipole polarizability of
+208Pb [9, 10], nuclear masses [11], and isovector
+skins (IAS+∆R) [12]. Also shown are constraints
+from χEFT (GP-B) [13], microscopic neutron-
+matter calculations (H, G) [14, 15], and from the
+unitary gas limit (UG) [16]. Figure from [13].
+The last decade has brought tremendous progress
+in extracting the EOS as a function of baryon den-
+sity nB, temperature T, and the isospin asymme-
+try δ (or, equivalently, the proton fraction) from
+a variety of experimental and astronomical data as
+well as theoretical calculations. Many-body theory,
+based on sophisticated approaches with input from
+nucleon scattering or nuclear structure data, can
+now state the EOS below and near the saturation
+density n0 with meaningful uncertainties (see Sec-
+tion II B, “Microscopic calculations of the EOS”).
+New classes of experiments have extracted the thick-
+ness of neutron skins in nuclei, shedding light on the
+isospin-dependence of the EOS (or, equivalently, the
+symmetry energy) near or below n0.
+High-energy
+heavy-ion collisions have constrained the EOS of the
+quark-gluon plasma at high temperatures and small
+baryon densities, while ongoing experimental efforts
+worldwide focus on the EOS of nearly-symmetric
+dense baryonic matter, probed in collisions at in-
+termediate energies. Meanwhile, collisions at lower
+energies have led to experimental constraints on the
+symmetry energy at sub- and suprasaturation den-
+sities.
+Most remarkably, a revolution in the qual-
+ity and breadth of astronomical observations, high-
+lighted by the first simultaneous detection of grav-
+itational waves and electromagnetic signals from a
+neutron-star merger, ushered in a new era of multi-
+messenger astronomy (see Section II C, “Neutron
+star theory”). Together with the newly available ex-
+perimental capabilities at the Facility for Rare Iso-
+tope Beams (FRIB), there are unprecedented oppor-
+tunities to probe the isospin-dependence of the EOS
+through astronomical and terrestrial measurements.
+Among the experimental efforts discussed above, heavy-ion collisions probe the widest range
+of baryon densities and, moreover, represent the only means to address the EOS away from n0
+in controlled terrestrial experiments, see Fig. 1.
+Indeed, heavy-ion reactions at beam energies
+from a few tens of MeV/nucleon to about 25 GeV/nucleon in the fixed-target frame probe the
+EOS of hadronic matter at baryon densities from a few tenths to about 5 times n0. Controlling the
+properties of matter produced in these experiments is possible by varying the beam energy, collision
+geometry, and isotopic composition of the target and projectile. Insights and constraints obtained
+from transport model analyses of these experiments are relevant both for our understanding of
+nuclear matter as found on Earth and for our understanding of neutron stars from crust to core.
+Within ongoing efforts, the STAR experiment’s Beam Energy Scan (BES) fixed-target (FXT)
+program at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory
+(BNL), which collided gold nuclei at intermediate beam energies and which completed data taking
+in 2022, leads the US effort to constrain the EOS of nearly-symmetric nuclear matter at high
+
+100
+Constraints on S-L
+HIC
+80
+△R
+X
+AS
+60
+GP-B
+G
+40
+H
+Masses
+pb
+20
+ Skin
+UG
+Analytic
+UG
+GDR
+0
+26
+28
+30
+32
+34
+Symmetry Energy S [MeV9
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Prior
+Astro + HIC
+Pressure P (MeV fm–3)
+100
+101
+102
+Number density n (nsat)
+FIG. 3.
+Pressure in neutron star matter as
+a function of density from a Bayesian analysis
+combining nuclear theory and data from multi-
+messenger neutron-star observations and heavy-
+ion collisions [17]: the dark blue and light blue
+region corresponds to the 68% and 95% credible
+interval, respectively, while the gray dashed line
+shows the 95% bound obtained in χEFT calcu-
+lations and used as a prior. Figure from [17].
+baryon densities up to around 5n0, corresponding
+to densities present in the deep interiors of neu-
+tron stars.
+Among comparable efforts in Europe,
+the HADES experiment at GSI, Germany, probes
+matter at densities up to 2.5n0. Preliminary results
+from these contemporary efforts, as well as measure-
+ments from other heavy-ion collision experiments
+in the past, have led to competitive constraints on
+the EOS of symmetric nuclear matter, with future
+measurements expected to shed more light on its
+high-density behavior. Detailed constraints on the
+isospin-dependence of the EOS can be obtained by
+varying the isospin content of the target and pro-
+jectile nuclei.
+Here, the ability to use radioactive
+isotopes, as in, e.g., intermediate-energy heavy-ion
+collision experiments at RIKEN and FRIB, is cru-
+cial to resolve the subtle effects arising from changes
+in the isospin asymmetry of the colliding systems.
+Above all, obtaining constraints on the
+EOS from heavy-ion measurements would not
+have been possible if not for advances in theory, and in particular for the collaborative
+effort to test the robustness and quantify the uncertainties of hadronic transport sim-
+ulations (see Section II A, “Transport model simulations of heavy-ion collisions”). At the same
+time, much remains to be learned, as tight constraints on both the symmetric and asymmetric
+EOS at higher densities have so far remained elusive. This is predominantly due to model uncer-
+tainties, which themselves are rooted in the inherent complexity of nucleus-nucleus collisions and
+the challenging task of describing all processes contributing to the final state observables.
+B.
+Connections to fundamental questions in nuclear physics
+The wealth of data from efforts conducted in recent years not only helps to get a better grasp
+on the nuclear matter EOS, but also has brought forward fascinating questions challenging our
+understanding of strong interactions.
+Following the successful BES-I campaign at RHIC, questions remain about the structure of the
+QCD phase diagram at finite baryon densities, where the sign problem prevents obtaining predic-
+tions with lattice QCD calculations. Surprisingly, the expected disappearance of the quark-gluon
+plasma signatures has not been observed in BES-I, with some observables suggesting that the
+QCD first-order phase transition may be located within the region probed by BES-II experiments,
+including the region probed by the currently analyzed BES FXT data. If this is the case, then
+constraining the EOS at lower densities and describing the approach to the transition from the
+hadronic side, which would manifest as a softening of the EOS, will be crucial for a robust inter-
+pretation BES-II measurements. Importantly, due to the largely out-of-equilibrium evolution of
+collision systems probing that region of the QCD phase diagram, hadronic transport simulations
+will play a dominant role in describing the dynamics of the collisions, and therefore in constraining
+the EOS of nearly-symmetric dense nuclear matter.
+Understanding the physics of neutron-rich matter across a range of densities is necessary not only
+to explain the properties of rare neutron-rich isotopes and the structure of neutron stars, but also
+to constrain microscopic interactions in isospin-asymmetric nuclear matter. At low densities, this
+
+10
+0.1
+−
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+'=0
+y'y
+d
+/
+1
+v
+d
+(AuAu) Protons (10-30%)
+HADES 
+=0.25-0.45)
+0
+(AuAu) Protons (b
+FOPI 
+(AuAu) Z=1 (b=2-5.5fm)
+FOPI 
+(AuAu) Z=1 
+Plastic Ball 
+(AuAu) Z=1 (b=2-5.5fm)
+INDRA 
+(AuAu) Protons (12-25%)
+E895 
+(AuAu) Protons
+E877 
+�
+(AuAu) h
+E877 
+(AuAu) Protons (10-40%)
+Star FXT 
+(AuAu) Protons (10-25%)
+Star FXT 
+(AuAu) Protons (10-40%)
+Star BES 
+(PbPb) Protons (12.5-33.5%)
+NA49 
+(PbPb) Protons (15-35%)
+NA61/SHINE 
+1
+−
+10
+1
+10
+2
+10
+(GeV)
+N
+-2m
+NN
+s
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+2
+v
+out-of-plane
+in-plane
+(AuAu) Protons (10-30%)
+HADES 
+(AuAu) Protons (15-29%)
+FOPI 
+(AuAu) Z=1 (20-30%)
+FOPI 
+(AuAu) Z=1 (b=5.5-7.5fm)
+INDRA 
+(AuAu) Protons
+EOS 
+(AuAu) Protons (12-25%)
+E895 
+(AuAu) Protons
+E877 
+�
+(AuAu) h
+E877 
+(AuAu) Protons (10-40%)
+Star FXT 
+(AuAu) Protons (0-30%)
+Star FXT 
+(AuAu) Protons (10-40%)
+Star BES 
+(10-20%)
+�
+(AuAu) h
+Star BES 
+(0-60%)
+�
+(AuAu) h
+Star 
+(0-60%)
+�
+(AuAu) h
+PHOBOS 
+(PbPb) Protons (12.5-33.5%)
+NA49 
+(10-30%)
+�
+(PbPb) h
+WA98 
+�
+(PbAu) h
+CERES 
+FIG. 4. Compilation of the world data on the slope of
+the directed flow at mid-rapidity (dv1/dy|y′=0, top) and
+the elliptic flow (v2, bottom) as functions of the reduced
+center-of-mass energy √sNN − 2mN for protons, Z = 1
+nuclei, and inclusive charged particles. Figure from [18].
+challenge is addressed by experimental and
+theoretical analyses of nuclear structure ob-
+servables. An important objective of nuclear
+many-body theorists is to accurately calcu-
+late these observables and reliably deduce
+the EOS using microscopic interactions de-
+rived within the framework of chiral effective
+field theory (χEFT). Probing the symmetry
+energy over a range of densities wider than
+found in nuclei is possible through heavy-ion
+collisions and neutron star studies. Often,
+knowledge of the isospin-asymmetric EOS is
+encoded in terms of constraints on the Tay-
+lor expansion coefficients of the symmetry
+energy around n0. Numerous analyses yield
+consistent constraints on the first few expan-
+sion coefficients (see, e.g., Fig. 2), although
+they rely on an assumption that the expan-
+sion remains accurate away from n0. The re-
+cent advent of Bayesian inference techniques
+allows one to pursue a different approach,
+within which the isospin-asymmetric EOS is
+described in terms of the dependence of the
+pressure on baryon density (see, e.g., Fig. 3).
+Moreover, Bayesian analyses can shed more
+light on densities at which measurements
+constrain the symmetry energy and quan-
+tify the uncertainties of the extracted EOS.
+As a result, combining diverse measurements
+and using advanced analysis techniques can
+lead to significantly tighter constraints, es-
+pecially on the high-density behavior of the
+symmetry energy (or, equivalently, on the
+higher-order symmetry energy expansion co-
+efficients), which is so far poorly known.
+Constraints on the EOS of neutron-rich
+matter at high densities have been dramat-
+ically affected by discoveries of heavy neutron stars. Combined with the properties of all known
+compact stars, these observations indicate that while the EOS of neutron-rich matter is relatively
+soft around (1–2)n0, the pressure steeply rises with density for nB >∼ 2n0. In fact, multiple analyses
+show that describing the known population of neutron stars is only possible for EOSs in which the
+speed of sound in neutron-star matter breaks the conformal limit at high densities, that is exceeds
+1/
+√
+3 of the speed of light c for nB >∼ 2n0. This striking behavior remains to be understood. In par-
+ticular, it is currently not known whether the speed of sound exceeds c/
+√
+3 above certain densities
+at all isospin fractions of nuclear matter or, alternatively, only in neutron-rich matter. Importantly,
+robust constraints on the symmetric matter EOS at nB >∼ 2n0, obtained from heavy-ion collisions
+at intermediate to high beam energies, would also put constraints on the isospin-dependent part of
+the EOS through comparisons with the EOS inferred from neutron star studies, thus uncovering
+the magnitude of isospin-related effects at high baryon density.
+
+11
+C.
+Upcoming opportunities
+The next decade will be an era of high-luminosity heavy-ion collision experiments at high baryon
+density with modern detector and analysis procedures, as well as detailed studies of the symmetry
+energy with collisions of proton- and neutron-rich isotopes.
+Many of the discoveries of the BES program in ultra-relativistic heavy-ion collisions at RHIC,
+e.g., the discovery of the triangular flow and elliptic flow fluctuations, illustrate that modern
+analyses of heavy-ion collisions bring new quality to the understanding of the underlying processes.
+Because of this, revisiting the intermediate to high beam energies, previously explored at the
+AGS at BNL as well as at SIS18 at GSI and now explored by the STAR FXT program and
+the HADES experiment, is imperative to enable putting tighter constraints on the EOS of dense
+nuclear matter. Moreover, the future CBM experiment at the Facility for Antiproton and Ion
+Research (FAIR), Germany, will be able to measure interaction rates exceeding those currently
+used by several orders of magnitude, allowing for exploration of multiple high-statistics observables.
+Furthermore, the explored beam energy range is where lower-order flow observables, reflecting the
+collective motion of the colliding system due to the underlying hadronic EOS, are particularly
+0.4 0.6 0.8 1.0 1.2 1.4 1.6
+1.8
+2.0150
+200
+250
+300
+350
+400
+450
+0.2
+0.3
+0.4
+0.5
+0.6
+Incompressibility (MeV)
+dv1/dy'|y
+'=0
+In-medium Xsection modification factor
+ free protons
+ free neutrons
+Au+Au, Ebeam/A=1.23 GeV
+b=6-9 fm
+HADES data: 0.46+0.03
+−0.03
+0.4 0.6 0.8 1.0 1.2 1.4
+1.6
+1.8
+2.0150
+200
+250
+300
+350
+400
+450
+-0.08
+-0.06
+-0.04
+-0.02
+0.00
+Incompressibility (MeV)
+v2
+In-medium Xsection modification factor
+ free protons
+ free neutrons
+Au+Au, Ebeam/A=1.23 GeV
+b=6-9 fm, |ycm|<0.05, pt>0.3 GeV/c
+HADES data: -0.06+0.01
+−0.01
+FIG. 5. Predicted slope of the directed flow at mid-
+rapidity (dv1/dy|y′=0, top) and elliptic flow (v2, bot-
+tom) as functions of the incompressibility and the in-
+medium nucleon-nucleon scattering cross section mod-
+ification factor, generated in simulations of Au+Au
+reactions using the isospin-dependent BUU (IBUU)
+transport model [19, 20]. Figure from Ref. [21].
+prominent (see Fig. 4).
+Therefore, the cor-
+responding precision measurements carry with
+them the opportunity to bring a richer perspec-
+tive and a better understanding of the physics
+underlying the complex dynamics of nuclear
+matter at extreme conditions (see Section III A,
+“Experiments to extract the EOS of symmetric
+nuclear matter”). This advancement can only
+occur provided a simultaneous development of
+hadronic transport simulations, as only a de-
+tailed understanding of various factors affect-
+ing the dynamics of heavy-ion collisions can
+lead to meaningful descriptions of the exper-
+imental data, and, consequently, more robust
+constraints on the EOS of nearly-symmetric
+nuclear matter (see Section II A, “Transport
+model simulations of heavy-ion collisions”). As
+an example of the sensitivity of observables to
+various details of the underlying physics, Fig. 5
+shows the dependence of the slope of the di-
+rected flow (top panel) and of the elliptic flow
+at midrapidity (bottom panel) on the stiffness
+of the EOS, parametrized by the incompress-
+ibility, and on the in-medium nucleon-nucleon
+scattering cross-section modification factor.
+Unprecedented possibilities are on the hori-
+zon for studies of the isospin-dependence of the
+EOS, which is critical for connecting heavy-ion
+physics measurements to astrophysical obser-
+vations.
+The difficulties in using nuclei with
+significant variations in the isospin asymme-
+try, along with the paucity of neutron measure-
+ments at midrapidity, have in the past greatly
+
+12
+restricted the capability to put tight constrains on the EOS of asymmetric nuclear matter. Fortu-
+nately, at this time modern neutron detectors are available for heavy-ion measurements in many
+facilities, including at accelerators performing collisions at high beam energies such as GSI, while
+radioactive beam measurements are entering a new era at RIKEN and FRIB. FRIB will provide
+proton- and neutron-rich beams of not only the highest-intensity worldwide, but also characterized
+by the widest currently accessible range of the isospin asymmetry. Establishing a strong heavy-ion
+program at FRIB will therefore enable previously inaccessible exploration of the symmetry en-
+ergy (see Section III B, “Experiments to extract the symmetry energy”). Moreover, the proposed
+FRIB400 beam energy upgrade would not only allow exploration of densities up to around 2n0,
+but it would also provide increased resolution of the isospin-dependence of the EOS. In particular,
+among observables sensitive to the symmetry energy, both charged pion yields and the absolute
+magnitude of the elliptic flow (see Fig. 4) significantly increase between the current top FRIB
+energy of 200 MeV/nucleon and the proposed 400 MeV/nucleon [22].
+The increase in available computing power and advances in statistical methods make it possible
+to perform wide-ranging comparisons of heavy-ion collision simulations with experimental data
+(e.g., using Bayesian analysis), allowing one to vary multiple model assumptions at the same
+time as well as to put robust uncertainties on the obtained constraints. Furthermore, given the
+wealth of the upcoming independent data, e.g., from heavy-ion collision experiments, neutron star
+observations, and microscopic nuclear theory calculations, global analyses of complementary efforts
+have likewise a strong potential for putting tight constraints on the EOS (see Section IV, “The
+EOS from combined constraints”).
+Beyond the much-needed interpretation of intermediate energy heavy-ion collisions, advances
+in transport theory can lead to significant contributions to other areas of nuclear physics.
+In
+particular, recently attention has been given to cross-cutting opportunities for employing state-
+of-the-art hadronic transport codes in studies supporting space exploration and advanced medical
+treatments (see Section V A, “Applications of hadronic transport”). Transport theories may also be
+used in tests of extensions of hydrodynamic approaches supporting far-from-equilibrium evolution
+(see Section V B, “Hydrodynamics”), which are a focus of intense studies due to their importance
+for modeling heavy-ion collisions at high energies. Finally, constraining the dense nuclear matter
+EOS through interpretations of heavy-ion collision measurements may have other profound conse-
+quences, including helping to answer fundamental questions about the possible existence of dark
+matter in the cores of neutron stars or providing the impetus for studies of nuclear systems in
+fractional dimensions (see Section VI, “Exploratory directions”).
+D.
+Needs
+The next-generation experimental measurements of observables sensitive to the nuclear matter
+EOS are imminent, and further progress in resolving the nuclear matter EOS is contingent on
+enhanced theory support. In particular, the development of transport theories based on
+microscopic hadronic degrees of freedom, which are the only means of interpreting
+measurements from heavy-ion collision experiments at intermediate to high beam
+energies, must be strengthened and expanded to fully realize the potential of the
+U.S. facilities leading the exploration of the dense nuclear matter EOS. Support for
+both individual scientists and collaborations, and in particular for viable career pathways for early
+career researchers, is imperative to maintain the health of and diversify the U.S. hadronic transport
+community, and to fully capitalize on the U.S. efforts exploring the dense nuclear matter EOS.
+
+13
+II.
+THE EQUATION OF STATE FROM 0 TO 5n0
+Efforts to determine the equation of state (EOS) of nuclear matter are at the forefront of nuclear
+physics. An EOS contains fundamental information about the properties of a many-body system
+(see, e.g., Section I B), and is, in essence, any nontrivial relation between the thermodynamic
+properties of a given type of matter. In nuclear physics, the form of the EOS that is most often
+pursued is the relation between energy per baryon or pressure and baryon density nB, isospin
+excess δ, and temperature T. For symmetric matter, the isospin excess vanishes (δ = 0), and
+for asymmetric matter the energy per baryon or pressure are commonly partitioned into a part
+corresponding to symmetric matter and the remainder, which contains all information about the
+isospin-dependence of the EOS. Due to the charge invariance of strong interactions, the latter
+part is (to a very good accuracy) quadratic in the isospin excess δ at densities relevant to nuclear
+experiments and astrophysical observations. The quadratic coefficients in the expansion around
+δ = 0 are independent of δ, and are often referred to as the symmetry energy (denoted as S(nB)
+at T = 0) or symmetry pressure, respectively. These, together with the EOS of symmetric matter,
+are then sufficient to describe the EOS of nuclear matter at any isospin asymmetry.
+While many approaches to constraining the nuclear matter EOS are pursued, here we describe
+three research areas which have the capability to constrain the EOS over wide ranges of density:
+inferences of the EOS from comparisons of experimental measurements to model simulations of
+heavy-ion collisions (Section II A), microscopic calculations of the EOS using chiral effective field
+theory (Section II B), and EOS inferences from neutron star studies (Section II C).
+A.
+Transport model simulations of heavy-ion collisions
+soft	EOS
+hard	EOS
+temperature	[MeV]
+0
+50
+100
+150
+200
+250
+300
+density	nB/n0
+0
+2
+4
+6
+8
+12.8	AGeV
+		6.4	AGeV
+		3.2	AGeV
+		1.6	AGeV
+		0.8	AGeV
+		0.4	AGeV
+		0.2	AGeV
+FIG. 6. Phase diagram trajectories of the central re-
+gion in Au+Au collisions at zero impact parameter,
+obtained from UrQMD simulations with a soft or a hard
+(characterized by K0 = 200 or K0 = 380 MeV, respec-
+tively) EOS [23, 24]. The trajectories follow the evolu-
+tion at times when temperature is fairly well-defined,
+from the moment of the highest compression to densi-
+ties around 0.5n0.
+Heavy-ion collisions at very low to interme-
+diate beam energies provide the means to probe
+nuclear matter at different densities (from sub-
+saturation to several times the saturation den-
+sity), temperatures (from a few MeV to well
+above one hundred), and neutron to proton
+ratios (from near symmetric nuclear matter,
+where Nn/Np ≈ 1, up to Nn/Np ≈ 2); see Fig. 6
+for an illustrative calculation of heavy-ion col-
+lision trajectories in the T-nB phase diagram
+from simulations using two schematic EOSs.
+These wide ranges of system properties ac-
+cessed in heavy-ion collisions position them as a
+perfect tool to extract the nuclear matter EOS,
+test predictions and extrapolations from regions
+of the QCD phase diagram accessed by other
+approaches, and provide a necessary input to
+nuclear theory and nuclear astrophysics calcu-
+lations. For example, the density-dependence
+of both the symmetric and asymmetric EOS
+can shed light on modeling effective nuclear in-
+teractions in the medium [15, 25–27] or con-
+strain approaches using the density functional
+theory [28–30].
+
+14
+However, systems created in heavy-ion collisions are short-lived, and their dynamic evolution
+is out of equilibrium over significant fractions of the total collision time.
+The evolution of a
+colliding system depends on the energy and centrality of the collision, and progresses through initial
+compression, growth of the compression zone, development of flows, and overall decompression
+with a gradual local equilibration during the process, see Fig. 7.
+The inherent complexity of
+the evolution means that the corresponding transport equations cannot be solved directly due to
+their high non-linearity, and therefore detailed inferences from heavy-ion collision experiments,
+where the non-equilibrium evolution probes nuclear matter over substantial ranges of density,
+require comparisons to results of collision simulations in transport models. Beyond modeling the
+dynamics of the collisions, transport models provide a connection to the equilibrium limit allowing
+for inferring the EOS [31], transport coefficients [32], as well as the in-medium properties and
+cross-sections of hadrons [33–35].
+1.
+Transport theory
+At its core, transport theory aims to describe the time evolution of the one-body phase-space
+distribution function in a semi-classical approximation for a dissipative system composed of a large
+number of particles, here in particular for a system of two heavy nuclei colliding at an energy per
+nucleon which is typically larger than the Fermi energy. The theoretical foundations of transport
+theory include the BBGKY hierarchy of coupled equations for reduced density matrices [36] as well
+as the equations of the nonequilibrium Green’s function theory [37, 38] such as obtained in Martin-
+Schwinger (also known as Schwinger-Keldysh) formalism for non-equilibrium Green’s function (see
+also Section V B).
+To arrive at transport equations, one employs (among others) a Wigner transformation and
+coarse-graining as well as a gradient expansion. The Wigner transformation and coarse-graining
+nB
+FIG. 7. Contour plots of the system-frame baryon density nB (top row) and local excitation energy E∗/A
+(bottom row) at times t = 0, 5, 10, 15, and 20 fm/c (columns from left to right), obtained from a transport
+simulation [39] of a 124Sn+124Sn reaction at beam energy Elab = 800 AMeV (√sNN = 2.24 GeV) and impact
+parameter b = 5 fm. The contour lines for the density use increments of 0.4n0, starting from 0.1n0, while
+the contour lines for the local excitation energy correspond to the values of E∗/A = {5, 20, 40, 80, 120} MeV;
+for statistical reasons, contour plots for the energy have been suppressed for baryon densities nB < 0.1n0.
+
+15
+-0.5
+0.0
+0.5
+0
+100
+200
+300
+dN/dy
+y
+<px/A> (GeV/c)
+132Sn+
+124Sn
+-0.5
+0.0
+0.5
+-0.1
+0.0
+0.1
+y
+ IBUU
+ pBUU
+ RVUU
+ SMF
+ IQMD
+ IQMD-BNU
+ IQMD-IMP
+ TuQMD
+-0.5
+0.0
+0.5
+0
+100
+200
+300
+dN/dy
+y
+<px/A> (GeV/c)
+132Sn+
+124Sn
+-0.5
+0.0
+0.5
+-0.1
+0.0
+0.1
+y
+ IBUU
+ pBUU
+ RVUU
+ SMF
+ IQMD
+ IQMD-BNU
+ IQMD-IMP
+ TuQMD
+-0.5
+0.0
+0.5
+0
+100
+200
+300
+dN/dy
+y
+<px/A> (GeV/c)
+132Sn+
+124Sn
+-0.5
+0.0
+0.5
+-0.1
+0.0
+0.1
+y
+ IBUU
+ pBUU
+ RVUU
+ SMF
+ IQMD
+ IQMD-BNU
+ IQMD-IMP
+ TuQMD
+-0.5
+0.0
+0.5
+0
+100
+200
+300
+dN/dy
+y
+<px/A> (GeV/c)
+132Sn+
+124Sn
+-0.5
+0.0
+0.5
+-0.1
+0.0
+0.1
+y
+ IBUU
+ pBUU
+ RVUU
+ SMF
+ IQMD
+ IQMD-BNU
+ IQMD-IMP
+ TuQMD
+-0.5
+0.0
+0.5
+0
+100
+200
+300
+dN/dy
+y
+<px/A> (GeV/c)
+132Sn+
+124Sn
+-0.5
+0.0
+0.5
+-0.1
+0.0
+0.1
+y
+ IBUU
+ pBUU
+ RVUU
+ SMF
+ IQMD
+ IQMD-BNU
+ IQMD-IMP
+ TuQMD
+FIG. 8.
+Comparison of results for rapidity distri-
+butions (top) and transverse flow of nucleons (bot-
+tom) as functions of the scaled rapidity, obtained
+with different transport codes (identified in the leg-
+end) within the TMEP initiative. The results shown
+were obtained for 132Sn+124Sn collisions at Elab =
+270 AMeV (√sNN = 2.01 GeV) and impact param-
+eter b = 4 fm, using controlled input models for the
+EOS and the cross sections as well as identically ini-
+tialized nuclei [40].
+lead to positive-definite phase-space distribu-
+tions [41] that can be efficiently sampled with
+Monte-Carlo techniques, while the gradient ex-
+pansion yields, for each particle species, the force
+acting on a particle and the particle’s veloc-
+ity as gradients of its total energy with respect
+to the spatial position and momentum, respec-
+tively. Knowledge of the kinematics of all parti-
+cles, together with the elementary collision rates,
+drives the evolution in the phase space. Finally,
+to arrive at a set of Vlasov-Boltzmann–like equa-
+tions, one employs the quasi-particle approxima-
+tion, neglecting details of the spectral functions
+and treating all particles as on-shell (we note
+here that while there are some transport codes
+with off-shell particle treatment, e.g., [42, 43],
+this approach is still an outstanding challenge in
+the transport theory, as will be discussed further
+below). Alternative approaches to arriving at a
+transport theory for heavy-ion collisions include
+using the relativistic Landau quasiparticle the-
+ory [44] or, in approaches starting from a molec-
+ular picture, representing the global wavefunc-
+tion as a product (sometimes antisymmetrized)
+of single-particle Gaussian wavepackets [45].
+The particle species considered in transport
+theory depend on the collision energy and may
+range from nucleons, through pions and the delta
+resonances, to higher resonances, kaons, and hy-
+perons. Some transport formulations further in-
+corporate light clusters (e.g., deuterons, tritons,
+and 3He nuclei) as independent degrees of free-
+dom, with recent extensions also including alpha
+particles [46] which appear abundantly in exper-
+iments and are of particular importance for colli-
+sions at fixed-target beam energies on the order
+of hundreds of MeV/nucleon. In some of these approaches, clusters are produced through multi-
+particle reactions, as discussed further below.
+For the lowest energy collisions, nonrelativistic
+formulations of the transport theory may be employed, but the majority of the available codes are
+relativistic, with many addressing collisions at energies from tens of MeV/nucleon to at least a few
+GeV/nucleon (see [35, 47, 48] for reviews).
+Transport approaches can be generally divided into those concentrating on a single-particle
+characterization of the colliding system and those attempting to describe many-particle correla-
+tions. Both types of approaches are highly complex and nonlinear, and the relevant equations are
+solved by simulations. The single-particle approaches typically solve a set of Boltzmann-Vlasov–
+type equations [47, 49] (also known as the Boltzmann-Uehling-Uhlenbeck, or BUU equations) in
+which the evolution of the system is governed by a mean-field evolution of the phase space distribu-
+tion (Vlasov equations) and a collision term which drives the dissipation (the Boltzmann collision
+term). While, in principle, the Boltzmann-Vlasov equation is deterministic, numerical solutions
+
+16
+contain numerically-induced fluctuations due to the fact that the evolution is obtained using the
+method of test particles, in which the continuous distribution function is represented by a large,
+but finite, number of test particles sampling the phase space. To include fluctuations of a physi-
+cal origin, one can add a fluctuation term to the two-particle collision term, thus arriving at the
+Boltzmann-Langevin formulation [35, 50].
+In contrast, quantum molecular dynamics (QMD) approaches include classical many-body cor-
+relations in the ansatz of the many-body wave function [47, 51], which is postulated as a product of
+single-particle wave packets of a fixed width, with the width regulating the amount of fluctuations
+and correlations in QMD. In Anti-Symmetrized Molecular Dynamics (AMD) [45], the product wave
+function is anti-symmetrized and the formulation includes Pauli correlations in the propagation as
+well as in, to a certain extent, the collision term.
+The fact that hadronic transport approaches are built on firm theoretical foundations has been
+crucial for the continued development of simulation frameworks. Reaching back to the roots of
+the nuclear transport theory has made it possible to resolve ambiguities which would be otherwise
+hard to tackle by purely phenomenological means, including descriptions of cluster production [52],
+low relative-velocity correlations (Hanbury–Brown-Twiss correlations) [53], and off-shell transport
+[42, 49, 54, 55].
+The strong theoretical foundation of transport theory has also been effective
+in ensuring covariance of the theory and preserving conservation laws in case of interactions that
+stray beyond outcomes of field-theoretic models, in particular interactions employing energy density
+functionals [44, 56–58] which are often needed for realistic descriptions of bulk properties of nuclear
+matter.
+An important effort to validate conclusions reached from comparing transport model results
+to data has been recently intensified by the formation of the Transport Model Evaluation Project
+(TMEP) [47]. Within this endeavor, predictions from different models are compared in controlled
+settings (e.g., ensuring the same physical input such as the EOS, initial densities, and cross sec-
+tions), oftentimes with comparisons to known results that can be achieved analytically or by other
+methods. Similar controlled comparisons of complex simulations have been done in other fields
+of physics: from atomic traps, through ultra-relativistic heavy-ion collisions, to core-collapse su-
+pernova calculations [59–62], and they are known to be very fruitful for their respective fields.
+The TMEP analyses not only enable identifying models that produce outlier predictions, but also
+determine details of implementation or physical assumptions behind the diverging results. An ex-
+ample of such a comparison of codes for simulations of heavy-ion collisions at lower energies, with
+controlled input, can be seen in Fig. 8, showing results for rapidity distributions (left) and the
+transverse flow (right) [40]. In general, the codes agree with each other reasonably well, however,
+differences between the codes are visible and, moreover, can be traced to specific model choices
+in the simulations. For example, the generally lower values of the transverse flow in the case of
+QMD codes are a result of an approximation used in the evaluation of a non-linear term in the
+mean-fields, which becomes relevant when density fluctuations become large, as often occurs in
+QMD. Beyond identifying this and similar problems, the Project has yielded recommendations
+for optimal algorithms used in transport codes, e.g., for ensuring obeying the Pauli principle in
+elementary two-body collisions [63] or for integration of equations of motion with mean-fields [64].
+Moreover, the project has identified a set of tests for transport codes that ensure their credibility
+when addressing different heavy-ion collision observables. Stringent tests of hadronic transport
+codes are especially important for studies aimed at constraining the nuclear symmetry energy,
+which, compared to other model parameters, has a comparatively weak effect on heavy-ion observ-
+ables and which therefore demands maximal precision from transport simulations. Below, we will
+also discuss the role that such comparisons can play in determining the uncertainty of transport
+model investigations.
+
+17
+2.
+Selected constraints on the EOS obtained from heavy-ion collisions
+A selection of important constraints on the EOS obtained from heavy-ion collisions can be found
+in Fig. 9 for both symmetric matter (pressure as a function of density, left panel) and asymmetric
+matter (symmetry energy as a function of density, right panel).
+We note here that while many results are reported in terms of constraints on the incompress-
+ibility K0, in the context of heavy-ion collision studies of the EOS, K0 should be understood as
+a parameter which specifies the behavior of the EOS in the range of densities probed by a given
+study. For example, in the case of experiments probing mostly densities above 2n0, constraints on
+K0 are only indicative of the behavior of the EOS above 2n0, and in particular do not constrain
+the behavior of the EOS around n0. This subtle, and often confusing, point is a consequence of
+simple parametrizations of the EOS used in many transport codes, where the only parameter con-
+trolling the behavior of the EOS both around n0 and at higher densities is K0. Recently, flexible
+parametrizations of the EOS have been developed (see, e.g., [57, 58]) and implemented (e.g., in
+hadronic transport code SMASH [75, 76]) which allow one to vary the incompressibility K0 and the
+high-density behavior of the EOS independently.
+The collective behavior of matter created in the collisions, especially the directed and elliptic
+flow, has been shown to be a very sensitive probe of the EOS [31, 67, 77–79].
+In contrast to
+collisions at the Fermi energies, where all nucleons within nuclei participate in the collisions, and
+unlike in collisions at ultrarelativistic energies, where the evolution of the colliding nuclei can be
+understood in terms of participant nucleons, at intermediate energies the interplay between the
+expanding collision zone and the dynamics of the spectators are key ingredients to understanding
+Le	Fèvre	et	al.
+Lynch	et	al.	from	Fuchs	et	al.
+Oliinychenko	et	al.
+Danielewicz	et	al.
+Walecka	model
+Fermi	gas
+pressure	[MeV/fm3]
+1
+10
+100
+baryon	density	nB/n0
+1
+2
+3
+4
+5
+HIC(isodiff)
+HIC(n/p)
+mass(Skyrme)
+IAS
+mass(DFT)
+PREX	II
+HIC(π)
+Tsang	et	al.
+ASY-EOS
+FOPI-LAND
+symmetry	energy	S(nB)	[MeV]
+0
+20
+40
+60
+80
+baryon	density	nB/n0
+0
+0.5
+1
+1.5
+2
+FIG. 9. Left: Selected constraints on the symmetric EOS obtained from comparisons of experimental data to
+hadronic transport simulations in [31] (region with black horizontal stripes), [65, 66] (region with red forward
+stripes), [67] (region with blue backward stripes), and [58] (region with green vertical stripes); see text for
+more details. Also shown are results of analytical calculations for the free Fermi gas (green dotted line)
+and in the linear Walecka model (pink dashed line). Right: Selected constraints on the symmetry energy
+obtained from comparisons of hadronic transport simulations to experimental data in [6] (region with purple
+forward stripes), [68] (region with green backward stripes), [69] (the solid orange region), and [70] (the red
+circle, square, and triangle symbols). Also shown are symmetry energy constraints obtained in [70] based on
+a novel interpretation of analyses of nuclear masses in DFTs [11, 71] (cyan diamond symbol) and in Skyrme
+models [72] (cyan star symbol), of Isobaric Analog States (IAS) energies [73] (magenta plus symbol), and of
+PREX-II experiment [74] (blue inverted triangle symbol).
+
+18
+experimental results.
+A seminal constraint on the symmetric nuclear matter EOS [31] in the
+density range (2–4.5)n0 was obtained by comparing measurements of collective flow from heavy-
+ion collisions [80–83] at beam energies Elab = 0.15–10 AGeV (corresponding to nucleon-nucleon
+center-of-mass energies √sNN = 1.95–4.72 GeV) with results from hadronic transport simulations
+using EOSs with different values of the incompressibility at saturation density K0. The outcome of
+this study suggests a symmetric-matter EOS to lie between those labeled with K0 = 210 MeV and
+K0 = 300 MeV (see the region with black horizontal stripes in the left panel of Fig. 9). For densities
+in the range (1.0–2.5) n0, probed in collisions below Elab <∼ 1.5 AGeV (√sNN <∼ 2.5 GeV), the EOS
+may be inferred from meson yields [84–86]. Indeed, subthreshold production of strange mesons
+(specifically, K+ and K0), which interact weakly with nuclear matter, depends on the highest
+densities sampled in the collision, which in turn depend on the stiffness of the EOS [87]. In [65],
+ratios of experimentally measured kaon yields in Au+Au and C+C collisions have been reproduced
+in hadronic transport simulations with soft mean-field interactions yielding K0 = 200 MeV and
+an EOS [66] consistent with the constraint from [31] (see the region with red forward stripes
+in the left panel of Fig. 9).
+In [67], the elliptic flow data measured at Elab = 0.4–1.5 AGeV
+(√sNN = 2.07–2.52 GeV) by the FOPI collaboration [88] were used together with simulations
+from Isospin Quantum Molecular Dynamics (IQMD) [23, 89] to constrain the incompressibility at
+K0 = 190 ± 30MeV, again indicating a rather soft EOS (see the region with blue backward stripes
+in the left panel of Fig. 9). Recently, new measurements by the STAR collaboration from the fixed
+target (FXT) program at RHIC have become available, providing an opportunity to expand the
+set of world data utilized to deduce the baryonic EOS. A Bayesian analysis study [58], in which the
+speed of sound was independently varied in specified intervals of baryon density (thus providing
+a more flexible EOS at higher densities), suggests a tension between the E895 [83, 90–92] and
+STAR [93, 94] data. Using only the STAR measurements, the study [58] further found that EOSs
+which simultaneously describe the slope of the directed flow and the elliptic flow, in the considered
+energy range of Elab = 2.9–9 AGeV (√sNN = 3.0–4.5 GeV), are relatively stiff at lower densities
+and relatively soft at higher densities (see the region with green vertical stripes in the left panel of
+Fig. 9). However, the model used in that work did not include the momentum dependence of the
+EOS, which likely results in a spuriously stiff EOS at intermediate densities. As such, the study
+should be treated as a proof of principle that a tight constraint on the EOS at high densities can
+be achieved by using a combination of precise data, flexible forms of the EOS used in simulations,
+state-of-the-art models, and advances in analysis techniques.
+The symmetry energy contribution to the EOS can be studied at low collision energies Elab <∼
+1.0 AGeV (√sNN <∼ 2.32 GeV), where in particular observables such as charged pion yields [95] or
+neutron and proton flow [96, 97] have been proposed as sensitive to the asymmetric contribution
+to the EOS. Some of the constraints derived from such studies are shown in the right panel of
+Fig. 9, where, in addition to the usual EOS constraint bands, symbols with uncertainty bars
+represent results from analyses in which the symmetry energy has been determined for the most
+sensitive density of a given measurement. At incident energies below Elab = 100 AMeV (√sNN =
+1.93 GeV), low densities are probed after the initial impact and compression of the projectile and
+target [6, 98]. Since the symmetry potentials for neutrons and protons have opposite signs, emission
+of a particular nucleon type is enhanced or suppressed depending on the asymmetry. A comparison
+of the experimental measurements of isospin diffusion and the ratio of neutron and proton spectra in
+collisions of 112Sn+124Sn at Elab = 50 AMeV (√sNN = 1.90 GeV) to results from ImQMD simulations
+produced a constraint on the symmetry energy for densities (0.3–1) n0 [6] (see the region with purple
+forward stripes in the right panel of Fig. 9). Collisions at higher energies (Elab > 200 AMeV, or
+√sNN > 1.97 GeV) probe the EOS at n > n0. In the FOPI-LAND experiment, constraints on
+the symmetry energy were obtained from studies of the ratio of the elliptic flow of neutrons and
+hydrogen nuclei in Au+Au collisions at Elab = 0.4 AGeV (√sNN = 2.07GeV) [68], while the ASY-
+
+19
+EOS experiment used neutron to charged fragments ratios measured in Au+Au collisions [69] (see
+the region with green backward stripes and the solid orange region, respectively, in the right panel
+of Fig. 9). In [70], a comprehensive analysis was performed with the goal of identifying the values of
+the symmetry energy at densities to which given experiments are most sensitive. Using the isospin
+diffusion in collision systems with different proton to neutron ratios [99], neutron to proton energy
+spectra in Sn+Sn systems [100], and spectral pion ratios measured by the SπRIT collaboration in
+Sn+Sn collisions at Elab = 270 AMeV (√sNN = 2.01 GeV) [101, 102], that work [70] put constraints
+on the values of the symmetry energy at about 0.2n0, 0.4n0, and 1.5n0, respectively (see the red
+circle, square, and triangle symbols in the right panel of Fig. 9). Also shown in the right panel
+of Fig. 9 are symmetry energy constraints obtained in [70] based on a novel interpretation of the
+analyses of nuclear masses in DFTs [11, 71] (cyan diamond symbol) and in Skyrme models [72]
+(cyan star symbol), of the Isobaric Analog State (IAS) energies [73] (magenta plus symbol), and
+of the PREX-II experiment result [74] (blue inverted triangle symbol).
+3.
+Challenges and opportunities
+Selected results presented in Fig. 9 showcase significant achievements in determining the EOS
+and, simultaneously, the need to develop improved transport models to obtain tighter and more
+reliable constraints. Answering this need will require support for a sustained collaborative effort
+within the community to address remaining challenges in modeling collisions, in particular in the
+intermediate energy range (Elab ≈ 0.1–25 AGeV, or √sNN ≈ 1.9–7.1 GeV). In the following, we
+will address selected areas where we see the need for such developments: (1) comprehensive treat-
+ment of both mean-field potentials and the collision term in transport codes, (2) use of microscopic
+information on mean fields and in-medium cross sections, such as discussed in Section II B, in trans-
+port, (3) better description of the initial state of heavy-ion collisions in hadronic transport codes,
+(4) deeper understanding of fluctuations in transport approaches, which affect many aspects of
+simulations, (5) inclusion of correlations beyond the mean field into transport, which is crucial for
+a realistic description of light-cluster production, (6) treatment of short-range-correlations (SRCs)
+in transport, which are tightly connected to multi-particle collisions as well as off-shell transport,
+(7) sub-threshold particle production, (8) the study of new observables, e.g., azimuthally resolved
+spectra, to obtain tighter constraints on the EOS, (9) the question of quantifying the uncertainty of
+results obtained in transport simulations, and (10) the use of emulators and flexible parametriza-
+tions for wide-ranging explorations of all possible EOSs. Fortunately, advances in transport theory
+as well as the greater availability of high-performance computing make many of these improvements
+possible. Support for these developments will lead to a firm control and greater understanding of
+multiple complex aspects of the collision dynamics, allowing comparisons of transport model cal-
+culations and heavy-ion experiment measurements to provide an important contribution to the
+determination of the EOS of dense nuclear matter, which, in particular, cannot be determined by
+any other method at intermediate densities (1–5)n0.
+Comprehensive treatment of mean-field potentials and the collision term
+Notably, driven by specific experimental needs over the last two decades, the refinement of
+hadronic transport codes has diverged into two complementary branches: Codes which were ap-
+plied to describing experiments at very low to low energies (Elab <∼ 1.5 AGeV, or √sNN <∼ 2.5 GeV),
+such as IQMD, AMD and pBUU, have become progressively better at describing the momentum- and
+isospin-dependence of the interaction, while codes which were primarily used as afterburners for
+simulations of ultra-relativistic heavy-ion collisions (Elab >∼ 25 AGeV, or √sNN >∼ 7 GeV), such
+as SMASH [75] or UrQMD, were developed to offer a fully relativistic evolution as well as scattering
+
+20
+and decay modes taking into account all established particle and resonance species. As heavy-ion
+collisions are entering an era of precision data on symmetric nuclear matter at higher densities
+(e.g., in experiments at HADES, BES FXT, and future CBM) and on asymmetric nuclear mat-
+ter at normal and supranormal densities (e.g., at FRIB and future FRIB400), where features of
+both diverging branches of hadronic transport codes are important, a vigorous development of
+transport models is needed. In particular, numerous studies show the importance of including
+the momentum-dependence of the interactions, which is observed in elastic scattering of hadrons
+off nuclei.
+Moreover, momentum-dependence naturally occurs in microscopic effective interac-
+tions [38, 103] where it contributes to the calculated mean fields, whether near or away from sat-
+uration density. Incorporating single-particle energies with momentum dependence different than
+that in free space, which is often quantified with effective masses, is crucial in hadronic transport
+both for studies of symmetric nuclear matter [31, 79, 104, 105] as well as studies of the symmetry
+energy and its relation to effects such as the neutron-proton effective mass splitting [106–108] (see
+also Section VI E for more discussion on effective masses and the nuclear symmetry energy). Some
+of the theoretical and implementation solutions have already been established, while others will
+require devising new approaches. When possible, the best practices need to be carried over across
+the domains, as has been exemplified in, e.g., the development of the SMASH code, which uses many
+implementation solutions from pBUU.
+Microscopic input to transport
+One of the most prominent opportunities for improvement in transport models concerns imple-
+mentations of the EOS informed by state-of-the-art many-body studies. Such efforts are especially
+timely given that sophisticated microscopic calculations of the properties of nuclear matter are
+currently becoming available for large ranges of baryon density, temperature, and isospin fraction
+(see Section II B for more details). To incorporate the effects of the resulting EOSs in hadronic
+transport calculations, the corresponding Lorentz-covariant single-particle potentials as well as the
+in-medium interactions (both as functions of density, asymmetry, and momentum) are needed. A
+particular challenge is to determine the connection between the EOS inferred from a transport
+calculation and the zero-temperature EOS obtained from microscopic calculations [109], or even
+the finite-temperature EOSs that are becoming increasingly available [110, 111]. In a heavy-ion
+collision, the medium progresses through a set of non-equilibrium states that relax toward a local
+equilibrium, however, the nature of the local equilibrium also evolves during the collision due to the
+system expansion, so that even if the system approaches a local equilibrium at any given moment
+of the evolution, that agreement is only temporary. Errors incurred due to differences between
+non-equilibrium and equilibrium states of high-density matter contribute to the systematic error in
+inferring the EOS when comparing transport to experimental data (see Fig. 9 and [31]). Here, the
+availability of state-of-the-art microscopic calculations at finite temperature could reduce system-
+atic errors in connecting the finite- and zero-temperature EOSs. Moreover, the use of microscopic
+input would provide a consistency between the effective in-medium cross sections in the collision
+term and the mean fields used in the propagation of the phase space distribution. It could also
+help address the question of the extent to which nonlocalities in the microscopic theory should
+be reflected in the propagation and the collision term [112, 113] (where, in particular, departures
+from standard approaches modify the entropy to take a form different than that obtained in the
+Landau quasiparticle theory [44, 114]). To accelerate progress at the interface of the transport
+description of heavy-ion collisions and microscopic nuclear matter theory, direct collaboration of
+practitioners in the two research areas is required to assess how the needs of transport simulations
+can be answered by what can be currently calculated in microscopic theories. Conversely, the use
+of microscopic interactions in transport could validate the many-body theory results in regions of
+density and temperature which are only accessible by heavy-ion collisions [115].
+
+21
+Initial state
+Numerous studies point toward the dependence of outcomes of heavy-ion collision experiments
+on details of the initial conditions. In ultrarelativistic heavy-ion collisions, understanding these
+effects have led to the discovery of higher order flow harmonics [116, 117] and flow fluctuations [118].
+(Interestingly, the importance of the initial state for experimental outcomes also positions heavy-ion
+collisions at high energies as an unusual, but complementary probe of nuclear structure, see, e.g., a
+white paper on Imaging the initial condition of heavy-ion collisions and nuclear structure across the
+nuclide chart [119].) Given the high sensitivity of flow observables to both the EOS and the initial
+state of collisions, the impact of the initial conditions on outcomes of heavy-ion collisions needs to
+be thoroughly understood in order to narrow the constraints on the EOS of both symmetric and
+asymmetric matter. Aspects of initial conditions that need to be considered include event-by-event
+fluctuations of the initial state [116–118], relative distributions of neutrons and protons and shell
+effects [120], and correlations tied to deformation [121] or short-range correlations [122]. Some of
+these elements will be further discussed below in the context of the dynamics of heavy-ion collisions.
+Fluctuations
+Fluctuations of the phase space distribution are an important ingredient of transport simula-
+tions. In particular, fluctuations of the one-body density are important for including the conse-
+quences of the dissipation-fluctuation theorem in the reaction dynamics as well as for describing
+effects due to the largely unknown, neglected many-body correlations, thus going beyond the mean-
+field description. The question of how to include them properly and of their consistency with the
+nucleon-nucleon correlations explicitly implemented in transport theories, however, has not been
+completely clarified. As discussed above, fluctuations are included in a different manner in the two
+families of transport approaches. While in the BUU transport fluctuations can be introduced by
+the Langevin extension of the Boltzmann-Vlasov equation, which adds a fluctuation term to the
+collision term (and which is still rarely implemented), in the molecular dynamics approach fluctu-
+ations are introduced in a classical way by using finite-size particles, the width of which regulates
+the amount of fluctuations. Fluctuations then affect the outcome of simulations in many ways, in-
+cluding by regulating the formation of intermediate-mass fragments (IMFs) which appear through
+the growth of fluctuations in regions of spinodal instability. It was also shown in box calculations
+that fluctuations have a strong influence on the efficiency of Pauli-blocking [63] and even on the
+calculation of the force in the Vlasov term for QMD codes in which non-linear parametrizations of
+the fields are used [64].
+Correlations
+Correlations in transport simulations strive to address intermediate-range correlations beyond
+the mean-field picture. Physically, such correlations are also a source of fluctuations, but at the
+same time have other additional impacts, including, e.g., influencing the production of light clusters
+(LCs), that is light nuclei up to the alpha particle which are copiously produced in heavy-ion
+collisions. The mean-field models used in transport calculations are usually not detailed enough to
+realistically describe very light nuclei with their particular spin-isospin structure reflecting strong
+quantum effects. An additional complication results from the fact that in a collision, clusters often
+appear in the nuclear medium where their properties are drastically changed (e.g., the binding
+energy of clusters is reduced with increasing density until the Mott point, at which they dissolve).
+Currently, most codes describe the production of clusters by using a cluster-finding algorithm,
+based on particle proximity in coordinate and/or momentum space (coalescence) toward the end
+of the evolution, which in more advanced versions also takes into account criteria related to the
+binding energy of the produced clusters [123]. However, these late-stage algorithms do not take into
+account the dynamic role played by both correlations and LCs in the evolution of the collision. One
+
+22
+of the known approaches to this problem has been to consider LCs as separate degrees of freedom,
+with their own distribution functions and corresponding transport equations, where the collision
+terms can lead to creation or destruction of clusters (pBUU, SMASH) and which in particular can
+also take into account the in-medium modifications of clusters. However, this approach becomes
+increasingly complex as heavier clusters are characterized by more and more production channels,
+and consequently it is significantly challenging to include, e.g., alpha particles. Another approach
+is to modify the phase space of the correlated nucleons according to the Wigner function of the
+cluster, but then to propagate them after the collision again as nucleons (as is done in, e.g., AMD
+[41]), which still requires using a cluster-finding step at the end. In both cases, the production and
+destruction of clusters necessarily requires multi-particle collisions to ensure energy-momentum
+conservation. Finally, at lower incident energies the LC production can also be described in terms
+of the catalyzing effect of spectator nucleons in few-particle collisions [46, 124]. To explain LC
+production in high-energy collisions, where LCs are produced in numbers that cannot be obtained
+through nucleon catalysis due to the relatively few nucleons present in the final stages of these
+collisions, a similar mechanism of catalysis by pions [52, 125, 126] can be invoked.
+Short-range correlations
+A particular aspect of describing correlations in transport simulations is the treatment of short-
+range-correlations (SRCs), which have been measured in nucleon knock-out experiments [127–130].
+Along with the experiments, microscopic many-body calculations show that SRCs introduce a
+high-momentum tail (HMT) into the nucleon momentum distribution and, moreover, reduce the
+kinetic symmetry energy relative to the Fermi gas kinetic energy, which is a consequence of the fact
+that SRCs are more pronounced in symmetric relative to asymmetric matter [131–138] (see also
+Section VI C). Phenomenological methods have been used to include SRCs in transport models,
+e.g., by initializing nuclei with a HMT, but such a procedure does not take into account the dynamic
+role of SRCs in the initial state, which in the case of the on-shell semiclassical equations of motion
+results in obtaining nonstationary, excited states of nuclei. In on-shell transport approaches, three-
+and many-body collisions, incorporated into transport codes within varying approximations, have
+been suggested as a way of treating SRCs. In particular, in an investigation [139] of three-body
+collisions for pion production processes (e.g., NNN → NN∆), it was found that SRCs between
+two of the incident nucleons give a noticeable contribution to pion yields. Another approach [140],
+based on a mean-free-path approximation to the collision integral, observed large effects also on
+bulk observables. The incorporation of n-body collisions in transport equations within a schematic
+cluster approximation was also studied [141], however, there the effects were found to be rather
+small.
+So far, none of these methods have been widely exploited in the description of heavy-
+ion reactions.
+Since HMTs are tied to the tails of the nucleon spectral functions (away from
+the quasiparticle peaks), a consistent description of SRCs should involve an off-shell transport
+formulation. Dynamical spectral functions of all considered particles, including those which are
+stable in free space like nucleons, have been accounted for in the off-shell transport approaches
+implemented, with some differences in detail, in the codes GiBUU [42] and PHSD [43]. A subsequent
+study [142] demonstrated that the momentum distribution automatically develops a HMT within
+the approach used in GiBUU. Differences in the results from the two approaches have yet to be
+investigated systematically, including the impact on symmetry energy inferences from heavy-ion
+collision data based on, e.g., charged pion subthreshold production yields. Fully quantum transport
+approaches with SRCs (or equivalent content), without any semi-classical expansions as are present
+in current off-shell transport approaches, remain a long-term goal, and progress in this area has
+not ventured yet beyond schematic models [143, 144]. However, increasing computational power
+combined with emulation techniques may make such efforts more realistic and enable, e.g., a
+seamless integration of the treatment of shell effects in the initial state and collision dynamics.
+
+23
+Threshold effects
+An important influence of mean-field potentials in heavy-ion transport appears in the form
+of threshold shifts and the related subthreshold production of particles. Thresholds of particle
+production are modified in a medium since the mean-field potentials have to be taken into account in
+the energy-momentum balance of a two-body collision. Specifically, when the mean-field potentials
+are momentum-dependent and/or as a consequence of other model assumptions for the mean-field
+potentials of the produced particles, the thresholds are shifted away from their free-space values.
+This may strongly change the production rates of particles. Moreover, the threshold shifts make
+it necessary to involve other nucleons, besides the two collision partners in the process, to ensure
+the energy-momentum conservation. Various schemes to achieve this locally or globally have been
+in use [115, 145]. Indeed, explaining recent heavy-ion collision subthreshold pion yields, measured
+by the SπRIT Collaboration [102], required invoking many-body elementary effects in the form of
+mean-field effects on thresholds in two-particle collisions [86, 101]. However, because the physics
+invoked in describing the threshold effects is similar to that invoked for other multi-particle effects,
+alternative multi-particle options remain to be investigated, including producing pion degrees of
+freedom in multi-particle collisions or in the aftermath of an off-shell propagation between binary
+collisions. (We note here that there is a physics overlap between these mechanisms and the impact
+of SRCs on pion production [42, 122, 139].) Notably, theoretical explorations find sequences of
+on-shell binary processes to dominate the production at higher beam energies [43, 55, 139], and
+no comparable difficulties have been encountered in describing the data [146, 147] by transport
+models without multi-particle effects.
+The contrasting struggles of transport models which do
+not include threshold or other multi-particle effects of this type [102] , together with expected
+further theoretical explorations and future measurements of the subthreshold production in heavy-
+ion collisions, offer exciting possibilities for gaining understanding of the more exotic in-medium
+processes.
+New observables
+Upcoming precision data will further bring unprecedented observables that could be previously
+considered only in theory, such as triple-differential spectra tied to a fixed orientation of the reaction
+plane [18, 148–150] not only for protons and most abundant mesons, but also for deuterons,tritons,
+light nuclei, and hypernuclei. The potential of such spectra for the determination of the EOS
+is still to be fully explored, but a preliminary investigation [149] indicates a rich structure with
+spectra which exhibit a maximum away from the beam direction, characterized by slopes dependent
+on azimuthal angle and slope discontinuities. Models that might have agreed with each other in
+describing low-order Fourier coefficients of flow will likely find describing such detailed observables
+difficult. Challenges remain even at the level of the low-order coefficients, as many models now
+reproduce proton flow, but not Lambda or pion flow (see, e.g., Fig. 14). Understanding the relations
+between observables for various particle species will lead to constraints on the physics driving
+the evolution of heavy-ion collisions in simulations and, through that, to understanding cluster
+formation, hyperon yields, in-medium interactions with of strange hadrons, and more (see also the
+white paper on QCD Phase Structure and Interactions at High Baryon Density: Continuation of
+BES Physics Program with CBM at FAIR [151]).
+Quantifying uncertainties of transport predictions
+In the era of multi-messenger physics, where information on the EOS is derived from different
+areas of physics such as nuclear structure, nuclear reactions, and astrophysics, the ability to assess
+the uncertainty of a particular result is of crucial importance. This problem is especially relevant for
+evaluations of constraints on the EOS from transport simulations of heavy-ion reactions, since it has
+been found that using different transport models to describe the same data can lead to very different
+
+24
+conclusions. As found in the TMEP comparisons (see [47] for a review), even with controlled input
+the results from different models may vary considerably due to different implementation strategies
+which in themselves are not dictated by the underlying physics. In such a situation, calculating
+the mean and variance of different model predictions is not a reliable way of determining the
+uncertainties. An approach currently considered for ensuring a robust quality control in combining
+inferences from different models is to weigh the models with a Bayesian weight which could be
+based, e.g., on the performance of a given model in benchmark tests and/or its ability to reproduce
+all key observables of a given reaction (for example, flow observables, particle multiplicities, and
+spectra). Bayesian analysis can be also used for model selection through a comparison of results
+from a list of available models with data, during which one assigns to each model a probability
+of being correct based on the quality of the fit. However, this approach implicitly assumes that
+among the considered models there is at least one “true” model (also known as the M-closed
+assumption), which is often not fulfilled. Efforts have been taken to analyze data with an M-open
+assumption, where the existence of a perfect model is not assumed. For nuclear physics efforts,
+this is being attempted within the Bayesian Analysis of Nuclear Dynamics (BAND) group [152] by
+using Bayesian model mixing, where information from different models is combined for inference.
+Emulators and flexible EOS parametrizations
+Robust explorations of the possible physics underlying various observables often necessitate
+repeating the calculations many times for different combinations of physics parameters. When high
+event statistics is needed, the computational task can easily overwhelm the available computational
+resources.
+An additional computational strain often arises from assessing Bayesian probability
+distributions for any conclusions. Increasingly, emulators are going to be used for this task, with
+some steps having been already made [58, 102, 153]. Notably, similar issues emerge in the area of
+applications of hadronic transport [154] (see also Section V A).
+For explorations focused on the EOS, it may be of advantage to fit various possible EOSs with
+flexible relativistic density functionals as suggested in [57, 76]. This approach, given the complete
+freedom in varying both the functional form of the EOS as well as the EOS parameters, is particu-
+larly amenable to Bayesian analyses (see, e.g., [58] for a Bayesian analysis with a parametrization
+of the EOS in terms of the functional dependence of the speed of sound on density).
+The above list of issues facing the application of transport theory to heavy-ion collisions high-
+lights the fact that this approach to putting tighter constraints on the EOS rests on overcoming
+certain challenges. In simple terms, one attempts here to use a very dynamic and complex non-
+equilibrium process to obtain information describing a relatively simple and well-defined system,
+namely the equilibrated EOS of nuclear matter for different densities, temperatures, and isospin
+asymmetries. To achieve this in a reliable way, multiple complex issues of many-body physics have
+to be well controlled. On the other hand, several of the needed improvements are relatively well-
+understood, and tackling some of the unresolved problems poses an exciting intellectual challenge.
+As a reward for undertaking this effort, one gains the opportunity to obtain information on the EOS
+in a region which cannot be accessed through any other means: For densities below saturation,
+there is strongly constraining information from nuclear structure, with significant contributions
+coming also from low-energy heavy-ion collisions. Astrophysical observations on neutron stars and
+neutron star mergers are mainly sensitive to densities above about 3n0. The gap between these
+domains can only be filled with intermediate energy heavy-ion collisions, and transport studies are
+the essential tool to extract the information on the EOS from experimental data.
+
+25
+B.
+Microscopic calculations of the EOS
+Over the past decade, many-body nuclear theory has made significant progress in deriving
+microscopic constraints on the nuclear EOS at low densities from chiral effective field theory
+(χEFT) [155–158]. The progress has been driven by improved two-nucleon (NN) and three-nucleon
+(3N) interactions, rigorous uncertainty quantification, and algorithmic and computational advances
+in the frameworks used to solve the many-body Schr¨odinger equation with these interactions (see
+also the recent white paper on Dense matter theory for heavy-ion collisions and neutron stars [159]).
+1.
+Status
+Chiral EFT [160–164] provides a systematic way to construct nuclear interactions consistent with
+the low-energy symmetries of QCD, using nucleons (N’s), pions (π’s), and (in the case of delta-full
+χEFT), ∆-resonances (∆’s) as the relevant effective degrees of freedoms. Nuclear interactions in
+χEFT are expanded in powers of momenta or the pion mass over a hard scale at which χEFT breaks
+down; this breakdown scale is expected to be of the order of the ρ-meson mass, Λb ≈ 600 MeV.
+At each order in the EFT expansion, only a finite number of diagrams enter the description of the
+interaction according to a chosen power counting scheme, of which the Weinberg power counting
+has been predominant.
+For example, at the leading-order (LO) in Weinberg’s power counting
+one includes contribution from the one-π exchange between two nucleons as well as momentum-
+independent contact interactions, which allow one to describe key features of the nuclear interaction
+already at the lowest order. At next-to-leading-order (NLO), two-π exchanges are included as well
+as momentum-dependent contact interactions, and similarly, more involved terms appear at higher
+orders. The various low-energy coupling constants are determined from fits to experimental data,
+e.g., the π-N couplings are fit to π-N scattering, while those describing NN short-range interactions
+are fit to NN scattering data. The advantage of χEFT over phenomenological approaches is that
+multi-nucleon interactions, such as the important 3N interactions, naturally emerge in the EFT
+expansion and, moreover, are consistent with the NN sector. Forces involving increasingly more
+nucleons are correspondingly more suppressed, e.g., the leading contribution to 3N forces (four-
+nucleon (4N) forces) appears at N2LO (at N3LO) in Weinberg’s power counting. Furthermore, there
+are only two new low-energy couplings appearing in the three- and four-body forces to N3LO, which
+govern the strengths of the intermediate- and short-range contribution to the leading 3N forces,
+respectively. Consequently, χEFT 3N and 4N interactions at N3LO are completely determined by
+constraints on the coupling constants obtained from NN and π-N scattering”, usually resulting in
+tight constraints on very neutron-rich matter from χEFT.
+Another key feature of χEFT is that order-by-order calculations in the χEFT expansion have
+enabled estimation of theoretical uncertainties due to truncating the chiral expansion at a fi-
+nite order [13, 158, 165, 166]. Quantifying and propagating these EFT truncation errors enables
+meaningful comparisons between competing nuclear theory predictions, see Fig. 10, and/or con-
+straints from nuclear experiments and neutron-star observations in the multi-messenger astron-
+omy era [167]. Such comparisons are facilitated by Bayesian methods in a statistically rigorous
+way [158, 167, 168] to take full advantage of the great variety of empirical EOS constraints we
+anticipate in the next decade.
+Chiral EFT also provides nuclear Hamiltonians governing the interactions in nuclear systems.
+However, to calculate properties of a many-body system, computational methods able to solve the
+Schroedinger equation for this system are necessary. Among various frameworks used to solve the
+nuclear many-body problem in dense matter, quantum Monte Carlo (QMC) methods and many-
+body perturbation theory (MBPT) have been the main tools employed to study the physics of
+
+26
+FIG. 10. Comparison of the energy per particle E/N (left) and the pressure P (right) as functions of density
+for pure neutron matter in different many-body calculations using interactions from χEFT. The left panel
+also shows low-density QMC results of Ref. [169] and the conjectured unitary-gas lower bound on the energy
+per particle of pure neutron matter from Ref. [16]. Figure from Ref. [170].
+neutron-star matter in recent years. Both methods have recently made tremendous advances in
+predicting properties of nuclei and calculating the nuclear matter EOS [156, 158, 171–175].
+QMC frameworks, such as the auxiliary field Diffusion Monte Carlo (AFDMC) method, are
+based on imaginary-time propagation of a many-body wave function and enable us to extract
+ground-state properties of a nuclear many-body system with high statistical precision [156, 171].
+Their nonperturbative nature also allows for the treatment of nuclear interactions at high mo-
+mentum cutoffs, providing important insights into nuclear interactions at relatively short distances
+that may help to improve the modelling of χEFT interactions. QMC calculations of binding en-
+ergies, radii, and electroweak transitions of nuclei up to A = 16 [176–182] using χEFT NN and
+3N interactions are in very good agreement with experimental data [183–186].
+QMC methods
+were also used to calculate the EOSs of matter up to about twice the nuclear saturation density
+n ≈ 2 n0 [187–191]. The calculated EOSs include estimates of systematic truncation uncertainties,
+and are commonly used to constrain properties of neutron stars [188, 192, 193].
+The past decade has also seen a renaissance for many-body perturbation theory (MBPT) calcu-
+lations in nuclear physics [158, 175]. Key to this development has been the discovery that nuclear
+potentials with momentum-space cutoffs in the range 400 MeV <∼ Λ <∼ 500 MeV (not to be confused
+with the breakdown scale of χEFT, Λb) are sufficiently soft to justify the use of perturbation theory
+methods [194] (see [195] for a Weinberg eigenvalue analysis). Such low-momentum potentials can
+be obtained from renormalization group methods [196] or by directly constructing chiral effective
+field theory potentials at a coarse resolution scale. Furthermore, recent advances in automatic
+diagram generation [197] combined with automatic code generation [198] and high-performance
+computing have led to a fully automated approach to MBPT calculations in nuclear physics [158],
+in which chiral two- and multi-nucleon forces can be included to high orders in the chiral and
+MBPT expansions. MBPT has been demonstrated to be a computationally efficient and versatile
+tool for studying the nuclear EOS as a function of baryon number density nB, isospin asymmetry
+δ = (nn −np)/(nn +np), and temperature T [110, 111, 199, 200] with implications for neutron star
+structure [158] and astrophysical simulations [201]; here, nn and np correspond to the neutron and
+
+25
+5
+Hebeler et al.,ApJ (2013)
+Hebeler et al.,ApJ(2013)
+Tews et al.,PRL(2013)
+Tewsetal.,PRL(2013)
+Lynn et al.,PRL (2016)
+Lynn et al., PRL (2016)
+20
+4 
+Drischler et al.,PRL (2019)
+Drischler et al.,PRL (2019)
+Drischler et al.,GP-B (2020)
+Drischler et al.,GP-B (2020)
+Gezerlis, Carlson, PRC (2010)
+Unitary gas (s = 0.376)
+3
+fm
+[MeV
+10
+P2
+5
+1 
+0
+0.
+0
+0.05
+0.1
+0.15
+0.2
+0
+0.05
+0.1
+0.15
+0.2
+n [fm-3]
+n [fm-3]27
+proton densities, respectively. In particular, MBPT allows us to compute the EOS of neutron-star
+(i.e., β-equilibrated) matter explicitly, which can help improve isospin asymmetry expansions of
+the low-density nuclear EOS such as the standard quadratic expansion [199, 202–206]. MBPT also
+allows us to study nuclear properties other than the nuclear EOS, including the linear response
+and transport coefficients that could be used to inform more accurate numerical simulations of
+supernovae and neutron-star mergers [207]. Furthermore, MBPT for (infinite) nuclear matter has
+been used to construct a microscopic global optical potential with quantified uncertainties based
+on χEFT NN and 3N interactions [208, 209].
+Altogether, MBPT calculations of nuclear matter
+properties can provide important constraints that enable microscopic interpretations of future nu-
+clear reaction experiments [210] (e.g., at the Facility for Rare Isotope Beams) and neutron star
+observations.
+To date, theoretical predictions for the nuclear EOS, optical potentials, and in-medium NN
+scattering cross sections have been computed at finite temperature at various levels of approxi-
+mation starting from fundamental two- and multi-nucleon forces. These quantities are inputs to
+transport model simulations [89, 211] of heavy-ion collisions used to extract constraints on the
+properties of hot and dense nuclear matter (see Section II A for more details). In transport simu-
+lations, the EOS, single-particle potentials, and in-medium NN cross sections are usually obtained
+from effective phenomenological interactions [212, 213] that are fitted to the properties of finite
+nuclei and cold nuclear matter, and then extrapolated into the finite-temperature regime. Recently,
+some effort has been devoted to benchmarking [109] the temperature dependence of these effective
+interactions against predictions from χEFT or directly using EFT constraints in fitting effective
+interactions [207, 214, 215]. To enable such comparisons, the free energy of homogeneous nuclear
+matter as a function of temperature, baryon number density, and isospin asymmetry has been cal-
+culated using χEFT interactions up to second order in many-body perturbation theory [110] and
+within the Self-Consistent Green’s Function (SCGF) approach [216], which resums particle-particle
+and hole-hole ladder diagrams to all orders. The resulting EOS has been shown to be consistent
+with the critical endpoint of the symmetric nuclear matter liquid-gas phase transition [110, 216] as
+well as the low-density/high-temperature pure neutron matter EOS from the virial expansion [204].
+Furthermore, single-particle potentials have been computed at finite temperature at the Hartree-
+Fock level [217], from G-matrix effective interactions [218], and in SCGF theory [201, 219]. Of
+particular importance is the associated nucleon effective mass, which is obtained from a momen-
+tum derivative of the single-particle energy. The nucleon effective mass is directly related to the
+density of states and hence governs entropy generation at finite temperature, with consequences for
+the dynamical evolution of core-collapse supernovae and neutron star mergers. Finally, in-medium
+NN scattering cross sections have been computed at finite density and zero [220] as well as at
+finite [218] temperature using high-precision nuclear forces. In the next decade, the use of effective
+field theory methods will enable a consistent framework for describing all of these quantities with
+uncertainty estimates for input into transport simulations of heavy-ion collisions and astrophysical
+simulations.
+2.
+Challenges and opportunities
+To fully capitalize on experimental and observational data and extract key information on fun-
+damental questions in nuclear physics, continued progress in nuclear theory is crucial. The combi-
+nation of χEFT with modern computational approaches like machine learning, artifical intelligence,
+emulators, and Bayesian inference have provided EOS results for a wide range of densities, and at
+various proton-to-neutron asymmetries and temperatures, with quantified uncertainties [111, 166].
+Future progress in the development of fundamental interactions, combined with these tools, will
+
+28
+increase the precision of the results and enable us to answer open problems in chiral EFT. Among
+these, the most pressing is at which densities and how χEFT breaks down [166, 188]. In particular,
+for studies of neutron-star mergers it is of great importance to describe dense matter at finite
+temperatures [200, 201, 204], however, these might influence the breakdown of the theory in dense
+matter. In the next decade, it will be crucial to reliably determine how far one can push the χEFT
+approach in nucleonic matter.
+While microscopic calculations have been very successful in calculating properties of nuclei and
+homogeneous matter at densities up to 1-2 times the nuclear saturation density, we need improved
+microscopic descriptions of neutron-rich dense matter beyond that regime, at a few times nuclear
+saturation density and finite temperatures, with quantified uncertainties. This can be achieved
+by employing models derived within relativistic mean-field or density functional theory that are
+firmly rooted in microscopic theory at lower densities. Such models will be very important to
+connect theoretical calculations within the framework of χEFT to heavy-ion collision experiments
+at accelerator facilities around the world. Heavy-ion collision experiments at intermediate beam
+energies bridge the low- and high-density regimes of the EOS and provide complimentary informa-
+tion to that obtained from nuclear structure or neutron-star studies [17] (see Section II A). Robust
+inferences from the experimental data will require more accurate predictions from transport the-
+ory, which strongly depend on, among others, mean-field or density functional models. It will be
+imperative to test and constrain such models for the EOS with more rigorous microscopic calcu-
+lations. Beyond their use in hadronic transport simulations, these models are also a crucial input
+for calculations of properties of neutron star crusts (see Section II C).
+Additional theoretical constraints might be provided by high-density calculations within the
+framework of perturbative QCD (pQCD) [221], which can be applied at very high densities of
+the order of 40 times the nuclear saturation density, where the strong interactions among quarks
+become perturbative. Constraints on the EOS based on pQCD, together with assumptions on
+causality and stability, have been used to constrain the EOS at lower densities probed in the core
+of neutron stars [222–225].
+However, it has been found that the constraining power of pQCD
+calculations is strongly dependent on the way in which they are implemented [225, 226]. Future
+studies have to establish to what extent pQCD constraints are robust at densities of the order of
+several times nuclear saturation density, and how constraining future higher-order calculations may
+become. In this regard, improved microscopic calculations of the nuclear EOS using the functional
+renormalization group [227, 228] will provide important insights.
+C.
+Neutron star theory
+1.
+Status
+Measurements of the EOS, masses of neutron-rich isotopes far from the band of stability, and
+experimental constraints on nucleon effective masses provide essential input into neutron star mod-
+els, progressing our understanding of the structure and dynamics of these astronomically important
+objects. Several properties of neutron stars, including the mass-radius relation and their tidal de-
+formabilities, can be calculated once the EOS is provided. This, in turn, enables us to constrain
+the EOS once those properties are observed [229].
+Nuclear EOSs for neutron stars can be constructed from, for example, ab initio calculations and
+density functionals [230–233] or, more schematically, from meta-models [234–236] parameterized
+by nuclear matter parameters, which can be used to make contact with heavy-ion collisions [17].
+Ab initio calculations take into account more fundamental properties of the nuclear force (see Sec-
+tion II B), but prohibit the calculation of large ensembles of EOSs spanning the nuclear parameter
+
+29
+FIG. 11. Impact of nuclear physics theory and experiment, and
+different astrophysical measurements on constraining the cold
+neutron-star EOS. Blue lines show a family of EOS that are con-
+strained by chiral EFT at low densities. At higher densities, the
+EOS can then be constrained using GWs from inspirals of neutron
+star mergers, data from radio and X-ray observations of pulsars,
+and electromagnetic signals associated with neutron star mergers.
+The indicated boundaries between regions affected by these mea-
+surements are not strict and depend on the EOS and properties
+of the astrophysical system. Figure from [237].
+space.
+Meta-models allow rapid
+computation of such large ensem-
+bles, but encode mainly bulk prop-
+erties of nuclear matter, which ex-
+cludes them from being used to
+model finite nuclei.
+Density func-
+tionals represent a compromise, al-
+lowing both rapid computation of
+EOSs and use in finite nuclear mod-
+els, and thus are more suited to
+combining nuclear experimental and
+astrophysical information.
+Many
+of these models can be smoothly
+extrapolated from the saturation-
+density to arbitrarily high density,
+in which case astronomical obser-
+vations can be used to constrain
+the saturation-density nuclear mat-
+ter parameters and their density de-
+pendence [236, 238]. This extrapo-
+lation, however, is model-dependent,
+as different density functionals have
+different dependence on density. Ad-
+ditionally, this extrapolation might
+not be physically well-founded.
+As densities inside neutron stars
+can reach up to several times nuclear saturation density, at some (as-yet not determined) density a
+description in terms of purely nucleonic degrees of freedom is expected to break down. Heavy-ion
+collisions can help us constrain that point, and the nature of any phase transitions that occur above
+saturation density. The nuclear EOSs can be then combined with models describing the EOS at
+higher densities. Models that explicitly include a range of possible high-density degrees of freedom,
+such as hyperons and quarks, can be constructed; the predicted neutron star compositions are then
+dependent on the particular model used. Another approach is to use more general models that give
+up the explicit dependence on the underlying degrees of freedom, thus losing information on, e.g.,
+appearance of exotic particles at high densities, in favor of spanning the full space of physically con-
+sistent EOSs, reducing the model dependence of inferences from astrophysical observations [239].
+These schemes include piecewise polytropes [14, 240–242], line segments [192, 243], speed-of-sound
+models [188, 189, 244–246], spectral models [247] and non-parametric models generated from Gaus-
+sian processes (GPs) [168, 248–251] or machine learning techniques [252]. If these more general
+approaches are used down to the nuclear saturation density, extra modeling is required to connect
+them to the microscopic nuclear EOS and nuclear observables [253].
+Once the EOS is specified, the solution of the Tolman-Oppenheimer-Volkov equations and their
+extensions including rotation, determining the structure of a neutron star through balancing the
+attractive force of gravity and the repulsion coming from the EOS, provide predictions for bulk
+properties of the neutron star such as radii, tidal deformabilities, moments of inertia, and break-up
+frequencies of neutron stars as a function of their mass. All of these properties can be compared
+with multi-messenger observations, including gravitational waves and electromagnetic signals from
+neutron-star mergers and isolated neutron stars [193].
+The systematic construction of neutron star EOS models and statistical inference of EOS pa-
+
+103
+8
+102
+[Mev fm
+EM: Kilonovae / GRB
+GWs (post-merger)
+Pressure
+101
+GWs (inspiral)
+Radio and X-ray pulsars
+100
+Nuclear Physics
+Experiment and Theory
+2
+4
+6
+8
+Number density [nsat30
+rameters from data is an endeavor that is just over a decade old [14, 240–242]. This effort has
+matured in the current era of multi-messenger astronomy with a large push to explore the model-
+dependence of EOS inferences [244, 254] and ways of connecting the EOS with astrophysical and
+nuclear data [17, 167, 193, 245, 246, 255]. Different choices of which observables to include or infer
+can be made. For example, astrophysical observations can be used to infer the EOS, which can
+then be connected to nuclear models to inform their parameters and predict nuclear observables.
+Conversely, nuclear observables can be used to infer nuclear parameters, which can then inform
+the neutron star models and predict astrophysical observables. The future lies in combining more
+and more sets of data of both types to understand nuclear and neutron star models better.
+Exciting progress has been made in gathering astrophysical data to constrain our dense matter
+theories (see Fig. 11 for an illustration of density regions affected by different observables). Neutron-
+star data from the last 5 years identified the heaviest neutron star known to date with a mass of
+2.08(7)M⊙ [256, 257] (where M⊙ is the solar mass), while the kilonova AT2017gfo, associated with
+GW170817, has placed an upper limit on the maximum mass to be on the order of 2.3M⊙ [258,
+259].
+The detection of GW170817 by the LIGO-Virgo Collaboration has enabled us to place
+constraints on the tidal deformability of this system, ˜ΛGW170817 ≤ 720 [260, 261]. Neutron Star
+Interior Composition Explorer Mission (NICER) has provided two mass-radius measurements by
+observing X-ray emission from several hot spots on the neutron star surface, finding a radius of
+13.02+1.24
+−1.06km for a star with mass 1.44+0.15
+−0.14M⊙ (PSR J0030+0451) and 13.7+2.6
+−1.5km for a star with
+mass 2.08(7)M⊙ (PSR J0740+6620) in the analyses of Refs. [255, 262–264]. X-ray observations of
+the temperature of the neutron star in the Cas A supernova remnant have revealed core cooling
+on the timescale of years, hinting at the possible superfluid properties of the core [265]. These
+observations have enabled meaningful constraints on the EOS to be set and have already allowed us
+fascinating glimpses into the possible properties of high-density matter. For example, perturbative
+QCD predicts that the speed of sound squared approaches the conformal limit of 1/3 from below
+as the density becomes arbitrarily high.
+Meanwhile, inferences of the neutron star EOS from
+observational data indicate that the speed of sound rises in the core to significantly above c2
+s =
+1/3 [188, 266–269]. Consequently, this suggests that the speed of sound has a non-trivial behavior
+with increasing density [188, 221, 270]. At the same time, tentative evidence for quark matter in
+neutron star cores, which in turn indicates a softening of the EOS, has likewise been suggested [246].
+If we want to leverage the substantial data we have on neutron star cooling and dynamical
+evolution, additional EOS quantities need to be supplied consistently for each EOS model, such
+as the effective masses (see also Section VI E) and superfluid neutron and proton gaps, essential
+for modeling thermal and dynamical properties of neutron stars. For example, the mutual friction
+of the core – the strength of the coupling between the charged particles (electrons, protons) and
+superfluid neutrons – depends on the effective neutron mass and the proton fraction [271], which
+both also correlate with the symmetry energy [108]. A consistent extraction of both symmetry
+energy parameters and effective masses from heavy-ion collision data is therefore required.
+In contrast to efforts devoted to systematic, statistically meaningful inferences of the EOS in the
+cores of neutron stars, modeling the neutron star crust is still in its infancy: The first calculations
+of large ensembles of systematically parameterized crust models and their use in statistical analysis
+have only been carried out recently [272–276]. However, much more nuclear experimental data can
+be brought to directly bear on crust physics, and we have entered an era where we can access
+information about the crust with unprecedented fidelity. For example, we have now observed the
+same neutron-star crust as it first cooled, then became heated by accreted matter, and then cooled
+again [277–281]. We have followed a pulsar through a glitch – a sudden change in the rotation period
+of the pulsar – and glitch recovery with a resolution of a few seconds [282]. These observations
+have provided very strong evidence that the crust is solid, that there exist superfluid neutrons in
+the inner crust which can be decoupled from the nuclei in the crustal lattice, and that nuclear
+
+31
+reactions from accreted material sinking into the crust provide deep crustal heating [279, 283, 284].
+Additionally, models of the neutron star crust predict that, prior to the transition to homoge-
+neous matter, isolated nuclei in the crust fuse to form cylindrical, planar, and more exotic shapes,
+termed “nuclear pasta”, that can affect neutron-star observations [285–287]. This crust-core bound-
+ary region, often referred to as the mantle, is likely a complex fluid. Density functional theory and
+molecular dynamics calculations of these structures reveal a complex energy landscape with many
+coexisting shapes, and correspondingly complex mechanical and transport properties [288–294],
+which are strongly influenced by the EOS at around 0.5n0 through the pressure, proton fraction,
+and surface energy of the structures.
+These properties can also be studied in multifragmenta-
+tion reactions, which probe, among others, the competition between nuclear surface energy and
+Coulomb energy at sub-saturation density [295–297].
+Inhomogeneous matter in the crust of a neutron star, including the dripped neutrons expected in
+the inner crust, can be modeled using a variety of nuclear theory techniques. These usually involve
+calculations within a single, repeating unit (Wigner-Seitz cell) of matter, typically containing a sin-
+gle nucleus [298–300]. The compressible liquid drop model (CLDM) treats the nuclear matter inside
+and outside of nuclei as homogeneous and described by the bulk matter EOS, while the surface
+energy is specified by a separate function with additional parameters [288, 300–303]. The surface
+parameters and those that define the dimensions of the cell and nucleus are minimized to obtain
+the ground state. The Thomas-Fermi model employs the local density approximation, modeling
+matter with a specified form of the inhomogeneous nuclear matter density in the unit cell; here,
+the parameters of the density distribution are varied to obtain the ground state configuration [304].
+Microscopic approaches to describing inhomogeneous nuclear matter, in which individual neutrons
+and protons are the degrees of freedom, include quantum Hartree-Fock or Relativistic Mean Field
+models [305–310], and semi-classical molecular dynamics approaches [292, 311].
+There is a great need for nuclear physics input into models of the neutron star crust, which
+analyses of heavy-ion collision data can provide. For example, the thickness, mass and moment
+of inertia of the crust depend on the higher-order symmetry energy parameters L, Ksym, and
+Qsym [272, 274, 312]. Thus measurements of the symmetry energy parameters up to third order
+in heavy-ion collision experiments are essential to understand the properties of the crust. The
+symmetry energy, effective masses, and surface energies of nuclear clusters strongly affect the
+proton fraction on either side of the crust-core transition density, the extent of nuclear pasta near
+the crust-core boundary, the mechanical and transport properties, the thermal conductivity and
+specific heat, the electrical conductivity, and the shear modulus of the crust [298, 304, 309, 313].
+Nuclear experiment can thus constrain neutron star crust models, and astrophysical observables
+associated with the crust can measure nuclear observables as well as measurements of neutron star
+bulk properties. For example, the symmetry energy can be constrained by combining nuclear data
+with crust and core observables, e.g., through a potential multi-messenger measurement of the
+resonant frequency of crust-core interface oscillations [276].
+2.
+Challenges and opportunities
+The next decade will provide a wealth of new data on neutron stars, as the LIGO-VIRGO-
+KAGRA detectors are expected to observe many new binary neutron-star mergers, some of them
+with electromagnetic counterparts [314–316]. As NICER continues to measure more neutron star
+masses and radii, next-generation X-ray timing missions such as Strobe-X [317] and radio tele-
+scopes such as the Square-Kilometer Array will increase the number of pulsars we see and are able
+to measure by an order of magnitude. Long-timescale observations of individual pulsars (using
+radio timing) and persistent gravitational waves from deformations of neutron stars will lead to
+
+32
+measurements of their moments of inertia. These new data points might enable us to pin down
+the nuclear matter EOS, to discover or rule out the existence of phase transitions to exotic forms
+of matter in the cores of neutron stars, and to reliably constrain microscopic interactions between
+fundamental particles.
+Although model-agnostic extrapolations to higher densities such as through the use of poly-
+tropes [194, 240], speed of sound schemes [188, 244, 318], Gaussian processes [249, 250] and spectral
+methods [247], combined with robust data analysis, will eventually allow us to pin down the dense-
+matter EOS, they cannot answer the question about the relevant microscopic degrees of freedom at
+high densities. Hence, it is crucial to develop improved microscopic models with well-quantified un-
+certainties in this regime. At the same time, creating ensembles of outer core and crust models that
+allow for inclusion of astrophysical and nuclear data requires underlying nuclear models to have
+enough freedom to explore a large region of parameter space, and allow fast computation of relevant
+quantities that also capture the essential physics. Currently, it is energy density functionals like
+Skyrme, Gogny, and Relativistic Mean Field models that provide these properties. Consequently,
+progress could be made by making a stronger connection between these models and microscopic
+approaches, e.g., connecting energy-density functionals to ab initio calculations allowing a more
+direct link to χEFT [299, 319, 320]. In the same spirit, EFT calculations of the EOS can be used as
+a “low-density limit” to calibrate higher-density models for neutron stars and heavy-ion collisions.
+The crust can be modeled consistently with nucleonic matter in the core using density functional
+theory to model both. When choosing a model, a compromise must be made between accurate
+modeling of microscopic quantum effects, such as shell effects in the nucleus and surrounding
+neutron gas, and the computational expediency required to construct large ensembles of crust
+models needed for statistical inference. For example, quantum shell effects strongly determine the
+evolution of the mass and charge number of nuclei with density, alter the effective mass of dripped
+neutrons, and drive the complex energy landscape of nuclear pasta. Fully microscopic quantum
+calculations include shell effects self-consistently, but are computationally expensive. The CLDM
+approach can be used to construct large numbers of crust models, but requires shell effects to be
+added by hand. Future work needs to develop schemes of incorporating such microscopic effects in
+large ensembles of crust models. The method that may allow that is the Extended Thomas-Fermi
+method, incorporating shell effects through the Strutinsky Integral: see, e.g., [321].
+Models should also incorporate nuclear pasta, as its extended structures may contribute to the
+mechanical and thermal properties of matter at the crust-core boundary. It is computationally
+demanding to model transport and mechanical properties of the crust microscopically or in simula-
+tions [322], particularly in the nuclear pasta phases, and it is unrealistic to include these quantities
+in large ensembles of crust models. Simpler schemes that extrapolate the mechanical and trans-
+port properties across the parameters space based on microscopic models could be developed. Also,
+representative crust models inferred from data can be used to calculate these crust properties.
+There is also a need for a balance between accuracy and precision. A model can be accurate
+but not precise (predicting the correct value of a physical quantity but having large error bars),
+or precise but not accurate predicting very small error bars, but not predicting the correct value
+of some physical observable).
+Individual crust models can be created from mass models that
+are precisely fit to data and which predict precise values for, for example, the symmetry energy
+parameters. However, to make accurate inferences of nuclear matter parameters from astrophysical
+observables, and to include their experimentally measured ranges, ensembles of models spanning
+the parameter space should be employed.
+Both strategies are important, and the precision-fit
+models can act as benchmarks against which we assess the outcomes of statistical inferences.
+When older neutron stars accrete matter in the crust the matter gets gradually pushed down
+into the core and replaced by the accreted matter. The temperatures in the crust are well below
+the nuclear potential energies, so the replacement crust cannot easily attain nuclear statistical
+
+33
+equilibrium. Ensembles of accreted crust models are yet to be constructed, but are necessary to
+correctly account for deep crustal heating and therefore to fully utilize the observations of cooling
+of accreted crusts in low mass X-ray binaries.
+In all this work, effort must be made to calculate the different observables consistently as well
+as to combine different data sets in a well-controlled way. This is expanded upon in Section IV.
+III.
+HEAVY-ION COLLISION EXPERIMENTS
+Establishing the equation of state (EOS) of nuclear matter has been a major focus of heavy-ion
+collision experiments. While very low energy collisions can probe nuclear matter at densities smaller
+than the saturation density n0, highly-compressed nuclear matter is produced in the laboratory
+by colliding heavy nuclei at relativistic velocities. At even higher energies, in the ultra-relativistic
+regime, quarks in the colliding nuclei become almost transparent to each other and therefore escape
+the collision region, which means that matter measured at midrapidity is characterized by a nearly-
+zero net baryon number. Heavy-ion collision experiments at top beam energies at the Relativistic
+Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) provided convincing evidence that
+at high temperatures and near-zero baryon density, nuclear matter becomes a quark-gluon plasma
+(QGP) [323–329], a deconfined but strongly-interacting state composed of color charges, confirming
+Lattice QCD (LQCD) calculations of the EOS at zero density [330–332].
+While the region of the QCD phase diagram explored in ultra-relativistic heavy-ion collisions
+is relatively well understood, the EOS of dense nuclear matter at moderate-to-high temperatures
+and moderate-to-high baryon densities is not known well due to the break-down of first-principle
+approaches in this regime. Answering pressing questions about the QCD EOS in this region, such as
+whether the quark-hadron transition becomes of first-order at high densities or what is the minimal
+energy required to produce the QGP, is the driving force behind Phase II of the Beam Energy Scan
+(BES) program at RHIC, the HADES experiment at GSI, and the future Compressed Baryonic
+Matter (CBM) experiment at the Facility for Antiproton and Ion Research (FAIR), Germany.
+This renewed interest in the nuclear matter EOS at high densities, accessible in heavy-ion
+collisions at intermediate energies, coincides with an increased effort to constrain the EOS of
+neutron-rich matter, probed in studies of neutron stars and neutron star mergers (see Section II C
+as well as recent white papers on QCD Phase Structure and Interactions at High Baryon Den-
+sity: Continuation of BES Physics Program with CBM at FAIR [151] and Dense matter theory for
+heavy-ion collisions and neutron stars [159]). Furthermore, studies show that heavy-ion collisions
+in this regime and neutron star mergers probe similar temperatures and baryon densities [333, 334].
+However, while matter created in collisions of heavy-ions has comparable numbers of protons and
+neutrons, matter inside neutron stars is neutron-rich. Establishing the much needed connection
+between the studies of the nuclear EOS as probed in heavy-ion collisions and as inferred from
+neutron star observations is possible by leveraging the experimental capabilities of the newly com-
+missioned Facility for Rare Ion Beams (FRIB), where energetic beams of proton- and neutron-rich
+nuclei can be produced. Heavy-ion collision experiments at FRIB can put tight constraints on the
+dependence of the nuclear matter EOS on the relative proton and neutron abundances [22], and
+thus enable a description of both dense nuclear and dense neutron-rich matter within a unified
+framework.
+Indeed, if we assume that the core of a neutron star is composed of mostly uniform nucleonic
+matter, then nuclear matter and neutron stars should be described by a common EOS, specifying
+the relationship between the pressure and the temperature, density, and isospin content.
+The
+theoretical construct of symmetric nuclear matter consisting of equal amounts of neutrons and
+
+34
+Number density 
+Astro
+HIC(asym)
+Nuclei properties
+Theory
+Crust
+HIC(SNM)
+FIG. 12. This schematic plot illustrates the approximate density
+ranges that are explored in the studies of chiral effective field
+theory, nuclei properties, heavy-ion collision experiments, and
+observations of neutron stars and their crusts in astronomy.
+protons has been successful to derive
+properties of symmetric matter such
+as the saturation density and bind-
+ing energy, however, an additional
+term in the EOS is needed to de-
+scribe nuclear matter with unequal
+neutron-proton composition.
+This
+second term depends on the asymme-
+try δ, defined as δ = (nn − np)/nB,
+where nn, np, and nB are the neu-
+tron, proton, and total baryon densi-
+ties, respectively. Consequently, one
+can view the asymmetry as the neu-
+tron excess fraction. Mathematically,
+the energy per nucleon can be then
+expressed as a sum of two terms:
+ϵ(nn, np) = ϵSNM(n) + S(n)δ2. Here, the first term represents the energy per nucleon of symmetric
+nuclear matter, while the second term accounts for the correction needed when δ ̸= 0. Therefore,
+δ is a crucial parameter that distinguishes neutron stars (with δ >∼ 0.8) from most nuclei (with
+δ <∼ 0.25). Given the relatively small values of the asymmetry δ for nuclei, in heavy-ion collision
+experiments it is easier to constrain the coefficients of the EOS of symmetric matter, ϵSNM(nB). In
+contrast, the energy contribution from the asymmetric term, also known as the symmetry energy,
+constitutes a small fraction of the total energy of a nucleus even for neutron-rich heavy radioactive
+isotopes (< 5% in the liquid drop model), and its determination requires precise measurements.
+Furthermore, because the isospin effects in any observable tend to diminish with temperature, it
+may be difficult to measure the symmetry energy at very high densities, which require high-energy
+heavy-ion reactions. Therefore, symmetry energy is best probed in heavy-ion collisions of highly
+asymmetric isotopes at low to intermediate energies.
+Fig. 12 shows schematically the baryon density regions explored by different areas in nuclear
+physics studies. Recent breakthroughs in astronomical observations with state-of-the-art instru-
+ments led to the first detection of a binary neutron-star merger and the unprecedented radii mea-
+surements of neutron stars with accurately known masses (see Section II C). The neutron star
+mass-radius relationship provides an insight into the EOS at high densities above twice saturation
+density (>∼ 2n0), as represented by the red arrow (labelled “Astro”) in the upper right corner. Labo-
+ratory experiments, especially those using heavy-ion collisions, are essential to provide information
+on the dependence of the EOS on density and the asymmetry (see also Section II A). High-energy
+heavy-ion collisions can provide insight into the symmetric nuclear matter EOS as represented
+by the gold right-pointing arrow (labeled “HIC(SNM)”), while current probes of the symmetry
+energy are more suited for measurements of lower energy heavy-ion reactions (<∼ 600 AMeV) as
+represented by the left-pointing gold arrow (labeled “HIC(asym)”). Many properties of nuclei,
+such as masses and radii, have been shown to be mainly sensitive to densities around (2/3)n0,
+however, with a careful selection of nuclear observables, the symmetry energy has been probed
+over densities of 0.3n0 < nB < n0 using Pearson correlation methods [12, 335] (green left-pointing
+arrow). Recent advances in chiral effective field theory (see Section II B) enabled extrapolations
+of the EOS to be extended up to ≈ 1.5n0 [13], but the uncertainty increases exponentially with
+density for densities that are higher than n0. It is not clear what is the maximum density up to
+which such extrapolations can succeed. Finally, one of the most interesting regions is at very low
+densities (<∼ 0.5n0), corresponding to the crust of a neutron star where matter is not uniform (see
+Section II C). There, matter changes with increasing density from a Coulomb-dominated lattice to
+
+L
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.035
+nuclear pasta and, ultimately, to uniform matter. The density and nature of these transformations
+are again dictated largely by the EOS.
+Measurements made in heavy-ion collisions at intermediate energies, probing high densities
+or, equivalently, small nucleon separations, will yield key insights into the nature of the nuclear
+force, including the density-dependence of the nuclear symmetry energy. Experimental efforts to
+determine the EOS for symmetric matter and the symmetry energy are described in Section III A
+and III B, respectively.
+Please note that all beam energies Elab quoted in this section are the single-beam kinetic energies
+per nucleon, in units of AMeV or AGeV. (Alternatively, Elab is also sometimes denoted by other
+authors as E/A, with units of MeV or GeV). Additionally, while many results are reported in
+terms of their constraints on the incompressibility K0, one should refrain from interpreting them
+as constraining the behavior of the EOS around the saturation density (see Section II A 2 for more
+details).
+A.
+Experiments to extract the EOS of symmetric nuclear matter
+Heavy-ion collision experiments worldwide have extensively studied the EOS of symmetric
+nuclear matter at supra-saturation densities over the past four decades. Experiments based at
+the Schwerionensynchrotron-18 (SIS-18) ring accelerator at the GSI Helmholtz Centre for Heavy
+Ion Research (GSI) have probed Au+Au collisions at energies between Elab = 0.09–1.5 AGeV
+(√sNN = 1.92–2.52 GeV), corresponding to fireball densities 1–2.5n0. Further experimental efforts
+with Au+Au collisions were carried out at higher energies, Elab = 2–10 AGeV (√sNN = 2.70–
+4.72 GeV), at the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory
+(BNL) to probe fireball densities 2.5–5n0. Complementing the densities reached at AGS-BNL is the
+Beam Energy Scan (BES) program of the Solenoidal Tracker at RHIC (STAR) experiment at RHIC
+in BNL, where high-statistics Au+Au collisions were performed at energies between Elab = 2.9–
+30.0 AGeV (√sNN = 3–7.7 GeV) in the fixed-target mode. A selection of constraints on the EOS
+extracted from the above experiments is shown in Fig. 9. Below, we describe the observables stud-
+ied to extract the symmetric nuclear matter EOS, experiments probing the aforementioned density
+ranges, and inferences for the hadronic transport codes.
+1.
+Measurements sensitive to the EOS
+Collisions of heavy nuclei at relativistic energies lead to a rapid compression and heating of
+matter trapped in the collision region, followed by its dynamic expansion and cooling (see Fig. 7).
+The EOS governs both the compression as well as the expansion of the hot and dense nuclear matter,
+which in turn affect measured particle distributions. For example, a stiffer EOS (characterizing
+matter that is more incompressible) leads to a relatively smaller compression and, consequently,
+smaller heating, but a faster transverse expansion. The smaller temperatures reached in the fireball
+lead to smaller thermal dilepton and photon yields (see, e.g., [336–338]), while the faster expansion
+manifests itself in relatively higher mean transverse momenta (see, e.g., [24]) and a shorter lifetime
+of the fireball, the latter of which can be probed by a combination of the femtoscopic radii, R2
+out −
+R2
+side, shown to be proportional to the duration of particle emission [339, 340].
+The EOS also plays a large role in the interplay between the initial geometry of the system,
+the expansion of matter originating from nucleons trapped in the collision zone (participants),
+and the propagation of nucleons which are either still incoming into the collision region or whose
+trajectories do not directly cross the collision region (spectators). In systems colliding at beam
+energies for which the speed of the fireball expansion is comparable with the speed of the spectators,
+
+36
+the resulting complex dynamical evolution affects the transverse expansion of the system and,
+therefore, the angular particle distributions in the transverse plane dN/dφ. In particular, moments
+of the angular momentum distribution, known as the collective flow coefficients and defined as
+vn =
+� dφ cos(nφ) (dN/dφ)/
+� dφ (dN/dφ), describe the collective motion of the system and are
+highly sensitive to the EOS, as shown in numerous hydrodynamic [77, 341–346] and hadronic
+transport [31, 67, 78, 79, 347, 348] models. At the same time, collective flow observables can be
+measured with high precision, making them primary observables used to constrain the EOS.
+In off-central collisions, the initial collision zone has an approximately elliptical shape, and the
+pressure gradients within the collision zone are larger along its short axis. If the spectator nucleons
+move out of the way before the fireball expands, the pressure gradients in the collision zone lead to
+particle distributions around midrapidity which have maxima coincident with the reaction plane
+(“in-plane” emission). If, however, the spectators stand in the way of the fireball expansion, this
+leads to a preferential emission along the long axis of the collision zone (“out-of-plane” emission,
+also referred to as “squeeze-out” due to the role that the spectators play in the expansion). The
+preferential emission in either in-plane or out-of-plane direction is described by the second Fourier
+coefficient of flow v2, also known as the elliptic flow, which is positive in the former case and
+negative in the latter case (see the lower panel of Fig. 4). The magnitude of the elliptic flow,
+as well as the energy at which v2 changes sign, are intrinsically connected to the stiffness of the
+EOS: for example, a stiffer EOS results in both a faster expansion and a more forceful blocking by
+spectators, which leads to a larger squeeze-out and a more negative v2.
+The rapidity-dependence of the first Fourier coefficient of flow, the directed flow v1, is also
+sensitive to the EOS as it measures the degree of spectator deflection due to the interaction with
+the collision zone [349]. In the center-of-mass frame, the spectators from a nucleus moving in the
+positive beam direction will be deflected to one side, while the spectators from the other nucleus,
+moving in the negative beam direction, will be deflected to the opposite side, resulting in a positive
+v1 at positive rapidities and a negative v2 at negative rapidities (here, the sign of v1 is a matter of
+convention; see [58] for a more detailed explanation). The magnitude of the directed flow in each
+region and, therefore, its slope at midrapidity are directly related to the EOS: for example, a softer
+EOS leads to a smaller deflection and a smaller slope of v1 at midrapidity, where in particular a
+sufficiently soft EOS can even lead to a negative slope of v1 [343, 350]. We note that spectators
+are necessary to obtain substantial magnitudes of the slope of the directed flow, as can be seen by
+its small values at high collision energies (see the upper panel of Fig. 4).
+Beyond the collective flow phenomena, the EOS also has an effect on hadron production. In
+particular, much attention has been given to production of hadrons in heavy-ion collision at energies
+below the nominal production threshold in NN reactions (“sub-threshold” production), which
+requires multiple sequential hadron-hadron collisions to occur. The probability of these collisions
+is significantly higher in the high-density regions, and consequently the yield of sub-threshold
+probes is expected to be substantially enhanced if higher densities are reached in the collision.
+Of particular importance for the EOS studies is sub-threshold production of K+ mesons, which
+undergo few final-state interactions with the nuclear medium and therefore mostly leave the fireball
+unperturbed, making them a sensitive probe of the highest densities reached and, consequently, of
+the nuclear EOS [87].
+2.
+Experiments probing densities between 1–2.5n0
+As described above, sub-threshold particle yields can be used as probes of the EOS. In partic-
+ular, due to their low in-medium cross-section, K+ mesons produced at energies lower than the
+production threshold of Elab = 1.58 GeV (√sNN = 2.55 GeV) can carry unperturbed informa-
+
+37
+0.8
+1.0
+1.2
+1.4
+1.6
+Elab [GeV]
+1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+(MK+/A)Au+Au / (MK+/A)C+C
+0.8
+1.0
+1.2
+1.4
+1.6
+Elab [GeV]
+1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+(MK+/A)Au+Au / (MK+/A)C+C
+soft EOS, pot ChPT
+hard EOS, pot ChPT
+soft EOS, IQMD, pot RMF 
+hard EOS, IQMD, pot RMF
+KaoS
+soft EOS, IQMD, Giessen cs
+hard EOS, IQMD, Giessen cs 
+❑HM
+▲ SM
+● FOPI
+Au+Au
+protons
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+ 0.25
+beam energy (A GeV)
+<b0<0.45 ut0>0.4
+v2n(0.8)
+FIG. 13. Left panel: Beam energy dependence of K+ yield ratios in inclusive Au+Au collisions and C+C
+collisions between Elab = 0.8–1.5 AGeV (√sNN = 2.24–2.52 GeV). A comparison of RQMD [84] and IQMD [354]
+model calculations indicates a soft EOS (K0 = 200 MeV, red symbols) instead of a hard EOS (K0 =
+380 MeV, blue symbols) when compared to KaoS data [353] (black symbols). Figure from [351]. Right
+panel: Beam-energy dependence of the elliptic flow for protons in Au+Au collisions at Elab = 0.4 AGeV
+(√sNN = 2.07 GeV) (black symbols) as measured by the FOPI experiment [88].
+Comparison to IQMD
+transport calculations with momentum dependence prefers a soft EOS (blue triangles) over a hard EOS (red
+squares), yielding K0 = 190 ± 30 MeV. Figure from [67].
+tion on the fireball density and the stiffness of the EOS [351]. The Kaon Spectrometer (KaoS)
+Experiment [352] at SIS18 in GSI studied the subthreshold production of K+ mesons at beam
+energies between Elab = 0.6–2.0 AGeV (√sNN = 2.16–2.70 GeV), and established it as a sensitive
+probe to the underlying EOS of the hot and dense nuclear matter. To reduce the experimental
+and model uncertainties, the production of K+ mesons in a heavier Au+Au system was compared
+with the production in a lighter C+C system [353]. Analyzing the experimental results together
+with transport model calculations in the RQMD [84] and IQMD [354] model enabled extraction of the
+EOS of symmetric nuclear matter characterized by an incompressibility of K0 = 200 MeV (see also
+Fig. 9). Both models included effects due to the momentum-dependence of the EOS by including
+K+/−N potentials, i.e., a repulsive mean field for K+ and an attractive mean field for K−, which
+are required to reproduce the K+ and K− emission pattern [355] (see Fig. 13).
+Collective behavior in heavy-ion collisions is likewise a very sensitive probe of the underlying
+EOS and has been extensively studied since its discovery by the Plastic Ball spectrometer at the
+Bevalac in Lawrence Berkeley National Laboratory [356, 357]. In particular, the elliptic flow v2 is
+highly sensitive to both the initial geometry of the collisions and pressure gradients experienced
+throughout the evolution of the created systems [77, 358]. The Four Pi (FOPI) Experiment at SIS18
+in GSI carried out extensive measurements of the beam energy dependence of the elliptic flow of
+protons and light fragments (such as deuterons, tritons, and 3He) over the entire range of SIS18
+energies, Elab = 0.09–1.5 AGeV (√sNN = 1.92–2.52 GeV) [88, 359]. The nucler EOS extracted from
+a comparison to IQMD simulations [67] is characterized by an incompressibility K0 = 190 ± 30 MeV
+when momentum-dependent interactions are taken into consideration. This constraint is consistent
+with the KaoS incompressibility inferences and suggests a soft EOS for symmetric nuclear matter
+at 1-2.5n0 (see Fig. 13 and also Fig. 9).
+
+38
+K+	SMASH
+K-	SMASH
+K+	UrQMD
+K+	JAM
+v2
+−0.08
+−0.04
+0
+0.04
+y	-	ycm
+−1
+−0.5
+0
+0.4	<	pT	<	1.6	GeV/c
+K+
+K-
+−0.4
+−0.2
+0
+π+	SMASH
+π-	SMASH
+π+	UrQMD
+π+	JAM
+v2
+−0.08
+−0.04
+0
+0.04
+y	-	ycm
+−1
+−0.5
+0
+0.2	<	pT	<	1.6	GeV/c
+π+
+π-
+−0.4
+−0.2
+0
+p	SMASH
+p	UrQMD
+p	JAM
+Λ	SMASH
+Λ	UrQMD
+Λ	JAM
+v2
+−0.08
+−0.04
+0
+0.04
+y	-	ycm
+−1
+−0.5
+0
+0.4	<	pT	<	2.0	GeV/c
+√sNN	=	3	GeV	10-40%
+Au+Au	collisions
+p
+Λ
+v1
+−0.4
+−0.2
+0
+FIG. 14. Directed (v1, top) and elliptic (v2, bottom) flow of protons and lambda baryons (left panels), pions
+(middle panels), and kaons (right panels) as a function of rapidity. Measurements from STAR [94] (symbols)
+were performed with Au+Au collisions at √sNN = 3 GeV (Elab = 2.91 AGeV) and 10–40% centrality.
+Results from UrQMD (blue bands), JAM (green bands), and SMASH (orange bands) hadronic transport models
+were obtained using a relatively hard EOS at moderate densities (characterized by K0 = 300 in SMASH
+and K0 = 380 MeV in JAM and UrQMD), with the EOS used in SMASH becoming significantly softer at high
+densities (see [58, 94] for more details).
+3.
+Experiments probing densities above 2.5n0
+Pioneering proton directed and elliptic flow measurements were performed in Au+Au colli-
+sions for beam energies Elab = 2–10 AGeV (√sNN = 2.70–4.72 GeV) by the E895 [83, 90] and
+E877 [82] experiments at AGS-BNL. Notably, it was observed that around Elab ≈ 4 AGeV
+(√sNN ≈ 3.32 GeV), the proton v2 changes from a preferential out-of-plane emission, reflecting
+a complex interplay between the spectators, the expanding collision zone, and the EOS, to an in-
+plane emission (see the lower panel of Fig. 4). The experimental results were used in a comparison
+with the pBUU transport model to extract the EOS for densities between 2–5n0, which constrained
+the EOS to those described by values of the nuclear incompressibility between K0 = 210–300 MeV,
+ruling out extremely soft and extremely hard EOSs [31] (see Fig. 9). This rather broad constraint
+on K0 reflects the fact that the experimental results for the collective flow could not be reproduced
+with one EOS.
+The STAR Experiment at RHIC-BNL with its Beam Energy Scan (BES) program [360, 361]
+performed Au+Au collisions for √sNN = 3–200 GeV.
+In terms of the freeze-out temperature
+and chemical potential, (Tfo, µfo), this allowed STAR to comprehensively scan the QCD phase
+diagram from (80, 760) MeV to (166, 25) MeV, respectively. Probing the phase diagram at high
+densities was possible at RHIC in part due to STAR’s capability to shift from a standard collider
+to a fixed-target (FXT) mode, which was used to scan through the lower energies √sNN = 3–
+13.7 GeV (Elab = 2.91–99.06 AGeV), thereby establishing a substantial overlap with the previously
+
+39
+discussed AGS experiments [362]. Recently, STAR measured collective flow (v1, v2) in collisions at
+√sNN = 3.0 GeV [94] and √sNN = 4.5 GeV [93]. A comparison of results from the √sNN = 3.0 GeV
+data (see Fig. 14) with UrQMD and JAM simulations indicates a relatively hard EOS (characterized by
+K0 = 380 MeV) [94]; similarly, a recent Bayesian analysis of the STAR flow data based on a flexible
+parametrization of the EOS used in the SMASH transport code results in a relatively hard EOS at
+moderate densities (characterized by K0 = 300 ± 60 MeV) with a substantial softening at higher
+densities [58]. However, both UrQMD and SMASH do not currently include momentum-dependent
+interactions, which are crucial for a correct description of the transverse-momentum-dependence
+of the elliptic flow [31]. Moreover, while the above models reproduce the proton v1, v2 well, none
+of the models can simultaneously describe the flow of Lambda baryons and mesons (see Fig. 14).
+4.
+Challenges and opportunities
+Experiments probing densities between 1–2.5n0
+The High Acceptance Di-Electron Spectrometer (HADES) Experiment [363] at SIS-18 in GSI
+has performed collective flow measurements in Au+Au collisions at Elab = 1.23 AGeV (√sNN =
+2.42 GeV). The high acceptance and high statistics of HADES measurements allow one to per-
+form multi-differential studies of flow harmonics, ranging from v1 up to v6, which in turn enables
+reconstruction of a full 3D-picture of the emission pattern in the momentum space [18, 148] (see
+Fig. 15). In addition to the collective flow measurement capabilities, HADES can also precisely
+measure the dielectron excess yield, which was used to extract the fireball temperature, finding
+it to be 71.8 ± 2.1 MeV/kB [364]. These precise measurements of the fireball temperature and
+the underlying dielectron spectra allow HADES to investigate the presence of a first-order phase
+transition at SIS-18 energies and look for signs of a potential change of degrees of freedom [365].
+During its 2024 beam campaign, HADES will be in a unique position to measure the fireball
+caloric curve and the beam energy dependence of the collective flow from Au+Au collisions at
+Elab = 0.4–0.8 AGeV (√sNN = 2.07–2.24 GeV). Furthermore, there are ongoing efforts to establish
+systematic consistency between results from FOPI and HADES, including understanding various
+1
+−
+0.5
+−
+0
+0.5
+1
+cm
+y
+0.4
+−
+0.3
+−
+0.2
+−
+0.1
+−
+0
+0.1
+0.2
+2
+v
+Protons
+Centrality 20-30%
+)
+c
+(GeV/
+tp
+0.35 - 0.40
+0.55 - 0.60
+0.75 - 0.80
+0.95 - 1.00
+1.15 - 1.20
+1.35 - 1.40
+1.55 - 1.60
+1.75 - 1.80
+1.95 - 2.00
+= 2.4 GeV
+NN
+s
+Au+Au 
+HADES
+Protons
+1.0 < p
+t < 1.5 GeV/c
+Centrality 20-30%
+ = 2.4 GeV
+NN
+s
+HADES
+Au+Au
+ 0.0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ycm
+0.5
+1.0
+1.5
+2.0
+ = 0
+φ
+π
+2
+1
+π
+π
+3
+2
+  0.0
+  0.2
+- 0.2
+- 0.4
+- 0.6
+  0.4
+  0.6
+  0.8
+ = 0
+φ
+π
+2
+1
+π
+π
+2
+3
+ycm
+FIG. 15. Left: Rapidity-dependence of proton elliptic flow (v2) in semi-central Au+Au collisions at Elab =
+1.23 AGeV (√sNN = 2.42 GeV) for various pT bins (see legend) as measured by the HADES experiment.
+Figure from [18].
+Right: 3D-picture of the proton angular emission pattern in momentum space (flow
+coefficients from v1 to v6) for HADES data in semi-central Au+Au collisions at Elab = 1.23 AGeV (√sNN =
+2.42 GeV). Figure from [148].
+
+40
+detector-related effects.
+This is highlighted by the observed discrepancy in pion multiplicities
+between FOPI and HADES, which could be partially explained by different methods used by the
+respective experiments to estimate the number of participant nucleons [366].
+The abundance of available and future data presents an opportunity to benchmark transport
+model simulations with measurements from KaoS, FOPI, and HADES experiments by enabling
+systematic studies of the symmetric nuclear matter EOS. A recent comparison between FOPI
+measurements and dcQMD transport code [86], using the transverse rapidity [367] and flow spec-
+tra [88] of protons and light clusters at Elab = 0.15–0.80 AGeV (√sNN = 1.95–2.24 GeV), has
+further tightened the constraints on the nuclear EOS at the probed densities to one characterized
+by an incompressibility K0 = 236 ± 6 MeV. The dcQMD analysis for the FOPI data is planned
+to be extended up to Elab = 1.5 AGeV (√sNN = 2.52 GeV), probing densities above 2n0, by
+taking into account an improved description of reaction dynamics through using more accurate
+approximations for 3-body terms in the interaction and considering multi-pion decay channels for
+the resonances [368].
+Moreover, perfect-fluid hydrodynamic calculations for binary-neutron-star mergers and heavy-
+ion collisions at SIS-18 energies show that comparable temperatures (T ≈ 50 MeV) and densities
+(nB ≈ 2n0) are reached in both systems [333, 334]. This has led to increasing efforts to use the
+existing constraints on the EOS of symmetric nuclear matter from KaoS and FOPI experiments
+in a multi-physics effort to constraint neutron star properties [17, 369, 370]. Such multi-physics
+constraints are discussed in detail in Section IV.
+Experiments probing densities above 2.5n0
+While collective flow can be used to deduce the geometry of the colliding system and its prop-
+erties in an indirect way (see Figs. 14 and 15), a more direct method – femtoscopy – can provide
+a direct handle on the space-time evolution of the fireball [371]. Here, the time of the particles’
+emission ∆τ is also a probe of the underlying EOS, with larger values of ∆τ corresponding to a
+softer EOS [338] (see Fig. 16). Access to this information is provided by measurements of femto-
+scopic radii Rlong, Rout, Rside [372], where the relation between Rout to Rside is strongly correlated
+with ∆τ. The sensitivity of pion emission to the EOS has already been studied [338, 373], however,
+experimental uncertainties are still too big to make precise comparisons with model calculations.
+Ongoing studies of proton femtoscopy at STAR are expected to bring new, substantial references
+for such investigations of the EOS.
+In addition to studying the EOS of dense nuclear matter, the STAR BES program also aims to
+search for a potential first-order phase transition from hadronic to partonic phase at higher baryonic
+densities. This search can provide an input on collision energies at which hadronic transport models
+should take into consideration new degrees of freedom. Among the explored observables, number-
+of-constituent-quark (NCQ) scaling was used as an important evidence of creation of QGP at the
+highest RHIC energy of √sNN = 200 GeV (Elab = 21, 300.0 AGeV) [374]. Recent results point to
+the breaking of the NCQ scaling in Au+Au collisions at √sNN = 3 GeV (Elab = 2.91 AGeV) [94].
+Other observables that can hint at the possible existence of a first-order phase transition include
+the thermodynamic susceptibilities of pressure, which are predicted to fluctuate in the vicinity of a
+critical point and manifest as a specific behavior of higher-order moments of conserved quantities
+(such as baryon number, strangeness, and electrical charge) with the beam energy [375, 376].
+STAR [377, 378] and HADES [379] have observed tentative non-monotonic behavior in the beam-
+energy-dependence of the fourth-order net-proton cumulant (a proxy for the net-baryon number
+cumulant) in Au+Au collisions at √sNN = 7.7–27.0 GeV (Elab = 2.91–400 AGeV).
+The experimental effort to uncover the symmetric nuclear matter EOS will be further strength-
+ened by the Compressed Baryonic Matter (CBM) experiment [49, 380] at the currently-under-
+construction Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany. The CBM
+
+41
+16
+18
+20
+22
+ 
+Freeze-out time 〈t〉 (fm/c)
+Au+Au 0%-10% π
+2
+3
+4
+5
+6
+7
+8
+9
+-
+ Cascade
+ Hard EoS
+ CMF EoS
+ CMF_PT2 EoS
+ CMF_PT3 EoS
+√s NN (GeV)
+2
+3
+4
+5
+6
+7
+8
+9
+-12
+-8
+-4
+0
+4
+8
+12
+16
+Au+Au 
+0%-10%
+|yππ|<0.35
+kT=(275±25) MeV/c
+HADES π-π-
+E895 π-π-
+E866 π-π-
+STAR π-π-
+STAR π-π-+π+π+
+RO
+2-RS
+2 (fm2)
+√sNN (GeV)
+FIG. 16. Left: Beam energy dependence of the π− freeze-out time as extracted from UrQMD simulations for
+different EOSs. The CMF PT2 and CMF PT3 EOS soften at low and high baryon density, respectively, by
+introducing a first-order phase transition, and the pure cascade mode can be considered as an extremely
+soft EOS. Figure from [338]. Right: Comparison of the beam energy dependence of R2
+out − R2
+side for π−
+extracted from experiments and UrQMD simulations with different EOSs. Figure from [338].
+experiment, which at the time of writing is expected to become operational in 2028-29, aims to
+use nucleus-nucleus collisions to precisely explore the QCD phase diagram with Au+Au collisions
+in the energy range of √sNN = 2.9–4.9 GeV (Elab = 2.49–11.1 AGeV). Other particle beams,
+such as Z = N species and protons can also be used at Elab = 15 AGeV (√sNN = 5.62 GeV)
+and Elab = 30 AGeV (√sNN = 7.73 GeV), respectively. This will be enabled by using primary
+heavy-ion beams from the Schwerionensynchrotron-100 (SIS-100) ring accelerator operating at an
+intensity of 109 ions/s [381]. CBM will operate at unprecedentedly high peak interaction rates of up
+to 10 MHz, which will be further complemented by a novel trigger-less data acquisition scheme and
+online event selection. This will allow CBM to perform systematic, multi-differential measurements
+of the dependence of observables on the beam energy and system size. The most promising observ-
+ables to explore are: (i) event-by-event fluctuations, (ii) thermal radiation (photons and dileptons),
+(iii) (multi-)strangeness, (iv) hypernuclei, and (v) charm production (recent physics performance
+results can be found in [382, 383]). Moreover, the HADES Experiment will be moved to the SIS-100
+beamline in the CBM experimental cave to complement the overarching CBM physics program in
+2031 [384]. The HADES detector, given its large polar angle acceptance (18◦ ≤ θ ≤ 85◦), will
+perform reference measurements for CBM at lower SIS-100 energies. This will be done with light
+collision systems, e.g., proton beams and heavy-ion beams with moderate particle multiplicities
+(such as Ni+Ni or Ag+Ag collisions) [380]. Altogether, CBM represents an opportunity to link
+the physics programs at SIS-18 and RHIC, thereby leading to a continuation of the Beam Energy
+Scan program (see also the white paper on QCD Phase Structure and Interactions at High Baryon
+Density: Continuation of BES Physics Program with CBM at FAIR [151]).
+Overall, STAR-FXT and CBM-FAIR are capable of performing high-statistics multi-differential
+measurements of the relevant EOS observables. However, a successful inference of the EOS depends
+on comparisons to transport simulations. Although many transport codes are available for describ-
+ing heavy-ion collisions in different energy ranges and extracting the underlying EOS (see [47] for
+a review), currently none of the available codes can reproduce all proposed experimental observ-
+ables (see, e.g., Fig. 14). A meaningful description of experimental data in the STAR-FXT and
+
+O
+米O
+米42
+CBM-FAIR range will require transport codes to incorporate physics allowing reproducing all of
+the above-mentioned key measurements and more, see Section II A.
+B.
+Experiments to extract the symmetry energy
+The energy contribution from the isospin dependence term, also known as the symmetry energy,
+is a small fraction of the total energy of a nucleus even for neutron-rich heavy radioactive isotopes
+(< 5% in the liquid drop model). However, due to the large isospin asymmetry in neutron stars,
+the density dependence of the symmetry energy is very important, determining many neutron star
+properties, including their size and the cooling pathways via neutrino emission. While experimental
+inferences of the symmetry energy pose significant challenges, researchers have developed methods
+to elucidate the relatively small effects that the asymmetry has on isospin-dependent observables,
+e.g., by measuring ratios of neutron and proton observables or charged pion observables. Exper-
+imental as well as theoretical systematic errors are further minimized by taking double ratios of
+the same observable using two reactions that differ mainly in the neutron/proton content, as in
+the measurement of isoscaling. To reach the widest range of asymmetry between reactions, intense
+radioactive beams are necessary. Large experiments designed to measure symmetry energy can re-
+quire large collaborations. However, small-scale experiments can likewise have an impact on some
+of the outstanding problems. Consequently, many groups contribute to the diverse experimental
+results.
+1.
+Experiments that probe low densities
+At beam energies below Elab = 100 AMeV (√sNN = 1.93 GeV), the colliding nuclei overlap
+briefly and then expand, with most of the detected particles being emitted during the expansion
+stage. The rates of emission of neutrons and protons during the expansion are influenced by the
+symmetry energy. Some nucleons emerge within fragments or clusters that are formed and emitted
+throughout the reactions. Nearly all theory studies require the symmetry energy to be zero at zero
+density. However, before matter reaches zero density, at low densities of (0.002 ≤ n/n0 ≤ 0.02)
+many nucleons combine into clusters and preserve the information about the symmetry energy at
+those low densities. In the estimation of Wada et al. [385, 386], clustering has a significant impact
+on the symmetry energy below 0.03 nucleons/fm3, see Fig. 17 (we note here that this conclusion
+depends on the definition of the symmetry energy). Overall, the presence of clusters changes the
+characterization of the symmetry energy. Nonetheless, low-density clusterization is an important
+ingredient in supernova matter and for the EOS in the neutrino sphere. It is also relevant to the
+nature of proto-neutron star matter as it cools and the crust crystallizes [387].
+2.
+Measurements to extract symmetry energy up to 1.5n0
+In the past decade, many studies have been conducted to extract the symmetry energy and
+symmetry pressure [70], mostly at low densities. Since the nuclear EOS should give a good de-
+scription of the properties of the nuclei, including the masses or binding energies, the large nuclear
+mass data base provides a great resource to determine the symmetry energy at about (2/3)n0 from
+(1) masses of double magic nuclei using Skyrme density functional [72], (2) nuclear masses using
+density functional theory [71], and (3) the energies of isobaric analogue states [12].
+By nature, a heavy nucleus has excess neutrons which are needed to overcome the Coulomb
+repulsion from the protons inside the nucleus. The symmetry-energy forces the excess neutrons
+
+43
+FIG. 17. The symmetry energy of clustered matter at very
+low densities. Figure from [385].
+to the surface.
+This surface layer of ex-
+cess neutrons is referred to as a neutron
+skin.
+Thickness of this neutron skin re-
+flects the symmetry pressure, or equiv-
+alently the slope of the symmetry en-
+ergy at the saturation density. The long-
+awaited measurements of the neutron skin
+of the 208Pb nucleus inferred from par-
+ity violation in electron scattering were
+recently published by the PREX collab-
+oration [74, 388].
+The measured value
+of 0.283 ± 0.071 fm, corresponding to the
+symmetry pressure of 2.38±0.75 MeV/fm3
+at (2/3)n0, is rather large and disagrees
+with most theoretical predictions. Subse-
+quently, the PREX/CREX collaboration
+measured the neutron skin of 48Ca to be
+0.121 ± 0.026 (exp) ± 0.024 (model) fm
+[389], which is much thinner than the
+208Pb skin.
+However, the 48Ca value is
+much closer to the theoretical predictions.
+The discrepancies between the two results and the expectations from models have not been re-
+solved, and the CREX collaboration has not released official values for the slope of the symmetry
+energy or symmetry pressure, even though there have been many attempts by outside groups to
+resolve the apparent discrepancies between the two skin measurements [390–392].
+In the last few years, an alternative way to measure skin thickness has been proposed [393].
+In the limit of an exact charge symmetry, the proton radius of a given nucleus is identical to the
+neutron radius of its mirror partner. Thus the neutron skin for a given nucleus may be determined
+from the difference in proton radii measured in these mirror pairs. In reality, there are relativistic
+and finite-size corrections, as well as corrections from the Coulomb force which breaks the isospin
+symmetry. In principle, these corrections can be calculated within the energy density functional
+theory. While larger neutron skins are expected in heavier nuclei due to the larger neutron excess,
+making them a better probe, proton-rich mirrors of heavy nuclei is typically far beyond the limits
+of existence. Thus this technique is limited to species of relatively low mass and isospin. Even with
+the use of high-intensity isotope beams near the proton driplines, it is still a challenge to do such
+experiments. The most recent result with this technique is from the 54Ni-Fe mirror pair [394].
+Complementary to structure experiments, heavy-ion collisions have probed the symmetry en-
+ergy and pressure over a wide density range.
+At incident energies below Elab = 100 AMeV
+(√sNN ≤ 1.93 GeV), low densities (estimated to be around (1/3)n0) are reached when matter
+expands after the initial impact and compression of the projectile and target.
+Therefore, the
+corresponding experimental observables primarily reflect the symmetry energy at sub-saturation
+densities [6, 98]. The transport of neutrons and protons allows systems with isospin gradients to
+equilibrate, where the degree of equilibration depends on the strength of the potential experienced
+by the nucleons and the duration of transport. The technique of equilibration chronometry al-
+lows the visualization of the time evolution of the neutron excess. Signatures of neutron-proton
+equilibration obeying first-order kinetics are observed both in experimental data [395–398] and in
+transport calculations
+[399]. Since the equilibration depends on the neutron and proton chem-
+ical potentials, this technique offers new experimental data to constrain the sub-saturation EOS
+through comparisons with simulations [6, 47, 400].
+
+25
+20
+15
+(MeV)
+10
+5
+0
+10-3
+10-2
+10-1
+p(N
+/fm3)
+nucleon44
+Isoscaling was first observed in central 124Sn+124Sn and central 112Sn+112Sn collisions at beam
+energy Elab = 50 AMeV (√sNN = 1.90 GeV) [401, 402]. Isoscaling describes a simple scaling law
+governing the ratios of isotope yields from two systems which differ mainly in their neutron-proton
+composition. It arises from the differences in the neutron and proton chemical potentials of the
+two reactions and is, therefore, sensitive to the symmetry energy. The isospin diffusion, derived
+from the isoscaling observable, reflects the driving forces arising from the asymmetry term of the
+EOS [6, 400, 403, 404] and provides a measurement of the symmetry energy at around (1/3)n0 [70].
+Other observables used to study the symmetry energy with light charged particles include both
+n/p and t/3He ratios and their double ratios obtained from two reactions with different isospin
+content [47, 95, 96, 100, 405]. Due to the difficulties in measuring neutrons, neutron data is not
+widely available. However, recent isoscaling measurements have allowed the construction of “pseudo
+neutrons”, that is a reconstruction of neutron yields from light particle ratios such as t/3He [406]. In
+particular, this method allows for a reconstruction of low-energy neutrons. However, due to the lack
+of high-energy charged particles data, it is a challenge to reconstruct high-energy neutron spectra
+in this way. Therefore, to study the symmetry energy at supranormal densities, neutron arrays
+constructed with new advanced materials will be needed in the next generation of experiments.
+In experiments utilizing central 124Sn+124Sn and central 112Sn+112Sn collisions at Elab =
+120 AMeV (√sNN = 1.94 GeV) [407], the spectra of neutrons emitted to 90 degrees in the center-
+of-mass frame are compared to the corresponding proton spectra. Transport calculations predict
+that if the effective masses of neutrons and protons satisfy m∗
+n < m∗
+p, then fast neutrons coming
+from the compressed participant region experience a more repulsive potential and a higher accel-
+eration than do fast protons at the same momentum, resulting in an enhanced ratio of neutron
+over proton (n/p) spectra at high energies. In contrast, calculations for m∗
+n > m∗
+p predict that the
+effective masses enhance the acceleration of protons relative to neutrons, resulting in a lower n/p
+spectral ratio. Bayesian analysis of the experimental results [98] compared to ImQMD calculations
+shows that the values of the first two Taylor expansion coefficients of the symmetry energy, S0
+and L, depend on both the symmetry energy and to the effective mass splitting. More examples
+of Bayesian analyses used to simultaneously constrain multiple parameters will be discussed in
+Section IV B, where methods to extract multiple transport model input parameters are discussed.
+3.
+Selected constraints on the symmetry energy around 1.5n0
+Current constraints on the symmetry energy above saturation are obtained with large uncer-
+tainties, mainly at densities around 1.5n0. This is the area of future opportunities, and we discuss
+this in more detail here to illustrate the complexity of the experiments and analyses as well as the
+central role played by transport models.
+The nucleon elliptic flow is sensitive to the pressure generated in nuclear collisions and, there-
+fore, to the EOS. Since a higher symmetry pressure will yield a larger magnitude of the elliptic flow
+at midrapidity for neutrons than for protons, comparisons of the neutron and proton elliptic flows
+provide sensitivity to the density-dependence of the symmetry energy [95]. The neutron and hydro-
+gen elliptic flow from Au+Au collisions at a beam energy of Elab = 0.4 AGeV (√sNN = 2.07 GeV)
+were measured in the FOPI-LAND and Asymmetric-Matter EOS (ASY-EOS) experiments, using
+the Land Area Neutron Detector (LAND) for the measurement of the neutron flow. A comparison
+of data to UrQMD simulations, shown in Fig. 18, was used to extract the dependence of the sym-
+metry energy on density, parametrized as proportional to (nB/n0)γasy, and the symmetry energy
+slope parameter L. The FOPI-LAND experiment reported γasy = 0.9 ± 0.4 and L = 83 ± 26 MeV
+[68], whereas the ASY-EOS obtained γasy = 0.72 ± 0.19 and L = 72 ± 13 MeV [69], indicating
+a moderately soft symmetry energy (see Fig. 18 and also Fig. 9). The analysis also illustrates
+
+45
+FIG. 18. Ratio of the elliptic flows of neutrons over
+charged particles vn
+2 /vch
+2
+as a function of transverse
+momentum per number of constituent nucleons pt/A
+in Au+Au collisions at Elab = 0.4 AGeV (√sNN =
+2.07 GeV). The comparison between ASY-EOS mea-
+surements (square black symbols) and UrQMD trans-
+port model calculations with a soft (pink dots) and
+hard (green triangles) symmetry potentials shows a
+preference for a soft symmetry energy; solid red line
+indicates γasy = 0.75 ± 0.10. Figure from [69].
+the dependence of S0 and L on other in-
+put parameters of the EOS, such as γasy.
+A
+subsequent comparison of data with dcQMD
+model [408] gives a value of L = 85 ± 32 MeV
+at n = 1.5n0.
+In addition to the ASY-EOS experiment,
+another effort that explores this density region
+is the SAMURAI Pion-Reconstruction and Ion-
+Tracker (SπRIT) experiment, performed with
+radioactive tin isotopes at RIKEN, Japan. For
+constraining the symmetry energy at supra-
+saturation densities, pion yield ratios are con-
+sidered as a unique observable since they do
+not form composite particles with other parti-
+cles.
+This makes their yields independent of
+clusterization processes which can affect the
+symmetry energy (see Section III B 1). Further-
+more, pion observables are predicted to be sen-
+sitive to the nuclear EOS at high densities due
+to their unique production mechanism: Above
+Elab = 200 AMeV (√sNN = 1.97 GeV), some of
+the interactions occurring in central collisions
+are energetic enough to form excited ∆(1232) baryon resonances (through the NN ↔ N∆ scatter-
+ing process), which then promptly decay into pions and nucleons. The high production threshold of
+the ∆(1232) resonance ensures that pions originate from the early stages of the reaction, and there-
+fore from regions characterized by a high density. The SπRIT collaboration measured charged pion
+emission from systems characterized by a wide range of asymmetry [102] by colliding tin isotope
+beams of 108,112,124,132Sn with isotopically enriched targets of 112,124Sn).
+The production of π− strongly depends on n-n collisions in the high-density region, while π+
+production largely depends on p-p collisions (the production of π− and π+ is equally likely in n-p
+collisions). It follows that the relative production of π− and π+ depends on the relative numbers
+of neutrons and protons and, therefore, is sensitive to the symmetry energy in the high-density
+region. Assuming a ∆-resonance model for pion production, one would expect that the pion yield
+ratio Y (π−)/Y (π+) follows a (N/Z)2 dependence [95, 367]. However, the measured total pion yield
+ratio follows N/Z with a best-fitted power index of 3.4, as shown in Fig. 19, where yield ratios
+without a transverse momentum cut are depicted by yellow crosses with circle markers. The radius
+of the circle in the center of each cross represents the experimental uncertainty, showcasing very
+good experimental accuracy of the measurement in which systematic errors are reduced by taking
+pion yield ratios. Moreover, comparisons of systems with different N/Z measured in the same ex-
+periment reduces systematic errors [409]. The discrepancy between the theoretical expectation and
+experimental data indicates the presence of dynamical factors beyond a simple ∆-resonance model,
+while the large measured exponent suggests that the ratios are strongly affected by the symmetry
+energy. When a transverse momentum cut of pT > 180 MeV/c is imposed, the result (represented
+in Fig. 19 by blue crosses with circle markers) still shows the same (N/Z)3.4 dependence, suggesting
+that effects due to the symmetry energy persist in high-momentum pions. Interestingly, current
+transport models do not seem to be able to reproduce the strong N/Z dependence [102].
+While Fig. 19 shows the total yield, the left panel of Fig. 20 focuses on the pT -dependence of the
+single ratio spectrum SR(π−/π+) = [dN(π−)/dpT ]/[dN(π+)/dpT ] for two extreme cases: reactions
+of neutron rich (132Sn+124Sn) and of near-symmetric (108Sn+112Sn) systems. The data is compared
+
+E
+1.3
+1.2
+y=0.75±0.10
+1.1
+2
+0.9
+stiff
+0.8
+T
+0.7
+0.6
+0.5
+soft
+0.4
+0.3
+0.4
+0.5
+0.6
+0.7
+p/A (GeV/c)46
+1.2
+1.3
+1.4
+1.5
+1.6
+N/Z
+1
+2
+3
+4
+5
+6
+Y(
+)/Y(
++ )
+pT > 0 MeV/c
+pT > 180 MeV/c
+y = 1.1(N/Z)3.4
+y = 0.5(N/Z)3.4
+FIG. 19. Ratios of yields of π− over yields of π+ in
+central (b < 3 fm) events for pions with pz > 0 in the
+center-of-mass frame, plotted as a function of N/Z.
+The yellow crosses show yield ratios with no transverse
+momentum cut, while the blue crosses show yield ra-
+tios for pT > 180 MeV/c.
+The radius of the circle
+inside each cross represents the statistical uncertainty
+of the ratio. The dashed blue and dotted blue line cor-
+responds to the best-fitted power functions of N/Z for
+pT > 0 and pT > 180 MeV/c pion ratios, respectively.
+Figure from [410].
+with the dcQMD model [145, 408], a Quantum
+Molecular Dynamic transport model that in-
+cludes total energy conservation and other ad-
+vanced features. To extract the EOS, the dcQMD
+model was used to predict single ratios with
+12 different parameter sets in the (L, ∆m∗
+np)
+space, forming a regular lattice; here, L is the
+slope of the symmetry energy and ∆m∗
+np is the
+neutron-proton effective mass splitting.
+The
+value of L in the lattice is either 15, 60, 106,
+or 151 MeV and ∆m∗
+np/δ is either -0.33, 0, or
+0.33. All other input parameters in the dcQMD
+have been fixed by comparing to FOPI data,
+as well as by comparing the predictions to the
+total yield of the charged pions and the aver-
+age pT obtained from the pion spectra.
+De-
+tails of the comparison can be found in Ref.
+[101].
+The left panel in Fig. 20 shows a few
+selected calculations and the measured single
+ratios.
+The (L, ∆m∗
+np) values for the solid
+blue line are (60, −0.33δ), for the dashed blue
+line are (60, 0.33δ), for the solid red line are
+(151, −0.33δ), and for the dashed red line are
+(151, 0.33δ). Coulomb effects dominate the low pT region, causing a steep rise in the measured
+ratios at pT < 200 MeV/c. All calculations at pT < 200 MeV/c disagree with data, which could be
+caused by inaccuracies in the simulation of Coulomb interactions or of the pion optical potential
+above the saturation density. At pT > 200 MeV/c, the Coulomb and pion potential effects diminish
+and the ratios should be good probes of the symmetry energy effects.
+The predicted single ratios at pT > 200 MeV/c are interpolated with 2D cubic splines over
+the (L, ∆m∗
+np) space, and the interpolated predictions are then compared to experimental mea-
+surements through a chi-square analysis. The resultant multivariate constraint on L and ∆m∗
+np is
+shown in the right panel of Fig. 20, where the green shaded region is the 1σ confidence interval
+and the area enclosed by the two blue dashed curves is the 2σ confidence interval. The corre-
+lation between ∆m∗
+np and L occurs because both parameters influence the nucleon momenta; L
+influences the momenta through its isospin-dependent contribution to the nucleon potential en-
+ergy, and ∆m∗
+np influences the momenta via its isospin-dependent impact on the nucleonic kinetic
+energy. Either increasing L or decreasing ∆m∗
+np will increase the energies of neutrons relative to
+protons. This increases the numbers of n-n collisions relative to p-p collisions at energies above
+the pion production threshold and enhances the production of π− relative to that for π+.
+4.
+Challenges and opportunities
+Experiment and theory
+Currently, there are few experiments that aim at inferring the symmetry energy and symmetry
+pressure from heavy-ion collisions probing densities of 1–2n0. Furthermore, the available constraints
+have very large uncertainties, especially for the symmetry pressure. It is worth noting that heavy-
+ion collision experiments do not measure the symmetry energy or pressure directly, but rather
+they depend on comparisons with transport model simulations that describe the dynamics of the
+
+47
+0
+100
+200
+300
+400
+pT (MeV/c)
+100
+101
+Single Ratio (SR)
+132Sn + 124Sn, E/A = 270 MeV
+0
+100
+200
+300
+400
+pT (MeV/c)
+108Sn + 112Sn, E/A = 270 MeV
+       L(MeV)   m *
+np
+60          -0.33
+60           0.33
+151        -0.33
+151         0.33
+50
+100
+150
+L (MeV)
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+∆m∗
+np/δ
+1-σ
+2-σ
+FIG. 20. Left panel: Single pion spectral ratios for 132Sn+124Sn (top) and 108Sn+112Sn (bottom) reac-
+tions with four selected dcQMD predictions overlaid [86]. Right panel: Correlation constraint between L
+and ∆m∗
+np/δ, extracted from pion single ratios at pT > 200 MeV/c in collisions of both neutron-deficient
+108Sn+112Sn and neutron-rich 132Sn+124Sn systems. The light blue shaded region (dashed blue lines) cor-
+responds to 68% (95%) confidence interval [101].
+collisions [47].
+The large uncertainties in available constraints mainly arise from the intrinsic
+uncertainties of the transport models and the accuracy of determining the parameters used as an
+input in these models. For example, a general feature of low-energy heavy-ion collisions is that
+more nucleons are emitted in light clusters than are emitted as free neutrons and protons, while
+the reverse is true of most transport model simulations of these reactions. Theoretical approaches
+to this issue have been proposed (see Section II A), but are rarely implemented to model the
+coalescence of nucleons in the medium into the observed distribution of clusters, and therefore it is
+not clear to what extent these approaches are valid. The current inaccuracy in cluster production
+complicates and limits the scientific conclusions that can be drawn by comparing data to transport
+theory, and therefore improving the accuracy of cluster production in transport theory would be a
+very significant achievement, enabling more stringent constraints on the symmetry energy.
+It is important to quantify major sources of systematic uncertainties in the transport model
+implementations and in the model parameters. Due to the quality as well as technical details of
+solutions adopted in different models, it may not be realistic to establish all uncertainties for all
+transport models. Nonetheless, developing methods to validate transport models and performing
+these validations remains a primary goal for the Transport Model Evaluation Project (TMEP)
+collaboration, and it is essential to extracting reliable constraints on the EOS from heavy-ion
+collisions (see Section II A).
+The current capabilities at FRIB, using beam energies up to Elab = 200 AMeV (√sNN =
+1.97 GeV), allow for exploration of densities up to 1.5n0, and the neutron excess can be varied
+over a wide range by changing the composition of the rare isotope beams and targets, allowing
+to more closely recreate the matter found in extreme astrophysical environments (e.g., neutron
+stars). From the dense collision region in heavy-ion collisions, pions and free nucleons are emitted
+with high transverse momentum. The relative yields of these particles, especially as a function of
+energy, as well as particle elliptic flow contain information about the dense collision zone and thus
+can be used to constrain the EOS that governs supra-saturation matter. Individual efforts based
+on small-scale experiments, which are the strength of the field, have provided a diversity of results.
+However, in order to take advantage of multiple-parameter Bayesian analyses, described below,
+and given the tight allotment of the expensive (and coveted) beam time, future experiments should
+utilize detectors that provide large coverage. The development of the time projection chamber
+
+48
+(TPC) detector at FRIB is essential to measure both pions and charged particles. The detector
+can be coupled with an upgraded or a new Large Neutron Array (LANA) to measure both charged
+particles and neutrons. Additionally, putting the TPC detector at the target position of the High
+Rigidity Spectrometer (HRS) enables coincident measurement of projectile-like fragments. The
+determination of the centrality and the reaction plane, required, e.g., for the elliptic flow studies,
+would benefit from a construction of a 4π detector placed around the target when the silicon
+detectors are used to identify the charged particles before the completion of the HRS. Such a
+detector would have to measure the energies of the emitted fragments and nucleons as well as their
+multiplicities with minimal energy losses.
+Reaching higher densities requires the energy upgrade to Elab = 400 AMeV (√sNN = 2.07 GeV).
+With the capability for producing high-intensity rare isotope beams with a wide range of asymme-
+tries, FRIB400 is essential for the U.S. effort to lead in the determination of the density-dependence
+of the symmetry energy [22].
+The beams available at FRIB, being complementary to those that can be accelerated at Eu-
+ropean facilities, may represent a unique opportunity to conduct nuclear transport investigations
+also by the international nuclear physics community. As described above, the development of de-
+tector arrays with high isotopic resolution over a wide dynamic range, from light particles to heavy
+fragments, provides the prospect of measuring observables (especially in the context of isospin
+diffusion and drift as well as in collective motion phenomena) that can amplify the sensitivity to
+the symmetry energy. Coupled with its capability to use high-quality radioactive beams, FRIB
+may represent a focus of interest for a wider community, stimulating the need for international
+discussions and collaborations in the coming years. Such an interest may concern also theoretical
+physicists that have been collaborating with FRIB colleagues within the Transport Model Evalu-
+ation Project (TMEP) initiative, aimed at improving investigations of the isospin-dependent EOS
+with comparisons to experimental observables (see also Section II A).
+Multiple Parameter Bayesian analysis
+The EOS is only one of many input parameters in transport models used to simulate heavy-ion
+collisions. Often, multiple measurements probing different parts of the collisions are needed to
+constrain other parameters of these models, such as the momentum-dependence of the isovector
+mean-field potential, or the in-medium isospin-dependent cross sections. However, constraining
+transport model parameters with experimental results is a delicate endeavor. The outcomes of
+nuclear collisions are influenced by a multitude of processes, and therefore the experimentally
+measured final stage observables can depend simultaneously on values of multiple parameters.
+However, carefully chosen observables may only be sensitive to just a few specific parameters. The
+full extent of the dependence of a given transport model on input parameters can only be tested
+empirically after performing a complete series of simulations of heavy-ion collisions.
+Bayesian statistical methods provide means to quantify the relation between observable values
+and physical parameters. They also provide a systematic way of constraining multiple nuclear
+properties and utilizing prior knowledge from different experiments, prior constraints from other
+sources, and results from new experimental measurements. For example, in the n/p ratio exper-
+iment mentioned above, measuring the yield ratios of neutron and protons spectra, a Bayesian
+analysis comparing the experimental results [98] to ImQMD calculations determines both ∆m∗
+np and
+the relationship between S0 and L, even though the uncertainties are large. More precise measure-
+ments in the future will enable a better resolution.
+In the long term, it is important to develop Bayesian analyses of multiple observables to de-
+termine multiple parameters simultaneously.
+As an example, in the SπRIT experiment many
+observables have been measured with four reaction systems. Eight observables in total, including
+the directed flow, elliptical flow, and the stopping observable from different reactions, are fitted si-
+
+49
+FIG. 21. Posterior distribution obtained from a Bayesian analysis of ImQMD simulation results and experi-
+mental data from SπRIT experiments [410]. Eight available observables are used for Bayesian analysis. The
+values for median and 68% confidence interval of the marginalized distribution are tabulated on the upper
+right-hand side of the figure. Figure from [410].
+multaneously by varying five transport model input parameters (two pertaining to the shape of the
+symmetry energy term in the nuclear matter EOS, two pertaining to the nuclear effective masses,
+and one pertaining to the nuclear in-medium cross-section). The posterior distribution shows a
+weak constraining power on the symmetry energy terms, but a strong sensitivity to effective masses
+and in-medium cross-section [410], see Fig. 21.
+The posterior parameter distributions are generated from repeated sampling of transport model
+predictions for hundreds of thousands of times, each with different parameter values. If carried out
+directly, this process would consume an unreasonable amount of computational resources. This can
+be alleviated with an effective, efficient, and capable model emulator which emulates the behavior
+of transport models at all points of the allowed parameter space from predictions at just a few tens
+of parameter values. Gaussian processes are readily available and commonly used in emulators,
+but the procedures for tuning hyperparameters vary across analyses. Numerous heuristics and cost
+functions are proposed for the optimization of hyperparameters, and one can also marginalize over
+all nuisance parameters with a Markov chain Monte Carlo.
+
+50
+(MeV)
+S
+30
+150
+(MeV)
+100
+50
+1.0
+Nu/
+0.8
+Nu/
+1.00
+0.75
+0.25
+0.00
+n
+-0.25
+30
+40
+50
+50 100 150
+0.8
+1.0 0.75 1.00 -0.25 0.00
+0.25
+So (MeV)
+L (MeV)
+ms /mN
+m,/mN
+n50
+IV.
+THE EQUATION OF STATE FROM COMBINED CONSTRAINTS
+Nuclear
+Neutron star
+Isospin diffusion in HICs
+Masses and radii
+Dipole polarizability
+Tidal deformability
+Spectral ratios of light clusters
+Moment of inertia
+Nuclear masses and radii
+Gravitational binding energy
+Isobaric analog states
+Cooling of young neutron stars
+n/p ratios in HICs
+Bulk oscillation modes
+Neutron skins
+Crust cooling
+Mirror nuclei
+Pulsar glitches
+Giant resonances
+Lower and upper limits on neutron star spin periods
+Flow of particles in HICs
+Torsional crust oscillations
+Charged pion ratios in HICs
+Crust-core interface modes
+TABLE I. Illustrative list of nuclear and astrophysical observables.
+There have been many attempts to extract the equation of state (EOS) as a function of density
+from both nuclear experiments and astronomical observations. In Table I, we provide an illustrative
+list of relevant experimental and observational measurements. Importantly, these observables probe
+the EOS at different densities: a few probe the EOS near the saturation density n0, but many probe
+densities that are significantly higher or lower. For example, nuclear structure typically probes
+densities that are somewhat lower than n0, while analyses of heavy-ion collisions or properties of
+neutron stars probe larger density ranges, as schematically illustrated in Figs. 12 and 22.
+Comparing constraints based on different measurements allows one to test their consistency and
+ultimately find tight constraints on the EOS over the full range of densities that can be probed
+either by experiments or by astronomical observations. Techniques of Bayesian inference or Pearson
+correlation analyses are well-suited to this endeavor and can provide more readily useful and
+Astro (M,R,Λ)�
+HIC�
+Astro: crust observables�
+Chiral EFT�
+Nuclear masses, �
+Δrnp,  αD, ….�
+FIG. 22. An ensemble of EOSs that range over crust and
+core uncertainties consistently. Ranges over which differ-
+ent nuclear and astrophysical probes provide information
+about the neutron star EOS are indicated. Figure modified
+from [411].
+testable
+information
+on
+the
+density-
+dependence of the EOS than, e.g., statis-
+tical comparisons or combining the Taylor
+expansion coefficients (such as S0, L, and
+Ksym) obtained from individual analyses.
+Key to this approach is the determination
+of the density that each experimental ob-
+servable most accurately probes.
+Away
+from that density, weaker constraints on
+the EOS are possible, but the analysis is
+more complex.
+In this section, we review the variety of
+observables that have been used to place
+constraints on the EOS; heavy-ion collision
+experiments, which produce many of these
+constraints, are described in Section III.
+We then discuss recent attempts at com-
+bining various constraints that result in
+meaningful EOSs with quantified uncer-
+tainties.
+
+？???
+Outer Crust
+Outer
+Inner Core
+Neutron Drip
+Inner Crust
+Crust/Core Transition
+Core
+1036.
+pressure (Dyne/cm2)
+1034
+1032
+1030
+J, L, Ksym
+in1!
+n2
+1011
+1012
+1013
+1014
+1015
+1016
+energy density (g/cm3)51
+A.
+Constraints
+As discussed in Section III, experiments are often designed to explore certain aspects of the EOS.
+Accordingly, we classify the constraints obtained from laboratory measurements as sensitive to
+either the symmetric nuclear matter EOS or the symmetry energy. In addition to the experimental
+inferences, constraints on the EOS can be also obtained from neutron star observations as well as
+from chiral EFT theory at low densities. The list of constraints discussed here is not exhaustive.
+Rather, it represents a slice of widely acknowledged constraints at the moment of writing. We note
+that some of the constraints reviewed here have already been presented in Sections II and III, to
+which we refer when appropriate.
+Symmetric matter constraints from laboratory experiments
+Some properties of the symmetric nuclear matter are fairly well-known near n0. For exam-
+ple, the generally accepted values of n0 and binding energy at saturation E0 are 0.16 fm−3 and
+−16 MeV [198, 203], respectively, to within 4%. The incompressibility parameter, K0, has been
+determined from giant monopole resonance (GMR) experiments [412] to be 231±5 MeV. However,
+subsequent GMR measurements of the Sn isotopes cast larger uncertainties on K0 [213]. While
+these larger uncertainties are consistent with values of K0 determined from heavy-ion collision
+experiments [67, 353, 413], we note here that these experiments derive their constraints on K0
+based on density functionals that are parametrized with K0, but used to describe the high-density
+behavior of the EOSs (i.e., these experiments do not probe the incompressibility at saturation; see
+also a similar discussion in Section II A 2). Measurements of the collective flow from high energy
+Au+Au collisions have constrained the EOS for symmetric nuclear matter at densities spanning
+(1–4.5)n0 [31, 58, 66, 67] (see the left panel in Fig. 9), as described in Sections II A 2 and III A.
+Symmetry energy constraints from laboratory experiments
+In the past decade, many studies have been conducted to extract the symmetry energy and
+the symmetry pressure, and some of the widely-known constraints are plotted in the right panel
+of Fig. 9, which includes both the usual EOS constraint bands as well as symbols located at
+densities which a novel analysis in Ref. [70] identified as the most sensitive densities for a given
+measurement. At (2/3)n0, precise symmetry energy constraints have been obtained from studies
+on nuclear masses using Skyrme density functional forms for the EOS. These are labeled in the
+right panel of Fig. 9 as “mass(Skyrme)” [72] and “mass(DFT)” [71], respectively. In this density
+region there are also precise constraints obtained from the energies of isobaric analogue states [12],
+indicated in the right panel of Fig. 9 by a data point labelled as “IAS”. The dipole polarizability
+αD, reflecting the response of a nucleus to the presence of an external electric field, also helps to
+constrain the symmetry energy at low densities. Constraints on the symmetry pressure Psym, which
+is proportional to the derivative of the symmetry energy with respect to density, have been recently
+provided by the measurements of the neutron skin of 208Pb in the Lead Radius EXperiment (PREX
+and PREX-II) [74, 388, 414] and of the neutron skin of 48Ca in the Calcium Radius EXperiment
+(CREX) [389–391], both at Jefferson Lab, which use parity-violating weak neutral interactions to
+probe the neutron distribution in 208Pb and 48Ca. A range of other scattering experiments have
+measured the neutron skins of a number of neutron-rich isotopes and likewise used them to constrain
+the symmetry energy [7, 415, 416]. Giant dipole resonances and polarizabilities [9, 417, 418] in
+neutron-rich isotopes provide another source of information about the symmetry energy [335, 419–
+424], as do mirror nuclei [393, 394]. At densities far from (2/3)n0, heavy-ion collisions have been
+used to probe the symmetry energy, as is described in Sections II A 2 and III B 3, and shown in the
+right panel of Fig. 9.
+
+52
+Constraints from astronomical observations
+The bulk properties of neutron stars (such as their maximum mass, radii, tidal deformabilities,
+moments of inertia, limits on the rotation frequency, and binding energy) depend strongly on the
+distribution of matter throughout the star, therefore providing a measure of the EOS integrated
+over the range of densities present in the star. The mass-radius relationship has a one-to-one corre-
+spondence to the neutron star EOS [425], and it is known that the radius, the tidal deformability,
+and the moment of inertia provide the strongest constraints on the EOS above 2n0 [251, 426],
+while the maximum measured mass of neutron stars constrains the EOS at the highest densities.
+Together, the tidal deformability measured in GW170817, the mass of J0740+6620, and the two
+mass-radius measurements of NICER, discussed in Section II C, form the current gold standard in
+measuring the neutron star properties using astronomical observations.
+A number of astronomical observables also probe the neutron star crust physics, which results
+in constraints on the pure neutron matter EOS, and in particular on the symmetry energy. This
+is because the neutrons provide the hydrostatic pressure that supports the inner crust, and the
+interplay between these neutrons and the lattice of nuclei that makes up the crust determines the
+crust-core boundary as well as the possible nuclear pasta shapes that appear near that boundary.
+The crust physics also depends, more weakly, on the symmetric matter EOS. The nuclear EOS
+at subsaturation densities, down to where the neutron drip begins (nB ≈ 10−4n0), is therefore an
+essential ingredient in crust models.
+Due to the complexity of crust physics, extracting rigorous EOS constraints from observations
+of crust-associated neutron star behavior is in its early stages, and it is an area where substantial
+progress can be made over the next decade. Here we list some constraints on the symmetry energy
+as an illustration of this potential, but, at the same time, we note that they are very tentative and
+do not have well-quantified errors; indeed, some of them are mutually exclusive, emphasizing the
+need to make progress in applying microscopic nuclear physics models to these observations.
+Constraints on the symmetry energy and its slope can be obtained from studying the following
+phenomena: A study of the cooling of the neutron star in the Cas A supernova remnant [312],
+which has been observed to cool on a timescale of decades, implies that the neutron star core may
+have superfluid properties [265, 427, 428]. Studying the temperatures of the population of neutron
+stars whose surface X-ray emission is observable leads to constraints on the neutron star masses
+and radii and the composition of the core [429]. Constraints from quasi-periodic oscillations in
+the X-ray tail of gamma-ray flares from soft gamma-ray repeaters [285, 430, 431], which could be
+a signature of torsional oscillations of the crust. Potential measurements of the crustal moment
+of inertia from glitches – sudden changes in rotation frequency – of radio pulsars and some X-ray
+pulsars [432–437].
+Limits on the longest and shortest observed periods of neutron stars probe
+physics such as the magnetic field evolution in the crust [438] and the development of rotation-
+induced instabilities in oscillations such as r-modes [439, 440]. During the last few seconds of an
+inspiral prior to the merger of two neutron stars or a neutron star with a black hole, tidal forces
+may shatter the crust, causing a gamma-ray flare: in this scenario, coincident timing of the flare
+with the gravitational wave signal measures the resonant frequency of crust-core interface modes
+and sets constrains on the symmetry energy [276, 441]. The cooling of the crusts of quiescent
+low-mass X-ray binaries promises to provide a source of constraints on the composition and size of
+the neutron star crust and the extent of nuclear pasta phases therein [283, 442–444]. The expected
+accurate measurement of the moment of inertia of pulsar J0737-3039a [445] will set constraints on
+the EOS competitive with the current radius constraints [446–450]. The heat capacity of a neutron
+star core can be measured by using inferences of the core temperature of transiently-accreting
+neutron stars, and strongly suggests that a core dominated by a color-flavor-locked quark phase is
+ruled out [451]. Some such objects are observed to have efficient cooling in the core, constraining
+superfluid gap models and the symmetry energy [452].
+
+53
+Constraints from nuclear theory
+In recent years, many-body nuclear theory such as chiral effective field theory (χEFT), discussed
+in Section II B, has made significant progress to be considered as the canonical nuclear matter
+EOS at low densities with rigorous uncertainty quantification [155–158]. Even though the theory
+is developed mainly for densities below saturation, it has been extended to 2n0, and it is a popular
+constraint for studies that focus mainly on astronomical observations.
+B.
+EOS obtained by combining various constraint sets
+Each of the nuclear and astrophysical observables discussed above provides vital information
+about the EOS over some density range, that can be combined with other constraints to globally
+constrain the density-dependence of the EOS from sub-saturation to supra-saturation densities.
+Such analysis techniques are relatively new, but several of such global constraints now exist, and
+a selection of studies is briefly described below to illustrate their potential.
+Beloin et al. [429] used relativistic mean-field models of the nuclear interaction to model the
+structure and cooling of neutron stars.
+This analysis consistently combines nuclear data, neu-
+tron star mass-radius measurements, and neutron star cooling measurements within a Bayesian
+framework.
+Legred et al. [251] performed nonparametric EOS inference based on Gaussian processes. It
+combines information from X-ray, radio, and gravitational wave observations of neutron stars.
+Their results are plotted in Fig. 23 and labelled as “Legred et al.”.
+These Bayesian analyses
+incorporate astrophysical data and provide constraints on the neutron star EOS at higher densities.
+Drischler et al. [453] performed a Bayesian analysis of correlated effective field theory truncation
+errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the χEFT
+expansion. The neutron star matter pressure calculated with these EOS is shown in Fig. 23 and
+labeled as “Drischler et al.”.
+Huth et al. [17] combined nuclear theory via χEFT calculations (constraining the EOS below
+1.5n0), EOS inferences from heavy-ion collisions via the FOPI (constraining the symmetric matter
+EOS up to 2n0) and ASY-EOS (constraining the symmetry energy at around 1.5n0) experiments,
+Huth	et	al.
+Drischler	et	al.
+Legred	et	al.
+Pressure	[MeV/fm3]
+0.1
+1
+10
+100
+baryon	density	nB/n0
+0.5
+1
+1.5
+2
+2.5
+3
+FIG. 23. The pressure of neutron star matter as a func-
+tion of number density nB, as obtained by Huth et al. [17],
+Drischler et al. [453], and Legred et al. [251] at 95%, 95%,
+and 90% confidence interval, respectively.
+and astrophysical data on bulk neutron star
+properties (constraining the total neutron
+star EOS above 2n0). The EOS models were
+extended to high densities using a speed-
+of-sound model.
+The results are shown in
+Fig. 23 and labeled as “Huth et al.”.
+Yue et al. [454] constructed neutron star
+models using a Skyrme energy-density func-
+tional, which allowed them to consistently
+calculate the neutron skin of
+208Pb and
+combine constraints from heavy-ion colli-
+sions, measurements of the neutron skin, and
+astrophysical constraints within the same
+model.
+Neill et al. [411] followed a similar strat-
+egy as the example above, using Skyrme
+models which were extended to the crust.
+This allowed them to combine neutron skin
+measurements, NICER/LIGO observations,
+
+54
+a crust observable (the resonant frequency of the crustal i-mode), and nuclear mass data to con-
+strain both the core and crust properties as well as the EOS. By calculating all these quantities
+using the same underlying Skyrme energy density functional (and polytrope parametrizations at
+the highest densities), some poorly controlled modeling uncertainties were eliminated. This work
+demonstrated the complementarity of different observables: within the particular model used, nu-
+clear masses constrain mainly the zeroth and first symmetry energy expansion coefficients, S0
+and L, the crust observable has the largest impact on the inferred values of L and the second
+expansion coefficient Ksym, and the neutron star radius and tidal deformability have the largest
+impact on the inferred values of Ksym and two polytrope parameters. Thus, when combined, dif-
+ferent observables provide complementary information that can contribute to a complete picture
+of the EOS. The ranges of these overlapping data are depicted in Fig. 22.
+Without crust observables, neutron star radii and tidal deformabilities tend to give weaker
+constraints on Ksym and stronger constraints on L (see the analysis of a large number of studies
+in [455]).
+However, the relative constraints on the symmetry energy parameters change when
+the used priors include the criterion that the crust is stable and incorporate potential data from
+crust observables [411]. While this result is model-dependent and correlations with higher-order
+symmetry parameters need to be investigated, it demonstrates the way in which crust observables
+could significantly contribute to constraining the EOS, and motivates the need for improving models
+to consistently combine crust and core observables with nuclear data.
+One of the defining strengths of the global constraint analysis is that one or more additional
+constraint(s) can be always included as long as an assessment of the corresponding statistical and
+systematic uncertainties is also provided. Moreover, the more data from nuclear and astrophysical
+observables can be meaningfully included in such EOS inferences, the greater the ability to deliver
+a robust EOS. Therefore, constraining the EOS by combining various inferences is highly promising
+and, furthermore, well-suited for the coming era of multi-differential observables from heavy-ion
+collisions and multi-messenger astronomical observations.
+V.
+CONNECTIONS TO OTHER AREAS OF NUCLEAR PHYSICS
+A.
+Applications of hadronic transport
+In addition to the use of transport codes to study fundamental nuclear physics, their ability
+to describe the transport and interactions of particles in a material also make them valuable for
+applications that benefit society.
+Examples include the design of nuclear physics experiments,
+detector development and simulations of detector performance, as well as medical applications and
+radiation shielding in accelerator and space exploration. Some of these uses are outlined here.
+Transport models are widely used to simulate particle emission from nucleus-nucleus collisions.
+In these simulations, the four-momentum of every emitted particle is tracked, making it possible
+to generate double differential distributions, the particle spectra at various emission angles. These
+distributions are particularly important for applications.
+Most transport models are optimized for describing physics in certain energy ranges. The type
+of code required can be tailored to the desired application. For example, some models perform best
+at energies of a few hundred MeV and below, which is near the peak of the cosmic ray flux [456],
+while others are more applicable for GeV-scale energies and above, in the tail of the cosmic ray
+flux. Over the entire experimental energy range, from intermediate energies through the highest
+collider energies, transport codes have been successfully employed to design complex detectors,
+optimize experimental setups, and carry out analyses of experimental data, including assessing the
+detector efficiencies and background contributions.
+
+55
+1.
+Detector design
+In high-energy experiments, the code packages most commonly used for detector development
+and data analysis are Geant3 [457], Geant4 [458], and FLUKA [459]. Most of these simulation
+packages use cascade codes, that is codes without mean-field potentials, to describe particle trans-
+port through matter. Modern transport codes that can cover a wide range of energies such as
+PHITS (Particle and Heavy-Ion Transport code System) can, however, provide a more complex
+description of particle transport [47].
+Aside from heavy-ion collisions, transport simulations play an important role for a variety of fun-
+damental physics experiments. For example, long-baseline neutrino experiments need to determine
+the incoming neutrino energy in order to extract the neutrino mixing parameters, CP violating
+phases, and neutrino mass ordering [460]. However, because the neutrino beam is generated from
+fixed-target proton-nucleus interactions producing secondary π and K mesons with neutrino decay
+products, there are large uncertainties on the energy of the interacting neutrinos. The neutrino
+energy must be reconstructed from the measurement of the final state [461, 462] which is often mod-
+eled by simple Monte-Carlo cascade approaches. Reliable transport descriptions could significantly
+improve these studies for the Deep Underground Neutrino Experiment (DUNE) [463], as well as
+for the ongoing experiments NuMI Off-axis νe Appearance (NOvA) [464] and Tokai to Kamioka
+(T2K) [465]. Other experiments that require transport simulations of backgrounds include dark
+matter searches [466], semi-inclusive electron scattering such as (e,e′p) on nuclear targets at Jef-
+ferson Lab to search for color transparency, short-range correlations [467], and hadronization in a
+nuclear medium [468].
+2.
+Space exploration, radiation therapy, and nuclear data
+Transport models can also be used in applications relevant to space exploration to understand
+and mitigate the harmful effects of the space radiation environment on electronics and astronauts.
+Collisions of galactic cosmic rays (GCRs) with nuclei, whether in the Earth’s atmosphere or in
+the material of a spacecraft above it, can generate showers of particles, including pions, muons,
+neutrinos, electrons, and photons as well as protons and neutrons. GCRs cover a wide range of
+energies, from tens or hundreds of MeV up to the TeV scale, and ion species, spanning elements
+1 ≤ Z < 28 [475], making it challenging to determine all their potential effects in a given material.
+The penetrating power of the initial GCRs and the secondaries generated by their interaction
+with matter can have a serious impact on the safety and viability of space exploration. The 1%
+of GCR primaries which are heavier than Helium nuclei can pose an especially serious problem,
+given that the damage they inflict scales as Z2. The secondary particles generated from GCR
+interactions with spacecraft materials [476] such as aluminum, polyethylene, and composites can
+harm astronauts and disrupt or even disable electronic systems. Moreover, spacecraft shielding
+designed to reduce the GCR flux is itself a target that can increase the secondary flux. Because
+of the wide variety of possible shielding materials and thicknesses, transport models are essential
+to determine the sensitivity of the secondaries (regarding both their flux and composition) to
+different shielding configurations, as well as the subsequent harmful impact of those secondaries on
+electronic systems [477] and humans [478]. A pictorial overview of applications transport modelling
+and nuclear data in space missions is given in Fig. 24.
+Due to the lack of data at the appropriate energies, simulations of space radiation effects have
+large uncertainties. The space research community has generally relied on phenomenological nu-
+clear reaction models such as the Double Differential Fragmentation model (DDFRG) [479], which
+consists of a sum of multiple exponential distributions with parameters fit to data. Many of these
+
+56
+models employ abrasion-ablation models [480, 481] (where abrasion and ablation refer to parti-
+cle removal in ion-ion interactions and nuclear de-excitation following abrasion, respectively) or
+semi-empirical parametrizations, see Ref. [482].
+Researchers modeling these interactions could
+benefit from transport codes discussed in this White Paper. The use of hadronic transport models
+such as the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) code [482], which was shown
+to correctly predict proton and deuteron yields from the BNL Alternating Gradient Synchrotron
+measurements of collisions of protons on Be and Au targets at Elab = 15 AGeV [483, 484], see
+Fig. 25, could significantly advance simulations of collisions relevant for space exploration. For
+further information about the needs for space applications, see Refs. [485, 486].
+Similar transport modeling needs arise in charged particle therapy for medical applications such
+as cancer treatment. In this case, the ion beam is tuned to penetrate the tissue at the tumor location
+so that the Bragg peak, or maximal dose, is delivered to the tumor site while minimizing the spread
+of the charged particle beams into surrounding tissue due to target fragmentation and secondary
+scattering [487]. Transport models can play an important role in improving the effectiveness and
+safety of charged particle therapy in cancer treatment [487, 488]. Moreover, if ions such as carbon
+are used instead of protons, the beam may also fragment and spread in the body. These interactions
+are also studied employing abrasion-ablation models. Better models of projectile fragmentation
+are needed to determine the effect of ion beams on normal tissue. Recently, the Stochastic Mean
+Field (SMF) [489] and Boltzmann-Langevin One Body (BLOB) [490] models have been coupled with
+Geant4 for studies related to radiation therapy [491].
+The nuclear information required for applications falls under the general umbrella term of
+FIG. 24. Some of the applications of hadronic transport calculations and nuclear data in space exploration
+research (counterclockwise from top left): energy production in outer space, such as with the TOPAZ
+nuclear reactor [469] or the proposed fission surface power system on the Moon KRUSTY [470]; nuclear
+thermal rocket propulsion [471]; planetary exploration [472]; and dose and shielding calculations of ions
+passing through electronics and humans (left: a heavy-ion interaction with a shielding structure [473], right:
+particle spectra calculated for an incident solar minimum GCR iron spectrum [474]).
+
+Nuclear reactors
+Dose/Shielding calculations for electronics & humans
+in outer space
+Cosmic ray track 
+Geant4
+104
+FLUKA
+PHITS
+Metal
+ Metal
+neutrons
+10
+3DHZETRN (N=1)
+3DHZETRN (N=22)
+ Glass
+102
+0 g/cm?
+Gate
+10
+Field
+10°
+oxide
+-Drain
+Source
+protons (xo.1)
+Depletion
+10°
+Positive region
+10
+He (x0.05)
+well
+10°
+- Funneling
+10
+TOPAZ
+region
+Substrate
+105
+10-2
+10-1
+100
+10l
+102
+103
+104
+Kinetic energy (MeV/n)
+Nuclear propulsion
+Planetary exploration
+Passive (n, xy)
+rays
+Cosmic
+Fast
+Natural
+KRUSTY
+neutrons
+radioactivity57
+−1.0−0.50.0 0.5 1.0 1.5 2.0 2.5 3.0
+y
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+dN
+dy
+p+Be
+UrQMD, protons
+UrQMD, deuterons
+E802, protons
+E802, deuterons
+−1.0−0.50.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
+y
+p+Au, minb., 14.6 A GeV
+FIG. 25. Rapidity distributions of protons and deuterons in
+minimum bias p+Be (left) and p+Au (right) collisions at
+a beam energy of Elab = 14.6 AGeV. Blue dashed and red
+solid lines show results for protons and deuterons, respec-
+tively, obtained from the UrQMD model, compared to data
+from the E892 experiment (blue and red dots for protons
+and deuterons, respectively) [484]. Figure from Ref. [483].
+“nuclear data”. The Geant3, Geant4, and
+FLUKA codes all utilize information taken
+directly from nuclear data libraries. How-
+ever, standard nuclear databases cover
+almost exclusively neutron-induced reac-
+tions, while few charged-particle data are
+available.
+In addition, the energy range
+covered by these databases typically only
+extends to 20 MeV. In higher energy
+databases such as the GSI-ESA-NASA
+database [482], there are essentially no
+data for light ions beyond Elab = 3 AGeV
+and scant data for heavy ions beyond a
+few hundred AMeV [482]. Transport mod-
+els such as Quantum Molecular Dynamics
+(QMD) [492] and PHITS [493, 494] have
+been used to simulate higher-energy colli-
+sions to fill the gaps in data. Experiments
+at nuclear accelerators are needed to verify
+these calculations.
+The
+US
+Nuclear
+Science
+Advisory
+Committee (NSAC) has been charged to “assess challenges, opportunities and priorities for ef-
+fective stewardship of nuclear data”.
+As part of the development of the Long Range Plan for
+nuclear science, town halls involving different sub-fields of the US nuclear physics community have
+adopted nuclear data resolutions, including a recommendation to identify cross-cutting opportuni-
+ties with other programs. We suggest that one of these opportunities is the use of transport codes
+to advance and enhance high-energy applications, such as space research and advanced medical
+treatments.
+B.
+Hydrodynamics
+Relativistic hydrodynamics (Landau-Lifshitz hydrodynamics [495]) can be defined as the effec-
+tive field theory (EFT) describing fluids on energy scales much smaller than the fluid tempera-
+ture [496]. Hydrodynamic equations encode the evolution of conserved charges, such as energy
+density and electric charge density, in spacetime. Solving the hydrodynamic equations requires
+the EOS as a crucial ingredient. Thus, in turn, hydrodynamics can be used to constrain the EOS.
+For example, this has been done or is of relevance for the quark-gluon plasma (QGP) generated
+in heavy-ion collisions [329, 497, 498] (recently, prospects of constraints on the EOS for nuclear
+matter forming the primordial QGP emerged [499]), for (proto)neutron stars [500–505], as well as
+for neutron star mergers [506–511].
+1.
+Status
+Hydrodynamics has had a great success describing nuclear matter generated in heavy-ion col-
+lisions over a wide range of energies [512, 513]. Remarkably, hydrodynamics applies to various
+system sizes accessible in heavy-ion collisions [514], with ALICE, ATLAS, and CMS experiments
+showing collective fluid behavior in proton-ion [515–517] and even proton-proton collisions [518–
+520], which was also successfully reproduced hydrodynamically [521]. Collective behavior in small
+
+58
+systems was also observed at RHIC by the PHENIX and STAR experiments [522, 523].
+It was realized early on that first-order hydrodynamics (in Landau or Eckart frame) is causality
+violating and unstable [524]. At this time, the standard solution to this problem is the M¨uller-
+Israel-Stewart (MIS) theory [525–527], or versions thereof [528, 529], which are used in most hy-
+drodynamic codes modeling heavy-ion collisions. In the MIS theory, transient modes are added as
+regulators ensuring a causal time evolution [530]. The behavior of such transient modes depends on
+the way they are introduced [528, 530]. Since they are generally not associated with any conserved
+quantities, their behavior is not what hydrodynamics aims to describe. MIS thus relies on these
+transient modes to decay sufficiently fast for the observables to behave hydrodynamically. This
+poses a problem for MIS at early times in a heavy-ion collision, when the regulator transient modes
+are still present, because observables sensitive to the early times may reflect the physics of these
+regulators. In addition, the causality violation [531] and stability [532] in these setups has to be
+monitored when modeling, for example, heavy-ion collisions.
+Alternatively, a more direct approach to constructing causal viscous hydrodynamics is based
+on the realization that hydrodynamics is causal when considered in a general frame (and not,
+e.g., Landau frame or Eckart frame). In that case it is not necessary to introduce any regulator
+or auxiliary fields, as the differential equations governing the hydrodynamic fields (temperature,
+fluid velocity, and chemical potential) are hyperbolic (i.e., there exists a solution for all times) and
+their time evolution is causal by construction. This leads to the Bemfica-Disconzi-Noronha-Kovtun
+(BDNK) theory [533–538], which is capable of, for example, modeling neutron star mergers [537].
+BDNK also has a practical use in constructing manifestly causal numerical codes solving hyperbolic
+equations. Note, however, that BDNK is merely a causal formulation of hydrodynamics, and thus
+BDNK is still not expected to be a good approximation at early times.
+Finally, a rigorous field-theory formulation of hydrodynamics was achieved, which expresses
+it as an EFT based on a generating functional [539–545], for a summary see Ref. [546].
+This
+approach employs the Schwinger-Keldysh formalism of thermal field theory. As applications of
+this formulation, effects of stochastic interactions on hydrodynamic correlation functions [547]
+as well as a theory of non-linear diffusion were derived, taking into account large hydrodynamic
+fluctuations (for example, leading to the dependence of transport coefficients on fluctuations of the
+hydrodynamic fields) [548, 549].
+2.
+Range of applicability
+Many factors influence whether a system may be described hydrodynamically. Most impor-
+tantly, like any EFT, hydrodynamics requires a separation of scales between the microscopic physics
+and the scales on which the system is described. Let us focus on two remarkable results regarding
+the range of applicability of hydrodynamics:
+1) The unreasonable effectiveness of hydrodynamics far away from equilibrium.
+2) Systematic extensions of hydrodynamics, extending its range of applicability.
+Regarding 1), the applicability of hydrodynamics has been historically tied to a requirement
+of a near-equilibrium state, near-isotropy, and small gradients. Astonishingly, heavy-ion collisions,
+where neither of these three conditions is met, were successfully described hydrodynamically, which
+is often referred to as the unreasonable effectiveness of hydrodynamics [550]. As a possible expla-
+nation, hydrodynamic attractors were proposed [551] in a supersymmetric conformal theory, and
+subsequently studied for heavy-ion collisions [552–561].
+The underlying reason for the attrac-
+tor behavior is proposed to be kinematic, i.e., owed to a fast expansion in the boost-invariant
+plasma [554, 562, 563]. Since systems cease to be boost invariant as the collision energy is lowered,
+
+59
+FIG. 26. Example of an extension of the regime of applica-
+bility of hydrodynamics: spin hydrodynamics. While stan-
+dard hydrodynamics is valid at small frequencies and mo-
+menta (labeled as “pure hydro regime”, indicated by cyan
+blue region), in the presence of spin degrees of freedom spin
+hydrodynamics is valid in an extended regime (labeled as
+“spin hydro regime”, indicated by a pink region). This is
+facilitated by adding the slowly relaxing spin modes (green
+curves) to the spectrum of standard hydrodynamic shear
+(red solid curve) and sound (blue solid curve) modes. HY-
+DRO+ is constructed in a similar way by adding modes
+which relax slower and slower when approaching the criti-
+cal point in the QCD phase diagram, bearing implications
+for the EOS [564, 565]. Figure adapted from [566].
+this may pose a challenge to the develop-
+ment of hydrodynamic attractors in nu-
+clear matter at low to intermediate ener-
+gies.
+Within a certain class of models (in-
+spired by the gauge/gravity correspon-
+dence), the position-space hydrodynamic
+expansion (in proper time) around the
+Bjorken flow within the MIS theory was
+shown to diverge factorially [551].
+In
+contrast, for the same theory Fourier-
+transformed into the momentum space
+there is a finite convergence radius lim-
+ited by the branch point singularity closest
+to the origin in the complex momentum
+plane [567–571]. This inspired the formu-
+lation of hydrodynamics far from equilib-
+rium via resummation [572].
+Regarding 2), a standard method to ex-
+tend the regime of validity of hydrodynam-
+ics is to add one or several mode(s). In
+fact, promoting the shear tensor to an aux-
+iliary field (regulator) adds a mode to the
+spectrum of first-order hydrodynamic for-
+mulation yielding the MIS model. In an-
+other crucial example, critical fluctuations
+need to be taken into account near the crit-
+ical point in the QCD phase diagram, and
+a set of slow modes can be added to the hydrodynamic modes yielding HYDRO+ [573, 574], which
+in turn bears implications for the EOS and the speed of sound [564, 565]. Furthermore, Lambda
+hyperon polarization data [575] indicates that the QGP is highly vortical and polarized [576–
+578], which motivated the inclusion of spin in various hydrodynamic descriptions [566, 579–588]
+(see, e.g., Fig. 26). Within a different systematic extension of hydrodynamics, dynamical elec-
+tromagnetic fields can be added, leading to versions of magnetohydrodynamics which couple the
+hydrodynamic conservation equations to Maxwell’s equations [589–591], and which can also include
+the chiral anomaly [592–594], relevant for neutron stars [594]. Finally, another natural extension of
+hydrodynamics is the simultaneous inclusion of multiple conserved charges, in particular, baryon
+number B, strangeness S, and electric charge Q (BSQ charges) [532]. This renders transport coef-
+ficients matrix-valued, which means that gradients in one charge may lead to diffusion of another
+charge [595, 596].
+Modern hydrodynamics has been developed in close relation to the gauge/gravity correspon-
+dence (a.k.a., AdS/CFT or holography). This development, which notably yielded the only consis-
+tent theoretical description of fluids with η/s as low as found in heavy-ion data [597–599], began
+with the insight that a lower bound on entropy production per degree of freedom (η/s) for a cer-
+tain class of theories is related to black branes in the Anti de Sitter (AdS) spacetime [600]. The
+fluid/gravity correspondence [601] as a systematic construction tool led to the discovery of the
+chiral vortical effect [602, 603] and the re-discovery of the chiral magnetic effect [603, 604]. Holo-
+graphic models are also suitable for exploration of plasmas at high densities [605], phase transitions
+(in particular a holographic version of the QCD critical point [606]), neutron stars [607], taking
+
+[w(k)l = frequency scale
+non-conserved
+quantities
+disappear fast
+fast modes
+(transient)
+Non-hydro regime
+T
+IWsound(k)l
+[Wshear(k)|
+approximately
+[Wspin,I (k)
+conserved
+charges diffuse
+[W spin,(k)I
+himois
+spin
+relaxdtion
+rate
+Spin hydro regime
+S
+conserved charges
+diffuse slowly
+Pure hydro regime
+0
+k = wave number60
+into account finite coupling [608, 609], and for investigating the far-from-equilibrium regime of
+holographic plasmas [610–612].
+3.
+Challenges and opportunities
+Given the recent developments described above, there are strong reasons to assume that hydro-
+dynamics either is valid or can be extended to be valid for the description of dense nuclear matter
+at intermediate energy scales, even in small systems with large gradients, far from equilibrium,
+and near the QCD critical point. Such (extended) versions of hydrodynamics may well overlap
+with the regime of validity of hadronic transport simulations, which needs to be studied. Here,
+in particular, further development of hybrid approaches using both hydrodynamics and hadronic
+transport will contribute to a better description of intermediate energy heavy-ion collisions.
+The way ahead will require pushing forward the development of the rigorous theoretical for-
+mulation of hydrodynamics, as well as testing its applicability with exactly solvable models (e.g.,
+constructed using the gauge/gravity correspondence) and, most importantly, against experimental
+data. By continuing the development of hydrodynamics in parallel with gauge/gravity models,
+the proposed versions of spin hydrodynamics can be tested and constructed rigorously using the
+correspondence; the same statement also applies to versions of magnetohydrodynamics. In the
+context of (magneto)hydrodynamics, one may also explore the interplay of multiple conserved
+charge currents and anomalous currents, leading to novel transport phenomena [593, 613–617].
+For an efficient modeling of heavy-ion collisions (as well as neutron stars and neutron star merg-
+ers), the BDNK approach needs to be developed and implemented in standard codes for data
+analysis. At high densities, it becomes necessary to describe the propagation of multiple conserved
+charges, in particular, the BSQ charges [532]. Consequently, the initial state used in numerical
+hydrodynamic simulations must be modified to include BSQ degrees of freedom [618–620]. Sim-
+ilarly, the EOS [621, 622] and the exact charge conservation when particles are formed (see, e.g.,
+Ref. [623, 624]) need to take into account BSQ charges. Beyond describing all conserved charges,
+theoretical consistency on one hand and the need to describe systems far-from-equilibrium on the
+other hand both necessitate a rigorous treatment of hydrodynamic fluctuations, which has been
+done using a deterministic approach to fluctuations [625–628]. As a viable future complementary
+approach, hydrodynamic fluctuations can be included using the Schwinger-Keldysh formulation
+of hydrodynamics [546]. These goals are in line with two recent white papers: Snowmass The-
+ory Frontier: Effective Field Theory Topical Group Summary [629] and Snowmass White Paper:
+Effective Field Theories for Condensed Matter Systems [630].
+VI.
+EXPLORATORY DIRECTIONS
+A.
+Dense nuclear matter EOS meeting extreme gravity and dark matter in supermassive
+neutron stars
+Do we need an independent determination of the nuclear EOS using terrestrial experiments in
+the era of high-precision multi-messenger astronomy? While it is often emphasized that combined
+data analyses of heavy-ion reactions and neutron star observations within a unified EOS theory
+framework are a powerful tool to study the EOS (see Section IV), the independent extraction of
+the nuclear EOS from heavy-ion reactions alone is fundamentally important. This assertion is
+motivated by a well-known degeneracy [631] between the EOS of dense matter (including hadronic
+and/or quark matter, and dark matter) and strong-field gravity in studies aimed at understanding
+
+61
+properties of super-massive neutron stars, the minimum mass of black holes, and properties of dark
+matter [632–639].
+In the Astro2020 Science White Paper on Extreme Gravity and Fundamental Physics [640],
+future gravitational wave (GW) observations are envisioned to enable unprecedented and unique
+science related to
+• The nature of gravity: Can we prove Einstein wrong? What building-block principles and
+symmetries in nature invoked in the description of gravity can be challenged?
+• The nature of dark matter: Is dark matter composed of particles, dark objects, or modifica-
+tions of gravitational interactions?
+• The nature of compact objects: Are black holes and neutron stars the only astrophysical
+extreme compact objects in the Universe? What is the EOS of densest matter?
+An independent determination of the EOS of dense nuclear matter from terrestrial experiments,
+which are free from gravitational effects, will address the question of whether exotic physics, such
+as modified gravity, is necessary to describe the behavior and phenomena of supermassive stars.
+Thus constraining the EOS from heavy-ion collision experiments will help realize the astrophysical
+science goals.
+The fundamental questions listed above are among the eleven greatest physics questions for the
+new century identified by the U.S. National Research Council in 2003 [641]. While gravity was the
+first force discovered in nature, the quest to unify it with other fundamental forces remains elusive,
+partially because of its apparent weakness at short distances [642, 643]. Moreover, while Einstein’s
+general relativity (GR) theory for gravity has successfully passed all observational tests so far, it is
+still not fully tested in the strong-field domain [644]. Searches for evidence of possible deviations
+from GR are at the forefront of several fields in natural sciences. It is fundamentally important to
+test whether GR will break down at the strongest possible gravitational fields reachable. For this
+goal, supermassive neutron stars are among the ideal testing sites [645, 646]. However, as already
+mentioned above, their properties can be accounted for by either modifying gravity, adding dark
+matter, and/or adjusting the nuclear EOS. Thus, an independent inference of the nuclear EOS
+from terrestrial experiments is fundamentally important for breaking the degeneracy between the
+EOS of supermassive neutron stars and the strong-field gravity.
+There are already some indications that the EOS of dense neutron-rich matter may play a
+significant role in understanding the nature of gravity [647–650]. Effects of the nuclear symmetry
+energy on the gravitational binding energy [651], surface curvature, and red shift [652], which are
+normally used to measure the strength of gravity of massive stars in GR, as well as examples of
+mass-radius relations in several classes of modified gravity theories are reviewed briefly in Ref. [653].
+More precise information about the dense nuclear matter EOS from terrestrial experiments will
+enable further progress in this direction.
+B.
+Nuclear EOS with reduced spatial dimensions
+Nuclear systems under constraints, with high degrees of symmetry and/or collectivity, may
+be considered as effectively moving in spaces with reduced spatial dimensions.
+Historically, in
+developing modern methodologies, the spatial dimension d has been considered to be either a
+continuous or a discrete variable. Many exciting and fundamentally new experimental discoveries
+in reduced dimensions have been made in recent years in material sciences (e.g., the graphene [654]
+and topological insulator [655, 656]) and cold atom physics (e.g., the superfluidity in a strongly
+correlated 2D Fermi gas [657] and the generalized hydrodynamics in a strongly interacting 1D Bose
+gas [658]).
+
+62
+The EOS of neutron-rich matter in spaces with reduced dimensions can be linked to that in the
+conventional 3-dimensional (3D) space by the ϵ-expansion (ϵ = d − 4) [659–661]. The latter is a
+perturbative approach that has been successfully used in treating second-order phase transitions
+and related critical phenomena in solid state physics and, more recently, in studying the EOS of
+cold atoms in 1D, 2D, and two-species Fermi and/or Bose gases with mixed dimensions [662, 663].
+The energy per nucleon E(nB, δ, d) in cold nuclear matter of dimension d at density nB and isospin
+asymmetry δ can be expanded around nB = n0, δ = 0, and d = 3. In cold symmetric nuclear
+matter, the E(nB, δ = 0, d) is predicted to decrease with decreasing d, indicating that nuclear
+matter with a smaller d tends to be more bound but, at the same time, saturates at a higher
+3D-equivalent density. The symmetry energy was also found to become smaller in spaces with
+lower dimensions compared to the conventional 3D case [664].
+Can we find or make 1D and/or 2D nuclear systems in our 3D world? Can nucleons in neutron-
+skins of heavy nuclei be considered as living in spaces with reduced spatial dimensions, and if so,
+can we discover the related effects in heavy-ion collisions? Can some of the objects (e.g., lasagna) in
+the predicted pasta phase [291, 312, 665] of the neutron star crust be described as nuclear systems
+with 1D, 2D, or fractional dimensions? What are the roles of the dimension-dependent EOS in
+multi-dimensional models of late stellar evolution? If 1D/2D simulations using 3D EOS do not
+lead to supernova explosions, what will happen if the corresponding 1D/2D EOSs are used instead?
+Answers to these questions may provide new perspectives on the EOS of neutron-rich matter in
+3D and help solve some of the unresolved puzzles.
+C.
+Interplay between nucleonic and partonic degrees of freedom: SRC effects on nuclear
+EOS, heavy-ion reactions, and neutron stars
+Short-range correlations (SRCs) in nuclei, that is correlations in the nuclear ground-state wave
+function, are mostly due to isosinglet neutron-proton pairs that have temporally fluctuated into
+a high-relative-momentum state with approximately zero total center-of mass-momentum and a
+spatial separation of about 1 fm [128, 129, 666–669]. Subnucleonic degrees of freedom are expected
+to play an important role in understanding SRC-related phenomena. Altered quark momentum
+distributions in nucleons embedded in nuclei with respect to those in free nucleons, known as
+the EMC effect, have been studied extensively since 1983 [670].
+SRCs have been proposed as
+one of the two leading causes of the EMC effect [671, 672]. Recent experiments found that the
+strength of SRCs and the EMC effect are strongly correlated [673, 674] and that they both depend
+strongly on the isospin asymmetry of the nuclei. Moreover, strong evidence was found that only
+the momentum distributions of quarks in SRC nucleon pairs in nuclei are modified with respect
+to free nucleons. Furthermore, the distributions of quarks in protons of neutron-rich nuclei are
+modified more than in neutrons, implying that, on average, u quarks are modified more than d
+quarks in neutron-rich nuclei [674], in analogy to an earlier finding that SRCs make protons mover
+faster than neutrons in neutron-rich nuclei [675]. These phenomena reflect profound QCD effects
+in the nuclear medium. Studying the flavour- and spin-dependence of nucleon structure functions
+is at the forefront of QCD and is a major science driver of future EIC experiments. An example
+of a correlation formed on short-range QCD length scales are quark-quark correlations known as
+diquarks. It was recently proposed that diquark formation across two nucleons via the attractive
+QCD quark-quark potential is the underlying QCD-level source of SRCs in nuclear matter and the
+cause of the EMC effect [676].
+The SRC-related effects have consequences for the nuclear structure, high-energy quark dis-
+tributions (the EMC effect), and high-density nuclear matter, including its EOS and in-medium
+nucleon-nucleon scattering cross sections. Better understanding of SRC effects in dense neutron-
+
+63
+rich medium through heavy-ion reactions may have important ramifications. The profound conse-
+quences of SRCs on the nuclear matter EOS can be easily seen when one considers the well-known
+Hugenholtz-Van Hove (HVH) theorem, which was derived by assuming there are sharp Fermi sur-
+faces for nucleons. The theorem provides intrinsic connections among the nuclear symmetry energy,
+momentum dependences of both isoscalar and isovector nucleon potentials, and the corresponding
+nucleon isoscalar effective mass and neutron-proton effective mass splitting in neutron-rich mat-
+ter [677]. However, due to the SRCs nucleons do not have sharp Fermi surfaces, but extended
+high-momentum tails, as evidenced by many experiments at the Jefferson Laboratory (JLAB) and
+Brookhaven National Laboratory (BNL), see, e.g., Refs. [127–130] for recent reviews. Therefore,
+the above relations may be completely altered by the isospin-dependence of SRCs induced by the
+nuclear tensor force, which is much stronger in the symmetric nuclear matter than in pure neu-
+tron matter [135, 678, 679]. Consequently, the composition of the symmetry energy itself (e.g.,
+the ratio of its kinetic over potential parts) may also be very different from the one without con-
+sidering the SRCs [131]. Most of the parametrizations of the nuclear symmetry energy used so
+far in both nuclear physics and astrophysics adopt the kinetic symmetry energy predicted by the
+free Fermi gas model. However, such parametrizations neglect SRC effects that may lead to a
+reduced or even negative kinetic symmetry energy. This effect originates in the fact that SRCs
+are dominated by isosinglet neutron-proton pairs. As the system becomes more neutron-rich, an
+increasingly larger fraction of protons, compared to neutrons, are found in the high momentum
+tail [128, 129, 674]. Consequently, the kinetic symmetry energy is reduced compared to the free
+Fermi gas model prediction [131–138].
+Interesting indications have been found very recently of SRC effects on the cooling of protoneu-
+tron stars, the formation of baryon resonances, dark matter, and nuclear pasta as well as on tidal
+deformation and mass-radius correlation in neutron stars [680–687], and also on several features of
+nuclear matter and heavy-ion reactions [688–695]. However, much more work remains to be done
+to systematically and consistently address the SRC-related issues in hadronic transport simulations
+(see Section II A). Investigations of SRC effects on the nuclear EOS using heavy-ion collisions at
+FRIB and FRIB400 will complement the ongoing and planned SRC research programs at JLAB,
+GSI, and EIC at BNL. Together, these efforts will reveal new knowledge about the spin-isospin
+dependence of three-body and tensor forces in dense neutron-rich matter. At short distances, these
+forces are mostly due to the ρ-meson exchange [696]. The in-medium ρ-meson mass, determined
+by QCD, may be significantly different from its free-space value [696–698]. Such modification in
+the ρ-meson mass has been found to significantly affect the high-density behavior of the nuclear
+symmetry energy [131, 699]. However, effects of the QCD quark-quark potential and the modified
+tensor or three-body force on in-medium nucleon-nucleon cross sections remain to be explored.
+D.
+High-density symmetry energy above 2n0
+Section III B has primarily focused on the physics of nuclear symmetry energy up to ≈ 2n0. This
+is because of substantial experimental challenges for measuring the symmetry energy using more
+energetic beams.
+However, at higher densities, but below the hadron-quark transition density,
+there are also many interesting issues to be addressed [97, 705]. Therefore, it is worthwhile to
+explore possible future directions to attack this problem (see also the white paper on QCD Phase
+Structure and Interactions at High Baryon Density: Continuation of BES Physics Program with
+CBM at FAIR [151]).
+Experiments at FRIB400, FAIR, and other high-energy rare isotope beam facilities around the
+world are expected to provide tremendous resolving power for determining the symmetry energy
+at densities >∼ 2n0. While both the magnitude Esym(n0) and slope L of the symmetry energy
+
+64
+1
+2
+3
+0
+30
+60
+90
+120
+150
+ Neutron Star 
+          Observations
+Esym (MeV)
+ρ�
+ρ0
+ NL-RMF(18)
+ PC-RMF(3)
+ RHF(2)
+ DD-RMF(2)
+ Gogny-HF(2)
+ SHF(33)
+�
+�
+�
+�
+ BHF (Vidana)
+ BHF (Z.H. Li)
+ DBHF (Fuchs)
+ DBHF (Sammarruca)
+ Chiral EFT-N3LO450
+ Chiral EFT-N3LO600
+ VMB-APR
+ VMB-FP
+ VMB-WFF1
+ VMB-WFF2
+ VMB-WFF3
+FIG. 27. Left: Symmetry energy as a function of baryon density as obtained within 60 example models,
+selected from 6 classes of over 520 phenomenological models and/or energy density functionals.
+Right:
+Symmetry energy as a function of baryon density as obtained within 11 examples from microscopic and/or
+ab initio theories. Thick blue lines are the upper and lower boundaries of symmetry energy from analyses
+of neutron star observables. Figure from Refs. [700–704].
+at n0 have been relatively well determined (Esym(n0) ≈ 31.7 ± 3.2 MeV and L ≈ 58 ± 19 MeV
+[506, 700, 706, 707], in very good agreement with χEFT calculations [13, 166, 206, 253]), the
+curvature Ksym and skewness Jsym of the nuclear symmetry energy are still poorly known. In
+particular, Ksym is most critical for determining the crust-core transition density and pressure in
+neutron stars [708–710]. Besides the importance for astrophysics, an experimental determination
+20
+30
+40
+50
+60
+70
+80
+Esym(2ρ0) (MeV)
+FOPI-LAND (2011)
+ASY-EOS (2016)
+Xie & Li (2019)
+Zhang & Li (2019)
+Zhou et al. (2019)
+d’Etivaux (2019)
+Nakazat & Suzuki (2019)
+Symmetry energy at 2ρ0 from analyzing terrestrial and astrophysical data
+2021 fiducial value=51 ± 13 MeV
+Xie & Li (2020)
+Tsang et al. (2020)
+Yue et al.(2021)
+Heavy-ion reactions
+Neutron stars
+Zhang et al. (2020)
+FIG. 28.
+Symmetry energy at twice the saturation
+density from both heavy-ion reactions and neutron
+stars. Figure modified from figures in Refs. [700, 711].
+of the high-density behavior of the nuclear sym-
+metry energy will provide important guidance
+for developing high-density nuclear many-body
+theories.
+Indeed, the density region explored
+in heavy-ion reactions at BES, HADES, and in
+the future at FRIB400 and FAIR is mostly be-
+yond the current validity range of χEFT, and
+it is also where the EOSs predicted by vari-
+ous nuclear many-body theories, especially the
+symmetry energy contributions, start to diverge
+broadly (see Fig. 27).
+Recent neutron star observations have led
+to some progress in constraining the symme-
+try energy at suprasaturation densities. Shown
+in Fig. 28 is a compilation of recent results on
+the symmetry energy at 2n0 from two analyses
+
+65
+of heavy-ion reactions at GSI and nine independent analyses of neutron star properties by sev-
+eral groups. These analyses give a mean value of Esym(2n0) ≈ 51 ± 13 MeV at 68% confidence
+level, as indicated by the green line. Interestingly, χEFT+MBPT calculations predict a value of
+Esym(2n0) ≈ 45±3 MeV [158]. Similarly, quantum Monte Carlo calculations using local interactions
+derived from the χEFT up to next-to-next-to-leading order predict a value of Esym(2n0) ≈ 46 ± 4
+MeV [184]. Evidently, the mean value of Esym(2n0) from the analyses mentioned above is consis-
+tent with the χEFT predictions, albeit with large uncertainties. As noted before, 2n0 is near the
+upper validity limit of the current χEFT theories. Thus, more precise measurements of Esym(2n0)
+will help to test χEFT predictions.
+Inspecting the results shown in Fig. 28 shows clearly that more work is necessary to reduce the
+error bars. Most of the neutron star constraints are extracted from radii and tidal deformations of
+canonical neutron stars with masses around 1.4M⊙. These observables are known to be sensitive
+mostly to the values of pressure around (1-2)n0 in neutron stars, and therefore their constraints
+on Esym(nB) around and above 2n0 are not strong.
+Observables from more massive neutron stars were expected to place stronger constraints on
+the high-density symmetry energy. To illustrate how the recent NICER+XMM-Newton’s mea-
+surements of both the radius and mass of PSR J0740+6620 can influence the constraint on the
+symmetry energy at densities above 2n0, the upper panel of Fig. 29 shows the extracted lower
+limits of Esym(nB) obtained from directly inverting the TOV equation within a 3-dimensional
+high-density EOS parameter space [238] for two cases: for the case where only the mass is ob-
+served (green line), and for the case where both the mass and radius are observed (red line using
+the 68% confidence lower radius limit reported by Riley et al. [264] and blue line using the radius
+reported by Miller et al. [255]). The orange line is the upper limit of the symmetry energy from
+0
+20
+40
+60
+80
+100
+120
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+0.00
+0.03
+0.06
+0.09
+0.12
+ Causality & M
+max
+=2.01 M
+sun
+ Causality & R
+2.01
+=11.41 km
+ Causality & R
+2.01
+=12.2 km
+ Causality & L
+1.4
+=427
+ 
+ 
+E
+sym
+ [MeV]
+11.1%
+ 
+ 
+X
+p
+r/r0
+FIG. 29. Constraints on the high-density sym-
+metry energy and proton fraction in neutron
+stars from analyzing the tidal polarizability of
+GW170817 and NICER’s observation of PSR
+J0740+6620. Figure from Ref. [238].
+analyzing the upper limit (68% confidence) of tidal
+deformation of GW170817 [236]. The upper limits
+of the symmetry energy from the upper radius limits
+reported by both Riley et al. and Miller et al. are
+far above the upper limit of symmetry energy from
+GW170817.
+The lower panel in Fig. 29 shows the correspond-
+ing proton fractions in PSR J0740+6620. The in-
+fluence of knowing both the mass and radius of
+this most massive neutron star currently known is
+seen by comparing the green line with the red or
+blue line, while the difference between the red and
+blue lines indicates the systematic error from the
+two independent analyses of the same observational
+data. Although the estimates of Esym(nB) around
+(2-3)n0 from these analyses are useful compared to
+the model predictions shown in Fig. 27, much more
+precise constraints on the Esym(nB) above 2n0 are
+needed.
+Pinning down the symmetry energy above 2n0
+will be very challenging, but achieving this goal will
+bring a great reward. For example, without a reli-
+able knowledge of the symmetry energy at suprasat-
+uration densities, the density profile of the proton
+fraction in the core of neutron stars (which has to
+be higher than about 11% for the fast cooling to oc-
+
+66
+cur) at β−equilibrium is not determined. Consequently, whether the fast cooling of protoneutron
+stars occurs through the direct URCA process remains uncertain. Heavy-ion reactions, especially
+with high-energy radioactive beams, will provide the much-needed data to calibrate nuclear many-
+body theories and constrain nuclear symmetry energy at densities >∼ 2n0. These efforts, in concert
+with astrophysical research using high-precision X-rays from massive neutron stars (e.g., NICER
+and STROBE-X [317]), GWs from new LIGO/VIRGO runs, and future detection of post-merger
+high-frequency GWs, will better constrain Esym(nB) at densities around and above 2n0. For these
+efforts to be fruitful, it is imperative to explore potential observables carrying undistorted infor-
+mation on the symmetry energy above 2n0 from neutron stars and their mergers as well as in
+high-energy heavy-ion reactions.
+E.
+Density-dependence of neutron-proton effective mass splitting in neutron-rich matter
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+<ρ/ρ0>
+0.6
+0.7
+0.8
+0.9
+1
+<m
+*
+τ> (GeV)
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+m
+*
+τ (GeV)
+0
+200
+400
+600
+dN/dm (GeV)
+−1
+n
+p
+n
+p
+132Sn+
+124Sn, E/A=50 MeV, b=5 fm
+t=10 fm/c
+FIG. 30.
+Correlation between the average nu-
+cleon effective mass and the average nucleon
+density (top), and the distribution of nucleon
+effective masses (bottom) in the reaction of
+132Sn+124 Sn at 10 fm/c with a beam energy
+Elab = 50 AMeV and an impact parameter
+b = 5 fm, as simulated within the IBUU trans-
+port model with an explicitly isospin-dependent
+single-nucleon potential. Figure from Ref. [712].
+The nucleon effective mass is a fundamental
+quantity characterizing the propagation of a nucleon
+in a nuclear medium [713–716], accounting (to lead-
+ing order) for effects such as the space-time non-
+locality of the effective nuclear interactions or Pauli
+exchange effects.
+The magnitude and sign of the
+difference (splitting) between the effective masses
+of neutrons and protons ∆m∗
+np have essential con-
+sequences for cosmology, astrophysics, and nuclear
+physics through influencing, e.g., the equilibrium
+neutron to proton ratio in the early universe and
+primordial nucleosynthesis [717], properties of mir-
+ror nuclei [718], and the location of drip-lines [719].
+In heavy-ion reactions, ∆m∗
+np is of importance for
+isospin-sensitive observables [100, 107, 720–725].
+The momentum-dependence of the single-nucleon
+potential is normally characterized by the nucleon
+effective mass m∗
+τ that can be decomposed into an
+isoscalar and an isovector component [108, 726, 727].
+Due to our poor knowledge of the momentum de-
+pendence of isovector interactions, the isovector nu-
+cleon effective mass measured by using the neutron-
+proton effective mass splitting ∆m∗
+np [706] has not
+been constrained well [108, 728]. Based on the HVH
+theorem, ∆m∗
+np was found approximately propor-
+tional to the isospin asymmetry δ of the medium,
+with a coefficient depending on the density as well
+as momentum-dependence of both the isoscalar and
+isovector nucleon potential [706]. Over the last decade, significant efforts have been made to extract
+this coefficient at n0. A recent survey [729] of model analyses using data from mostly nucleon-
+nucleus scattering and giant resonances of heavy nuclei suggests that the ∆m∗
+np, scaled by the
+average nucleon mass in free space, ranges from 0 to about 0.5δ [208, 677, 706, 730–735].
+While experimental efforts to better constrain ∆m∗
+np at n0 using heavy-ion reactions with in-
+termediate energy stable beams are ongoing (see, e.g., Ref. [98]), future experiments at FRIB
+and FRIB400 will enable more sensitive probes of not only ∆m∗
+np at n0, but also of its density-
+
+67
+FIG. 31. Top: Density dependence of the nuclear sym-
+metry energy for two typical symmetry energy func-
+tionals used in the IBUU simulations. Bottom: Density
+dependence of the isospin asymmetry δ in 132Sn+124
+Sn collisions at 20 fm/c with a beam energy of 400
+MeV/A and an impact parameter of 1 fm, and in the
+core of neutron stars at β-equilibrium (inset). Taken
+from Refs. [95, 736].
+dependence (which cannot be probed by the
+nucleon-nucleus
+scattering
+and
+giant
+reso-
+nances) up to about 2n0.
+As an illustration,
+shown in Fig. 30 are the density dependence
+of the average nucleon effective mass (top)
+and the distribution of the nucleon effective
+masses (bottom) during a typical FRIB reac-
+tion as simulated [712] within the IBUU trans-
+port model with an explicitly isospin-dependent
+single-nucleon potential [106, 720].
+From the
+top panel, it is seen that the neutron-proton ef-
+fective mass splitting is positive and increases
+with the density up to about 1.3n0, consistent
+with recent χEFT calculations [208]. To reach
+higher densities, more energetic beams are re-
+quired.
+Heavy-ion reactions at FRIB400 will extend
+the ranges of both density and isospin asymme-
+try of the medium formed. Shown in the lower
+panel of Fig. 31 are the isospin asymmetry δ
+as a function of density during a typical reac-
+tion at FRIB400 (main) and in neutron stars
+at β-equilibrium (inset), calculated using the
+same two typical symmetry energy functionals,
+shown in the upper panel. The δ-nB relations in
+both systems show the same isospin fractiona-
+tion phenomena, e.g., reaching a higher isospin
+asymmetry when a density functional with a
+lower symmetry energy is used. One can also
+see that generally, the low-density regions are
+more neutron-rich than the high-density regions. These δ-nB relations are the fundamental origins
+of all isospin-sensitive observables in both heavy-ion reactions and neutron stars.
+A number of observables in heavy-ion reactions have been proposed as promising messengers of
+the underlying momentum-dependence of the isovector potential and the corresponding neutron-
+proton effective mass splitting, see, e.g., [20, 737, 738] for reviews. The momentum dependence of
+the single-nucleon potential affects the reaction dynamics directly through the equations of motion
+and indirectly through the scattering term of nucleons. As the in-medium nucleon-nucleon cross
+section is proportional to the square of the reduced effective mass of the two colliding nucleons,
+the nucleon effective mass will affect the nuclear stopping power (which is also described in the
+literature in terms of the nucleon mean free path especially for nucleon-nucleus scattering) [739].
+Consequently, the reaction dynamics and observables of heavy-ion reactions are expected to bear
+useful information about the density-dependence of the neutron-proton effective mass splitting in
+neutron-rich matter. The challenge is to find such observables that are both robust and sensitive
+to the variations of the neutron-proton effective mass splitting with density. The nucleon effective
+mass affects also transport properties of neutron stars, see, e.g., Refs. [233, 740–744]. Neutron
+star observables, e.g., neutrino emission and torsional oscillations of neutron stars, may also pro-
+vide useful information about the density-dependence of neutron-proton effective mass splitting in
+neutron-rich matter. Explorations of these issues are invaluable.
+
+68
+ACKNOWLEDGMENTS
+This White Paper has benefited from talks and discussions at the workshop on Dense nuclear
+matter equation of state in heavy-ion collisions that took place at the Institute for Nuclear Theory,
+University of Washington (December 5-9, 2022).
+K.A. thanks Hans-Rudolf Schmidt and Arnaud Le F`evre, and M.S. thanks Daniel Cebra for
+insightful discussions. P.D. and B.T. thank Abdou Chbihi, Maria Colonna, Arnaud Le F`evre, and
+Giuseppe Verde for discussing complementary international efforts.
+K.A. and M.S. thank Peter Senger and Richard Seto for helpful comments on the draft of
+Section III A. M.K. thanks Navid Abbasi, David Blaschke, Casey Cartwright, Saso Grozdanov,
+and Misha Stephanov for helpful comments on the draft of Section V B.
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+page_content=' Lawrence Livermore  National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Livermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' CA 94551,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' USA and  Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Davis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' CA 95616,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' USA  Hermann Wolter  Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' D-85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Germany  Hanna Zbroszczyk  Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Warsaw University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Koszykowa 75,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Poland  Endorsing authors:  Anton Andronic  Westf¨alische Wilhelms-Universit¨at M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Institut f¨ur Kernphysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 48149 M¨unster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Germany  Steffen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Bass  Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Durham NC 27708  Abdelouahad Chbihi  GANIL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Boulevard Henri Becquerel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' F-14076 Caen Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' France  Maria Colonna  INFN-LNS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Laboratori Nazionali del Sud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 95123 Catania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Italy  Mircea Dan Cozma  IFIN-HH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Reactorului 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 077125 Mˇagurele-Bucharest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Romania  Veronica Dexheimer  Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Kent State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Kent OH 44242 USA  Xin Dong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Jørgen Randrup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and Nu Xu  Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' CA 94720  Travis Dore  Fakult¨at f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Universit¨at at Bielefeld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' D-33615 Bielefeld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Germany  Lipei Du  Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' McGill University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Montreal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Quebec H3A 2T8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Canada  Steven P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Harris, Larry McLerran, and Sanjay Reddy  Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA  Huan Zhong Huang  University of California, Los Angeles, CA 90095  3  Jos´e C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Jim´enez  Instituto de F´ısica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1371, 05508–090 S˜ao Paulo-SP, Brazil  Joseph Kapusta  School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 USA  Arnaud Le F`evre, Christian Sturm, and Wolfgang Trautmann  GSI Helmholtz Centre for Heavy-ion Research, Planckstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1, 64291 Darmstadt, Germany  Jacquelyn Noronha-Hostler  University of Illinois at Urbana-Champaign, Urbana, IL 61801  Christopher Plumberg  Natural Science Division, Pepperdine University, Malibu, CA 90263, USA  Hans-Rudolf Schmidt  Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany and  GSI Helmholtz Centre for Heavy-ion Research, Planckstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1, 64291 Darmstadt, Germany  Peter Senger  Facility for Antiproton and Ion Research, Planckstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1, Darmstadt, Germany  Richard Seto  University of California-Riverside, Riverside, California 92521, USA  Chun Shen  Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA and  RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA  Jan Steinheimer  Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1, D-60438 Frankfurt am Main, Germany  Joachim Stroth  Institut f¨ur Kernphysik, Goethe-Universit¨at, 60438 Frankfurt, Germany and  GSI Helmholtz Centre for Heavy-ion Research, Planckstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 64291 Darmstadt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Germany  Kai-Jia Sun  Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 200438,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='China  Giuseppe Verde  INFN Sezione di Catania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 64 Via Santa Sofia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' I-95123 Catania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Italy  Volodymyr Vovchenko  Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' University of Houston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Box 351550,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Houston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' TX 77204,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' USA  (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 2023)  4  Executive Summary  The nuclear equation of state (EOS) is at the center of numerous theoretical and experimental  efforts in nuclear physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' motivated by its crucial role in our understanding of the properties of  nuclear matter found on Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' in neutron stars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and in neutron-star mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' With advances in  microscopic theories for nuclear interactions, the availability of experiments probing nuclear matter  under conditions not reached before, and the advent of multi-messenger astronomy, the next decade  will bring new opportunities for determining the nuclear matter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Profound questions challenging our understanding of strong interactions remain  unanswered: It is still unknown whether the transition between a hadronic gas and a  quark-gluon plasma, which at zero baryon density is known to be consistent with a crossover  transition predicted by Lattice QCD, becomes of first order in the finite-density region of  the QCD phase diagram accessible in terrestrial experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The isospin-dependence of  the EOS, crucial to our understanding of both the structure of neutron-rich nuclei and the  properties of neutron stars, is poorly known above nuclear saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover,  recent observations of very heavy compact stars indicate that the EOS in neutron-rich mat-  ter becomes very stiff at densities of the order of a few times saturation density, leading to  values of the speed of sound exceeding 1/  √  3 of the speed of light (breaking the conformal  limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Not only is the mechanism behind this striking behavior not known, but it is also  unknown whether a similar stiffening occurs in symmetric or nearly-symmetric nuclear mat-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Resolving these and other questions about the properties of dense nuclear  matter is possible by taking advantage of the unique opportunities for studying  the nuclear matter EOS in heavy-ion collision experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Among controlled terrestrial experiments, collisions of heavy nuclei at interme-  diate beam energies (from a few tens of MeV/nucleon to about 25 GeV/nucleon  in the fixed-target frame) probe the widest ranges of baryon density and tem-  perature, enabling studies of nuclear matter from a few tenths to about 5 times the nuclear  saturation density and for temperatures from a few to well above a hundred MeV, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the next decade, numerous efforts worldwide will be devoted to uncovering the  dense nuclear matter EOS through heavy-ion collisions, including studies at FRIB where  the isospin-dependence of the EOS can be probed in energetic collisions of rare isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Modern detectors and refined analysis techniques will yield measurements that  will elucidate the dependence of the EOS on density, temperature, and isospin  asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Hadronic transport simulations are currently the only means of interpreting  observables measured in heavy-ion collision experiments at intermediate beam  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This means that capitalizing on the enormous scientific effort aimed at uncovering  the dense nuclear matter EOS, both at RHIC and at FRIB, depends on the continued  development of state-of-the-art hadronic transport simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Support for the hadronic  transport community, and in particular for viable career pathways for early  career researchers, is imperative to maintain the health of and diversify the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  hadronic transport community, and to fully realize the potential of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' efforts  leading the exploration of the dense nuclear matter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  5  CONTENTS  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Introduction  7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Constraining the nuclear matter EOS using heavy-ion collisions  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Connections to fundamental questions in nuclear physics  9  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Upcoming opportunities  11  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Needs  12  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The equation of state from 0 to 5n0  13  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Transport model simulations of heavy-ion collisions  13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Transport theory  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Selected constraints on the EOS obtained from heavy-ion collisions  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges and opportunities  19  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Microscopic calculations of the EOS  25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Status  25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges and opportunities  27  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Neutron star theory  28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Status  28  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges and opportunities  31  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Heavy-ion collision experiments  33  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments to extract the EOS of symmetric nuclear matter  35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Measurements sensitive to the EOS  35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments probing densities between 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments probing densities above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  38  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges and opportunities  39  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments to extract the symmetry energy  42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments that probe low densities  42  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Measurements to extract symmetry energy up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  42  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Selected constraints on the symmetry energy around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  44  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges and opportunities  46  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The equation of state from combined constraints  50  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Constraints  51  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' EOS obtained by combining various constraint sets  53  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Connections to other areas of nuclear physics  54  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Applications of hadronic transport  54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Detector design  55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Space exploration, radiation therapy, and nuclear data  55  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Hydrodynamics  57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Status  57  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Range of applicability  58  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges and opportunities  60  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Exploratory directions  60  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Dense nuclear matter EOS meeting extreme gravity and dark matter in supermassive  neutron stars  60  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Nuclear EOS with reduced spatial dimensions  61  6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Interplay between nucleonic and partonic degrees of freedom: SRC effects on nuclear  EOS, heavy-ion reactions, and neutron stars  62  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' High-density symmetry energy above 2n0  63  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Density-dependence of neutron-proton effective mass splitting in neutron-rich matter  66  Acknowledgments  68  References  68  7  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  INTRODUCTION  The equation of state (EOS) is a fundamental property of nuclear matter, describing its emergent  macroscopic properties originating from the underlying strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Around the saturation  density of nuclear matter, the EOS controls the structure of nuclei through the binding energy and  the incompressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The EOS also determines, among other things, the neutron-skin thickness  in neutron-rich nuclei as well as the properties of nuclear matter at extreme densities and/or tem-  peratures, corresponding to conditions produced in experiments colliding heavy nuclei or observed  in neutron stars and neutron star mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Far beyond describing the properties of matter com-  posed of only protons and neutrons, the EOS can also reflect the appearance of new degrees of  freedom, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', strange particles in the cores of neutron stars or quarks and gluons in ultrarelativistic  heavy-ion collisions, or the emergence of new states of matter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', chirally-restored matter, meson  condensates, or quarkyonic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In heavy-ion collision experiments, the EOS is studied by detecting particles emerging from the  collision zone and measuring observables sensitive to the properties of nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Crucially,  any interpretation of these observables, including quantitative constraints on the EOS, requires  comparisons of experimentally measured observables to results obtained in dynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This white paper highlights the essential role of hadronic transport simulations of  heavy-ion collisions in advancing our understanding of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It also elucidates the  many connections between inferences of the EOS from heavy-ion collision data and  other efforts aiming to describe and understand the properties of nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Schematic depiction of the ranges of density and temperature probed in experiments and astronom-  ical observations sensitive to the EOS of nuclear matter (counterclockwise from bottom left): neutron star  crust physics, including nuclear pasta structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' properties of nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' structure of neutron stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' dynamics of  neutron star mergers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and outcomes of heavy-ion collisions which can probe both symmetric and asymmetric  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figures adapted from [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  100  HIC (sym)  temperature [MeV]  10  HIC (asym)  S  ROOKHAVEN  NS mergers  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='NS crust  nuclear  properties  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  Z  neutron stars (NS)  0  1  2  3  4  5  density np/no8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Constraining the nuclear matter EOS using heavy-ion collisions  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Constraints on the zeroth (Sv) and  first (L) coefficient of the symmetry energy ex-  pansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experimental constraints are derived  from heavy-ion collisions (HIC) [6], neutron-skin  thicknesses of Sn isotopes [7], giant dipole res-  onances (GDR) [8], the dipole polarizability of  208Pb [9, 10], nuclear masses [11], and isovector  skins (IAS+∆R) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Also shown are constraints  from χEFT (GP-B) [13], microscopic neutron-  matter calculations (H, G) [14, 15], and from the  unitary gas limit (UG) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The last decade has brought tremendous progress  in extracting the EOS as a function of baryon den-  sity nB, temperature T, and the isospin asymme-  try δ (or, equivalently, the proton fraction) from  a variety of experimental and astronomical data as  well as theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Many-body theory,  based on sophisticated approaches with input from  nucleon scattering or nuclear structure data, can  now state the EOS below and near the saturation  density n0 with meaningful uncertainties (see Sec-  tion II B, “Microscopic calculations of the EOS”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  New classes of experiments have extracted the thick-  ness of neutron skins in nuclei, shedding light on the  isospin-dependence of the EOS (or, equivalently, the  symmetry energy) near or below n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  High-energy  heavy-ion collisions have constrained the EOS of the  quark-gluon plasma at high temperatures and small  baryon densities, while ongoing experimental efforts  worldwide focus on the EOS of nearly-symmetric  dense baryonic matter, probed in collisions at in-  termediate energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Meanwhile, collisions at lower  energies have led to experimental constraints on the  symmetry energy at sub- and suprasaturation den-  sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Most remarkably, a revolution in the qual-  ity and breadth of astronomical observations, high-  lighted by the first simultaneous detection of grav-  itational waves and electromagnetic signals from a  neutron-star merger, ushered in a new era of multi-  messenger astronomy (see Section II C, “Neutron  star theory”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Together with the newly available ex-  perimental capabilities at the Facility for Rare Iso-  tope Beams (FRIB), there are unprecedented oppor-  tunities to probe the isospin-dependence of the EOS  through astronomical and terrestrial measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Among the experimental efforts discussed above, heavy-ion collisions probe the widest range  of baryon densities and, moreover, represent the only means to address the EOS away from n0  in controlled terrestrial experiments, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Indeed, heavy-ion reactions at beam energies  from a few tens of MeV/nucleon to about 25 GeV/nucleon in the fixed-target frame probe the  EOS of hadronic matter at baryon densities from a few tenths to about 5 times n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Controlling the  properties of matter produced in these experiments is possible by varying the beam energy, collision  geometry, and isotopic composition of the target and projectile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Insights and constraints obtained  from transport model analyses of these experiments are relevant both for our understanding of  nuclear matter as found on Earth and for our understanding of neutron stars from crust to core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Within ongoing efforts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' the STAR experiment’s Beam Energy Scan (BES) fixed-target (FXT)  program at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory  (BNL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' which collided gold nuclei at intermediate beam energies and which completed data taking  in 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' leads the US effort to constrain the EOS of nearly-symmetric nuclear matter at high  100  Constraints on S-L  HIC  80  △R  X  AS  60  GP-B  G  40  H  Masses  pb  20  Skin  UG  Analytic  UG  GDR  0  26  28  30  32  34  Symmetry Energy S [MeV9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  Prior  Astro + HIC  Pressure P (MeV fm–3)  100  101  102  Number density n (nsat)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Pressure in neutron star matter as  a function of density from a Bayesian analysis  combining nuclear theory and data from multi-  messenger neutron-star observations and heavy-  ion collisions [17]: the dark blue and light blue  region corresponds to the 68% and 95% credible  interval, respectively, while the gray dashed line  shows the 95% bound obtained in χEFT calcu-  lations and used as a prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  baryon densities up to around 5n0, corresponding  to densities present in the deep interiors of neu-  tron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Among comparable efforts in Europe,  the HADES experiment at GSI, Germany, probes  matter at densities up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Preliminary results  from these contemporary efforts, as well as measure-  ments from other heavy-ion collision experiments  in the past, have led to competitive constraints on  the EOS of symmetric nuclear matter, with future  measurements expected to shed more light on its  high-density behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Detailed constraints on the  isospin-dependence of the EOS can be obtained by  varying the isospin content of the target and pro-  jectile nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Here, the ability to use radioactive  isotopes, as in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', intermediate-energy heavy-ion  collision experiments at RIKEN and FRIB, is cru-  cial to resolve the subtle effects arising from changes  in the isospin asymmetry of the colliding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Above all, obtaining constraints on the  EOS from heavy-ion measurements would not  have been possible if not for advances in theory, and in particular for the collaborative  effort to test the robustness and quantify the uncertainties of hadronic transport sim-  ulations (see Section II A, “Transport model simulations of heavy-ion collisions”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At the same  time, much remains to be learned, as tight constraints on both the symmetric and asymmetric  EOS at higher densities have so far remained elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This is predominantly due to model uncer-  tainties, which themselves are rooted in the inherent complexity of nucleus-nucleus collisions and  the challenging task of describing all processes contributing to the final state observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Connections to fundamental questions in nuclear physics  The wealth of data from efforts conducted in recent years not only helps to get a better grasp  on the nuclear matter EOS, but also has brought forward fascinating questions challenging our  understanding of strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Following the successful BES-I campaign at RHIC, questions remain about the structure of the  QCD phase diagram at finite baryon densities, where the sign problem prevents obtaining predic-  tions with lattice QCD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Surprisingly, the expected disappearance of the quark-gluon  plasma signatures has not been observed in BES-I, with some observables suggesting that the  QCD first-order phase transition may be located within the region probed by BES-II experiments,  including the region probed by the currently analyzed BES FXT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' If this is the case, then  constraining the EOS at lower densities and describing the approach to the transition from the  hadronic side, which would manifest as a softening of the EOS, will be crucial for a robust inter-  pretation BES-II measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Importantly, due to the largely out-of-equilibrium evolution of  collision systems probing that region of the QCD phase diagram, hadronic transport simulations  will play a dominant role in describing the dynamics of the collisions, and therefore in constraining  the EOS of nearly-symmetric dense nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Understanding the physics of neutron-rich matter across a range of densities is necessary not only  to explain the properties of rare neutron-rich isotopes and the structure of neutron stars, but also  to constrain microscopic interactions in isospin-asymmetric nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At low densities, this  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content="6  '=0  y'y  d  /  1  v  d  (AuAu) Protons (10-30%)  HADES  =0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='45)  0  (AuAu) Protons (b  FOPI  (AuAu) Z=1 (b=2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5fm)  FOPI  (AuAu) Z=1  Plastic Ball  (AuAu) Z=1 (b=2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5fm)  INDRA  (AuAu) Protons (12-25%)  E895  (AuAu) Protons  E877  �  (AuAu) h  E877  (AuAu) Protons (10-40%)  Star FXT  (AuAu) Protons (10-25%)  Star FXT  (AuAu) Protons (10-40%)  Star BES  (PbPb) Protons (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5%)  NA49  (PbPb) Protons (15-35%)  NA61/SHINE  1  −  10  1  10  2  10  (GeV)  N  2m  NN  s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  2  v  out-of-plane  in-plane  (AuAu) Protons (10-30%)  HADES  (AuAu) Protons (15-29%)  FOPI  (AuAu) Z=1 (20-30%)  FOPI  (AuAu) Z=1 (b=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5fm)  INDRA  (AuAu) Protons  EOS  (AuAu) Protons (12-25%)  E895  (AuAu) Protons  E877  �  (AuAu) h  E877  (AuAu) Protons (10-40%)  Star FXT  (AuAu) Protons (0-30%)  Star FXT  (AuAu) Protons (10-40%)  Star BES  (10-20%)  �  (AuAu) h  Star BES  (0-60%)  �  (AuAu) h  Star  (0-60%)  �  (AuAu) h  PHOBOS  (PbPb) Protons (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5%)  NA49  (10-30%)  �  (PbPb) h  WA98  �  (PbAu) h  CERES  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Compilation of the world data on the slope of  the directed flow at mid-rapidity (dv1/dy|y′=0, top) and  the elliptic flow (v2, bottom) as functions of the reduced  center-of-mass energy √sNN − 2mN for protons, Z = 1  nuclei, and inclusive charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  challenge is addressed by experimental and  theoretical analyses of nuclear structure ob-  servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An important objective of nuclear  many-body theorists is to accurately calcu-  late these observables and reliably deduce  the EOS using microscopic interactions de-  rived within the framework of chiral effective  field theory (χEFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Probing the symmetry  energy over a range of densities wider than  found in nuclei is possible through heavy-ion  collisions and neutron star studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Often,  knowledge of the isospin-asymmetric EOS is  encoded in terms of constraints on the Tay-  lor expansion coefficients of the symmetry  energy around n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Numerous analyses yield  consistent constraints on the first few expan-  sion coefficients (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 2), although  they rely on an assumption that the expan-  sion remains accurate away from n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The re-  cent advent of Bayesian inference techniques  allows one to pursue a different approach,  within which the isospin-asymmetric EOS is  described in terms of the dependence of the  pressure on baryon density (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Moreover, Bayesian analyses can shed more  light on densities at which measurements  constrain the symmetry energy and quan-  tify the uncertainties of the extracted EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  As a result, combining diverse measurements  and using advanced analysis techniques can  lead to significantly tighter constraints, es-  pecially on the high-density behavior of the  symmetry energy (or, equivalently, on the  higher-order symmetry energy expansion co-  efficients), which is so far poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Constraints on the EOS of neutron-rich  matter at high densities have been dramat-  ically affected by discoveries of heavy neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Combined with the properties of all known  compact stars, these observations indicate that while the EOS of neutron-rich matter is relatively  soft around (1–2)n0, the pressure steeply rises with density for nB >∼ 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In fact, multiple analyses  show that describing the known population of neutron stars is only possible for EOSs in which the  speed of sound in neutron-star matter breaks the conformal limit at high densities, that is exceeds  1/  √  3 of the speed of light c for nB >∼ 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This striking behavior remains to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In par-  ticular, it is currently not known whether the speed of sound exceeds c/  √  3 above certain densities  at all isospin fractions of nuclear matter or, alternatively, only in neutron-rich matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Importantly,  robust constraints on the symmetric matter EOS at nB >∼ 2n0, obtained from heavy-ion collisions  at intermediate to high beam energies, would also put constraints on the isospin-dependent part of  the EOS through comparisons with the EOS inferred from neutron star studies, thus uncovering  the magnitude of isospin-related effects at high baryon density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  11  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Upcoming opportunities  The next decade will be an era of high-luminosity heavy-ion collision experiments at high baryon  density with modern detector and analysis procedures, as well as detailed studies of the symmetry  energy with collisions of proton- and neutron-rich isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Many of the discoveries of the BES program in ultra-relativistic heavy-ion collisions at RHIC,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the discovery of the triangular flow and elliptic flow fluctuations, illustrate that modern  analyses of heavy-ion collisions bring new quality to the understanding of the underlying processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Because of this, revisiting the intermediate to high beam energies, previously explored at the  AGS at BNL as well as at SIS18 at GSI and now explored by the STAR FXT program and  the HADES experiment, is imperative to enable putting tighter constraints on the EOS of dense  nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, the future CBM experiment at the Facility for Antiproton and Ion  Research (FAIR), Germany, will be able to measure interaction rates exceeding those currently  used by several orders of magnitude, allowing for exploration of multiple high-statistics observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Furthermore, the explored beam energy range is where lower-order flow observables, reflecting the  collective motion of the colliding system due to the underlying hadronic EOS, are particularly  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='23 GeV  b=6-9 fm, |ycm|<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05, pt>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3 GeV/c  HADES data: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Predicted slope of the directed flow at mid-  rapidity (dv1/dy|y′=0, top) and elliptic flow (v2, bot-  tom) as functions of the incompressibility and the in-  medium nucleon-nucleon scattering cross section mod-  ification factor, generated in simulations of Au+Au  reactions using the isospin-dependent BUU (IBUU)  transport model [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  prominent (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Therefore, the cor-  responding precision measurements carry with  them the opportunity to bring a richer perspec-  tive and a better understanding of the physics  underlying the complex dynamics of nuclear  matter at extreme conditions (see Section III A,  “Experiments to extract the EOS of symmetric  nuclear matter”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This advancement can only  occur provided a simultaneous development of  hadronic transport simulations, as only a de-  tailed understanding of various factors affect-  ing the dynamics of heavy-ion collisions can  lead to meaningful descriptions of the exper-  imental data, and, consequently, more robust  constraints on the EOS of nearly-symmetric  nuclear matter (see Section II A, “Transport  model simulations of heavy-ion collisions”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As  an example of the sensitivity of observables to  various details of the underlying physics, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 5  shows the dependence of the slope of the di-  rected flow (top panel) and of the elliptic flow  at midrapidity (bottom panel) on the stiffness  of the EOS, parametrized by the incompress-  ibility, and on the in-medium nucleon-nucleon  scattering cross-section modification factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Unprecedented possibilities are on the hori-  zon for studies of the isospin-dependence of the  EOS, which is critical for connecting heavy-ion  physics measurements to astrophysical obser-  vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The difficulties in using nuclei with  significant variations in the isospin asymme-  try, along with the paucity of neutron measure-  ments at midrapidity, have in the past greatly  12  restricted the capability to put tight constrains on the EOS of asymmetric nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Fortu-  nately, at this time modern neutron detectors are available for heavy-ion measurements in many  facilities, including at accelerators performing collisions at high beam energies such as GSI, while  radioactive beam measurements are entering a new era at RIKEN and FRIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' FRIB will provide  proton- and neutron-rich beams of not only the highest-intensity worldwide, but also characterized  by the widest currently accessible range of the isospin asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Establishing a strong heavy-ion  program at FRIB will therefore enable previously inaccessible exploration of the symmetry en-  ergy (see Section III B, “Experiments to extract the symmetry energy”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, the proposed  FRIB400 beam energy upgrade would not only allow exploration of densities up to around 2n0,  but it would also provide increased resolution of the isospin-dependence of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular,  among observables sensitive to the symmetry energy, both charged pion yields and the absolute  magnitude of the elliptic flow (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 4) significantly increase between the current top FRIB  energy of 200 MeV/nucleon and the proposed 400 MeV/nucleon [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The increase in available computing power and advances in statistical methods make it possible  to perform wide-ranging comparisons of heavy-ion collision simulations with experimental data  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', using Bayesian analysis), allowing one to vary multiple model assumptions at the same  time as well as to put robust uncertainties on the obtained constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, given the  wealth of the upcoming independent data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', from heavy-ion collision experiments, neutron star  observations, and microscopic nuclear theory calculations, global analyses of complementary efforts  have likewise a strong potential for putting tight constraints on the EOS (see Section IV, “The  EOS from combined constraints”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Beyond the much-needed interpretation of intermediate energy heavy-ion collisions, advances  in transport theory can lead to significant contributions to other areas of nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In  particular, recently attention has been given to cross-cutting opportunities for employing state-  of-the-art hadronic transport codes in studies supporting space exploration and advanced medical  treatments (see Section V A, “Applications of hadronic transport”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Transport theories may also be  used in tests of extensions of hydrodynamic approaches supporting far-from-equilibrium evolution  (see Section V B, “Hydrodynamics”), which are a focus of intense studies due to their importance  for modeling heavy-ion collisions at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Finally, constraining the dense nuclear matter  EOS through interpretations of heavy-ion collision measurements may have other profound conse-  quences, including helping to answer fundamental questions about the possible existence of dark  matter in the cores of neutron stars or providing the impetus for studies of nuclear systems in  fractional dimensions (see Section VI, “Exploratory directions”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Needs  The next-generation experimental measurements of observables sensitive to the nuclear matter  EOS are imminent, and further progress in resolving the nuclear matter EOS is contingent on  enhanced theory support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, the development of transport theories based on  microscopic hadronic degrees of freedom, which are the only means of interpreting  measurements from heavy-ion collision experiments at intermediate to high beam  energies, must be strengthened and expanded to fully realize the potential of the  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' facilities leading the exploration of the dense nuclear matter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Support for  both individual scientists and collaborations, and in particular for viable career pathways for early  career researchers, is imperative to maintain the health of and diversify the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' hadronic transport  community, and to fully capitalize on the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' efforts exploring the dense nuclear matter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  13  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  THE EQUATION OF STATE FROM 0 TO 5n0  Efforts to determine the equation of state (EOS) of nuclear matter are at the forefront of nuclear  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An EOS contains fundamental information about the properties of a many-body system  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Section I B), and is, in essence, any nontrivial relation between the thermodynamic  properties of a given type of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In nuclear physics, the form of the EOS that is most often  pursued is the relation between energy per baryon or pressure and baryon density nB, isospin  excess δ, and temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For symmetric matter, the isospin excess vanishes (δ = 0), and  for asymmetric matter the energy per baryon or pressure are commonly partitioned into a part  corresponding to symmetric matter and the remainder, which contains all information about the  isospin-dependence of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Due to the charge invariance of strong interactions, the latter  part is (to a very good accuracy) quadratic in the isospin excess δ at densities relevant to nuclear  experiments and astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The quadratic coefficients in the expansion around  δ = 0 are independent of δ, and are often referred to as the symmetry energy (denoted as S(nB)  at T = 0) or symmetry pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These, together with the EOS of symmetric matter,  are then sufficient to describe the EOS of nuclear matter at any isospin asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  While many approaches to constraining the nuclear matter EOS are pursued, here we describe  three research areas which have the capability to constrain the EOS over wide ranges of density:  inferences of the EOS from comparisons of experimental measurements to model simulations of  heavy-ion collisions (Section II A), microscopic calculations of the EOS using chiral effective field  theory (Section II B), and EOS inferences from neutron star studies (Section II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Transport model simulations of heavy-ion collisions  soft EOS  hard EOS  temperature [MeV]  0  50  100  150  200  250  300  density nB/n0  0  2  4  6  8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8 AGeV  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 AGeV  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2 AGeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6 AGeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8 AGeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 AGeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2 AGeV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Phase diagram trajectories of the central re-  gion in Au+Au collisions at zero impact parameter,  obtained from UrQMD simulations with a soft or a hard  (characterized by K0 = 200 or K0 = 380 MeV, respec-  tively) EOS [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The trajectories follow the evolu-  tion at times when temperature is fairly well-defined,  from the moment of the highest compression to densi-  ties around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Heavy-ion collisions at very low to interme-  diate beam energies provide the means to probe  nuclear matter at different densities (from sub-  saturation to several times the saturation den-  sity), temperatures (from a few MeV to well  above one hundred), and neutron to proton  ratios (from near symmetric nuclear matter,  where Nn/Np ≈ 1, up to Nn/Np ≈ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 6  for an illustrative calculation of heavy-ion col-  lision trajectories in the T-nB phase diagram  from simulations using two schematic EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  These wide ranges of system properties ac-  cessed in heavy-ion collisions position them as a  perfect tool to extract the nuclear matter EOS,  test predictions and extrapolations from regions  of the QCD phase diagram accessed by other  approaches, and provide a necessary input to  nuclear theory and nuclear astrophysics calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, the density-dependence  of both the symmetric and asymmetric EOS  can shed light on modeling effective nuclear in-  teractions in the medium [15, 25–27] or con-  strain approaches using the density functional  theory [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  14  However, systems created in heavy-ion collisions are short-lived, and their dynamic evolution  is out of equilibrium over significant fractions of the total collision time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The evolution of a  colliding system depends on the energy and centrality of the collision, and progresses through initial  compression, growth of the compression zone, development of flows, and overall decompression  with a gradual local equilibration during the process, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The inherent complexity of  the evolution means that the corresponding transport equations cannot be solved directly due to  their high non-linearity, and therefore detailed inferences from heavy-ion collision experiments,  where the non-equilibrium evolution probes nuclear matter over substantial ranges of density,  require comparisons to results of collision simulations in transport models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Beyond modeling the  dynamics of the collisions, transport models provide a connection to the equilibrium limit allowing  for inferring the EOS [31], transport coefficients [32], as well as the in-medium properties and  cross-sections of hadrons [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Transport theory  At its core, transport theory aims to describe the time evolution of the one-body phase-space  distribution function in a semi-classical approximation for a dissipative system composed of a large  number of particles, here in particular for a system of two heavy nuclei colliding at an energy per  nucleon which is typically larger than the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The theoretical foundations of transport  theory include the BBGKY hierarchy of coupled equations for reduced density matrices [36] as well  as the equations of the nonequilibrium Green’s function theory [37, 38] such as obtained in Martin-  Schwinger (also known as Schwinger-Keldysh) formalism for non-equilibrium Green’s function (see  also Section V B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  To arrive at transport equations, one employs (among others) a Wigner transformation and  coarse-graining as well as a gradient expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The Wigner transformation and coarse-graining  nB  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Contour plots of the system-frame baryon density nB (top row) and local excitation energy E∗/A  (bottom row) at times t = 0, 5, 10, 15, and 20 fm/c (columns from left to right), obtained from a transport  simulation [39] of a 124Sn+124Sn reaction at beam energy Elab = 800 AMeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='24 GeV) and impact  parameter b = 5 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The contour lines for the density use increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4n0, starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1n0, while  the contour lines for the local excitation energy correspond to the values of E∗/A = {5, 20, 40, 80, 120} MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  for statistical reasons, contour plots for the energy have been suppressed for baryon densities nB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='1  y  IBUU  pBUU  RVUU  SMF  IQMD  IQMD-BNU  IQMD-IMP  TuQMD  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Comparison of results for rapidity distri-  butions (top) and transverse flow of nucleons (bot-  tom) as functions of the scaled rapidity, obtained  with different transport codes (identified in the leg-  end) within the TMEP initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The results shown  were obtained for 132Sn+124Sn collisions at Elab =  270 AMeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01 GeV) and impact param-  eter b = 4 fm, using controlled input models for the  EOS and the cross sections as well as identically ini-  tialized nuclei [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  lead to positive-definite phase-space distribu-  tions [41] that can be efficiently sampled with  Monte-Carlo techniques, while the gradient ex-  pansion yields, for each particle species, the force  acting on a particle and the particle’s veloc-  ity as gradients of its total energy with respect  to the spatial position and momentum, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Knowledge of the kinematics of all parti-  cles, together with the elementary collision rates,  drives the evolution in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Finally,  to arrive at a set of Vlasov-Boltzmann–like equa-  tions, one employs the quasi-particle approxima-  tion, neglecting details of the spectral functions  and treating all particles as on-shell (we note  here that while there are some transport codes  with off-shell particle treatment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [42, 43],  this approach is still an outstanding challenge in  the transport theory, as will be discussed further  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Alternative approaches to arriving at a  transport theory for heavy-ion collisions include  using the relativistic Landau quasiparticle the-  ory [44] or, in approaches starting from a molec-  ular picture, representing the global wavefunc-  tion as a product (sometimes antisymmetrized)  of single-particle Gaussian wavepackets [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The particle species considered in transport  theory depend on the collision energy and may  range from nucleons, through pions and the delta  resonances, to higher resonances, kaons, and hy-  perons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some transport formulations further in-  corporate light clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', deuterons, tritons,  and 3He nuclei) as independent degrees of free-  dom, with recent extensions also including alpha  particles [46] which appear abundantly in exper-  iments and are of particular importance for colli-  sions at fixed-target beam energies on the order  of hundreds of MeV/nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In some of these approaches, clusters are produced through multi-  particle reactions, as discussed further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  For the lowest energy collisions, nonrelativistic  formulations of the transport theory may be employed, but the majority of the available codes are  relativistic, with many addressing collisions at energies from tens of MeV/nucleon to at least a few  GeV/nucleon (see [35, 47, 48] for reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Transport approaches can be generally divided into those concentrating on a single-particle  characterization of the colliding system and those attempting to describe many-particle correla-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Both types of approaches are highly complex and nonlinear, and the relevant equations are  solved by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The single-particle approaches typically solve a set of Boltzmann-Vlasov–  type equations [47, 49] (also known as the Boltzmann-Uehling-Uhlenbeck, or BUU equations) in  which the evolution of the system is governed by a mean-field evolution of the phase space distribu-  tion (Vlasov equations) and a collision term which drives the dissipation (the Boltzmann collision  term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While, in principle, the Boltzmann-Vlasov equation is deterministic, numerical solutions  16  contain numerically-induced fluctuations due to the fact that the evolution is obtained using the  method of test particles, in which the continuous distribution function is represented by a large,  but finite, number of test particles sampling the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To include fluctuations of a physi-  cal origin, one can add a fluctuation term to the two-particle collision term, thus arriving at the  Boltzmann-Langevin formulation [35, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In contrast, quantum molecular dynamics (QMD) approaches include classical many-body cor-  relations in the ansatz of the many-body wave function [47, 51], which is postulated as a product of  single-particle wave packets of a fixed width, with the width regulating the amount of fluctuations  and correlations in QMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In Anti-Symmetrized Molecular Dynamics (AMD) [45], the product wave  function is anti-symmetrized and the formulation includes Pauli correlations in the propagation as  well as in, to a certain extent, the collision term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The fact that hadronic transport approaches are built on firm theoretical foundations has been  crucial for the continued development of simulation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Reaching back to the roots of  the nuclear transport theory has made it possible to resolve ambiguities which would be otherwise  hard to tackle by purely phenomenological means, including descriptions of cluster production [52],  low relative-velocity correlations (Hanbury–Brown-Twiss correlations) [53], and off-shell transport  [42, 49, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The strong theoretical foundation of transport theory has also been effective  in ensuring covariance of the theory and preserving conservation laws in case of interactions that  stray beyond outcomes of field-theoretic models, in particular interactions employing energy density  functionals [44, 56–58] which are often needed for realistic descriptions of bulk properties of nuclear  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  An important effort to validate conclusions reached from comparing transport model results  to data has been recently intensified by the formation of the Transport Model Evaluation Project  (TMEP) [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Within this endeavor, predictions from different models are compared in controlled  settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', ensuring the same physical input such as the EOS, initial densities, and cross sec-  tions), oftentimes with comparisons to known results that can be achieved analytically or by other  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Similar controlled comparisons of complex simulations have been done in other fields  of physics: from atomic traps, through ultra-relativistic heavy-ion collisions, to core-collapse su-  pernova calculations [59–62], and they are known to be very fruitful for their respective fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The TMEP analyses not only enable identifying models that produce outlier predictions, but also  determine details of implementation or physical assumptions behind the diverging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An ex-  ample of such a comparison of codes for simulations of heavy-ion collisions at lower energies, with  controlled input, can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 8, showing results for rapidity distributions (left) and the  transverse flow (right) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In general, the codes agree with each other reasonably well, however,  differences between the codes are visible and, moreover, can be traced to specific model choices  in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, the generally lower values of the transverse flow in the case of  QMD codes are a result of an approximation used in the evaluation of a non-linear term in the  mean-fields, which becomes relevant when density fluctuations become large, as often occurs in  QMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Beyond identifying this and similar problems, the Project has yielded recommendations  for optimal algorithms used in transport codes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', for ensuring obeying the Pauli principle in  elementary two-body collisions [63] or for integration of equations of motion with mean-fields [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Moreover, the project has identified a set of tests for transport codes that ensure their credibility  when addressing different heavy-ion collision observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Stringent tests of hadronic transport  codes are especially important for studies aimed at constraining the nuclear symmetry energy,  which, compared to other model parameters, has a comparatively weak effect on heavy-ion observ-  ables and which therefore demands maximal precision from transport simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Below, we will  also discuss the role that such comparisons can play in determining the uncertainty of transport  model investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Selected constraints on the EOS obtained from heavy-ion collisions  A selection of important constraints on the EOS obtained from heavy-ion collisions can be found  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9 for both symmetric matter (pressure as a function of density, left panel) and asymmetric  matter (symmetry energy as a function of density, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  We note here that while many results are reported in terms of constraints on the incompress-  ibility K0, in the context of heavy-ion collision studies of the EOS, K0 should be understood as  a parameter which specifies the behavior of the EOS in the range of densities probed by a given  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, in the case of experiments probing mostly densities above 2n0, constraints on  K0 are only indicative of the behavior of the EOS above 2n0, and in particular do not constrain  the behavior of the EOS around n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This subtle, and often confusing, point is a consequence of  simple parametrizations of the EOS used in many transport codes, where the only parameter con-  trolling the behavior of the EOS both around n0 and at higher densities is K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recently, flexible  parametrizations of the EOS have been developed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [57, 58]) and implemented (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', in  hadronic transport code SMASH [75, 76]) which allow one to vary the incompressibility K0 and the  high-density behavior of the EOS independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The collective behavior of matter created in the collisions, especially the directed and elliptic  flow, has been shown to be a very sensitive probe of the EOS [31, 67, 77–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In contrast to  collisions at the Fermi energies, where all nucleons within nuclei participate in the collisions, and  unlike in collisions at ultrarelativistic energies, where the evolution of the colliding nuclei can be  understood in terms of participant nucleons, at intermediate energies the interplay between the  expanding collision zone and the dynamics of the spectators are key ingredients to understanding  Le Fèvre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Lynch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' from Fuchs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Oliinychenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Danielewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Walecka model  Fermi gas  pressure [MeV/fm3]  1  10  100  baryon density nB/n0  1  2  3  4  5  HIC(isodiff)  HIC(n/p)  mass(Skyrme)  IAS  mass(DFT)  PREX II  HIC(π)  Tsang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  ASY-EOS  FOPI-LAND  symmetry energy S(nB) [MeV]  0  20  40  60  80  baryon density nB/n0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Left: Selected constraints on the symmetric EOS obtained from comparisons of experimental data to  hadronic transport simulations in [31] (region with black horizontal stripes), [65, 66] (region with red forward  stripes), [67] (region with blue backward stripes), and [58] (region with green vertical stripes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' see text for  more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Also shown are results of analytical calculations for the free Fermi gas (green dotted line)  and in the linear Walecka model (pink dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Right: Selected constraints on the symmetry energy  obtained from comparisons of hadronic transport simulations to experimental data in [6] (region with purple  forward stripes), [68] (region with green backward stripes), [69] (the solid orange region), and [70] (the red  circle, square, and triangle symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Also shown are symmetry energy constraints obtained in [70] based on  a novel interpretation of analyses of nuclear masses in DFTs [11, 71] (cyan diamond symbol) and in Skyrme  models [72] (cyan star symbol), of Isobaric Analog States (IAS) energies [73] (magenta plus symbol), and of  PREX-II experiment [74] (blue inverted triangle symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  18  experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  A seminal constraint on the symmetric nuclear matter EOS [31] in the  density range (2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5)n0 was obtained by comparing measurements of collective flow from heavy-  ion collisions [80–83] at beam energies Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='15–10 AGeV (corresponding to nucleon-nucleon  center-of-mass energies √sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='95–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='72 GeV) with results from hadronic transport simulations  using EOSs with different values of the incompressibility at saturation density K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The outcome of  this study suggests a symmetric-matter EOS to lie between those labeled with K0 = 210 MeV and  K0 = 300 MeV (see the region with black horizontal stripes in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For densities  in the range (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5) n0, probed in collisions below Elab <∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV (√sNN <∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 GeV), the EOS  may be inferred from meson yields [84–86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Indeed, subthreshold production of strange mesons  (specifically, K+ and K0), which interact weakly with nuclear matter, depends on the highest  densities sampled in the collision, which in turn depend on the stiffness of the EOS [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In [65],  ratios of experimentally measured kaon yields in Au+Au and C+C collisions have been reproduced  in hadronic transport simulations with soft mean-field interactions yielding K0 = 200 MeV and  an EOS [66] consistent with the constraint from [31] (see the region with red forward stripes  in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In [67], the elliptic flow data measured at Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV  (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='52 GeV) by the FOPI collaboration [88] were used together with simulations  from Isospin Quantum Molecular Dynamics (IQMD) [23, 89] to constrain the incompressibility at  K0 = 190 ± 30MeV, again indicating a rather soft EOS (see the region with blue backward stripes  in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recently, new measurements by the STAR collaboration from the fixed  target (FXT) program at RHIC have become available, providing an opportunity to expand the  set of world data utilized to deduce the baryonic EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A Bayesian analysis study [58], in which the  speed of sound was independently varied in specified intervals of baryon density (thus providing  a more flexible EOS at higher densities), suggests a tension between the E895 [83, 90–92] and  STAR [93, 94] data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Using only the STAR measurements, the study [58] further found that EOSs  which simultaneously describe the slope of the directed flow and the elliptic flow, in the considered  energy range of Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9–9 AGeV (√sNN = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 GeV), are relatively stiff at lower densities  and relatively soft at higher densities (see the region with green vertical stripes in the left panel of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, the model used in that work did not include the momentum dependence of the  EOS, which likely results in a spuriously stiff EOS at intermediate densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As such, the study  should be treated as a proof of principle that a tight constraint on the EOS at high densities can  be achieved by using a combination of precise data, flexible forms of the EOS used in simulations,  state-of-the-art models, and advances in analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The symmetry energy contribution to the EOS can be studied at low collision energies Elab <∼  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 AGeV (√sNN <∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='32 GeV), where in particular observables such as charged pion yields [95] or  neutron and proton flow [96, 97] have been proposed as sensitive to the asymmetric contribution  to the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some of the constraints derived from such studies are shown in the right panel of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9, where, in addition to the usual EOS constraint bands, symbols with uncertainty bars  represent results from analyses in which the symmetry energy has been determined for the most  sensitive density of a given measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At incident energies below Elab = 100 AMeV (√sNN =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='93 GeV), low densities are probed after the initial impact and compression of the projectile and  target [6, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Since the symmetry potentials for neutrons and protons have opposite signs, emission  of a particular nucleon type is enhanced or suppressed depending on the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A comparison  of the experimental measurements of isospin diffusion and the ratio of neutron and proton spectra in  collisions of 112Sn+124Sn at Elab = 50 AMeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='90 GeV) to results from ImQMD simulations  produced a constraint on the symmetry energy for densities (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3–1) n0 [6] (see the region with purple  forward stripes in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Collisions at higher energies (Elab > 200 AMeV, or  √sNN > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='97 GeV) probe the EOS at n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the FOPI-LAND experiment, constraints on  the symmetry energy were obtained from studies of the ratio of the elliptic flow of neutrons and  hydrogen nuclei in Au+Au collisions at Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07GeV) [68], while the ASY-  19  EOS experiment used neutron to charged fragments ratios measured in Au+Au collisions [69] (see  the region with green backward stripes and the solid orange region, respectively, in the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In [70], a comprehensive analysis was performed with the goal of identifying the values of  the symmetry energy at densities to which given experiments are most sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Using the isospin  diffusion in collision systems with different proton to neutron ratios [99], neutron to proton energy  spectra in Sn+Sn systems [100], and spectral pion ratios measured by the SπRIT collaboration in  Sn+Sn collisions at Elab = 270 AMeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01 GeV) [101, 102], that work [70] put constraints  on the values of the symmetry energy at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2n0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4n0, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0, respectively (see the red  circle, square, and triangle symbols in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Also shown in the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9 are symmetry energy constraints obtained in [70] based on a novel interpretation of the  analyses of nuclear masses in DFTs [11, 71] (cyan diamond symbol) and in Skyrme models [72]  (cyan star symbol), of the Isobaric Analog State (IAS) energies [73] (magenta plus symbol), and  of the PREX-II experiment result [74] (blue inverted triangle symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Challenges and opportunities  Selected results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9 showcase significant achievements in determining the EOS  and, simultaneously, the need to develop improved transport models to obtain tighter and more  reliable constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Answering this need will require support for a sustained collaborative effort  within the community to address remaining challenges in modeling collisions, in particular in the  intermediate energy range (Elab ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1–25 AGeV, or √sNN ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' we  will address selected areas where we see the need for such developments: (1) comprehensive treat-  ment of both mean-field potentials and the collision term in transport codes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (2) use of microscopic  information on mean fields and in-medium cross sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' such as discussed in Section II B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' in trans-  port,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (3) better description of the initial state of heavy-ion collisions in hadronic transport codes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  (4) deeper understanding of fluctuations in transport approaches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' which affect many aspects of  simulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (5) inclusion of correlations beyond the mean field into transport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' which is crucial for  a realistic description of light-cluster production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (6) treatment of short-range-correlations (SRCs)  in transport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' which are tightly connected to multi-particle collisions as well as off-shell transport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  (7) sub-threshold particle production,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (8) the study of new observables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', azimuthally resolved  spectra, to obtain tighter constraints on the EOS, (9) the question of quantifying the uncertainty of  results obtained in transport simulations, and (10) the use of emulators and flexible parametriza-  tions for wide-ranging explorations of all possible EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Fortunately, advances in transport theory  as well as the greater availability of high-performance computing make many of these improvements  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Support for these developments will lead to a firm control and greater understanding of  multiple complex aspects of the collision dynamics, allowing comparisons of transport model cal-  culations and heavy-ion experiment measurements to provide an important contribution to the  determination of the EOS of dense nuclear matter, which, in particular, cannot be determined by  any other method at intermediate densities (1–5)n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Comprehensive treatment of mean-field potentials and the collision term  Notably, driven by specific experimental needs over the last two decades, the refinement of  hadronic transport codes has diverged into two complementary branches: Codes which were ap-  plied to describing experiments at very low to low energies (Elab <∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV, or √sNN <∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 GeV),  such as IQMD, AMD and pBUU, have become progressively better at describing the momentum- and  isospin-dependence of the interaction, while codes which were primarily used as afterburners for  simulations of ultra-relativistic heavy-ion collisions (Elab >∼ 25 AGeV, or √sNN >∼ 7 GeV), such  as SMASH [75] or UrQMD, were developed to offer a fully relativistic evolution as well as scattering  20  and decay modes taking into account all established particle and resonance species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As heavy-ion  collisions are entering an era of precision data on symmetric nuclear matter at higher densities  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', in experiments at HADES, BES FXT, and future CBM) and on asymmetric nuclear mat-  ter at normal and supranormal densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', at FRIB and future FRIB400), where features of  both diverging branches of hadronic transport codes are important, a vigorous development of  transport models is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, numerous studies show the importance of including  the momentum-dependence of the interactions, which is observed in elastic scattering of hadrons  off nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Moreover, momentum-dependence naturally occurs in microscopic effective interac-  tions [38, 103] where it contributes to the calculated mean fields, whether near or away from sat-  uration density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Incorporating single-particle energies with momentum dependence different than  that in free space, which is often quantified with effective masses, is crucial in hadronic transport  both for studies of symmetric nuclear matter [31, 79, 104, 105] as well as studies of the symmetry  energy and its relation to effects such as the neutron-proton effective mass splitting [106–108] (see  also Section VI E for more discussion on effective masses and the nuclear symmetry energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some  of the theoretical and implementation solutions have already been established, while others will  require devising new approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' When possible, the best practices need to be carried over across  the domains, as has been exemplified in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the development of the SMASH code, which uses many  implementation solutions from pBUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Microscopic input to transport  One of the most prominent opportunities for improvement in transport models concerns imple-  mentations of the EOS informed by state-of-the-art many-body studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such efforts are especially  timely given that sophisticated microscopic calculations of the properties of nuclear matter are  currently becoming available for large ranges of baryon density, temperature, and isospin fraction  (see Section II B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To incorporate the effects of the resulting EOSs in hadronic  transport calculations, the corresponding Lorentz-covariant single-particle potentials as well as the  in-medium interactions (both as functions of density, asymmetry, and momentum) are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A  particular challenge is to determine the connection between the EOS inferred from a transport  calculation and the zero-temperature EOS obtained from microscopic calculations [109], or even  the finite-temperature EOSs that are becoming increasingly available [110, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In a heavy-ion  collision, the medium progresses through a set of non-equilibrium states that relax toward a local  equilibrium, however, the nature of the local equilibrium also evolves during the collision due to the  system expansion, so that even if the system approaches a local equilibrium at any given moment  of the evolution, that agreement is only temporary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Errors incurred due to differences between  non-equilibrium and equilibrium states of high-density matter contribute to the systematic error in  inferring the EOS when comparing transport to experimental data (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9 and [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Here, the  availability of state-of-the-art microscopic calculations at finite temperature could reduce system-  atic errors in connecting the finite- and zero-temperature EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, the use of microscopic  input would provide a consistency between the effective in-medium cross sections in the collision  term and the mean fields used in the propagation of the phase space distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It could also  help address the question of the extent to which nonlocalities in the microscopic theory should  be reflected in the propagation and the collision term [112, 113] (where, in particular, departures  from standard approaches modify the entropy to take a form different than that obtained in the  Landau quasiparticle theory [44, 114]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To accelerate progress at the interface of the transport  description of heavy-ion collisions and microscopic nuclear matter theory, direct collaboration of  practitioners in the two research areas is required to assess how the needs of transport simulations  can be answered by what can be currently calculated in microscopic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Conversely, the use  of microscopic interactions in transport could validate the many-body theory results in regions of  density and temperature which are only accessible by heavy-ion collisions [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  21  Initial state  Numerous studies point toward the dependence of outcomes of heavy-ion collision experiments  on details of the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In ultrarelativistic heavy-ion collisions, understanding these  effects have led to the discovery of higher order flow harmonics [116, 117] and flow fluctuations [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  (Interestingly, the importance of the initial state for experimental outcomes also positions heavy-ion  collisions at high energies as an unusual, but complementary probe of nuclear structure, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', a  white paper on Imaging the initial condition of heavy-ion collisions and nuclear structure across the  nuclide chart [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=') Given the high sensitivity of flow observables to both the EOS and the initial  state of collisions, the impact of the initial conditions on outcomes of heavy-ion collisions needs to  be thoroughly understood in order to narrow the constraints on the EOS of both symmetric and  asymmetric matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Aspects of initial conditions that need to be considered include event-by-event  fluctuations of the initial state [116–118], relative distributions of neutrons and protons and shell  effects [120], and correlations tied to deformation [121] or short-range correlations [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some of  these elements will be further discussed below in the context of the dynamics of heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Fluctuations  Fluctuations of the phase space distribution are an important ingredient of transport simula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, fluctuations of the one-body density are important for including the conse-  quences of the dissipation-fluctuation theorem in the reaction dynamics as well as for describing  effects due to the largely unknown, neglected many-body correlations, thus going beyond the mean-  field description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The question of how to include them properly and of their consistency with the  nucleon-nucleon correlations explicitly implemented in transport theories, however, has not been  completely clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As discussed above, fluctuations are included in a different manner in the two  families of transport approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While in the BUU transport fluctuations can be introduced by  the Langevin extension of the Boltzmann-Vlasov equation, which adds a fluctuation term to the  collision term (and which is still rarely implemented), in the molecular dynamics approach fluctu-  ations are introduced in a classical way by using finite-size particles, the width of which regulates  the amount of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Fluctuations then affect the outcome of simulations in many ways, in-  cluding by regulating the formation of intermediate-mass fragments (IMFs) which appear through  the growth of fluctuations in regions of spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It was also shown in box calculations  that fluctuations have a strong influence on the efficiency of Pauli-blocking [63] and even on the  calculation of the force in the Vlasov term for QMD codes in which non-linear parametrizations of  the fields are used [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Correlations  Correlations in transport simulations strive to address intermediate-range correlations beyond  the mean-field picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Physically, such correlations are also a source of fluctuations, but at the  same time have other additional impacts, including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', influencing the production of light clusters  (LCs), that is light nuclei up to the alpha particle which are copiously produced in heavy-ion  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The mean-field models used in transport calculations are usually not detailed enough to  realistically describe very light nuclei with their particular spin-isospin structure reflecting strong  quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An additional complication results from the fact that in a collision, clusters often  appear in the nuclear medium where their properties are drastically changed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the binding  energy of clusters is reduced with increasing density until the Mott point, at which they dissolve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Currently, most codes describe the production of clusters by using a cluster-finding algorithm,  based on particle proximity in coordinate and/or momentum space (coalescence) toward the end  of the evolution, which in more advanced versions also takes into account criteria related to the  binding energy of the produced clusters [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, these late-stage algorithms do not take into  account the dynamic role played by both correlations and LCs in the evolution of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' One  22  of the known approaches to this problem has been to consider LCs as separate degrees of freedom,  with their own distribution functions and corresponding transport equations, where the collision  terms can lead to creation or destruction of clusters (pBUU, SMASH) and which in particular can  also take into account the in-medium modifications of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, this approach becomes  increasingly complex as heavier clusters are characterized by more and more production channels,  and consequently it is significantly challenging to include, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', alpha particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Another approach  is to modify the phase space of the correlated nucleons according to the Wigner function of the  cluster, but then to propagate them after the collision again as nucleons (as is done in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', AMD  [41]), which still requires using a cluster-finding step at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In both cases, the production and  destruction of clusters necessarily requires multi-particle collisions to ensure energy-momentum  conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Finally, at lower incident energies the LC production can also be described in terms  of the catalyzing effect of spectator nucleons in few-particle collisions [46, 124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To explain LC  production in high-energy collisions, where LCs are produced in numbers that cannot be obtained  through nucleon catalysis due to the relatively few nucleons present in the final stages of these  collisions, a similar mechanism of catalysis by pions [52, 125, 126] can be invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Short-range correlations  A particular aspect of describing correlations in transport simulations is the treatment of short-  range-correlations (SRCs), which have been measured in nucleon knock-out experiments [127–130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Along with the experiments, microscopic many-body calculations show that SRCs introduce a  high-momentum tail (HMT) into the nucleon momentum distribution and, moreover, reduce the  kinetic symmetry energy relative to the Fermi gas kinetic energy, which is a consequence of the fact  that SRCs are more pronounced in symmetric relative to asymmetric matter [131–138] (see also  Section VI C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Phenomenological methods have been used to include SRCs in transport models,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', by initializing nuclei with a HMT, but such a procedure does not take into account the dynamic  role of SRCs in the initial state, which in the case of the on-shell semiclassical equations of motion  results in obtaining nonstationary, excited states of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In on-shell transport approaches, three-  and many-body collisions, incorporated into transport codes within varying approximations, have  been suggested as a way of treating SRCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, in an investigation [139] of three-body  collisions for pion production processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', NNN → NN∆), it was found that SRCs between  two of the incident nucleons give a noticeable contribution to pion yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Another approach [140],  based on a mean-free-path approximation to the collision integral, observed large effects also on  bulk observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The incorporation of n-body collisions in transport equations within a schematic  cluster approximation was also studied [141], however, there the effects were found to be rather  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  So far, none of these methods have been widely exploited in the description of heavy-  ion reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Since HMTs are tied to the tails of the nucleon spectral functions (away from  the quasiparticle peaks), a consistent description of SRCs should involve an off-shell transport  formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Dynamical spectral functions of all considered particles, including those which are  stable in free space like nucleons, have been accounted for in the off-shell transport approaches  implemented, with some differences in detail, in the codes GiBUU [42] and PHSD [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A subsequent  study [142] demonstrated that the momentum distribution automatically develops a HMT within  the approach used in GiBUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Differences in the results from the two approaches have yet to be  investigated systematically, including the impact on symmetry energy inferences from heavy-ion  collision data based on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', charged pion subthreshold production yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Fully quantum transport  approaches with SRCs (or equivalent content), without any semi-classical expansions as are present  in current off-shell transport approaches, remain a long-term goal, and progress in this area has  not ventured yet beyond schematic models [143, 144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, increasing computational power  combined with emulation techniques may make such efforts more realistic and enable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', a  seamless integration of the treatment of shell effects in the initial state and collision dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  23  Threshold effects  An important influence of mean-field potentials in heavy-ion transport appears in the form  of threshold shifts and the related subthreshold production of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thresholds of particle  production are modified in a medium since the mean-field potentials have to be taken into account in  the energy-momentum balance of a two-body collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Specifically, when the mean-field potentials  are momentum-dependent and/or as a consequence of other model assumptions for the mean-field  potentials of the produced particles, the thresholds are shifted away from their free-space values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This may strongly change the production rates of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, the threshold shifts make  it necessary to involve other nucleons, besides the two collision partners in the process, to ensure  the energy-momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Various schemes to achieve this locally or globally have been  in use [115, 145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Indeed, explaining recent heavy-ion collision subthreshold pion yields, measured  by the SπRIT Collaboration [102], required invoking many-body elementary effects in the form of  mean-field effects on thresholds in two-particle collisions [86, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, because the physics  invoked in describing the threshold effects is similar to that invoked for other multi-particle effects,  alternative multi-particle options remain to be investigated, including producing pion degrees of  freedom in multi-particle collisions or in the aftermath of an off-shell propagation between binary  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (We note here that there is a physics overlap between these mechanisms and the impact  of SRCs on pion production [42, 122, 139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=') Notably, theoretical explorations find sequences of  on-shell binary processes to dominate the production at higher beam energies [43, 55, 139], and  no comparable difficulties have been encountered in describing the data [146, 147] by transport  models without multi-particle effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The contrasting struggles of transport models which do  not include threshold or other multi-particle effects of this type [102] , together with expected  further theoretical explorations and future measurements of the subthreshold production in heavy-  ion collisions, offer exciting possibilities for gaining understanding of the more exotic in-medium  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  New observables  Upcoming precision data will further bring unprecedented observables that could be previously  considered only in theory, such as triple-differential spectra tied to a fixed orientation of the reaction  plane [18, 148–150] not only for protons and most abundant mesons, but also for deuterons,tritons,  light nuclei, and hypernuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The potential of such spectra for the determination of the EOS  is still to be fully explored, but a preliminary investigation [149] indicates a rich structure with  spectra which exhibit a maximum away from the beam direction, characterized by slopes dependent  on azimuthal angle and slope discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Models that might have agreed with each other in  describing low-order Fourier coefficients of flow will likely find describing such detailed observables  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Challenges remain even at the level of the low-order coefficients, as many models now  reproduce proton flow, but not Lambda or pion flow (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Understanding the relations  between observables for various particle species will lead to constraints on the physics driving  the evolution of heavy-ion collisions in simulations and, through that, to understanding cluster  formation, hyperon yields, in-medium interactions with of strange hadrons, and more (see also the  white paper on QCD Phase Structure and Interactions at High Baryon Density: Continuation of  BES Physics Program with CBM at FAIR [151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Quantifying uncertainties of transport predictions  In the era of multi-messenger physics, where information on the EOS is derived from different  areas of physics such as nuclear structure, nuclear reactions, and astrophysics, the ability to assess  the uncertainty of a particular result is of crucial importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This problem is especially relevant for  evaluations of constraints on the EOS from transport simulations of heavy-ion reactions, since it has  been found that using different transport models to describe the same data can lead to very different  24  conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As found in the TMEP comparisons (see [47] for a review), even with controlled input  the results from different models may vary considerably due to different implementation strategies  which in themselves are not dictated by the underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In such a situation, calculating  the mean and variance of different model predictions is not a reliable way of determining the  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An approach currently considered for ensuring a robust quality control in combining  inferences from different models is to weigh the models with a Bayesian weight which could be  based, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', on the performance of a given model in benchmark tests and/or its ability to reproduce  all key observables of a given reaction (for example, flow observables, particle multiplicities, and  spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Bayesian analysis can be also used for model selection through a comparison of results  from a list of available models with data, during which one assigns to each model a probability  of being correct based on the quality of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, this approach implicitly assumes that  among the considered models there is at least one “true” model (also known as the M-closed  assumption), which is often not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Efforts have been taken to analyze data with an M-open  assumption, where the existence of a perfect model is not assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For nuclear physics efforts,  this is being attempted within the Bayesian Analysis of Nuclear Dynamics (BAND) group [152] by  using Bayesian model mixing, where information from different models is combined for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Emulators and flexible EOS parametrizations  Robust explorations of the possible physics underlying various observables often necessitate  repeating the calculations many times for different combinations of physics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' When high  event statistics is needed, the computational task can easily overwhelm the available computational  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  An additional computational strain often arises from assessing Bayesian probability  distributions for any conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Increasingly, emulators are going to be used for this task, with  some steps having been already made [58, 102, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Notably, similar issues emerge in the area of  applications of hadronic transport [154] (see also Section V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  For explorations focused on the EOS, it may be of advantage to fit various possible EOSs with  flexible relativistic density functionals as suggested in [57, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This approach, given the complete  freedom in varying both the functional form of the EOS as well as the EOS parameters, is particu-  larly amenable to Bayesian analyses (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [58] for a Bayesian analysis with a parametrization  of the EOS in terms of the functional dependence of the speed of sound on density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The above list of issues facing the application of transport theory to heavy-ion collisions high-  lights the fact that this approach to putting tighter constraints on the EOS rests on overcoming  certain challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In simple terms, one attempts here to use a very dynamic and complex non-  equilibrium process to obtain information describing a relatively simple and well-defined system,  namely the equilibrated EOS of nuclear matter for different densities, temperatures, and isospin  asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To achieve this in a reliable way, multiple complex issues of many-body physics have  to be well controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' On the other hand, several of the needed improvements are relatively well-  understood, and tackling some of the unresolved problems poses an exciting intellectual challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  As a reward for undertaking this effort, one gains the opportunity to obtain information on the EOS  in a region which cannot be accessed through any other means: For densities below saturation,  there is strongly constraining information from nuclear structure, with significant contributions  coming also from low-energy heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Astrophysical observations on neutron stars and  neutron star mergers are mainly sensitive to densities above about 3n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The gap between these  domains can only be filled with intermediate energy heavy-ion collisions, and transport studies are  the essential tool to extract the information on the EOS from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  25  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Microscopic calculations of the EOS  Over the past decade, many-body nuclear theory has made significant progress in deriving  microscopic constraints on the nuclear EOS at low densities from chiral effective field theory  (χEFT) [155–158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The progress has been driven by improved two-nucleon (NN) and three-nucleon  (3N) interactions, rigorous uncertainty quantification, and algorithmic and computational advances  in the frameworks used to solve the many-body Schr¨odinger equation with these interactions (see  also the recent white paper on Dense matter theory for heavy-ion collisions and neutron stars [159]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Status  Chiral EFT [160–164] provides a systematic way to construct nuclear interactions consistent with  the low-energy symmetries of QCD, using nucleons (N’s), pions (π’s), and (in the case of delta-full  χEFT), ∆-resonances (∆’s) as the relevant effective degrees of freedoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Nuclear interactions in  χEFT are expanded in powers of momenta or the pion mass over a hard scale at which χEFT breaks  down;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' this breakdown scale is expected to be of the order of the ρ-meson mass, Λb ≈ 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  At each order in the EFT expansion, only a finite number of diagrams enter the description of the  interaction according to a chosen power counting scheme, of which the Weinberg power counting  has been predominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  For example, at the leading-order (LO) in Weinberg’s power counting  one includes contribution from the one-π exchange between two nucleons as well as momentum-  independent contact interactions, which allow one to describe key features of the nuclear interaction  already at the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At next-to-leading-order (NLO), two-π exchanges are included as well  as momentum-dependent contact interactions, and similarly, more involved terms appear at higher  orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The various low-energy coupling constants are determined from fits to experimental data,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the π-N couplings are fit to π-N scattering, while those describing NN short-range interactions  are fit to NN scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The advantage of χEFT over phenomenological approaches is that  multi-nucleon interactions, such as the important 3N interactions, naturally emerge in the EFT  expansion and, moreover, are consistent with the NN sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Forces involving increasingly more  nucleons are correspondingly more suppressed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the leading contribution to 3N forces (four-  nucleon (4N) forces) appears at N2LO (at N3LO) in Weinberg’s power counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, there  are only two new low-energy couplings appearing in the three- and four-body forces to N3LO, which  govern the strengths of the intermediate- and short-range contribution to the leading 3N forces,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, χEFT 3N and 4N interactions at N3LO are completely determined by  constraints on the coupling constants obtained from NN and π-N scattering”, usually resulting in  tight constraints on very neutron-rich matter from χEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Another key feature of χEFT is that order-by-order calculations in the χEFT expansion have  enabled estimation of theoretical uncertainties due to truncating the chiral expansion at a fi-  nite order [13, 158, 165, 166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Quantifying and propagating these EFT truncation errors enables  meaningful comparisons between competing nuclear theory predictions, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 10, and/or con-  straints from nuclear experiments and neutron-star observations in the multi-messenger astron-  omy era [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such comparisons are facilitated by Bayesian methods in a statistically rigorous  way [158, 167, 168] to take full advantage of the great variety of empirical EOS constraints we  anticipate in the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Chiral EFT also provides nuclear Hamiltonians governing the interactions in nuclear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, to calculate properties of a many-body system, computational methods able to solve the  Schroedinger equation for this system are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Among various frameworks used to solve the  nuclear many-body problem in dense matter, quantum Monte Carlo (QMC) methods and many-  body perturbation theory (MBPT) have been the main tools employed to study the physics of  26  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Comparison of the energy per particle E/N (left) and the pressure P (right) as functions of density  for pure neutron matter in different many-body calculations using interactions from χEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The left panel  also shows low-density QMC results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [169] and the conjectured unitary-gas lower bound on the energy  per particle of pure neutron matter from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  neutron-star matter in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Both methods have recently made tremendous advances in  predicting properties of nuclei and calculating the nuclear matter EOS [156, 158, 171–175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  QMC frameworks, such as the auxiliary field Diffusion Monte Carlo (AFDMC) method, are  based on imaginary-time propagation of a many-body wave function and enable us to extract  ground-state properties of a nuclear many-body system with high statistical precision [156, 171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Their nonperturbative nature also allows for the treatment of nuclear interactions at high mo-  mentum cutoffs, providing important insights into nuclear interactions at relatively short distances  that may help to improve the modelling of χEFT interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' QMC calculations of binding en-  ergies, radii, and electroweak transitions of nuclei up to A = 16 [176–182] using χEFT NN and  3N interactions are in very good agreement with experimental data [183–186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  QMC methods  were also used to calculate the EOSs of matter up to about twice the nuclear saturation density  n ≈ 2 n0 [187–191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The calculated EOSs include estimates of systematic truncation uncertainties,  and are commonly used to constrain properties of neutron stars [188, 192, 193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The past decade has also seen a renaissance for many-body perturbation theory (MBPT) calcu-  lations in nuclear physics [158, 175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Key to this development has been the discovery that nuclear  potentials with momentum-space cutoffs in the range 400 MeV <∼ Λ <∼ 500 MeV (not to be confused  with the breakdown scale of χEFT, Λb) are sufficiently soft to justify the use of perturbation theory  methods [194] (see [195] for a Weinberg eigenvalue analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such low-momentum potentials can  be obtained from renormalization group methods [196] or by directly constructing chiral effective  field theory potentials at a coarse resolution scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, recent advances in automatic  diagram generation [197] combined with automatic code generation [198] and high-performance  computing have led to a fully automated approach to MBPT calculations in nuclear physics [158],  in which chiral two- and multi-nucleon forces can be included to high orders in the chiral and  MBPT expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' MBPT has been demonstrated to be a computationally efficient and versatile  tool for studying the nuclear EOS as a function of baryon number density nB, isospin asymmetry  δ = (nn −np)/(nn +np), and temperature T [110, 111, 199, 200] with implications for neutron star  structure [158] and astrophysical simulations [201];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' here, nn and np correspond to the neutron and  25  5  Hebeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',ApJ (2013)  Hebeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',ApJ(2013)  Tews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',PRL(2013)  Tewsetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',PRL(2013)  Lynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',PRL (2016)  Lynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', PRL (2016)  20  4  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',PRL (2019)  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',PRL (2019)  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',GP-B (2020)  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',GP-B (2020)  Gezerlis, Carlson, PRC (2010)  Unitary gas (s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='376)  3  fm  [MeV  10  P2  5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  n [fm-3]  n [fm-3]27  proton densities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, MBPT allows us to compute the EOS of neutron-star  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', β-equilibrated) matter explicitly, which can help improve isospin asymmetry expansions of  the low-density nuclear EOS such as the standard quadratic expansion [199, 202–206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' MBPT also  allows us to study nuclear properties other than the nuclear EOS, including the linear response  and transport coefficients that could be used to inform more accurate numerical simulations of  supernovae and neutron-star mergers [207].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, MBPT for (infinite) nuclear matter has  been used to construct a microscopic global optical potential with quantified uncertainties based  on χEFT NN and 3N interactions [208, 209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Altogether, MBPT calculations of nuclear matter  properties can provide important constraints that enable microscopic interpretations of future nu-  clear reaction experiments [210] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', at the Facility for Rare Isotope Beams) and neutron star  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  To date, theoretical predictions for the nuclear EOS, optical potentials, and in-medium NN  scattering cross sections have been computed at finite temperature at various levels of approxi-  mation starting from fundamental two- and multi-nucleon forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These quantities are inputs to  transport model simulations [89, 211] of heavy-ion collisions used to extract constraints on the  properties of hot and dense nuclear matter (see Section II A for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In transport simu-  lations, the EOS, single-particle potentials, and in-medium NN cross sections are usually obtained  from effective phenomenological interactions [212, 213] that are fitted to the properties of finite  nuclei and cold nuclear matter, and then extrapolated into the finite-temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recently,  some effort has been devoted to benchmarking [109] the temperature dependence of these effective  interactions against predictions from χEFT or directly using EFT constraints in fitting effective  interactions [207, 214, 215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To enable such comparisons, the free energy of homogeneous nuclear  matter as a function of temperature, baryon number density, and isospin asymmetry has been cal-  culated using χEFT interactions up to second order in many-body perturbation theory [110] and  within the Self-Consistent Green’s Function (SCGF) approach [216], which resums particle-particle  and hole-hole ladder diagrams to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The resulting EOS has been shown to be consistent  with the critical endpoint of the symmetric nuclear matter liquid-gas phase transition [110, 216] as  well as the low-density/high-temperature pure neutron matter EOS from the virial expansion [204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Furthermore, single-particle potentials have been computed at finite temperature at the Hartree-  Fock level [217], from G-matrix effective interactions [218], and in SCGF theory [201, 219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Of  particular importance is the associated nucleon effective mass, which is obtained from a momen-  tum derivative of the single-particle energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The nucleon effective mass is directly related to the  density of states and hence governs entropy generation at finite temperature, with consequences for  the dynamical evolution of core-collapse supernovae and neutron star mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Finally, in-medium  NN scattering cross sections have been computed at finite density and zero [220] as well as at  finite [218] temperature using high-precision nuclear forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the next decade, the use of effective  field theory methods will enable a consistent framework for describing all of these quantities with  uncertainty estimates for input into transport simulations of heavy-ion collisions and astrophysical  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Challenges and opportunities  To fully capitalize on experimental and observational data and extract key information on fun-  damental questions in nuclear physics, continued progress in nuclear theory is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The combi-  nation of χEFT with modern computational approaches like machine learning, artifical intelligence,  emulators, and Bayesian inference have provided EOS results for a wide range of densities, and at  various proton-to-neutron asymmetries and temperatures, with quantified uncertainties [111, 166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Future progress in the development of fundamental interactions, combined with these tools, will  28  increase the precision of the results and enable us to answer open problems in chiral EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Among  these, the most pressing is at which densities and how χEFT breaks down [166, 188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular,  for studies of neutron-star mergers it is of great importance to describe dense matter at finite  temperatures [200, 201, 204], however, these might influence the breakdown of the theory in dense  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the next decade, it will be crucial to reliably determine how far one can push the χEFT  approach in nucleonic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  While microscopic calculations have been very successful in calculating properties of nuclei and  homogeneous matter at densities up to 1-2 times the nuclear saturation density, we need improved  microscopic descriptions of neutron-rich dense matter beyond that regime, at a few times nuclear  saturation density and finite temperatures, with quantified uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This can be achieved  by employing models derived within relativistic mean-field or density functional theory that are  firmly rooted in microscopic theory at lower densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such models will be very important to  connect theoretical calculations within the framework of χEFT to heavy-ion collision experiments  at accelerator facilities around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Heavy-ion collision experiments at intermediate beam  energies bridge the low- and high-density regimes of the EOS and provide complimentary informa-  tion to that obtained from nuclear structure or neutron-star studies [17] (see Section II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Robust  inferences from the experimental data will require more accurate predictions from transport the-  ory, which strongly depend on, among others, mean-field or density functional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It will be  imperative to test and constrain such models for the EOS with more rigorous microscopic calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Beyond their use in hadronic transport simulations, these models are also a crucial input  for calculations of properties of neutron star crusts (see Section II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Additional theoretical constraints might be provided by high-density calculations within the  framework of perturbative QCD (pQCD) [221], which can be applied at very high densities of  the order of 40 times the nuclear saturation density, where the strong interactions among quarks  become perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Constraints on the EOS based on pQCD, together with assumptions on  causality and stability, have been used to constrain the EOS at lower densities probed in the core  of neutron stars [222–225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, it has been found that the constraining power of pQCD  calculations is strongly dependent on the way in which they are implemented [225, 226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Future  studies have to establish to what extent pQCD constraints are robust at densities of the order of  several times nuclear saturation density, and how constraining future higher-order calculations may  become.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In this regard, improved microscopic calculations of the nuclear EOS using the functional  renormalization group [227, 228] will provide important insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Neutron star theory  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Status  Measurements of the EOS, masses of neutron-rich isotopes far from the band of stability, and  experimental constraints on nucleon effective masses provide essential input into neutron star mod-  els, progressing our understanding of the structure and dynamics of these astronomically important  objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Several properties of neutron stars, including the mass-radius relation and their tidal de-  formabilities, can be calculated once the EOS is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This, in turn, enables us to constrain  the EOS once those properties are observed [229].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Nuclear EOSs for neutron stars can be constructed from, for example, ab initio calculations and  density functionals [230–233] or, more schematically, from meta-models [234–236] parameterized  by nuclear matter parameters, which can be used to make contact with heavy-ion collisions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Ab initio calculations take into account more fundamental properties of the nuclear force (see Sec-  tion II B), but prohibit the calculation of large ensembles of EOSs spanning the nuclear parameter  29  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Impact of nuclear physics theory and experiment, and  different astrophysical measurements on constraining the cold  neutron-star EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Blue lines show a family of EOS that are con-  strained by chiral EFT at low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At higher densities, the  EOS can then be constrained using GWs from inspirals of neutron  star mergers, data from radio and X-ray observations of pulsars,  and electromagnetic signals associated with neutron star mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The indicated boundaries between regions affected by these mea-  surements are not strict and depend on the EOS and properties  of the astrophysical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [237].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Meta-models allow rapid  computation of such large ensem-  bles, but encode mainly bulk prop-  erties of nuclear matter, which ex-  cludes them from being used to  model finite nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Density func-  tionals represent a compromise, al-  lowing both rapid computation of  EOSs and use in finite nuclear mod-  els, and thus are more suited to  combining nuclear experimental and  astrophysical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Many  of these models can be smoothly  extrapolated from the saturation-  density to arbitrarily high density,  in which case astronomical obser-  vations can be used to constrain  the saturation-density nuclear mat-  ter parameters and their density de-  pendence [236, 238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This extrapo-  lation, however, is model-dependent,  as different density functionals have  different dependence on density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Ad-  ditionally, this extrapolation might  not be physically well-founded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  As densities inside neutron stars  can reach up to several times nuclear saturation density, at some (as-yet not determined) density a  description in terms of purely nucleonic degrees of freedom is expected to break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Heavy-ion  collisions can help us constrain that point, and the nature of any phase transitions that occur above  saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The nuclear EOSs can be then combined with models describing the EOS at  higher densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Models that explicitly include a range of possible high-density degrees of freedom,  such as hyperons and quarks, can be constructed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' the predicted neutron star compositions are then  dependent on the particular model used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Another approach is to use more general models that give  up the explicit dependence on the underlying degrees of freedom, thus losing information on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',  appearance of exotic particles at high densities, in favor of spanning the full space of physically con-  sistent EOSs, reducing the model dependence of inferences from astrophysical observations [239].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  These schemes include piecewise polytropes [14, 240–242], line segments [192, 243], speed-of-sound  models [188, 189, 244–246], spectral models [247] and non-parametric models generated from Gaus-  sian processes (GPs) [168, 248–251] or machine learning techniques [252].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' If these more general  approaches are used down to the nuclear saturation density, extra modeling is required to connect  them to the microscopic nuclear EOS and nuclear observables [253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Once the EOS is specified, the solution of the Tolman-Oppenheimer-Volkov equations and their  extensions including rotation, determining the structure of a neutron star through balancing the  attractive force of gravity and the repulsion coming from the EOS, provide predictions for bulk  properties of the neutron star such as radii, tidal deformabilities, moments of inertia, and break-up  frequencies of neutron stars as a function of their mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' All of these properties can be compared  with multi-messenger observations, including gravitational waves and electromagnetic signals from  neutron-star mergers and isolated neutron stars [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The systematic construction of neutron star EOS models and statistical inference of EOS pa-  103  8  102  [Mev fm  EM: Kilonovae / GRB  GWs (post-merger)  Pressure  101  GWs (inspiral)  Radio and X-ray pulsars  100  Nuclear Physics  Experiment and Theory  2  4  6  8  Number density [nsat30  rameters from data is an endeavor that is just over a decade old [14, 240–242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This effort has  matured in the current era of multi-messenger astronomy with a large push to explore the model-  dependence of EOS inferences [244, 254] and ways of connecting the EOS with astrophysical and  nuclear data [17, 167, 193, 245, 246, 255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Different choices of which observables to include or infer  can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, astrophysical observations can be used to infer the EOS, which can  then be connected to nuclear models to inform their parameters and predict nuclear observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Conversely, nuclear observables can be used to infer nuclear parameters, which can then inform  the neutron star models and predict astrophysical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The future lies in combining more  and more sets of data of both types to understand nuclear and neutron star models better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Exciting progress has been made in gathering astrophysical data to constrain our dense matter  theories (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 11 for an illustration of density regions affected by different observables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Neutron-  star data from the last 5 years identified the heaviest neutron star known to date with a mass of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='08(7)M⊙ [256, 257] (where M⊙ is the solar mass), while the kilonova AT2017gfo, associated with  GW170817, has placed an upper limit on the maximum mass to be on the order of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3M⊙ [258,  259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The detection of GW170817 by the LIGO-Virgo Collaboration has enabled us to place  constraints on the tidal deformability of this system, ˜ΛGW170817 ≤ 720 [260, 261].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Neutron Star  Interior Composition Explorer Mission (NICER) has provided two mass-radius measurements by  observing X-ray emission from several hot spots on the neutron star surface, finding a radius of  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='02+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='24  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='06km for a star with mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='15  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='14M⊙ (PSR J0030+0451) and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5km for a star with  mass 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='08(7)M⊙ (PSR J0740+6620) in the analyses of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [255, 262–264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' X-ray observations of  the temperature of the neutron star in the Cas A supernova remnant have revealed core cooling  on the timescale of years, hinting at the possible superfluid properties of the core [265].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These  observations have enabled meaningful constraints on the EOS to be set and have already allowed us  fascinating glimpses into the possible properties of high-density matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, perturbative  QCD predicts that the speed of sound squared approaches the conformal limit of 1/3 from below  as the density becomes arbitrarily high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Meanwhile, inferences of the neutron star EOS from  observational data indicate that the speed of sound rises in the core to significantly above c2  s =  1/3 [188, 266–269].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, this suggests that the speed of sound has a non-trivial behavior  with increasing density [188, 221, 270].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At the same time, tentative evidence for quark matter in  neutron star cores, which in turn indicates a softening of the EOS, has likewise been suggested [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  If we want to leverage the substantial data we have on neutron star cooling and dynamical  evolution, additional EOS quantities need to be supplied consistently for each EOS model, such  as the effective masses (see also Section VI E) and superfluid neutron and proton gaps, essential  for modeling thermal and dynamical properties of neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, the mutual friction  of the core – the strength of the coupling between the charged particles (electrons, protons) and  superfluid neutrons – depends on the effective neutron mass and the proton fraction [271], which  both also correlate with the symmetry energy [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A consistent extraction of both symmetry  energy parameters and effective masses from heavy-ion collision data is therefore required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In contrast to efforts devoted to systematic, statistically meaningful inferences of the EOS in the  cores of neutron stars, modeling the neutron star crust is still in its infancy: The first calculations  of large ensembles of systematically parameterized crust models and their use in statistical analysis  have only been carried out recently [272–276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, much more nuclear experimental data can  be brought to directly bear on crust physics, and we have entered an era where we can access  information about the crust with unprecedented fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, we have now observed the  same neutron-star crust as it first cooled, then became heated by accreted matter, and then cooled  again [277–281].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' We have followed a pulsar through a glitch – a sudden change in the rotation period  of the pulsar – and glitch recovery with a resolution of a few seconds [282].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These observations  have provided very strong evidence that the crust is solid, that there exist superfluid neutrons in  the inner crust which can be decoupled from the nuclei in the crustal lattice, and that nuclear  31  reactions from accreted material sinking into the crust provide deep crustal heating [279, 283, 284].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Additionally, models of the neutron star crust predict that, prior to the transition to homoge-  neous matter, isolated nuclei in the crust fuse to form cylindrical, planar, and more exotic shapes,  termed “nuclear pasta”, that can affect neutron-star observations [285–287].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This crust-core bound-  ary region, often referred to as the mantle, is likely a complex fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Density functional theory and  molecular dynamics calculations of these structures reveal a complex energy landscape with many  coexisting shapes, and correspondingly complex mechanical and transport properties [288–294],  which are strongly influenced by the EOS at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0 through the pressure, proton fraction,  and surface energy of the structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  These properties can also be studied in multifragmenta-  tion reactions, which probe, among others, the competition between nuclear surface energy and  Coulomb energy at sub-saturation density [295–297].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Inhomogeneous matter in the crust of a neutron star, including the dripped neutrons expected in  the inner crust, can be modeled using a variety of nuclear theory techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These usually involve  calculations within a single, repeating unit (Wigner-Seitz cell) of matter, typically containing a sin-  gle nucleus [298–300].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The compressible liquid drop model (CLDM) treats the nuclear matter inside  and outside of nuclei as homogeneous and described by the bulk matter EOS, while the surface  energy is specified by a separate function with additional parameters [288, 300–303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The surface  parameters and those that define the dimensions of the cell and nucleus are minimized to obtain  the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The Thomas-Fermi model employs the local density approximation, modeling  matter with a specified form of the inhomogeneous nuclear matter density in the unit cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' here,  the parameters of the density distribution are varied to obtain the ground state configuration [304].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Microscopic approaches to describing inhomogeneous nuclear matter, in which individual neutrons  and protons are the degrees of freedom, include quantum Hartree-Fock or Relativistic Mean Field  models [305–310], and semi-classical molecular dynamics approaches [292, 311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  There is a great need for nuclear physics input into models of the neutron star crust, which  analyses of heavy-ion collision data can provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, the thickness, mass and moment  of inertia of the crust depend on the higher-order symmetry energy parameters L, Ksym, and  Qsym [272, 274, 312].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus measurements of the symmetry energy parameters up to third order  in heavy-ion collision experiments are essential to understand the properties of the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The  symmetry energy, effective masses, and surface energies of nuclear clusters strongly affect the  proton fraction on either side of the crust-core transition density, the extent of nuclear pasta near  the crust-core boundary, the mechanical and transport properties, the thermal conductivity and  specific heat, the electrical conductivity, and the shear modulus of the crust [298, 304, 309, 313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Nuclear experiment can thus constrain neutron star crust models, and astrophysical observables  associated with the crust can measure nuclear observables as well as measurements of neutron star  bulk properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, the symmetry energy can be constrained by combining nuclear data  with crust and core observables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', through a potential multi-messenger measurement of the  resonant frequency of crust-core interface oscillations [276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Challenges and opportunities  The next decade will provide a wealth of new data on neutron stars, as the LIGO-VIRGO-  KAGRA detectors are expected to observe many new binary neutron-star mergers, some of them  with electromagnetic counterparts [314–316].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As NICER continues to measure more neutron star  masses and radii, next-generation X-ray timing missions such as Strobe-X [317] and radio tele-  scopes such as the Square-Kilometer Array will increase the number of pulsars we see and are able  to measure by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Long-timescale observations of individual pulsars (using  radio timing) and persistent gravitational waves from deformations of neutron stars will lead to  32  measurements of their moments of inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These new data points might enable us to pin down  the nuclear matter EOS, to discover or rule out the existence of phase transitions to exotic forms  of matter in the cores of neutron stars, and to reliably constrain microscopic interactions between  fundamental particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Although model-agnostic extrapolations to higher densities such as through the use of poly-  tropes [194, 240], speed of sound schemes [188, 244, 318], Gaussian processes [249, 250] and spectral  methods [247], combined with robust data analysis, will eventually allow us to pin down the dense-  matter EOS, they cannot answer the question about the relevant microscopic degrees of freedom at  high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Hence, it is crucial to develop improved microscopic models with well-quantified un-  certainties in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At the same time, creating ensembles of outer core and crust models that  allow for inclusion of astrophysical and nuclear data requires underlying nuclear models to have  enough freedom to explore a large region of parameter space, and allow fast computation of relevant  quantities that also capture the essential physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Currently, it is energy density functionals like  Skyrme, Gogny, and Relativistic Mean Field models that provide these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently,  progress could be made by making a stronger connection between these models and microscopic  approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', connecting energy-density functionals to ab initio calculations allowing a more  direct link to χEFT [299, 319, 320].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the same spirit, EFT calculations of the EOS can be used as  a “low-density limit” to calibrate higher-density models for neutron stars and heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The crust can be modeled consistently with nucleonic matter in the core using density functional  theory to model both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' When choosing a model, a compromise must be made between accurate  modeling of microscopic quantum effects, such as shell effects in the nucleus and surrounding  neutron gas, and the computational expediency required to construct large ensembles of crust  models needed for statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, quantum shell effects strongly determine the  evolution of the mass and charge number of nuclei with density, alter the effective mass of dripped  neutrons, and drive the complex energy landscape of nuclear pasta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Fully microscopic quantum  calculations include shell effects self-consistently, but are computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The CLDM  approach can be used to construct large numbers of crust models, but requires shell effects to be  added by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Future work needs to develop schemes of incorporating such microscopic effects in  large ensembles of crust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The method that may allow that is the Extended Thomas-Fermi  method, incorporating shell effects through the Strutinsky Integral: see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [321].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Models should also incorporate nuclear pasta, as its extended structures may contribute to the  mechanical and thermal properties of matter at the crust-core boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It is computationally  demanding to model transport and mechanical properties of the crust microscopically or in simula-  tions [322], particularly in the nuclear pasta phases, and it is unrealistic to include these quantities  in large ensembles of crust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Simpler schemes that extrapolate the mechanical and trans-  port properties across the parameters space based on microscopic models could be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Also,  representative crust models inferred from data can be used to calculate these crust properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  There is also a need for a balance between accuracy and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A model can be accurate  but not precise (predicting the correct value of a physical quantity but having large error bars),  or precise but not accurate predicting very small error bars, but not predicting the correct value  of some physical observable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Individual crust models can be created from mass models that  are precisely fit to data and which predict precise values for, for example, the symmetry energy  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, to make accurate inferences of nuclear matter parameters from astrophysical  observables, and to include their experimentally measured ranges, ensembles of models spanning  the parameter space should be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Both strategies are important, and the precision-fit  models can act as benchmarks against which we assess the outcomes of statistical inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  When older neutron stars accrete matter in the crust the matter gets gradually pushed down  into the core and replaced by the accreted matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The temperatures in the crust are well below  the nuclear potential energies, so the replacement crust cannot easily attain nuclear statistical  33  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Ensembles of accreted crust models are yet to be constructed, but are necessary to  correctly account for deep crustal heating and therefore to fully utilize the observations of cooling  of accreted crusts in low mass X-ray binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In all this work, effort must be made to calculate the different observables consistently as well  as to combine different data sets in a well-controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This is expanded upon in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  HEAVY-ION COLLISION EXPERIMENTS  Establishing the equation of state (EOS) of nuclear matter has been a major focus of heavy-ion  collision experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While very low energy collisions can probe nuclear matter at densities smaller  than the saturation density n0, highly-compressed nuclear matter is produced in the laboratory  by colliding heavy nuclei at relativistic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At even higher energies, in the ultra-relativistic  regime, quarks in the colliding nuclei become almost transparent to each other and therefore escape  the collision region, which means that matter measured at midrapidity is characterized by a nearly-  zero net baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Heavy-ion collision experiments at top beam energies at the Relativistic  Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) provided convincing evidence that  at high temperatures and near-zero baryon density, nuclear matter becomes a quark-gluon plasma  (QGP) [323–329], a deconfined but strongly-interacting state composed of color charges, confirming  Lattice QCD (LQCD) calculations of the EOS at zero density [330–332].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  While the region of the QCD phase diagram explored in ultra-relativistic heavy-ion collisions  is relatively well understood, the EOS of dense nuclear matter at moderate-to-high temperatures  and moderate-to-high baryon densities is not known well due to the break-down of first-principle  approaches in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Answering pressing questions about the QCD EOS in this region, such as  whether the quark-hadron transition becomes of first-order at high densities or what is the minimal  energy required to produce the QGP, is the driving force behind Phase II of the Beam Energy Scan  (BES) program at RHIC, the HADES experiment at GSI, and the future Compressed Baryonic  Matter (CBM) experiment at the Facility for Antiproton and Ion Research (FAIR), Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This renewed interest in the nuclear matter EOS at high densities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' accessible in heavy-ion  collisions at intermediate energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' coincides with an increased effort to constrain the EOS of  neutron-rich matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' probed in studies of neutron stars and neutron star mergers (see Section II C  as well as recent white papers on QCD Phase Structure and Interactions at High Baryon Den-  sity: Continuation of BES Physics Program with CBM at FAIR [151] and Dense matter theory for  heavy-ion collisions and neutron stars [159]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, studies show that heavy-ion collisions  in this regime and neutron star mergers probe similar temperatures and baryon densities [333, 334].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, while matter created in collisions of heavy-ions has comparable numbers of protons and  neutrons, matter inside neutron stars is neutron-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Establishing the much needed connection  between the studies of the nuclear EOS as probed in heavy-ion collisions and as inferred from  neutron star observations is possible by leveraging the experimental capabilities of the newly com-  missioned Facility for Rare Ion Beams (FRIB), where energetic beams of proton- and neutron-rich  nuclei can be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Heavy-ion collision experiments at FRIB can put tight constraints on the  dependence of the nuclear matter EOS on the relative proton and neutron abundances [22], and  thus enable a description of both dense nuclear and dense neutron-rich matter within a unified  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Indeed, if we assume that the core of a neutron star is composed of mostly uniform nucleonic  matter, then nuclear matter and neutron stars should be described by a common EOS, specifying  the relationship between the pressure and the temperature, density, and isospin content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The  theoretical construct of symmetric nuclear matter consisting of equal amounts of neutrons and  34  Number density  Astro  HIC(asym)  Nuclei properties  Theory  Crust  HIC(SNM)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This schematic plot illustrates the approximate density  ranges that are explored in the studies of chiral effective field  theory, nuclei properties, heavy-ion collision experiments, and  observations of neutron stars and their crusts in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  protons has been successful to derive  properties of symmetric matter such  as the saturation density and bind-  ing energy, however, an additional  term in the EOS is needed to de-  scribe nuclear matter with unequal  neutron-proton composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This  second term depends on the asymme-  try δ, defined as δ = (nn − np)/nB,  where nn, np, and nB are the neu-  tron, proton, and total baryon densi-  ties, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, one  can view the asymmetry as the neu-  tron excess fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Mathematically,  the energy per nucleon can be then  expressed as a sum of two terms:  ϵ(nn, np) = ϵSNM(n) + S(n)δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Here, the first term represents the energy per nucleon of symmetric  nuclear matter, while the second term accounts for the correction needed when δ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Therefore,  δ is a crucial parameter that distinguishes neutron stars (with δ >∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8) from most nuclei (with  δ <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Given the relatively small values of the asymmetry δ for nuclei, in heavy-ion collision  experiments it is easier to constrain the coefficients of the EOS of symmetric matter, ϵSNM(nB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In  contrast, the energy contribution from the asymmetric term, also known as the symmetry energy,  constitutes a small fraction of the total energy of a nucleus even for neutron-rich heavy radioactive  isotopes (< 5% in the liquid drop model), and its determination requires precise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Furthermore, because the isospin effects in any observable tend to diminish with temperature, it  may be difficult to measure the symmetry energy at very high densities, which require high-energy  heavy-ion reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Therefore, symmetry energy is best probed in heavy-ion collisions of highly  asymmetric isotopes at low to intermediate energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 12 shows schematically the baryon density regions explored by different areas in nuclear  physics studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recent breakthroughs in astronomical observations with state-of-the-art instru-  ments led to the first detection of a binary neutron-star merger and the unprecedented radii mea-  surements of neutron stars with accurately known masses (see Section II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The neutron star  mass-radius relationship provides an insight into the EOS at high densities above twice saturation  density (>∼ 2n0), as represented by the red arrow (labelled “Astro”) in the upper right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Labo-  ratory experiments, especially those using heavy-ion collisions, are essential to provide information  on the dependence of the EOS on density and the asymmetry (see also Section II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' High-energy  heavy-ion collisions can provide insight into the symmetric nuclear matter EOS as represented  by the gold right-pointing arrow (labeled “HIC(SNM)”), while current probes of the symmetry  energy are more suited for measurements of lower energy heavy-ion reactions (<∼ 600 AMeV) as  represented by the left-pointing gold arrow (labeled “HIC(asym)”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Many properties of nuclei,  such as masses and radii, have been shown to be mainly sensitive to densities around (2/3)n0,  however, with a careful selection of nuclear observables, the symmetry energy has been probed  over densities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3n0 < nB < n0 using Pearson correlation methods [12, 335] (green left-pointing  arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recent advances in chiral effective field theory (see Section II B) enabled extrapolations  of the EOS to be extended up to ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0 [13], but the uncertainty increases exponentially with  density for densities that are higher than n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It is not clear what is the maximum density up to  which such extrapolations can succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Finally, one of the most interesting regions is at very low  densities (<∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0), corresponding to the crust of a neutron star where matter is not uniform (see  Section II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' There, matter changes with increasing density from a Coulomb-dominated lattice to  L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='035  nuclear pasta and, ultimately, to uniform matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The density and nature of these transformations  are again dictated largely by the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Measurements made in heavy-ion collisions at intermediate energies, probing high densities  or, equivalently, small nucleon separations, will yield key insights into the nature of the nuclear  force, including the density-dependence of the nuclear symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experimental efforts to  determine the EOS for symmetric matter and the symmetry energy are described in Section III A  and III B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Please note that all beam energies Elab quoted in this section are the single-beam kinetic energies  per nucleon, in units of AMeV or AGeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (Alternatively, Elab is also sometimes denoted by other  authors as E/A, with units of MeV or GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Additionally, while many results are reported in  terms of their constraints on the incompressibility K0, one should refrain from interpreting them  as constraining the behavior of the EOS around the saturation density (see Section II A 2 for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments to extract the EOS of symmetric nuclear matter  Heavy-ion collision experiments worldwide have extensively studied the EOS of symmetric  nuclear matter at supra-saturation densities over the past four decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments based at  the Schwerionensynchrotron-18 (SIS-18) ring accelerator at the GSI Helmholtz Centre for Heavy  Ion Research (GSI) have probed Au+Au collisions at energies between Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='09–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV  (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='92–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='52 GeV), corresponding to fireball densities 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Further experimental efforts  with Au+Au collisions were carried out at higher energies, Elab = 2–10 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='70–  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='72 GeV), at the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory  (BNL) to probe fireball densities 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5–5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Complementing the densities reached at AGS-BNL is the  Beam Energy Scan (BES) program of the Solenoidal Tracker at RHIC (STAR) experiment at RHIC  in BNL, where high-statistics Au+Au collisions were performed at energies between Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9–  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 AGeV (√sNN = 3–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7 GeV) in the fixed-target mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A selection of constraints on the EOS  extracted from the above experiments is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Below, we describe the observables stud-  ied to extract the symmetric nuclear matter EOS, experiments probing the aforementioned density  ranges, and inferences for the hadronic transport codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Measurements sensitive to the EOS  Collisions of heavy nuclei at relativistic energies lead to a rapid compression and heating of  matter trapped in the collision region, followed by its dynamic expansion and cooling (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The EOS governs both the compression as well as the expansion of the hot and dense nuclear matter,  which in turn affect measured particle distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, a stiffer EOS (characterizing  matter that is more incompressible) leads to a relatively smaller compression and, consequently,  smaller heating, but a faster transverse expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The smaller temperatures reached in the fireball  lead to smaller thermal dilepton and photon yields (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [336–338]), while the faster expansion  manifests itself in relatively higher mean transverse momenta (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [24]) and a shorter lifetime  of the fireball, the latter of which can be probed by a combination of the femtoscopic radii, R2  out −  R2  side, shown to be proportional to the duration of particle emission [339, 340].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The EOS also plays a large role in the interplay between the initial geometry of the system,  the expansion of matter originating from nucleons trapped in the collision zone (participants),  and the propagation of nucleons which are either still incoming into the collision region or whose  trajectories do not directly cross the collision region (spectators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In systems colliding at beam  energies for which the speed of the fireball expansion is comparable with the speed of the spectators,  36  the resulting complex dynamical evolution affects the transverse expansion of the system and,  therefore, the angular particle distributions in the transverse plane dN/dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, moments  of the angular momentum distribution, known as the collective flow coefficients and defined as  vn =  � dφ cos(nφ) (dN/dφ)/  � dφ (dN/dφ), describe the collective motion of the system and are  highly sensitive to the EOS, as shown in numerous hydrodynamic [77, 341–346] and hadronic  transport [31, 67, 78, 79, 347, 348] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At the same time, collective flow observables can be  measured with high precision, making them primary observables used to constrain the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In off-central collisions, the initial collision zone has an approximately elliptical shape, and the  pressure gradients within the collision zone are larger along its short axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' If the spectator nucleons  move out of the way before the fireball expands, the pressure gradients in the collision zone lead to  particle distributions around midrapidity which have maxima coincident with the reaction plane  (“in-plane” emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' If, however, the spectators stand in the way of the fireball expansion, this  leads to a preferential emission along the long axis of the collision zone (“out-of-plane” emission,  also referred to as “squeeze-out” due to the role that the spectators play in the expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The  preferential emission in either in-plane or out-of-plane direction is described by the second Fourier  coefficient of flow v2, also known as the elliptic flow, which is positive in the former case and  negative in the latter case (see the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The magnitude of the elliptic flow,  as well as the energy at which v2 changes sign, are intrinsically connected to the stiffness of the  EOS: for example, a stiffer EOS results in both a faster expansion and a more forceful blocking by  spectators, which leads to a larger squeeze-out and a more negative v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The rapidity-dependence of the first Fourier coefficient of flow, the directed flow v1, is also  sensitive to the EOS as it measures the degree of spectator deflection due to the interaction with  the collision zone [349].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the center-of-mass frame, the spectators from a nucleus moving in the  positive beam direction will be deflected to one side, while the spectators from the other nucleus,  moving in the negative beam direction, will be deflected to the opposite side, resulting in a positive  v1 at positive rapidities and a negative v2 at negative rapidities (here, the sign of v1 is a matter of  convention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' see [58] for a more detailed explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The magnitude of the directed flow in each  region and, therefore, its slope at midrapidity are directly related to the EOS: for example, a softer  EOS leads to a smaller deflection and a smaller slope of v1 at midrapidity, where in particular a  sufficiently soft EOS can even lead to a negative slope of v1 [343, 350].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' We note that spectators  are necessary to obtain substantial magnitudes of the slope of the directed flow, as can be seen by  its small values at high collision energies (see the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Beyond the collective flow phenomena, the EOS also has an effect on hadron production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In  particular, much attention has been given to production of hadrons in heavy-ion collision at energies  below the nominal production threshold in NN reactions (“sub-threshold” production), which  requires multiple sequential hadron-hadron collisions to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The probability of these collisions  is significantly higher in the high-density regions, and consequently the yield of sub-threshold  probes is expected to be substantially enhanced if higher densities are reached in the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Of particular importance for the EOS studies is sub-threshold production of K+ mesons, which  undergo few final-state interactions with the nuclear medium and therefore mostly leave the fireball  unperturbed, making them a sensitive probe of the highest densities reached and, consequently, of  the nuclear EOS [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments probing densities between 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  As described above, sub-threshold particle yields can be used as probes of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In partic-  ular, due to their low in-medium cross-section, K+ mesons produced at energies lower than the  production threshold of Elab = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='58 GeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='55 GeV) can carry unperturbed informa-  37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  Elab [GeV]  1  2  3  4  5  6  7  (MK+/A)Au+Au / (MK+/A)C+C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  Elab [GeV]  1  2  3  4  5  6  7  (MK+/A)Au+Au / (MK+/A)C+C  soft EOS, pot ChPT  hard EOS, pot ChPT  soft EOS, IQMD, pot RMF  hard EOS, IQMD, pot RMF  KaoS  soft EOS, IQMD, Giessen cs  hard EOS, IQMD, Giessen cs  ❑HM  ▲ SM  FOPI  Au+Au  protons  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25  beam energy (A GeV)  <b0<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='45 ut0>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  v2n(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Left panel: Beam energy dependence of K+ yield ratios in inclusive Au+Au collisions and C+C  collisions between Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='24–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='52 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A comparison of RQMD [84] and IQMD [354]  model calculations indicates a soft EOS (K0 = 200 MeV, red symbols) instead of a hard EOS (K0 =  380 MeV, blue symbols) when compared to KaoS data [353] (black symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [351].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Right  panel: Beam-energy dependence of the elliptic flow for protons in Au+Au collisions at Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 AGeV  (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07 GeV) (black symbols) as measured by the FOPI experiment [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Comparison to IQMD  transport calculations with momentum dependence prefers a soft EOS (blue triangles) over a hard EOS (red  squares), yielding K0 = 190 ± 30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  tion on the fireball density and the stiffness of the EOS [351].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The Kaon Spectrometer (KaoS)  Experiment [352] at SIS18 in GSI studied the subthreshold production of K+ mesons at beam  energies between Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='16–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='70 GeV), and established it as a sensitive  probe to the underlying EOS of the hot and dense nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To reduce the experimental  and model uncertainties, the production of K+ mesons in a heavier Au+Au system was compared  with the production in a lighter C+C system [353].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Analyzing the experimental results together  with transport model calculations in the RQMD [84] and IQMD [354] model enabled extraction of the  EOS of symmetric nuclear matter characterized by an incompressibility of K0 = 200 MeV (see also  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Both models included effects due to the momentum-dependence of the EOS by including  K+/−N potentials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', a repulsive mean field for K+ and an attractive mean field for K−, which  are required to reproduce the K+ and K− emission pattern [355] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Collective behavior in heavy-ion collisions is likewise a very sensitive probe of the underlying  EOS and has been extensively studied since its discovery by the Plastic Ball spectrometer at the  Bevalac in Lawrence Berkeley National Laboratory [356, 357].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In particular, the elliptic flow v2 is  highly sensitive to both the initial geometry of the collisions and pressure gradients experienced  throughout the evolution of the created systems [77, 358].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The Four Pi (FOPI) Experiment at SIS18  in GSI carried out extensive measurements of the beam energy dependence of the elliptic flow of  protons and light fragments (such as deuterons, tritons, and 3He) over the entire range of SIS18  energies, Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='09–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='92–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='52 GeV) [88, 359].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The nucler EOS extracted from  a comparison to IQMD simulations [67] is characterized by an incompressibility K0 = 190 ± 30 MeV  when momentum-dependent interactions are taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This constraint is consistent  with the KaoS incompressibility inferences and suggests a soft EOS for symmetric nuclear matter  at 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 13 and also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  38  K+ SMASH  K- SMASH  K+ UrQMD  K+ JAM  v2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='04  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='04  y - ycm  −1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 < pT < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6 GeV/c  K+  K-  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0  π+ SMASH  π- SMASH  π+ UrQMD  π+ JAM  v2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='04  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='04  y - ycm  −1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2 < pT < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6 GeV/c  π+  π-  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0  p SMASH  p UrQMD  p JAM  Λ SMASH  Λ UrQMD  Λ JAM  v2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='04  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='04  y - ycm  −1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 < pT < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 GeV/c  √sNN = 3 GeV 10-40%  Au+Au collisions  p  Λ  v1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Directed (v1, top) and elliptic (v2, bottom) flow of protons and lambda baryons (left panels), pions  (middle panels), and kaons (right panels) as a function of rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Measurements from STAR [94] (symbols)  were performed with Au+Au collisions at √sNN = 3 GeV (Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='91 AGeV) and 10–40% centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Results from UrQMD (blue bands), JAM (green bands), and SMASH (orange bands) hadronic transport models  were obtained using a relatively hard EOS at moderate densities (characterized by K0 = 300 in SMASH  and K0 = 380 MeV in JAM and UrQMD), with the EOS used in SMASH becoming significantly softer at high  densities (see [58, 94] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments probing densities above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  Pioneering proton directed and elliptic flow measurements were performed in Au+Au colli-  sions for beam energies Elab = 2–10 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='70–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='72 GeV) by the E895 [83, 90] and  E877 [82] experiments at AGS-BNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Notably, it was observed that around Elab ≈ 4 AGeV  (√sNN ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='32 GeV), the proton v2 changes from a preferential out-of-plane emission, reflecting  a complex interplay between the spectators, the expanding collision zone, and the EOS, to an in-  plane emission (see the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The experimental results were used in a comparison  with the pBUU transport model to extract the EOS for densities between 2–5n0, which constrained  the EOS to those described by values of the nuclear incompressibility between K0 = 210–300 MeV,  ruling out extremely soft and extremely hard EOSs [31] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This rather broad constraint  on K0 reflects the fact that the experimental results for the collective flow could not be reproduced  with one EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The STAR Experiment at RHIC-BNL with its Beam Energy Scan (BES) program [360, 361]  performed Au+Au collisions for √sNN = 3–200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In terms of the freeze-out temperature  and chemical potential, (Tfo, µfo), this allowed STAR to comprehensively scan the QCD phase  diagram from (80, 760) MeV to (166, 25) MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Probing the phase diagram at high  densities was possible at RHIC in part due to STAR’s capability to shift from a standard collider  to a fixed-target (FXT) mode, which was used to scan through the lower energies √sNN = 3–  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7 GeV (Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='91–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='06 AGeV), thereby establishing a substantial overlap with the previously  39  discussed AGS experiments [362].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recently, STAR measured collective flow (v1, v2) in collisions at  √sNN = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 GeV [94] and √sNN = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 GeV [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A comparison of results from the √sNN = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 GeV  data (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 14) with UrQMD and JAM simulations indicates a relatively hard EOS (characterized by  K0 = 380 MeV) [94];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' similarly, a recent Bayesian analysis of the STAR flow data based on a flexible  parametrization of the EOS used in the SMASH transport code results in a relatively hard EOS at  moderate densities (characterized by K0 = 300 ± 60 MeV) with a substantial softening at higher  densities [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, both UrQMD and SMASH do not currently include momentum-dependent  interactions, which are crucial for a correct description of the transverse-momentum-dependence  of the elliptic flow [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, while the above models reproduce the proton v1, v2 well, none  of the models can simultaneously describe the flow of Lambda baryons and mesons (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Challenges and opportunities  Experiments probing densities between 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  The High Acceptance Di-Electron Spectrometer (HADES) Experiment [363] at SIS-18 in GSI  has performed collective flow measurements in Au+Au collisions at Elab = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='23 AGeV (√sNN =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='42 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The high acceptance and high statistics of HADES measurements allow one to per-  form multi-differential studies of flow harmonics, ranging from v1 up to v6, which in turn enables  reconstruction of a full 3D-picture of the emission pattern in the momentum space [18, 148] (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In addition to the collective flow measurement capabilities, HADES can also precisely  measure the dielectron excess yield, which was used to extract the fireball temperature, finding  it to be 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1 MeV/kB [364].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These precise measurements of the fireball temperature and  the underlying dielectron spectra allow HADES to investigate the presence of a first-order phase  transition at SIS-18 energies and look for signs of a potential change of degrees of freedom [365].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  During its 2024 beam campaign, HADES will be in a unique position to measure the fireball  caloric curve and the beam energy dependence of the collective flow from Au+Au collisions at  Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='24 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, there are ongoing efforts to establish  systematic consistency between results from FOPI and HADES, including understanding various  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1  cm  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='2  2  v  Protons  Centrality 20-30%  )  c  (GeV/  tp  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='35 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='95 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='35 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='55 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='75 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='95 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='00  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 GeV  NN  s  Au+Au  HADES  Protons  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 < p  t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 GeV/c  Centrality 20-30%  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 GeV  NN  s  HADES  Au+Au  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  ycm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  = 0  φ  π  2  1  π  π  3  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  = 0  φ  π  2  1  π  π  2  3  ycm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Left: Rapidity-dependence of proton elliptic flow (v2) in semi-central Au+Au collisions at Elab =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='23 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='42 GeV) for various pT bins (see legend) as measured by the HADES experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Figure from [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Right: 3D-picture of the proton angular emission pattern in momentum space (flow  coefficients from v1 to v6) for HADES data in semi-central Au+Au collisions at Elab = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='23 AGeV (√sNN =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='42 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  40  detector-related effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This is highlighted by the observed discrepancy in pion multiplicities  between FOPI and HADES, which could be partially explained by different methods used by the  respective experiments to estimate the number of participant nucleons [366].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The abundance of available and future data presents an opportunity to benchmark transport  model simulations with measurements from KaoS, FOPI, and HADES experiments by enabling  systematic studies of the symmetric nuclear matter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A recent comparison between FOPI  measurements and dcQMD transport code [86], using the transverse rapidity [367] and flow spec-  tra [88] of protons and light clusters at Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='15–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='80 AGeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='95–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='24 GeV), has  further tightened the constraints on the nuclear EOS at the probed densities to one characterized  by an incompressibility K0 = 236 ± 6 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The dcQMD analysis for the FOPI data is planned  to be extended up to Elab = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='52 GeV), probing densities above 2n0, by  taking into account an improved description of reaction dynamics through using more accurate  approximations for 3-body terms in the interaction and considering multi-pion decay channels for  the resonances [368].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Moreover, perfect-fluid hydrodynamic calculations for binary-neutron-star mergers and heavy-  ion collisions at SIS-18 energies show that comparable temperatures (T ≈ 50 MeV) and densities  (nB ≈ 2n0) are reached in both systems [333, 334].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This has led to increasing efforts to use the  existing constraints on the EOS of symmetric nuclear matter from KaoS and FOPI experiments  in a multi-physics effort to constraint neutron star properties [17, 369, 370].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such multi-physics  constraints are discussed in detail in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments probing densities above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  While collective flow can be used to deduce the geometry of the colliding system and its prop-  erties in an indirect way (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 14 and 15), a more direct method – femtoscopy – can provide  a direct handle on the space-time evolution of the fireball [371].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Here, the time of the particles’  emission ∆τ is also a probe of the underlying EOS, with larger values of ∆τ corresponding to a  softer EOS [338] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Access to this information is provided by measurements of femto-  scopic radii Rlong, Rout, Rside [372], where the relation between Rout to Rside is strongly correlated  with ∆τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The sensitivity of pion emission to the EOS has already been studied [338, 373], however,  experimental uncertainties are still too big to make precise comparisons with model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Ongoing studies of proton femtoscopy at STAR are expected to bring new, substantial references  for such investigations of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In addition to studying the EOS of dense nuclear matter, the STAR BES program also aims to  search for a potential first-order phase transition from hadronic to partonic phase at higher baryonic  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This search can provide an input on collision energies at which hadronic transport models  should take into consideration new degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Among the explored observables, number-  of-constituent-quark (NCQ) scaling was used as an important evidence of creation of QGP at the  highest RHIC energy of √sNN = 200 GeV (Elab = 21, 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 AGeV) [374].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recent results point to  the breaking of the NCQ scaling in Au+Au collisions at √sNN = 3 GeV (Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='91 AGeV) [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Other observables that can hint at the possible existence of a first-order phase transition include  the thermodynamic susceptibilities of pressure, which are predicted to fluctuate in the vicinity of a  critical point and manifest as a specific behavior of higher-order moments of conserved quantities  (such as baryon number, strangeness, and electrical charge) with the beam energy [375, 376].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  STAR [377, 378] and HADES [379] have observed tentative non-monotonic behavior in the beam-  energy-dependence of the fourth-order net-proton cumulant (a proxy for the net-baryon number  cumulant) in Au+Au collisions at √sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 GeV (Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='91–400 AGeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The experimental effort to uncover the symmetric nuclear matter EOS will be further strength-  ened by the Compressed Baryonic Matter (CBM) experiment [49, 380] at the currently-under-  construction Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The CBM  41  16  18  20  22  Freeze-out time 〈t〉 (fm/c)  Au+Au 0%-10% π  2  3  4  5  6  7  8  9 Cascade  Hard EoS  CMF EoS  CMF_PT2 EoS  CMF_PT3 EoS  √s NN (GeV)  2  3  4  5  6  7  8  9  12  8  4  0  4  8  12  16  Au+Au  0%-10%  |yππ|<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='35  kT=(275±25) MeV/c  HADES π-π-  E895 π-π-  E866 π-π-  STAR π-π-  STAR π-π-+π+π+  RO  2-RS  2 (fm2)  √sNN (GeV)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Left: Beam energy dependence of the π− freeze-out time as extracted from UrQMD simulations for  different EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The CMF PT2 and CMF PT3 EOS soften at low and high baryon density, respectively, by  introducing a first-order phase transition, and the pure cascade mode can be considered as an extremely  soft EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [338].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Right: Comparison of the beam energy dependence of R2  out − R2  side for π−  extracted from experiments and UrQMD simulations with different EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [338].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  experiment, which at the time of writing is expected to become operational in 2028-29, aims to  use nucleus-nucleus collisions to precisely explore the QCD phase diagram with Au+Au collisions  in the energy range of √sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9 GeV (Elab = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='49–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1 AGeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Other particle beams,  such as Z = N species and protons can also be used at Elab = 15 AGeV (√sNN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='62 GeV)  and Elab = 30 AGeV (√sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='73 GeV), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This will be enabled by using primary  heavy-ion beams from the Schwerionensynchrotron-100 (SIS-100) ring accelerator operating at an  intensity of 109 ions/s [381].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' CBM will operate at unprecedentedly high peak interaction rates of up  to 10 MHz, which will be further complemented by a novel trigger-less data acquisition scheme and  online event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This will allow CBM to perform systematic, multi-differential measurements  of the dependence of observables on the beam energy and system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The most promising observ-  ables to explore are: (i) event-by-event fluctuations, (ii) thermal radiation (photons and dileptons),  (iii) (multi-)strangeness, (iv) hypernuclei, and (v) charm production (recent physics performance  results can be found in [382, 383]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, the HADES Experiment will be moved to the SIS-100  beamline in the CBM experimental cave to complement the overarching CBM physics program in  2031 [384].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The HADES detector, given its large polar angle acceptance (18◦ ≤ θ ≤ 85◦), will  perform reference measurements for CBM at lower SIS-100 energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This will be done with light  collision systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', proton beams and heavy-ion beams with moderate particle multiplicities  (such as Ni+Ni or Ag+Ag collisions) [380].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Altogether, CBM represents an opportunity to link  the physics programs at SIS-18 and RHIC, thereby leading to a continuation of the Beam Energy  Scan program (see also the white paper on QCD Phase Structure and Interactions at High Baryon  Density: Continuation of BES Physics Program with CBM at FAIR [151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Overall, STAR-FXT and CBM-FAIR are capable of performing high-statistics multi-differential  measurements of the relevant EOS observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, a successful inference of the EOS depends  on comparisons to transport simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Although many transport codes are available for describ-  ing heavy-ion collisions in different energy ranges and extracting the underlying EOS (see [47] for  a review), currently none of the available codes can reproduce all proposed experimental observ-  ables (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A meaningful description of experimental data in the STAR-FXT and  O  米O  米42  CBM-FAIR range will require transport codes to incorporate physics allowing reproducing all of  the above-mentioned key measurements and more, see Section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments to extract the symmetry energy  The energy contribution from the isospin dependence term, also known as the symmetry energy,  is a small fraction of the total energy of a nucleus even for neutron-rich heavy radioactive isotopes  (< 5% in the liquid drop model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, due to the large isospin asymmetry in neutron stars,  the density dependence of the symmetry energy is very important, determining many neutron star  properties, including their size and the cooling pathways via neutrino emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While experimental  inferences of the symmetry energy pose significant challenges, researchers have developed methods  to elucidate the relatively small effects that the asymmetry has on isospin-dependent observables,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', by measuring ratios of neutron and proton observables or charged pion observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Exper-  imental as well as theoretical systematic errors are further minimized by taking double ratios of  the same observable using two reactions that differ mainly in the neutron/proton content, as in  the measurement of isoscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To reach the widest range of asymmetry between reactions, intense  radioactive beams are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Large experiments designed to measure symmetry energy can re-  quire large collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, small-scale experiments can likewise have an impact on some  of the outstanding problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, many groups contribute to the diverse experimental  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments that probe low densities  At beam energies below Elab = 100 AMeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='93 GeV), the colliding nuclei overlap  briefly and then expand, with most of the detected particles being emitted during the expansion  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The rates of emission of neutrons and protons during the expansion are influenced by the  symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some nucleons emerge within fragments or clusters that are formed and emitted  throughout the reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Nearly all theory studies require the symmetry energy to be zero at zero  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, before matter reaches zero density, at low densities of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='002 ≤ n/n0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='02)  many nucleons combine into clusters and preserve the information about the symmetry energy at  those low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the estimation of Wada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [385, 386], clustering has a significant impact  on the symmetry energy below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='03 nucleons/fm3, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 17 (we note here that this conclusion  depends on the definition of the symmetry energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Overall, the presence of clusters changes the  characterization of the symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Nonetheless, low-density clusterization is an important  ingredient in supernova matter and for the EOS in the neutrino sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It is also relevant to the  nature of proto-neutron star matter as it cools and the crust crystallizes [387].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Measurements to extract symmetry energy up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  In the past decade, many studies have been conducted to extract the symmetry energy and  symmetry pressure [70], mostly at low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Since the nuclear EOS should give a good de-  scription of the properties of the nuclei, including the masses or binding energies, the large nuclear  mass data base provides a great resource to determine the symmetry energy at about (2/3)n0 from  (1) masses of double magic nuclei using Skyrme density functional [72], (2) nuclear masses using  density functional theory [71], and (3) the energies of isobaric analogue states [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  By nature, a heavy nucleus has excess neutrons which are needed to overcome the Coulomb  repulsion from the protons inside the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The symmetry-energy forces the excess neutrons  43  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The symmetry energy of clustered matter at very  low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [385].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This surface layer of ex-  cess neutrons is referred to as a neutron  skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Thickness of this neutron skin re-  flects the symmetry pressure, or equiv-  alently the slope of the symmetry en-  ergy at the saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The long-  awaited measurements of the neutron skin  of the 208Pb nucleus inferred from par-  ity violation in electron scattering were  recently published by the PREX collab-  oration [74, 388].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The measured value  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='283 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='071 fm, corresponding to the  symmetry pressure of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='75 MeV/fm3  at (2/3)n0, is rather large and disagrees  with most theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Subse-  quently, the PREX/CREX collaboration  measured the neutron skin of 48Ca to be  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='121 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='026 (exp) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='024 (model) fm  [389], which is much thinner than the  208Pb skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, the 48Ca value is  much closer to the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The discrepancies between the two results and the expectations from models have not been re-  solved, and the CREX collaboration has not released official values for the slope of the symmetry  energy or symmetry pressure, even though there have been many attempts by outside groups to  resolve the apparent discrepancies between the two skin measurements [390–392].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In the last few years, an alternative way to measure skin thickness has been proposed [393].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In the limit of an exact charge symmetry, the proton radius of a given nucleus is identical to the  neutron radius of its mirror partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus the neutron skin for a given nucleus may be determined  from the difference in proton radii measured in these mirror pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In reality, there are relativistic  and finite-size corrections, as well as corrections from the Coulomb force which breaks the isospin  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In principle, these corrections can be calculated within the energy density functional  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While larger neutron skins are expected in heavier nuclei due to the larger neutron excess,  making them a better probe, proton-rich mirrors of heavy nuclei is typically far beyond the limits  of existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus this technique is limited to species of relatively low mass and isospin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Even with  the use of high-intensity isotope beams near the proton driplines, it is still a challenge to do such  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The most recent result with this technique is from the 54Ni-Fe mirror pair [394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Complementary to structure experiments, heavy-ion collisions have probed the symmetry en-  ergy and pressure over a wide density range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  At incident energies below Elab = 100 AMeV  (√sNN ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='93 GeV), low densities (estimated to be around (1/3)n0) are reached when matter  expands after the initial impact and compression of the projectile and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Therefore, the  corresponding experimental observables primarily reflect the symmetry energy at sub-saturation  densities [6, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The transport of neutrons and protons allows systems with isospin gradients to  equilibrate, where the degree of equilibration depends on the strength of the potential experienced  by the nucleons and the duration of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The technique of equilibration chronometry al-  lows the visualization of the time evolution of the neutron excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Signatures of neutron-proton  equilibration obeying first-order kinetics are observed both in experimental data [395–398] and in  transport calculations  [399].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Since the equilibration depends on the neutron and proton chem-  ical potentials, this technique offers new experimental data to constrain the sub-saturation EOS  through comparisons with simulations [6, 47, 400].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  25  20  15  (MeV)  10  5  0  10-3  10-2  10-1  p(N  /fm3)  nucleon44  Isoscaling was first observed in central 124Sn+124Sn and central 112Sn+112Sn collisions at beam  energy Elab = 50 AMeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='90 GeV) [401, 402].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Isoscaling describes a simple scaling law  governing the ratios of isotope yields from two systems which differ mainly in their neutron-proton  composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It arises from the differences in the neutron and proton chemical potentials of the  two reactions and is, therefore, sensitive to the symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The isospin diffusion, derived  from the isoscaling observable, reflects the driving forces arising from the asymmetry term of the  EOS [6, 400, 403, 404] and provides a measurement of the symmetry energy at around (1/3)n0 [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Other observables used to study the symmetry energy with light charged particles include both  n/p and t/3He ratios and their double ratios obtained from two reactions with different isospin  content [47, 95, 96, 100, 405].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Due to the difficulties in measuring neutrons, neutron data is not  widely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, recent isoscaling measurements have allowed the construction of “pseudo  neutrons”, that is a reconstruction of neutron yields from light particle ratios such as t/3He [406].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In  particular, this method allows for a reconstruction of low-energy neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, due to the lack  of high-energy charged particles data, it is a challenge to reconstruct high-energy neutron spectra  in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Therefore, to study the symmetry energy at supranormal densities, neutron arrays  constructed with new advanced materials will be needed in the next generation of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In experiments utilizing central 124Sn+124Sn and central 112Sn+112Sn collisions at Elab =  120 AMeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='94 GeV) [407], the spectra of neutrons emitted to 90 degrees in the center-  of-mass frame are compared to the corresponding proton spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Transport calculations predict  that if the effective masses of neutrons and protons satisfy m∗  n < m∗  p, then fast neutrons coming  from the compressed participant region experience a more repulsive potential and a higher accel-  eration than do fast protons at the same momentum, resulting in an enhanced ratio of neutron  over proton (n/p) spectra at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In contrast, calculations for m∗  n > m∗  p predict that the  effective masses enhance the acceleration of protons relative to neutrons, resulting in a lower n/p  spectral ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Bayesian analysis of the experimental results [98] compared to ImQMD calculations  shows that the values of the first two Taylor expansion coefficients of the symmetry energy, S0  and L, depend on both the symmetry energy and to the effective mass splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' More examples  of Bayesian analyses used to simultaneously constrain multiple parameters will be discussed in  Section IV B, where methods to extract multiple transport model input parameters are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Selected constraints on the symmetry energy around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0  Current constraints on the symmetry energy above saturation are obtained with large uncer-  tainties, mainly at densities around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This is the area of future opportunities, and we discuss  this in more detail here to illustrate the complexity of the experiments and analyses as well as the  central role played by transport models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The nucleon elliptic flow is sensitive to the pressure generated in nuclear collisions and, there-  fore, to the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Since a higher symmetry pressure will yield a larger magnitude of the elliptic flow  at midrapidity for neutrons than for protons, comparisons of the neutron and proton elliptic flows  provide sensitivity to the density-dependence of the symmetry energy [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The neutron and hydro-  gen elliptic flow from Au+Au collisions at a beam energy of Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 AGeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07 GeV)  were measured in the FOPI-LAND and Asymmetric-Matter EOS (ASY-EOS) experiments, using  the Land Area Neutron Detector (LAND) for the measurement of the neutron flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A comparison  of data to UrQMD simulations, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 18, was used to extract the dependence of the sym-  metry energy on density, parametrized as proportional to (nB/n0)γasy, and the symmetry energy  slope parameter L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The FOPI-LAND experiment reported γasy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 and L = 83 ± 26 MeV  [68], whereas the ASY-EOS obtained γasy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='19 and L = 72 ± 13 MeV [69], indicating  a moderately soft symmetry energy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 18 and also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The analysis also illustrates  45  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Ratio of the elliptic flows of neutrons over  charged particles vn  2 /vch  2  as a function of transverse  momentum per number of constituent nucleons pt/A  in Au+Au collisions at Elab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 AGeV (√sNN =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The comparison between ASY-EOS mea-  surements (square black symbols) and UrQMD trans-  port model calculations with a soft (pink dots) and  hard (green triangles) symmetry potentials shows a  preference for a soft symmetry energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' solid red line  indicates γasy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  the dependence of S0 and L on other in-  put parameters of the EOS, such as γasy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  A  subsequent comparison of data with dcQMD  model [408] gives a value of L = 85 ± 32 MeV  at n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In addition to the ASY-EOS experiment,  another effort that explores this density region  is the SAMURAI Pion-Reconstruction and Ion-  Tracker (SπRIT) experiment, performed with  radioactive tin isotopes at RIKEN, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For  constraining the symmetry energy at supra-  saturation densities, pion yield ratios are con-  sidered as a unique observable since they do  not form composite particles with other parti-  cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This makes their yields independent of  clusterization processes which can affect the  symmetry energy (see Section III B 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Further-  more, pion observables are predicted to be sen-  sitive to the nuclear EOS at high densities due  to their unique production mechanism: Above  Elab = 200 AMeV (√sNN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='97 GeV), some of  the interactions occurring in central collisions  are energetic enough to form excited ∆(1232) baryon resonances (through the NN ↔ N∆ scatter-  ing process), which then promptly decay into pions and nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The high production threshold of  the ∆(1232) resonance ensures that pions originate from the early stages of the reaction, and there-  fore from regions characterized by a high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The SπRIT collaboration measured charged pion  emission from systems characterized by a wide range of asymmetry [102] by colliding tin isotope  beams of 108,112,124,132Sn with isotopically enriched targets of 112,124Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The production of π− strongly depends on n-n collisions in the high-density region, while π+  production largely depends on p-p collisions (the production of π− and π+ is equally likely in n-p  collisions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It follows that the relative production of π− and π+ depends on the relative numbers  of neutrons and protons and, therefore, is sensitive to the symmetry energy in the high-density  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Assuming a ∆-resonance model for pion production, one would expect that the pion yield  ratio Y (π−)/Y (π+) follows a (N/Z)2 dependence [95, 367].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, the measured total pion yield  ratio follows N/Z with a best-fitted power index of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 19, where yield ratios  without a transverse momentum cut are depicted by yellow crosses with circle markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The radius  of the circle in the center of each cross represents the experimental uncertainty, showcasing very  good experimental accuracy of the measurement in which systematic errors are reduced by taking  pion yield ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, comparisons of systems with different N/Z measured in the same ex-  periment reduces systematic errors [409].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The discrepancy between the theoretical expectation and  experimental data indicates the presence of dynamical factors beyond a simple ∆-resonance model,  while the large measured exponent suggests that the ratios are strongly affected by the symmetry  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' When a transverse momentum cut of pT > 180 MeV/c is imposed, the result (represented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 19 by blue crosses with circle markers) still shows the same (N/Z)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4 dependence, suggesting  that effects due to the symmetry energy persist in high-momentum pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Interestingly, current  transport models do not seem to be able to reproduce the strong N/Z dependence [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  While Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 19 shows the total yield, the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 20 focuses on the pT -dependence of the  single ratio spectrum SR(π−/π+) = [dN(π−)/dpT ]/[dN(π+)/dpT ] for two extreme cases: reactions  of neutron rich (132Sn+124Sn) and of near-symmetric (108Sn+112Sn) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The data is compared  E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9  stiff  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  soft  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7  p/A (GeV/c)46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  N/Z  1  2  3  4  5  6  Y(  )/Y(  + )  pT > 0 MeV/c  pT > 180 MeV/c  y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1(N/Z)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5(N/Z)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Ratios of yields of π− over yields of π+ in  central (b < 3 fm) events for pions with pz > 0 in the  center-of-mass frame, plotted as a function of N/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The yellow crosses show yield ratios with no transverse  momentum cut, while the blue crosses show yield ra-  tios for pT > 180 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The radius of the circle  inside each cross represents the statistical uncertainty  of the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The dashed blue and dotted blue line cor-  responds to the best-fitted power functions of N/Z for  pT > 0 and pT > 180 MeV/c pion ratios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Figure from [410].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  with the dcQMD model [145, 408], a Quantum  Molecular Dynamic transport model that in-  cludes total energy conservation and other ad-  vanced features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To extract the EOS, the dcQMD  model was used to predict single ratios with  12 different parameter sets in the (L, ∆m∗  np)  space, forming a regular lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' here, L is the  slope of the symmetry energy and ∆m∗  np is the  neutron-proton effective mass splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The  value of L in the lattice is either 15, 60, 106,  or 151 MeV and ∆m∗  np/δ is either -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33, 0, or  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' All other input parameters in the dcQMD  have been fixed by comparing to FOPI data,  as well as by comparing the predictions to the  total yield of the charged pions and the aver-  age pT obtained from the pion spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  De-  tails of the comparison can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 20 shows a few  selected calculations and the measured single  ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The (L, ∆m∗  np) values for the solid  blue line are (60, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33δ), for the dashed blue  line are (60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33δ), for the solid red line are  (151, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33δ), and for the dashed red line are  (151, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Coulomb effects dominate the low pT region, causing a steep rise in the measured  ratios at pT < 200 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' All calculations at pT < 200 MeV/c disagree with data, which could be  caused by inaccuracies in the simulation of Coulomb interactions or of the pion optical potential  above the saturation density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At pT > 200 MeV/c, the Coulomb and pion potential effects diminish  and the ratios should be good probes of the symmetry energy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The predicted single ratios at pT > 200 MeV/c are interpolated with 2D cubic splines over  the (L, ∆m∗  np) space, and the interpolated predictions are then compared to experimental mea-  surements through a chi-square analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The resultant multivariate constraint on L and ∆m∗  np is  shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 20, where the green shaded region is the 1σ confidence interval  and the area enclosed by the two blue dashed curves is the 2σ confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The corre-  lation between ∆m∗  np and L occurs because both parameters influence the nucleon momenta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' L  influences the momenta through its isospin-dependent contribution to the nucleon potential en-  ergy, and ∆m∗  np influences the momenta via its isospin-dependent impact on the nucleonic kinetic  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Either increasing L or decreasing ∆m∗  np will increase the energies of neutrons relative to  protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This increases the numbers of n-n collisions relative to p-p collisions at energies above  the pion production threshold and enhances the production of π− relative to that for π+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Challenges and opportunities  Experiment and theory  Currently, there are few experiments that aim at inferring the symmetry energy and symmetry  pressure from heavy-ion collisions probing densities of 1–2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, the available constraints  have very large uncertainties, especially for the symmetry pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It is worth noting that heavy-  ion collision experiments do not measure the symmetry energy or pressure directly, but rather  they depend on comparisons with transport model simulations that describe the dynamics of the  47  0  100  200  300  400  pT (MeV/c)  100  101  Single Ratio (SR)  132Sn + 124Sn, E/A = 270 MeV  0  100  200  300  400  pT (MeV/c)  108Sn + 112Sn, E/A = 270 MeV  L(MeV)   m *  np  60          -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33  60           0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33  151        -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33  151         0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='33  50  100  150  L (MeV)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3  ∆m∗  np/δ  1-σ  2-σ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Left panel: Single pion spectral ratios for 132Sn+124Sn (top) and 108Sn+112Sn (bottom) reac-  tions with four selected dcQMD predictions overlaid [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Right panel: Correlation constraint between L  and ∆m∗  np/δ, extracted from pion single ratios at pT > 200 MeV/c in collisions of both neutron-deficient  108Sn+112Sn and neutron-rich 132Sn+124Sn systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The light blue shaded region (dashed blue lines) cor-  responds to 68% (95%) confidence interval [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  collisions [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The large uncertainties in available constraints mainly arise from the intrinsic  uncertainties of the transport models and the accuracy of determining the parameters used as an  input in these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, a general feature of low-energy heavy-ion collisions is that  more nucleons are emitted in light clusters than are emitted as free neutrons and protons, while  the reverse is true of most transport model simulations of these reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Theoretical approaches  to this issue have been proposed (see Section II A), but are rarely implemented to model the  coalescence of nucleons in the medium into the observed distribution of clusters, and therefore it is  not clear to what extent these approaches are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The current inaccuracy in cluster production  complicates and limits the scientific conclusions that can be drawn by comparing data to transport  theory, and therefore improving the accuracy of cluster production in transport theory would be a  very significant achievement, enabling more stringent constraints on the symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  It is important to quantify major sources of systematic uncertainties in the transport model  implementations and in the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Due to the quality as well as technical details of  solutions adopted in different models, it may not be realistic to establish all uncertainties for all  transport models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Nonetheless, developing methods to validate transport models and performing  these validations remains a primary goal for the Transport Model Evaluation Project (TMEP)  collaboration, and it is essential to extracting reliable constraints on the EOS from heavy-ion  collisions (see Section II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The current capabilities at FRIB, using beam energies up to Elab = 200 AMeV (√sNN =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='97 GeV), allow for exploration of densities up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0, and the neutron excess can be varied  over a wide range by changing the composition of the rare isotope beams and targets, allowing  to more closely recreate the matter found in extreme astrophysical environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', neutron  stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' From the dense collision region in heavy-ion collisions, pions and free nucleons are emitted  with high transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The relative yields of these particles, especially as a function of  energy, as well as particle elliptic flow contain information about the dense collision zone and thus  can be used to constrain the EOS that governs supra-saturation matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Individual efforts based  on small-scale experiments, which are the strength of the field, have provided a diversity of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, in order to take advantage of multiple-parameter Bayesian analyses, described below,  and given the tight allotment of the expensive (and coveted) beam time, future experiments should  utilize detectors that provide large coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The development of the time projection chamber  48  (TPC) detector at FRIB is essential to measure both pions and charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The detector  can be coupled with an upgraded or a new Large Neutron Array (LANA) to measure both charged  particles and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Additionally, putting the TPC detector at the target position of the High  Rigidity Spectrometer (HRS) enables coincident measurement of projectile-like fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The  determination of the centrality and the reaction plane, required, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', for the elliptic flow studies,  would benefit from a construction of a 4π detector placed around the target when the silicon  detectors are used to identify the charged particles before the completion of the HRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such a  detector would have to measure the energies of the emitted fragments and nucleons as well as their  multiplicities with minimal energy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Reaching higher densities requires the energy upgrade to Elab = 400 AMeV (√sNN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='07 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  With the capability for producing high-intensity rare isotope beams with a wide range of asymme-  tries, FRIB400 is essential for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' effort to lead in the determination of the density-dependence  of the symmetry energy [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The beams available at FRIB, being complementary to those that can be accelerated at Eu-  ropean facilities, may represent a unique opportunity to conduct nuclear transport investigations  also by the international nuclear physics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As described above, the development of de-  tector arrays with high isotopic resolution over a wide dynamic range, from light particles to heavy  fragments, provides the prospect of measuring observables (especially in the context of isospin  diffusion and drift as well as in collective motion phenomena) that can amplify the sensitivity to  the symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Coupled with its capability to use high-quality radioactive beams, FRIB  may represent a focus of interest for a wider community, stimulating the need for international  discussions and collaborations in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such an interest may concern also theoretical  physicists that have been collaborating with FRIB colleagues within the Transport Model Evalu-  ation Project (TMEP) initiative, aimed at improving investigations of the isospin-dependent EOS  with comparisons to experimental observables (see also Section II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Multiple Parameter Bayesian analysis  The EOS is only one of many input parameters in transport models used to simulate heavy-ion  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Often, multiple measurements probing different parts of the collisions are needed to  constrain other parameters of these models, such as the momentum-dependence of the isovector  mean-field potential, or the in-medium isospin-dependent cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, constraining  transport model parameters with experimental results is a delicate endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The outcomes of  nuclear collisions are influenced by a multitude of processes, and therefore the experimentally  measured final stage observables can depend simultaneously on values of multiple parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, carefully chosen observables may only be sensitive to just a few specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The  full extent of the dependence of a given transport model on input parameters can only be tested  empirically after performing a complete series of simulations of heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Bayesian statistical methods provide means to quantify the relation between observable values  and physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' They also provide a systematic way of constraining multiple nuclear  properties and utilizing prior knowledge from different experiments, prior constraints from other  sources, and results from new experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, in the n/p ratio exper-  iment mentioned above, measuring the yield ratios of neutron and protons spectra, a Bayesian  analysis comparing the experimental results [98] to ImQMD calculations determines both ∆m∗  np and  the relationship between S0 and L, even though the uncertainties are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' More precise measure-  ments in the future will enable a better resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In the long term, it is important to develop Bayesian analyses of multiple observables to de-  termine multiple parameters simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  As an example, in the SπRIT experiment many  observables have been measured with four reaction systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Eight observables in total, including  the directed flow, elliptical flow, and the stopping observable from different reactions, are fitted si-  49  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Posterior distribution obtained from a Bayesian analysis of ImQMD simulation results and experi-  mental data from SπRIT experiments [410].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Eight available observables are used for Bayesian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The  values for median and 68% confidence interval of the marginalized distribution are tabulated on the upper  right-hand side of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from [410].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  multaneously by varying five transport model input parameters (two pertaining to the shape of the  symmetry energy term in the nuclear matter EOS, two pertaining to the nuclear effective masses,  and one pertaining to the nuclear in-medium cross-section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The posterior distribution shows a  weak constraining power on the symmetry energy terms, but a strong sensitivity to effective masses  and in-medium cross-section [410], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The posterior parameter distributions are generated from repeated sampling of transport model  predictions for hundreds of thousands of times, each with different parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' If carried out  directly, this process would consume an unreasonable amount of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This can  be alleviated with an effective, efficient, and capable model emulator which emulates the behavior  of transport models at all points of the allowed parameter space from predictions at just a few tens  of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Gaussian processes are readily available and commonly used in emulators,  but the procedures for tuning hyperparameters vary across analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Numerous heuristics and cost  functions are proposed for the optimization of hyperparameters, and one can also marginalize over  all nuisance parameters with a Markov chain Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  50  (MeV)  S  30  150  (MeV)  100  50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  Nu/  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  Nu/  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='00  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25  30  40  50  50 100 150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='25  So (MeV)  L (MeV)  ms /mN  m,/mN  n50  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='THE EQUATION OF STATE FROM COMBINED CONSTRAINTS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Nuclear  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Neutron star  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Isospin diffusion in HICs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Masses and radii  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Dipole polarizability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Tidal deformability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Spectral ratios of light clusters  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Moment of inertia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Nuclear masses and radii  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Gravitational binding energy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Isobaric analog states  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Cooling of young neutron stars  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='n/p ratios in HICs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Bulk oscillation modes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Neutron skins  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Crust cooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Mirror nuclei  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Pulsar glitches  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Giant resonances  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Lower and upper limits on neutron star spin periods  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Flow of particles in HICs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Torsional crust oscillations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Charged pion ratios in HICs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='Crust-core interface modes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Illustrative list of nuclear and astrophysical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  There have been many attempts to extract the equation of state (EOS) as a function of density  from both nuclear experiments and astronomical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In Table I, we provide an illustrative  list of relevant experimental and observational measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Importantly, these observables probe  the EOS at different densities: a few probe the EOS near the saturation density n0, but many probe  densities that are significantly higher or lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, nuclear structure typically probes  densities that are somewhat lower than n0, while analyses of heavy-ion collisions or properties of  neutron stars probe larger density ranges, as schematically illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 12 and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Comparing constraints based on different measurements allows one to test their consistency and  ultimately find tight constraints on the EOS over the full range of densities that can be probed  either by experiments or by astronomical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Techniques of Bayesian inference or Pearson  correlation analyses are well-suited to this endeavor and can provide more readily useful and  Astro (M,R,Λ)�  HIC�  Astro: crust observables�  Chiral EFT�  Nuclear masses, �  Δrnp,  αD, ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='�  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An ensemble of EOSs that range over crust and  core uncertainties consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Ranges over which differ-  ent nuclear and astrophysical probes provide information  about the neutron star EOS are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure modified  from [411].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  testable  information  on  the  density-  dependence of the EOS than, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', statis-  tical comparisons or combining the Taylor  expansion coefficients (such as S0, L, and  Ksym) obtained from individual analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Key to this approach is the determination  of the density that each experimental ob-  servable most accurately probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Away  from that density, weaker constraints on  the EOS are possible, but the analysis is  more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In this section, we review the variety of  observables that have been used to place  constraints on the EOS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' heavy-ion collision  experiments, which produce many of these  constraints, are described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  We then discuss recent attempts at com-  bining various constraints that result in  meaningful EOSs with quantified uncer-  tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  ？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='??' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Outer Crust  Outer  Inner Core  Neutron Drip  Inner Crust  Crust/Core Transition  Core  1036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  pressure (Dyne/cm2)  1034  1032  1030  J, L, Ksym  in1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  n2  1011  1012  1013  1014  1015  1016  energy density (g/cm3)51  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Constraints  As discussed in Section III, experiments are often designed to explore certain aspects of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Accordingly, we classify the constraints obtained from laboratory measurements as sensitive to  either the symmetric nuclear matter EOS or the symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In addition to the experimental  inferences, constraints on the EOS can be also obtained from neutron star observations as well as  from chiral EFT theory at low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The list of constraints discussed here is not exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Rather, it represents a slice of widely acknowledged constraints at the moment of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' We note  that some of the constraints reviewed here have already been presented in Sections II and III, to  which we refer when appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Symmetric matter constraints from laboratory experiments  Some properties of the symmetric nuclear matter are fairly well-known near n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For exam-  ple, the generally accepted values of n0 and binding energy at saturation E0 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='16 fm−3 and  −16 MeV [198, 203], respectively, to within 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The incompressibility parameter, K0, has been  determined from giant monopole resonance (GMR) experiments [412] to be 231±5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However,  subsequent GMR measurements of the Sn isotopes cast larger uncertainties on K0 [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While  these larger uncertainties are consistent with values of K0 determined from heavy-ion collision  experiments [67, 353, 413], we note here that these experiments derive their constraints on K0  based on density functionals that are parametrized with K0, but used to describe the high-density  behavior of the EOSs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', these experiments do not probe the incompressibility at saturation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' see  also a similar discussion in Section II A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Measurements of the collective flow from high energy  Au+Au collisions have constrained the EOS for symmetric nuclear matter at densities spanning  (1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5)n0 [31, 58, 66, 67] (see the left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9), as described in Sections II A 2 and III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Symmetry energy constraints from laboratory experiments  In the past decade, many studies have been conducted to extract the symmetry energy and  the symmetry pressure, and some of the widely-known constraints are plotted in the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9, which includes both the usual EOS constraint bands as well as symbols located at  densities which a novel analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [70] identified as the most sensitive densities for a given  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At (2/3)n0, precise symmetry energy constraints have been obtained from studies  on nuclear masses using Skyrme density functional forms for the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These are labeled in the  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9 as “mass(Skyrme)” [72] and “mass(DFT)” [71], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In this density  region there are also precise constraints obtained from the energies of isobaric analogue states [12],  indicated in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9 by a data point labelled as “IAS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The dipole polarizability  αD, reflecting the response of a nucleus to the presence of an external electric field, also helps to  constrain the symmetry energy at low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Constraints on the symmetry pressure Psym, which  is proportional to the derivative of the symmetry energy with respect to density, have been recently  provided by the measurements of the neutron skin of 208Pb in the Lead Radius EXperiment (PREX  and PREX-II) [74, 388, 414] and of the neutron skin of 48Ca in the Calcium Radius EXperiment  (CREX) [389–391], both at Jefferson Lab, which use parity-violating weak neutral interactions to  probe the neutron distribution in 208Pb and 48Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A range of other scattering experiments have  measured the neutron skins of a number of neutron-rich isotopes and likewise used them to constrain  the symmetry energy [7, 415, 416].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Giant dipole resonances and polarizabilities [9, 417, 418] in  neutron-rich isotopes provide another source of information about the symmetry energy [335, 419–  424], as do mirror nuclei [393, 394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At densities far from (2/3)n0, heavy-ion collisions have been  used to probe the symmetry energy, as is described in Sections II A 2 and III B 3, and shown in the  right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  52  Constraints from astronomical observations  The bulk properties of neutron stars (such as their maximum mass, radii, tidal deformabilities,  moments of inertia, limits on the rotation frequency, and binding energy) depend strongly on the  distribution of matter throughout the star, therefore providing a measure of the EOS integrated  over the range of densities present in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The mass-radius relationship has a one-to-one corre-  spondence to the neutron star EOS [425], and it is known that the radius, the tidal deformability,  and the moment of inertia provide the strongest constraints on the EOS above 2n0 [251, 426],  while the maximum measured mass of neutron stars constrains the EOS at the highest densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Together, the tidal deformability measured in GW170817, the mass of J0740+6620, and the two  mass-radius measurements of NICER, discussed in Section II C, form the current gold standard in  measuring the neutron star properties using astronomical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  A number of astronomical observables also probe the neutron star crust physics, which results  in constraints on the pure neutron matter EOS, and in particular on the symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This  is because the neutrons provide the hydrostatic pressure that supports the inner crust, and the  interplay between these neutrons and the lattice of nuclei that makes up the crust determines the  crust-core boundary as well as the possible nuclear pasta shapes that appear near that boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The crust physics also depends, more weakly, on the symmetric matter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The nuclear EOS  at subsaturation densities, down to where the neutron drip begins (nB ≈ 10−4n0), is therefore an  essential ingredient in crust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Due to the complexity of crust physics, extracting rigorous EOS constraints from observations  of crust-associated neutron star behavior is in its early stages, and it is an area where substantial  progress can be made over the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Here we list some constraints on the symmetry energy  as an illustration of this potential, but, at the same time, we note that they are very tentative and  do not have well-quantified errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' indeed, some of them are mutually exclusive, emphasizing the  need to make progress in applying microscopic nuclear physics models to these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Constraints on the symmetry energy and its slope can be obtained from studying the following  phenomena: A study of the cooling of the neutron star in the Cas A supernova remnant [312],  which has been observed to cool on a timescale of decades, implies that the neutron star core may  have superfluid properties [265, 427, 428].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Studying the temperatures of the population of neutron  stars whose surface X-ray emission is observable leads to constraints on the neutron star masses  and radii and the composition of the core [429].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Constraints from quasi-periodic oscillations in  the X-ray tail of gamma-ray flares from soft gamma-ray repeaters [285, 430, 431], which could be  a signature of torsional oscillations of the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Potential measurements of the crustal moment  of inertia from glitches – sudden changes in rotation frequency – of radio pulsars and some X-ray  pulsars [432–437].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Limits on the longest and shortest observed periods of neutron stars probe  physics such as the magnetic field evolution in the crust [438] and the development of rotation-  induced instabilities in oscillations such as r-modes [439, 440].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' During the last few seconds of an  inspiral prior to the merger of two neutron stars or a neutron star with a black hole, tidal forces  may shatter the crust, causing a gamma-ray flare: in this scenario, coincident timing of the flare  with the gravitational wave signal measures the resonant frequency of crust-core interface modes  and sets constrains on the symmetry energy [276, 441].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The cooling of the crusts of quiescent  low-mass X-ray binaries promises to provide a source of constraints on the composition and size of  the neutron star crust and the extent of nuclear pasta phases therein [283, 442–444].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The expected  accurate measurement of the moment of inertia of pulsar J0737-3039a [445] will set constraints on  the EOS competitive with the current radius constraints [446–450].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The heat capacity of a neutron  star core can be measured by using inferences of the core temperature of transiently-accreting  neutron stars, and strongly suggests that a core dominated by a color-flavor-locked quark phase is  ruled out [451].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some such objects are observed to have efficient cooling in the core, constraining  superfluid gap models and the symmetry energy [452].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  53  Constraints from nuclear theory  In recent years, many-body nuclear theory such as chiral effective field theory (χEFT), discussed  in Section II B, has made significant progress to be considered as the canonical nuclear matter  EOS at low densities with rigorous uncertainty quantification [155–158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Even though the theory  is developed mainly for densities below saturation, it has been extended to 2n0, and it is a popular  constraint for studies that focus mainly on astronomical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  EOS obtained by combining various constraint sets  Each of the nuclear and astrophysical observables discussed above provides vital information  about the EOS over some density range, that can be combined with other constraints to globally  constrain the density-dependence of the EOS from sub-saturation to supra-saturation densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Such analysis techniques are relatively new, but several of such global constraints now exist, and  a selection of studies is briefly described below to illustrate their potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Beloin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [429] used relativistic mean-field models of the nuclear interaction to model the  structure and cooling of neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This analysis consistently combines nuclear data, neu-  tron star mass-radius measurements, and neutron star cooling measurements within a Bayesian  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Legred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [251] performed nonparametric EOS inference based on Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It  combines information from X-ray, radio, and gravitational wave observations of neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Their results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 23 and labelled as “Legred et al.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  These Bayesian analyses  incorporate astrophysical data and provide constraints on the neutron star EOS at higher densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [453] performed a Bayesian analysis of correlated effective field theory truncation  errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the χEFT  expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The neutron star matter pressure calculated with these EOS is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 23 and  labeled as “Drischler et al.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [17] combined nuclear theory via χEFT calculations (constraining the EOS below  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0), EOS inferences from heavy-ion collisions via the FOPI (constraining the symmetric matter  EOS up to 2n0) and ASY-EOS (constraining the symmetry energy at around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5n0) experiments,  Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Legred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Pressure [MeV/fm3]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1  1  10  100  baryon density nB/n0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The pressure of neutron star matter as a func-  tion of number density nB, as obtained by Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [17],  Drischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [453], and Legred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [251] at 95%, 95%,  and 90% confidence interval, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  and astrophysical data on bulk neutron star  properties (constraining the total neutron  star EOS above 2n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The EOS models were  extended to high densities using a speed-  of-sound model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The results are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 23 and labeled as “Huth et al.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [454] constructed neutron star  models using a Skyrme energy-density func-  tional, which allowed them to consistently  calculate the neutron skin of  208Pb and  combine constraints from heavy-ion colli-  sions, measurements of the neutron skin, and  astrophysical constraints within the same  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Neill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [411] followed a similar strat-  egy as the example above, using Skyrme  models which were extended to the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This allowed them to combine neutron skin  measurements, NICER/LIGO observations,  54  a crust observable (the resonant frequency of the crustal i-mode), and nuclear mass data to con-  strain both the core and crust properties as well as the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' By calculating all these quantities  using the same underlying Skyrme energy density functional (and polytrope parametrizations at  the highest densities), some poorly controlled modeling uncertainties were eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This work  demonstrated the complementarity of different observables: within the particular model used, nu-  clear masses constrain mainly the zeroth and first symmetry energy expansion coefficients, S0  and L, the crust observable has the largest impact on the inferred values of L and the second  expansion coefficient Ksym, and the neutron star radius and tidal deformability have the largest  impact on the inferred values of Ksym and two polytrope parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus, when combined, dif-  ferent observables provide complementary information that can contribute to a complete picture  of the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The ranges of these overlapping data are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Without crust observables, neutron star radii and tidal deformabilities tend to give weaker  constraints on Ksym and stronger constraints on L (see the analysis of a large number of studies  in [455]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, the relative constraints on the symmetry energy parameters change when  the used priors include the criterion that the crust is stable and incorporate potential data from  crust observables [411].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While this result is model-dependent and correlations with higher-order  symmetry parameters need to be investigated, it demonstrates the way in which crust observables  could significantly contribute to constraining the EOS, and motivates the need for improving models  to consistently combine crust and core observables with nuclear data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  One of the defining strengths of the global constraint analysis is that one or more additional  constraint(s) can be always included as long as an assessment of the corresponding statistical and  systematic uncertainties is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, the more data from nuclear and astrophysical  observables can be meaningfully included in such EOS inferences, the greater the ability to deliver  a robust EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Therefore, constraining the EOS by combining various inferences is highly promising  and, furthermore, well-suited for the coming era of multi-differential observables from heavy-ion  collisions and multi-messenger astronomical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  CONNECTIONS TO OTHER AREAS OF NUCLEAR PHYSICS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Applications of hadronic transport  In addition to the use of transport codes to study fundamental nuclear physics, their ability  to describe the transport and interactions of particles in a material also make them valuable for  applications that benefit society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Examples include the design of nuclear physics experiments,  detector development and simulations of detector performance, as well as medical applications and  radiation shielding in accelerator and space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some of these uses are outlined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Transport models are widely used to simulate particle emission from nucleus-nucleus collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In these simulations, the four-momentum of every emitted particle is tracked, making it possible  to generate double differential distributions, the particle spectra at various emission angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These  distributions are particularly important for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Most transport models are optimized for describing physics in certain energy ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The type  of code required can be tailored to the desired application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, some models perform best  at energies of a few hundred MeV and below, which is near the peak of the cosmic ray flux [456],  while others are more applicable for GeV-scale energies and above, in the tail of the cosmic ray  flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Over the entire experimental energy range, from intermediate energies through the highest  collider energies, transport codes have been successfully employed to design complex detectors,  optimize experimental setups, and carry out analyses of experimental data, including assessing the  detector efficiencies and background contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Detector design  In high-energy experiments, the code packages most commonly used for detector development  and data analysis are Geant3 [457], Geant4 [458], and FLUKA [459].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Most of these simulation  packages use cascade codes, that is codes without mean-field potentials, to describe particle trans-  port through matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Modern transport codes that can cover a wide range of energies such as  PHITS (Particle and Heavy-Ion Transport code System) can, however, provide a more complex  description of particle transport [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Aside from heavy-ion collisions, transport simulations play an important role for a variety of fun-  damental physics experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, long-baseline neutrino experiments need to determine  the incoming neutrino energy in order to extract the neutrino mixing parameters, CP violating  phases, and neutrino mass ordering [460].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, because the neutrino beam is generated from  fixed-target proton-nucleus interactions producing secondary π and K mesons with neutrino decay  products, there are large uncertainties on the energy of the interacting neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The neutrino  energy must be reconstructed from the measurement of the final state [461, 462] which is often mod-  eled by simple Monte-Carlo cascade approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Reliable transport descriptions could significantly  improve these studies for the Deep Underground Neutrino Experiment (DUNE) [463], as well as  for the ongoing experiments NuMI Off-axis νe Appearance (NOvA) [464] and Tokai to Kamioka  (T2K) [465].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Other experiments that require transport simulations of backgrounds include dark  matter searches [466], semi-inclusive electron scattering such as (e,e′p) on nuclear targets at Jef-  ferson Lab to search for color transparency, short-range correlations [467], and hadronization in a  nuclear medium [468].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Space exploration, radiation therapy, and nuclear data  Transport models can also be used in applications relevant to space exploration to understand  and mitigate the harmful effects of the space radiation environment on electronics and astronauts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Collisions of galactic cosmic rays (GCRs) with nuclei, whether in the Earth’s atmosphere or in  the material of a spacecraft above it, can generate showers of particles, including pions, muons,  neutrinos, electrons, and photons as well as protons and neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' GCRs cover a wide range of  energies, from tens or hundreds of MeV up to the TeV scale, and ion species, spanning elements  1 ≤ Z < 28 [475], making it challenging to determine all their potential effects in a given material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The penetrating power of the initial GCRs and the secondaries generated by their interaction  with matter can have a serious impact on the safety and viability of space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The 1%  of GCR primaries which are heavier than Helium nuclei can pose an especially serious problem,  given that the damage they inflict scales as Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The secondary particles generated from GCR  interactions with spacecraft materials [476] such as aluminum, polyethylene, and composites can  harm astronauts and disrupt or even disable electronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, spacecraft shielding  designed to reduce the GCR flux is itself a target that can increase the secondary flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Because  of the wide variety of possible shielding materials and thicknesses, transport models are essential  to determine the sensitivity of the secondaries (regarding both their flux and composition) to  different shielding configurations, as well as the subsequent harmful impact of those secondaries on  electronic systems [477] and humans [478].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A pictorial overview of applications transport modelling  and nuclear data in space missions is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Due to the lack of data at the appropriate energies, simulations of space radiation effects have  large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The space research community has generally relied on phenomenological nu-  clear reaction models such as the Double Differential Fragmentation model (DDFRG) [479], which  consists of a sum of multiple exponential distributions with parameters fit to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Many of these  56  models employ abrasion-ablation models [480, 481] (where abrasion and ablation refer to parti-  cle removal in ion-ion interactions and nuclear de-excitation following abrasion, respectively) or  semi-empirical parametrizations, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [482].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Researchers modeling these interactions could  benefit from transport codes discussed in this White Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The use of hadronic transport models  such as the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) code [482], which was shown  to correctly predict proton and deuteron yields from the BNL Alternating Gradient Synchrotron  measurements of collisions of protons on Be and Au targets at Elab = 15 AGeV [483, 484], see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 25, could significantly advance simulations of collisions relevant for space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For  further information about the needs for space applications, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [485, 486].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Similar transport modeling needs arise in charged particle therapy for medical applications such  as cancer treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In this case, the ion beam is tuned to penetrate the tissue at the tumor location  so that the Bragg peak, or maximal dose, is delivered to the tumor site while minimizing the spread  of the charged particle beams into surrounding tissue due to target fragmentation and secondary  scattering [487].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Transport models can play an important role in improving the effectiveness and  safety of charged particle therapy in cancer treatment [487, 488].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, if ions such as carbon  are used instead of protons, the beam may also fragment and spread in the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These interactions  are also studied employing abrasion-ablation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Better models of projectile fragmentation  are needed to determine the effect of ion beams on normal tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recently, the Stochastic Mean  Field (SMF) [489] and Boltzmann-Langevin One Body (BLOB) [490] models have been coupled with  Geant4 for studies related to radiation therapy [491].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The nuclear information required for applications falls under the general umbrella term of  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Some of the applications of hadronic transport calculations and nuclear data in space exploration  research (counterclockwise from top left): energy production in outer space, such as with the TOPAZ  nuclear reactor [469] or the proposed fission surface power system on the Moon KRUSTY [470];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' nuclear  thermal rocket propulsion [471];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' planetary exploration [472];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and dose and shielding calculations of ions  passing through electronics and humans (left: a heavy-ion interaction with a shielding structure [473], right:  particle spectra calculated for an incident solar minimum GCR iron spectrum [474]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Nuclear reactors  Dose/Shielding calculations for electronics & humans  in outer space  Cosmic ray track  Geant4  104  FLUKA  PHITS  Metal  Metal  neutrons  10  3DHZETRN (N=1)  3DHZETRN (N=22)  Glass  102  0 g/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Gate  10  Field  10°  oxide  Drain  Source  protons (xo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1)  Depletion  10°  Positive region  10  He (x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='05)  well  10°  Funneling  10  TOPAZ  region  Substrate  105  10-2  10-1  100  10l  102  103  104  Kinetic energy (MeV/n)  Nuclear propulsion  Planetary exploration  Passive (n, xy)  rays  Cosmic  Fast  Natural  KRUSTY  neutrons  radioactivity57  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  y  10−5  10−4  10−3  10−2  10−1  100  101  102  103  dN  dy  p+Be  UrQMD, protons  UrQMD, deuterons  E802, protons  E802, deuterons  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  y  p+Au, minb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6 A GeV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Rapidity distributions of protons and deuterons in  minimum bias p+Be (left) and p+Au (right) collisions at  a beam energy of Elab = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6 AGeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Blue dashed and red  solid lines show results for protons and deuterons, respec-  tively, obtained from the UrQMD model, compared to data  from the E892 experiment (blue and red dots for protons  and deuterons, respectively) [484].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [483].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  “nuclear data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The Geant3, Geant4, and  FLUKA codes all utilize information taken  directly from nuclear data libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' How-  ever, standard nuclear databases cover  almost exclusively neutron-induced reac-  tions, while few charged-particle data are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In addition, the energy range  covered by these databases typically only  extends to 20 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In higher energy  databases such as the GSI-ESA-NASA  database [482], there are essentially no  data for light ions beyond Elab = 3 AGeV  and scant data for heavy ions beyond a  few hundred AMeV [482].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Transport mod-  els such as Quantum Molecular Dynamics  (QMD) [492] and PHITS [493, 494] have  been used to simulate higher-energy colli-  sions to fill the gaps in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Experiments  at nuclear accelerators are needed to verify  these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The  US  Nuclear  Science  Advisory  Committee (NSAC) has been charged to “assess challenges, opportunities and priorities for ef-  fective stewardship of nuclear data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  As part of the development of the Long Range Plan for  nuclear science, town halls involving different sub-fields of the US nuclear physics community have  adopted nuclear data resolutions, including a recommendation to identify cross-cutting opportuni-  ties with other programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' We suggest that one of these opportunities is the use of transport codes  to advance and enhance high-energy applications, such as space research and advanced medical  treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Hydrodynamics  Relativistic hydrodynamics (Landau-Lifshitz hydrodynamics [495]) can be defined as the effec-  tive field theory (EFT) describing fluids on energy scales much smaller than the fluid tempera-  ture [496].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Hydrodynamic equations encode the evolution of conserved charges, such as energy  density and electric charge density, in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Solving the hydrodynamic equations requires  the EOS as a crucial ingredient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus, in turn, hydrodynamics can be used to constrain the EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  For example, this has been done or is of relevance for the quark-gluon plasma (QGP) generated  in heavy-ion collisions [329, 497, 498] (recently, prospects of constraints on the EOS for nuclear  matter forming the primordial QGP emerged [499]), for (proto)neutron stars [500–505], as well as  for neutron star mergers [506–511].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Status  Hydrodynamics has had a great success describing nuclear matter generated in heavy-ion col-  lisions over a wide range of energies [512, 513].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Remarkably, hydrodynamics applies to various  system sizes accessible in heavy-ion collisions [514], with ALICE, ATLAS, and CMS experiments  showing collective fluid behavior in proton-ion [515–517] and even proton-proton collisions [518–  520], which was also successfully reproduced hydrodynamically [521].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Collective behavior in small  58  systems was also observed at RHIC by the PHENIX and STAR experiments [522, 523].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  It was realized early on that first-order hydrodynamics (in Landau or Eckart frame) is causality  violating and unstable [524].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At this time, the standard solution to this problem is the M¨uller-  Israel-Stewart (MIS) theory [525–527], or versions thereof [528, 529], which are used in most hy-  drodynamic codes modeling heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the MIS theory, transient modes are added as  regulators ensuring a causal time evolution [530].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The behavior of such transient modes depends on  the way they are introduced [528, 530].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Since they are generally not associated with any conserved  quantities, their behavior is not what hydrodynamics aims to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' MIS thus relies on these  transient modes to decay sufficiently fast for the observables to behave hydrodynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This  poses a problem for MIS at early times in a heavy-ion collision, when the regulator transient modes  are still present, because observables sensitive to the early times may reflect the physics of these  regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In addition, the causality violation [531] and stability [532] in these setups has to be  monitored when modeling, for example, heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Alternatively, a more direct approach to constructing causal viscous hydrodynamics is based  on the realization that hydrodynamics is causal when considered in a general frame (and not,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Landau frame or Eckart frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In that case it is not necessary to introduce any regulator  or auxiliary fields, as the differential equations governing the hydrodynamic fields (temperature,  fluid velocity, and chemical potential) are hyperbolic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', there exists a solution for all times) and  their time evolution is causal by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This leads to the Bemfica-Disconzi-Noronha-Kovtun  (BDNK) theory [533–538], which is capable of, for example, modeling neutron star mergers [537].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  BDNK also has a practical use in constructing manifestly causal numerical codes solving hyperbolic  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Note, however, that BDNK is merely a causal formulation of hydrodynamics, and thus  BDNK is still not expected to be a good approximation at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Finally, a rigorous field-theory formulation of hydrodynamics was achieved, which expresses  it as an EFT based on a generating functional [539–545], for a summary see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [546].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  This  approach employs the Schwinger-Keldysh formalism of thermal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As applications of  this formulation, effects of stochastic interactions on hydrodynamic correlation functions [547]  as well as a theory of non-linear diffusion were derived, taking into account large hydrodynamic  fluctuations (for example, leading to the dependence of transport coefficients on fluctuations of the  hydrodynamic fields) [548, 549].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Range of applicability  Many factors influence whether a system may be described hydrodynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Most impor-  tantly, like any EFT, hydrodynamics requires a separation of scales between the microscopic physics  and the scales on which the system is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Let us focus on two remarkable results regarding  the range of applicability of hydrodynamics:  1) The unreasonable effectiveness of hydrodynamics far away from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  2) Systematic extensions of hydrodynamics, extending its range of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Regarding 1), the applicability of hydrodynamics has been historically tied to a requirement  of a near-equilibrium state, near-isotropy, and small gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Astonishingly, heavy-ion collisions,  where neither of these three conditions is met, were successfully described hydrodynamically, which  is often referred to as the unreasonable effectiveness of hydrodynamics [550].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As a possible expla-  nation, hydrodynamic attractors were proposed [551] in a supersymmetric conformal theory, and  subsequently studied for heavy-ion collisions [552–561].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The underlying reason for the attrac-  tor behavior is proposed to be kinematic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', owed to a fast expansion in the boost-invariant  plasma [554, 562, 563].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Since systems cease to be boost invariant as the collision energy is lowered,  59  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Example of an extension of the regime of applica-  bility of hydrodynamics: spin hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While stan-  dard hydrodynamics is valid at small frequencies and mo-  menta (labeled as “pure hydro regime”, indicated by cyan  blue region), in the presence of spin degrees of freedom spin  hydrodynamics is valid in an extended regime (labeled as  “spin hydro regime”, indicated by a pink region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This is  facilitated by adding the slowly relaxing spin modes (green  curves) to the spectrum of standard hydrodynamic shear  (red solid curve) and sound (blue solid curve) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' HY-  DRO+ is constructed in a similar way by adding modes  which relax slower and slower when approaching the criti-  cal point in the QCD phase diagram, bearing implications  for the EOS [564, 565].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure adapted from [566].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  this may pose a challenge to the develop-  ment of hydrodynamic attractors in nu-  clear matter at low to intermediate ener-  gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Within a certain class of models (in-  spired by the gauge/gravity correspon-  dence), the position-space hydrodynamic  expansion (in proper time) around the  Bjorken flow within the MIS theory was  shown to diverge factorially [551].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In  contrast, for the same theory Fourier-  transformed into the momentum space  there is a finite convergence radius lim-  ited by the branch point singularity closest  to the origin in the complex momentum  plane [567–571].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This inspired the formu-  lation of hydrodynamics far from equilib-  rium via resummation [572].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Regarding 2), a standard method to ex-  tend the regime of validity of hydrodynam-  ics is to add one or several mode(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In  fact, promoting the shear tensor to an aux-  iliary field (regulator) adds a mode to the  spectrum of first-order hydrodynamic for-  mulation yielding the MIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In an-  other crucial example, critical fluctuations  need to be taken into account near the crit-  ical point in the QCD phase diagram, and  a set of slow modes can be added to the hydrodynamic modes yielding HYDRO+ [573, 574], which  in turn bears implications for the EOS and the speed of sound [564, 565].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, Lambda  hyperon polarization data [575] indicates that the QGP is highly vortical and polarized [576–  578], which motivated the inclusion of spin in various hydrodynamic descriptions [566, 579–588]  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Within a different systematic extension of hydrodynamics, dynamical elec-  tromagnetic fields can be added, leading to versions of magnetohydrodynamics which couple the  hydrodynamic conservation equations to Maxwell’s equations [589–591], and which can also include  the chiral anomaly [592–594], relevant for neutron stars [594].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Finally, another natural extension of  hydrodynamics is the simultaneous inclusion of multiple conserved charges, in particular, baryon  number B, strangeness S, and electric charge Q (BSQ charges) [532].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This renders transport coef-  ficients matrix-valued, which means that gradients in one charge may lead to diffusion of another  charge [595, 596].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Modern hydrodynamics has been developed in close relation to the gauge/gravity correspon-  dence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', AdS/CFT or holography).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This development, which notably yielded the only consis-  tent theoretical description of fluids with η/s as low as found in heavy-ion data [597–599], began  with the insight that a lower bound on entropy production per degree of freedom (η/s) for a cer-  tain class of theories is related to black branes in the Anti de Sitter (AdS) spacetime [600].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The  fluid/gravity correspondence [601] as a systematic construction tool led to the discovery of the  chiral vortical effect [602, 603] and the re-discovery of the chiral magnetic effect [603, 604].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Holo-  graphic models are also suitable for exploration of plasmas at high densities [605],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' phase transitions  (in particular a holographic version of the QCD critical point [606]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' neutron stars [607],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' taking  [w(k)l = frequency scale  non-conserved  quantities  disappear fast  fast modes  (transient)  Non-hydro regime  T  IWsound(k)l  [Wshear(k)|  approximately  [Wspin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='I (k)  conserved  charges diffuse  [W spin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='(k)I  himois  spin  relaxdtion  rate  Spin hydro regime  S  conserved charges  diffuse slowly  Pure hydro regime  0  k = wave number60  into account finite coupling [608,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 609],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and for investigating the far-from-equilibrium regime of  holographic plasmas [610–612].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Challenges and opportunities  Given the recent developments described above, there are strong reasons to assume that hydro-  dynamics either is valid or can be extended to be valid for the description of dense nuclear matter  at intermediate energy scales, even in small systems with large gradients, far from equilibrium,  and near the QCD critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such (extended) versions of hydrodynamics may well overlap  with the regime of validity of hadronic transport simulations, which needs to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Here,  in particular, further development of hybrid approaches using both hydrodynamics and hadronic  transport will contribute to a better description of intermediate energy heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The way ahead will require pushing forward the development of the rigorous theoretical for-  mulation of hydrodynamics, as well as testing its applicability with exactly solvable models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',  constructed using the gauge/gravity correspondence) and, most importantly, against experimental  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' By continuing the development of hydrodynamics in parallel with gauge/gravity models,  the proposed versions of spin hydrodynamics can be tested and constructed rigorously using the  correspondence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' the same statement also applies to versions of magnetohydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In the  context of (magneto)hydrodynamics, one may also explore the interplay of multiple conserved  charge currents and anomalous currents, leading to novel transport phenomena [593, 613–617].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  For an efficient modeling of heavy-ion collisions (as well as neutron stars and neutron star merg-  ers), the BDNK approach needs to be developed and implemented in standard codes for data  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At high densities, it becomes necessary to describe the propagation of multiple conserved  charges, in particular, the BSQ charges [532].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, the initial state used in numerical  hydrodynamic simulations must be modified to include BSQ degrees of freedom [618–620].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Sim-  ilarly, the EOS [621, 622] and the exact charge conservation when particles are formed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [623, 624]) need to take into account BSQ charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Beyond describing all conserved charges,  theoretical consistency on one hand and the need to describe systems far-from-equilibrium on the  other hand both necessitate a rigorous treatment of hydrodynamic fluctuations, which has been  done using a deterministic approach to fluctuations [625–628].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As a viable future complementary  approach, hydrodynamic fluctuations can be included using the Schwinger-Keldysh formulation  of hydrodynamics [546].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These goals are in line with two recent white papers: Snowmass The-  ory Frontier: Effective Field Theory Topical Group Summary [629] and Snowmass White Paper:  Effective Field Theories for Condensed Matter Systems [630].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  EXPLORATORY DIRECTIONS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Dense nuclear matter EOS meeting extreme gravity and dark matter in supermassive  neutron stars  Do we need an independent determination of the nuclear EOS using terrestrial experiments in  the era of high-precision multi-messenger astronomy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While it is often emphasized that combined  data analyses of heavy-ion reactions and neutron star observations within a unified EOS theory  framework are a powerful tool to study the EOS (see Section IV), the independent extraction of  the nuclear EOS from heavy-ion reactions alone is fundamentally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This assertion is  motivated by a well-known degeneracy [631] between the EOS of dense matter (including hadronic  and/or quark matter, and dark matter) and strong-field gravity in studies aimed at understanding  61  properties of super-massive neutron stars, the minimum mass of black holes, and properties of dark  matter [632–639].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In the Astro2020 Science White Paper on Extreme Gravity and Fundamental Physics [640],  future gravitational wave (GW) observations are envisioned to enable unprecedented and unique  science related to  The nature of gravity: Can we prove Einstein wrong?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' What building-block principles and  symmetries in nature invoked in the description of gravity can be challenged?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The nature of dark matter: Is dark matter composed of particles, dark objects, or modifica-  tions of gravitational interactions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The nature of compact objects: Are black holes and neutron stars the only astrophysical  extreme compact objects in the Universe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' What is the EOS of densest matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  An independent determination of the EOS of dense nuclear matter from terrestrial experiments,  which are free from gravitational effects, will address the question of whether exotic physics, such  as modified gravity, is necessary to describe the behavior and phenomena of supermassive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Thus constraining the EOS from heavy-ion collision experiments will help realize the astrophysical  science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The fundamental questions listed above are among the eleven greatest physics questions for the  new century identified by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' National Research Council in 2003 [641].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While gravity was the  first force discovered in nature, the quest to unify it with other fundamental forces remains elusive,  partially because of its apparent weakness at short distances [642, 643].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, while Einstein’s  general relativity (GR) theory for gravity has successfully passed all observational tests so far, it is  still not fully tested in the strong-field domain [644].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Searches for evidence of possible deviations  from GR are at the forefront of several fields in natural sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It is fundamentally important to  test whether GR will break down at the strongest possible gravitational fields reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For this  goal, supermassive neutron stars are among the ideal testing sites [645, 646].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, as already  mentioned above, their properties can be accounted for by either modifying gravity, adding dark  matter, and/or adjusting the nuclear EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus, an independent inference of the nuclear EOS  from terrestrial experiments is fundamentally important for breaking the degeneracy between the  EOS of supermassive neutron stars and the strong-field gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  There are already some indications that the EOS of dense neutron-rich matter may play a  significant role in understanding the nature of gravity [647–650].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Effects of the nuclear symmetry  energy on the gravitational binding energy [651], surface curvature, and red shift [652], which are  normally used to measure the strength of gravity of massive stars in GR, as well as examples of  mass-radius relations in several classes of modified gravity theories are reviewed briefly in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [653].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  More precise information about the dense nuclear matter EOS from terrestrial experiments will  enable further progress in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Nuclear EOS with reduced spatial dimensions  Nuclear systems under constraints, with high degrees of symmetry and/or collectivity, may  be considered as effectively moving in spaces with reduced spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Historically, in  developing modern methodologies, the spatial dimension d has been considered to be either a  continuous or a discrete variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Many exciting and fundamentally new experimental discoveries  in reduced dimensions have been made in recent years in material sciences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the graphene [654]  and topological insulator [655, 656]) and cold atom physics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the superfluidity in a strongly  correlated 2D Fermi gas [657] and the generalized hydrodynamics in a strongly interacting 1D Bose  gas [658]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  62  The EOS of neutron-rich matter in spaces with reduced dimensions can be linked to that in the  conventional 3-dimensional (3D) space by the ϵ-expansion (ϵ = d − 4) [659–661].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The latter is a  perturbative approach that has been successfully used in treating second-order phase transitions  and related critical phenomena in solid state physics and, more recently, in studying the EOS of  cold atoms in 1D, 2D, and two-species Fermi and/or Bose gases with mixed dimensions [662, 663].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The energy per nucleon E(nB, δ, d) in cold nuclear matter of dimension d at density nB and isospin  asymmetry δ can be expanded around nB = n0, δ = 0, and d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In cold symmetric nuclear  matter, the E(nB, δ = 0, d) is predicted to decrease with decreasing d, indicating that nuclear  matter with a smaller d tends to be more bound but, at the same time, saturates at a higher  3D-equivalent density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The symmetry energy was also found to become smaller in spaces with  lower dimensions compared to the conventional 3D case [664].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Can we find or make 1D and/or 2D nuclear systems in our 3D world?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Can nucleons in neutron-  skins of heavy nuclei be considered as living in spaces with reduced spatial dimensions, and if so,  can we discover the related effects in heavy-ion collisions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Can some of the objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', lasagna) in  the predicted pasta phase [291, 312, 665] of the neutron star crust be described as nuclear systems  with 1D, 2D, or fractional dimensions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' What are the roles of the dimension-dependent EOS in  multi-dimensional models of late stellar evolution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' If 1D/2D simulations using 3D EOS do not  lead to supernova explosions, what will happen if the corresponding 1D/2D EOSs are used instead?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Answers to these questions may provide new perspectives on the EOS of neutron-rich matter in  3D and help solve some of the unresolved puzzles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Interplay between nucleonic and partonic degrees of freedom: SRC effects on nuclear  EOS, heavy-ion reactions, and neutron stars  Short-range correlations (SRCs) in nuclei, that is correlations in the nuclear ground-state wave  function, are mostly due to isosinglet neutron-proton pairs that have temporally fluctuated into  a high-relative-momentum state with approximately zero total center-of mass-momentum and a  spatial separation of about 1 fm [128, 129, 666–669].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Subnucleonic degrees of freedom are expected  to play an important role in understanding SRC-related phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Altered quark momentum  distributions in nucleons embedded in nuclei with respect to those in free nucleons, known as  the EMC effect, have been studied extensively since 1983 [670].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  SRCs have been proposed as  one of the two leading causes of the EMC effect [671, 672].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Recent experiments found that the  strength of SRCs and the EMC effect are strongly correlated [673, 674] and that they both depend  strongly on the isospin asymmetry of the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Moreover, strong evidence was found that only  the momentum distributions of quarks in SRC nucleon pairs in nuclei are modified with respect  to free nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Furthermore, the distributions of quarks in protons of neutron-rich nuclei are  modified more than in neutrons, implying that, on average, u quarks are modified more than d  quarks in neutron-rich nuclei [674], in analogy to an earlier finding that SRCs make protons mover  faster than neutrons in neutron-rich nuclei [675].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These phenomena reflect profound QCD effects  in the nuclear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Studying the flavour- and spin-dependence of nucleon structure functions  is at the forefront of QCD and is a major science driver of future EIC experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' An example  of a correlation formed on short-range QCD length scales are quark-quark correlations known as  diquarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' It was recently proposed that diquark formation across two nucleons via the attractive  QCD quark-quark potential is the underlying QCD-level source of SRCs in nuclear matter and the  cause of the EMC effect [676].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The SRC-related effects have consequences for the nuclear structure, high-energy quark dis-  tributions (the EMC effect), and high-density nuclear matter, including its EOS and in-medium  nucleon-nucleon scattering cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Better understanding of SRC effects in dense neutron-  63  rich medium through heavy-ion reactions may have important ramifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The profound conse-  quences of SRCs on the nuclear matter EOS can be easily seen when one considers the well-known  Hugenholtz-Van Hove (HVH) theorem, which was derived by assuming there are sharp Fermi sur-  faces for nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The theorem provides intrinsic connections among the nuclear symmetry energy,  momentum dependences of both isoscalar and isovector nucleon potentials, and the corresponding  nucleon isoscalar effective mass and neutron-proton effective mass splitting in neutron-rich mat-  ter [677].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, due to the SRCs nucleons do not have sharp Fermi surfaces, but extended  high-momentum tails, as evidenced by many experiments at the Jefferson Laboratory (JLAB) and  Brookhaven National Laboratory (BNL), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [127–130] for recent reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Therefore,  the above relations may be completely altered by the isospin-dependence of SRCs induced by the  nuclear tensor force, which is much stronger in the symmetric nuclear matter than in pure neu-  tron matter [135, 678, 679].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, the composition of the symmetry energy itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=',  the ratio of its kinetic over potential parts) may also be very different from the one without con-  sidering the SRCs [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Most of the parametrizations of the nuclear symmetry energy used so  far in both nuclear physics and astrophysics adopt the kinetic symmetry energy predicted by the  free Fermi gas model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, such parametrizations neglect SRC effects that may lead to a  reduced or even negative kinetic symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This effect originates in the fact that SRCs  are dominated by isosinglet neutron-proton pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As the system becomes more neutron-rich, an  increasingly larger fraction of protons, compared to neutrons, are found in the high momentum  tail [128, 129, 674].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, the kinetic symmetry energy is reduced compared to the free  Fermi gas model prediction [131–138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Interesting indications have been found very recently of SRC effects on the cooling of protoneu-  tron stars, the formation of baryon resonances, dark matter, and nuclear pasta as well as on tidal  deformation and mass-radius correlation in neutron stars [680–687], and also on several features of  nuclear matter and heavy-ion reactions [688–695].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, much more work remains to be done  to systematically and consistently address the SRC-related issues in hadronic transport simulations  (see Section II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Investigations of SRC effects on the nuclear EOS using heavy-ion collisions at  FRIB and FRIB400 will complement the ongoing and planned SRC research programs at JLAB,  GSI, and EIC at BNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Together, these efforts will reveal new knowledge about the spin-isospin  dependence of three-body and tensor forces in dense neutron-rich matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' At short distances, these  forces are mostly due to the ρ-meson exchange [696].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The in-medium ρ-meson mass, determined  by QCD, may be significantly different from its free-space value [696–698].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Such modification in  the ρ-meson mass has been found to significantly affect the high-density behavior of the nuclear  symmetry energy [131, 699].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' However, effects of the QCD quark-quark potential and the modified  tensor or three-body force on in-medium nucleon-nucleon cross sections remain to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  High-density symmetry energy above 2n0  Section III B has primarily focused on the physics of nuclear symmetry energy up to ≈ 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' This  is because of substantial experimental challenges for measuring the symmetry energy using more  energetic beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  However, at higher densities, but below the hadron-quark transition density,  there are also many interesting issues to be addressed [97, 705].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Therefore, it is worthwhile to  explore possible future directions to attack this problem (see also the white paper on QCD Phase  Structure and Interactions at High Baryon Density: Continuation of BES Physics Program with  CBM at FAIR [151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Experiments at FRIB400, FAIR, and other high-energy rare isotope beam facilities around the  world are expected to provide tremendous resolving power for determining the symmetry energy  at densities >∼ 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' While both the magnitude Esym(n0) and slope L of the symmetry energy  64  1  2  3  0  30  60  90  120  150  Neutron Star  Observations  Esym (MeV)  ρ�  ρ0  NL-RMF(18)  PC-RMF(3)  RHF(2)  DD-RMF(2)  Gogny-HF(2)  SHF(33)  �  �  �  �  BHF (Vidana)  BHF (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Li)  DBHF (Fuchs)  DBHF (Sammarruca)  Chiral EFT-N3LO450  Chiral EFT-N3LO600  VMB-APR  VMB-FP  VMB-WFF1  VMB-WFF2  VMB-WFF3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Left: Symmetry energy as a function of baryon density as obtained within 60 example models,  selected from 6 classes of over 520 phenomenological models and/or energy density functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Right:  Symmetry energy as a function of baryon density as obtained within 11 examples from microscopic and/or  ab initio theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thick blue lines are the upper and lower boundaries of symmetry energy from analyses  of neutron star observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [700–704].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  at n0 have been relatively well determined (Esym(n0) ≈ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2 MeV and L ≈ 58 ± 19 MeV  [506, 700, 706, 707], in very good agreement with χEFT calculations [13, 166, 206, 253]), the  curvature Ksym and skewness Jsym of the nuclear symmetry energy are still poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' In  particular, Ksym is most critical for determining the crust-core transition density and pressure in  neutron stars [708–710].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Besides the importance for astrophysics, an experimental determination  20  30  40  50  60  70  80  Esym(2ρ0) (MeV)  FOPI-LAND (2011)  ASY-EOS (2016)  Xie & Li (2019)  Zhang & Li (2019)  Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (2019)  d’Etivaux (2019)  Nakazat & Suzuki (2019)  Symmetry energy at 2ρ0 from analyzing terrestrial and astrophysical data  2021 fiducial value=51 ± 13 MeV  Xie & Li (2020)  Tsang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (2020)  Yue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (2021)  Heavy-ion reactions  Neutron stars  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' (2020)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Symmetry energy at twice the saturation  density from both heavy-ion reactions and neutron  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure modified from figures in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [700, 711].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  of the high-density behavior of the nuclear sym-  metry energy will provide important guidance  for developing high-density nuclear many-body  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Indeed, the density region explored  in heavy-ion reactions at BES, HADES, and in  the future at FRIB400 and FAIR is mostly be-  yond the current validity range of χEFT, and  it is also where the EOSs predicted by vari-  ous nuclear many-body theories, especially the  symmetry energy contributions, start to diverge  broadly (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Recent neutron star observations have led  to some progress in constraining the symme-  try energy at suprasaturation densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 28 is a compilation of recent results on  the symmetry energy at 2n0 from two analyses  65  of heavy-ion reactions at GSI and nine independent analyses of neutron star properties by sev-  eral groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These analyses give a mean value of Esym(2n0) ≈ 51 ± 13 MeV at 68% confidence  level, as indicated by the green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Interestingly, χEFT+MBPT calculations predict a value of  Esym(2n0) ≈ 45±3 MeV [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Similarly, quantum Monte Carlo calculations using local interactions  derived from the χEFT up to next-to-next-to-leading order predict a value of Esym(2n0) ≈ 46 ± 4  MeV [184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Evidently, the mean value of Esym(2n0) from the analyses mentioned above is consis-  tent with the χEFT predictions, albeit with large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As noted before, 2n0 is near the  upper validity limit of the current χEFT theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Thus, more precise measurements of Esym(2n0)  will help to test χEFT predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Inspecting the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 28 shows clearly that more work is necessary to reduce the  error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Most of the neutron star constraints are extracted from radii and tidal deformations of  canonical neutron stars with masses around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These observables are known to be sensitive  mostly to the values of pressure around (1-2)n0 in neutron stars, and therefore their constraints  on Esym(nB) around and above 2n0 are not strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Observables from more massive neutron stars were expected to place stronger constraints on  the high-density symmetry energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To illustrate how the recent NICER+XMM-Newton’s mea-  surements of both the radius and mass of PSR J0740+6620 can influence the constraint on the  symmetry energy at densities above 2n0, the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 29 shows the extracted lower  limits of Esym(nB) obtained from directly inverting the TOV equation within a 3-dimensional  high-density EOS parameter space [238] for two cases: for the case where only the mass is ob-  served (green line), and for the case where both the mass and radius are observed (red line using  the 68% confidence lower radius limit reported by Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [264] and blue line using the radius  reported by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [255]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The orange line is the upper limit of the symmetry energy from  0  20  40  60  80  100  120  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='12  Causality & M  max  =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01 M  sun  Causality & R  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01  =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='41 km  Causality & R  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='01  =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2 km  Causality & L  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  =427  E  sym  [MeV]  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='1%  X  p  r/r0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Constraints on the high-density sym-  metry energy and proton fraction in neutron  stars from analyzing the tidal polarizability of  GW170817 and NICER’s observation of PSR  J0740+6620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  analyzing the upper limit (68% confidence) of tidal  deformation of GW170817 [236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The upper limits  of the symmetry energy from the upper radius limits  reported by both Riley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' are  far above the upper limit of symmetry energy from  GW170817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 29 shows the correspond-  ing proton fractions in PSR J0740+6620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The in-  fluence of knowing both the mass and radius of  this most massive neutron star currently known is  seen by comparing the green line with the red or  blue line, while the difference between the red and  blue lines indicates the systematic error from the  two independent analyses of the same observational  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Although the estimates of Esym(nB) around  (2-3)n0 from these analyses are useful compared to  the model predictions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 27, much more  precise constraints on the Esym(nB) above 2n0 are  needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Pinning down the symmetry energy above 2n0  will be very challenging, but achieving this goal will  bring a great reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For example, without a reli-  able knowledge of the symmetry energy at suprasat-  uration densities, the density profile of the proton  fraction in the core of neutron stars (which has to  be higher than about 11% for the fast cooling to oc-  66  cur) at β−equilibrium is not determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Consequently, whether the fast cooling of protoneutron  stars occurs through the direct URCA process remains uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Heavy-ion reactions, especially  with high-energy radioactive beams, will provide the much-needed data to calibrate nuclear many-  body theories and constrain nuclear symmetry energy at densities >∼ 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These efforts, in concert  with astrophysical research using high-precision X-rays from massive neutron stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', NICER  and STROBE-X [317]), GWs from new LIGO/VIRGO runs, and future detection of post-merger  high-frequency GWs, will better constrain Esym(nB) at densities around and above 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' For these  efforts to be fruitful, it is imperative to explore potential observables carrying undistorted infor-  mation on the symmetry energy above 2n0 from neutron stars and their mergers as well as in  high-energy heavy-ion reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Density-dependence of neutron-proton effective mass splitting in neutron-rich matter  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  <ρ/ρ0>  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='9  1  <m τ> (GeV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='2  m τ (GeV)  0  200  400  600  dN/dm (GeV)  −1  n  p  n  p  132Sn+  124Sn, E/A=50 MeV, b=5 fm  t=10 fm/c  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Correlation between the average nu-  cleon effective mass and the average nucleon  density (top), and the distribution of nucleon  effective masses (bottom) in the reaction of  132Sn+124 Sn at 10 fm/c with a beam energy  Elab = 50 AMeV and an impact parameter  b = 5 fm, as simulated within the IBUU trans-  port model with an explicitly isospin-dependent  single-nucleon potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Figure from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [712].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The nucleon effective mass is a fundamental  quantity characterizing the propagation of a nucleon  in a nuclear medium [713–716], accounting (to lead-  ing order) for effects such as the space-time non-  locality of the effective nuclear interactions or Pauli  exchange effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The magnitude and sign of the  difference (splitting) between the effective masses  of neutrons and protons ∆m∗  np have essential con-  sequences for cosmology, astrophysics, and nuclear  physics through influencing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', the equilibrium  neutron to proton ratio in the early universe and  primordial nucleosynthesis [717], properties of mir-  ror nuclei [718], and the location of drip-lines [719].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  In heavy-ion reactions, ∆m∗  np is of importance for  isospin-sensitive observables [100, 107, 720–725].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  The momentum-dependence of the single-nucleon  potential is normally characterized by the nucleon  effective mass m∗  τ that can be decomposed into an  isoscalar and an isovector component [108, 726, 727].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Due to our poor knowledge of the momentum de-  pendence of isovector interactions, the isovector nu-  cleon effective mass measured by using the neutron-  proton effective mass splitting ∆m∗  np [706] has not  been constrained well [108, 728].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Based on the HVH  theorem, ∆m∗  np was found approximately propor-  tional to the isospin asymmetry δ of the medium,  with a coefficient depending on the density as well  as momentum-dependence of both the isoscalar and  isovector nucleon potential [706].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Over the last decade, significant efforts have been made to extract  this coefficient at n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' A recent survey [729] of model analyses using data from mostly nucleon-  nucleus scattering and giant resonances of heavy nuclei suggests that the ∆m∗  np, scaled by the  average nucleon mass in free space, ranges from 0 to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='5δ [208, 677, 706, 730–735].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  While experimental efforts to better constrain ∆m∗  np at n0 using heavy-ion reactions with in-  termediate energy stable beams are ongoing (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [98]), future experiments at FRIB  and FRIB400 will enable more sensitive probes of not only ∆m∗  np at n0, but also of its density-  67  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Top: Density dependence of the nuclear sym-  metry energy for two typical symmetry energy func-  tionals used in the IBUU simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Bottom: Density  dependence of the isospin asymmetry δ in 132Sn+124  Sn collisions at 20 fm/c with a beam energy of 400  MeV/A and an impact parameter of 1 fm, and in the  core of neutron stars at β-equilibrium (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Taken  from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [95, 736].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  dependence (which cannot be probed by the  nucleon-nucleus  scattering  and  giant  reso-  nances) up to about 2n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  As an illustration,  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 30 are the density dependence  of the average nucleon effective mass (top)  and the distribution of the nucleon effective  masses (bottom) during a typical FRIB reac-  tion as simulated [712] within the IBUU trans-  port model with an explicitly isospin-dependent  single-nucleon potential [106, 720].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  From the  top panel, it is seen that the neutron-proton ef-  fective mass splitting is positive and increases  with the density up to about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='3n0, consistent  with recent χEFT calculations [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' To reach  higher densities, more energetic beams are re-  quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Heavy-ion reactions at FRIB400 will extend  the ranges of both density and isospin asymme-  try of the medium formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Shown in the lower  panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' 31 are the isospin asymmetry δ  as a function of density during a typical reac-  tion at FRIB400 (main) and in neutron stars  at β-equilibrium (inset), calculated using the  same two typical symmetry energy functionals,  shown in the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The δ-nB relations in  both systems show the same isospin fractiona-  tion phenomena, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', reaching a higher isospin  asymmetry when a density functional with a  lower symmetry energy is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' One can also  see that generally, the low-density regions are  more neutron-rich than the high-density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' These δ-nB relations are the fundamental origins  of all isospin-sensitive observables in both heavy-ion reactions and neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  A number of observables in heavy-ion reactions have been proposed as promising messengers of  the underlying momentum-dependence of the isovector potential and the corresponding neutron-  proton effective mass splitting, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', [20, 737, 738] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The momentum dependence of  the single-nucleon potential affects the reaction dynamics directly through the equations of motion  and indirectly through the scattering term of nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' As the in-medium nucleon-nucleon cross  section is proportional to the square of the reduced effective mass of the two colliding nucleons,  the nucleon effective mass will affect the nuclear stopping power (which is also described in the  literature in terms of the nucleon mean free path especially for nucleon-nucleus scattering) [739].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  Consequently, the reaction dynamics and observables of heavy-ion reactions are expected to bear  useful information about the density-dependence of the neutron-proton effective mass splitting in  neutron-rich matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The challenge is to find such observables that are both robust and sensitive  to the variations of the neutron-proton effective mass splitting with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' The nucleon effective  mass affects also transport properties of neutron stars, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' [233, 740–744].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Neutron  star observables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=', neutrino emission and torsional oscillations of neutron stars, may also pro-  vide useful information about the density-dependence of neutron-proton effective mass splitting in  neutron-rich matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Explorations of these issues are invaluable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  68  ACKNOWLEDGMENTS  This White Paper has benefited from talks and discussions at the workshop on Dense nuclear  matter equation of state in heavy-ion collisions that took place at the Institute for Nuclear Theory,  University of Washington (December 5-9, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' thanks Hans-Rudolf Schmidt and Arnaud Le F`evre, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' thanks Daniel Cebra for  insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' thank Abdou Chbihi, Maria Colonna, Arnaud Le F`evre, and  Giuseppe Verde for discussing complementary international efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Miller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FQT4oBgHgl3EQfIjX8/content/2301.13253v1.pdf'}
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+(Preprint) AAS 23-294
+DEEP MONOCULAR HAZARD DETECTION FOR
+SAFE SMALL BODY LANDING
+Travis Driver⋆*, Kento Tomita⋆†, Koki Ho‡, and Panagiotis Tsiotras§
+Hazard detection and avoidance is a key technology for future robotic small body
+sample return and lander missions. Current state-of-the-practice methods rely on
+high-fidelity, a priori terrain maps, which require extensive human-in-the-loop
+verification and expensive reconnaissance campaigns to resolve mapping uncer-
+tainties. We propose a novel safety mapping paradigm that leverages deep se-
+mantic segmentation techniques to predict landing safety directly from a single
+monocular image, thus reducing reliance on high-fidelity, a priori data products.
+We demonstrate precise and accurate safety mapping performance on real in-situ
+imagery of prospective sample sites from the OSIRIS-REx mission.
+INTRODUCTION
+Hazard detection and avoidance (HD&A) is a key technology for future robotic small body sam-
+ple return and lander missions. Current approaches rely on high-fidelity digital elevation maps
+(DEMs) derived from digital terrain models (DTMs), local topography and albedo maps, generated
+on the ground.1 However, DTM construction involves extensive human-in-the-loop verification,
+carefully designed image acquisition plans, and expensive reconnaissance campaigns to resolve
+mapping uncertainties.2,3 We, instead, propose a novel safety mapping paradigm that leverages
+Bayesian deep learning techniques to accurately predict landing safety maps directly from monocu-
+lar images in order to reduce reliance on expensive high-fidelity, a priori data products (i.e., DTMs).
+Safety mapping methodologies that leverage deep learning have demonstrated potential to im-
+prove the accuracy of onboard hazard detection. Previous works4,5 have leveraged deep semantic
+segmentation to classify safe and unsafe landing locations from digital elevation maps (DEMs)
+derived from simulated LiDAR scans. However, generating reliable DEMs from LiDAR scans is
+non-trivial and requires accurate state estimates and range measurements. Moreover, LiDARs typ-
+ically feature a relatively small effective operating range6 and increased size, weight, and power
+(SWaP) requirements relative to passive sensors such as monocular cameras. Thus, we propose to
+derive landing safety maps directly from monocular images without assuming any a priori data or
+relying on the fidelity of the current state estimate. Deep semantic segmentation has been previously
+employed for surface characterization of small bodies from simulated monocular images, primar-
+ily focusing on boulder detection.7,8 Conversely, we apply our models to real images and directly
+predict safety maps that conform to realistic landing parameters and constraints.
+⋆These authors contributed equally to this work.
+*PhD Student, Institute for Robotics and Intelligent Machines, School of Aerospace Engineering, Georgia Institute of
+Technology, Atlanta, GA 30332, USA.
+†PhD Student, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.
+‡Associate Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.
+§David & Andrew Lewis Chair, Professor, Institute for Robotics and Intelligent Machines, School of Aerospace Engineer-
+ing, Georgia Institute of Technology, Atlanta, GA 30332, USA.
+1
+arXiv:2301.13254v1  [cs.CV]  30 Jan 2023
+
+The contributions of this paper are as follows: first, we develop a novel safety mapping paradigm
+that leverages Bayesian deep learning techniques to predict landing safety directly from monoc-
+ular images; second, we construct a dataset of real monocular images and corresponding land-
+ing safety maps that conform to realistic landing parameters for training and testing our mod-
+els; third, we demonstrate precise and accurate safety mapping performance on real imagery of
+prospective sample sites from the recent OSIRIS-REx mission to Asteroid 101955 Bennu. Our
+code, data, and trained models will be made available to the public at https://github.com/
+travisdriver/deep_monocular_hd.
+RELATED WORK
+Current hazard detection methodologies for small body missions rely on high-fidelity digital el-
+evation maps (DEMs) derived from digital terrain models (DTMs), local topography and albedo
+maps.1 However, DTM construction typically involves extensive human-in-the-loop verification
+and carefully designed image acquisition plans to achieve optimal results.2,3 Consequently, au-
+tonomous hazard detection and avoidance (HD&A) has been identified as a high-priority technol-
+ogy9 to promote and enable new mission concepts to near-earth asteroids, comets, the Moon, Mars,
+and beyond.
+The Autonomous Landing Hazard Avoidance Technology (ALHAT)10,11 program was launched
+in 2005, followed by the Safe & Precise Landing—Integrated Capabilities Evolution (SPLICE)
+program12 in 2018, in order to develop autonomous landing technologies. These programs have fo-
+cused on developing HD&A algorithms that operate on DEMs generated from range measurements
+acquired by active sensors such as flash LiDARs. However, these methods are constrained by the
+relatively small effective operating range and the increased size, weight, and power (SWaP) require-
+ments of LiDARs relative to passive sensors such as monocular cameras. Indeed, the OSIRIS-REx
+Guidance, Navigation, and Control (GNC) flash LiDAR had a maximum operational range of ap-
+proximately 1 km,13,14 while preliminary testing of the Hazard Detection LiDAR of the SPLICE
+program demonstrated a 5 cm ground sample distance at 500 meters and near-nadir pointing.12
+Conversely, the OSIRIS-REx Camera Suite (OCAMS)15 was able to acquire 5 cm GSD images
+at almost 4 km, providing higher-resolution measurements earlier in the mission than the active
+sensors onboard and allowing for detailed surface characterization during the early phases of the
+mission.6 Moreover, constructing a DEM from LiDAR scans is non-trivial and requires accurate
+range measurements and precise knowledge of the spacecraft’s relative pose to the landing plane.
+Instead, we focus on estimating landing safety directly from a single monocular image.
+Safety mapping methodologies that leverage deep learning have demonstrated potential to im-
+prove hazard detection accuracy and have also been shown to offer competitive runtimes on flight
+relevant hardware.16 Previous works have leveraged deep semantic segmentation to classify safe
+and unsafe landing locations from high-resolution DEMs. Moghe and Zanetti4 leverage a deep neu-
+ral network architecture for predicting safety maps from a DEM and design a novel loss function
+specifically designed to decrease the false safe rate and encourage more precise safe predictions.
+Tomita et al.5 employ a Bayesian SegNet architecture17 for segmentation of input DEMs into safe
+and unsafe landing locations. The Bayesian architecture implemented by Tomita et al.5 enables
+uncertainty quantification of the predicted safety map through the predictive entropy of the model,
+allowing for more precise predictions through global thresholding with respect to this uncertainty
+measure. We build upon this work and demonstrate its efficacy on monocular imagery.
+Methods based on deep learning have also been developed for surface segmentation from monoc-
+2
+
+ular imagery. Pugliatti and Maestrini7 employ a custom U-Net architecture for classification of
+surface landmarks, i.e., boulders and crater rims. Caroselli et al.8 apply deep semantic segmen-
+tation to boulder detection on synthetic images of a fabricated small body model and post-process
+the network prediction to derive a landing safety map based on boulder density. Conversely, we
+apply our models to real images and directly predict safety maps that conform to realistic landing
+parameters and constraints.
+PROPOSED APPROACH
+Our novel safety mapping paradigm leverages Bayesian deep learning techniques to develop an
+uncertainty-aware semantic segmentation model to predict safety maps from monocular imagery.
+We train and test our model on real in-situ imagery from the OSIRIS-REx mission to asteroid
+101955 Bennu with corresponding ground truth safety maps generated using realistic landing pa-
+rameters and constraints.
+Bayesian Deep Learning
+Given training input data X = {x1, . . . , xN} with corresponding labels Y = {y1, . . . , yN},
+Bayesian deep learning employs Bayesian inference to maximize the posterior distribution of the
+network parameters θ given the training data X, Y:
+p(θ | X, Y) = p(Y | X, θ)p(θ)
+p(Y | X)
+.
+(1)
+The distribution above can then be used to predict the likelihood of an output y∗ for a new input x∗
+via
+p(y∗ | x∗, X, Y) = Ep(θ | X,Y)[p(y∗ | x∗, θ)],
+(2)
+where we may assume a softmax likelihood for p(y∗ | x∗, θ). However, computing p(θ | X, Y) is
+intractable and must be approximated using variational inference.
+Gal and Ghahramani18,19 showed that training a deep convolutional neural network (CNN) with
+dropout layers is equivalent to approximate variational inference with a variational distribution q(θ)
+which imposes a Bernoulli distribution over the model weights. Specifically, consider a convolu-
+tional layer i with ci−1 input channels, ci output channels, and kernel size k. Then dropout can be
+viewed as imposing a distribution over the layer weights Wi according to
+Wi = Mi diag([ϵj]ci
+j=1),
+ϵj ∼ Bern(pj),
+(3)
+where ϵj are Bernoulli distributed random variables with parameter pj (which we take to be 0.5),
+and Mi ∈ Rci−1×k×k×ci are the variational weight parameters optimized during training. Therefore,
+Equation (2) may be approximated by
+p(y∗ | x∗, X, Y) ≈ Eq(θ)[p(y∗ | x∗, θ)].
+(4)
+Finally, employing dropout at test time permits the use of the predictive entropy approximated
+through T stochastic forward passes through the network as a measure of uncertainty:20
+H[y | x, X, Y] ≈ −
+d
+�
+k=1
+�
+1
+T
+T
+�
+t=1
+p(y = k | x, θt)
+�
+log
+�
+1
+T
+T
+�
+t=1
+p(y = k | x, θt)
+�
+,
+(5)
+3
+
+CFF
+(1/2)
+CFF
+Entropy
+Mean
+Stochastic dropout
+samples
+Uncertainty
+Safety map
+32
+32
+64
+128
+2
+64
+128
+256
+128
+1024
+CFF
+Conv + BN + ReLU
+Dropout
+Max Pool
+Pyramid Pool
+(1/4)
+(1/8)
+(1/16)
+Bilinear downsampling
+(1/32)
+256
+256
+2x Bilinear Upsample
+Cascade Feature Fusion
+(1/2)
+(1/4)
+(1/4)
+(1/2)
+(1)
+Figure 1: Bayesian ICNet architecture. Multiscale fusion is conducted within the cascade feature
+fusion (CFF)21 modules. The ratio in parentheses denotes the relative magnitude of the spatial
+dimensions with respect to the original image.
+where θt corresponds to a realization of the network parameters, distributed according to q(θ),
+sampled during a forward pass through the network, and the average over the forward passes is
+taken to be the final prediction probabilities. This process is referred to as Monte Carlo (MC)
+dropout.18,19 For the task of semantic segmentation, assume x is a tuple (X, u) containing an
+image tensor X ∈ Rh×w×c and an image coordinate u ∈ R2, and k ∈ {1, . . . , d} is a pixel-wise
+class label for the pixel located at u. We will demonstrate that leveraging this uncertainty measure
+for our network predictions leads to increased precision and accuracy of safe landing locations.
+Uncertainty-Aware Semantic Segmentation
+We leverage an uncertainty-aware semantic segmentation architecture based on the image cas-
+cade network (ICNet),21 shown in Figure 1. ICNet is a highly efficient segmentation architecture
+that blends coarse prediction maps obtained from down-sampled inputs with high-resolution feature
+maps obtained from high throughput networks that operate on the full-resolution image, allowing
+for fast inference on high-resolution images while maintaining accuracy. Multiscale feature map
+fusion is conducted by the cascade feature fusion (CFF) modules, whereby a reduced-resolution
+segmentation map is computed from the two multiscale feature map inputs. The multiscale predic-
+tions are used to train the network via a weighted softmax cross-entropy loss.21
+We implement a Bayesian version of ICNet, which we denote as BICNet, where dropout layers
+are added to allow for stochastic sampling with respect to the model parameters using techniques
+from Bayesian deep learning, i.e., MC dropout, as described in the previous subsection. Ideally, a
+Bayesian NN would feature a dropout layer after every hidden layer of the network.18,19 However,
+as observed in previous works,17,20 adding dropout layers after every convolutional layer in more
+complex networks is too strong of a regularizer, resulting in underfitting. Therefore, we follow the
+4
+
+00(a) Global shape model
+Nightingale
+Osprey
+Kingfisher
+Sandpiper
+(b) DTM (left) and reconnaissance imagery (right) for each tag site
+Figure 2: OSIRIS-REx TAG site datasets. TAG site locations are indicated by the corresponding
+color in the global shape model.
+work of Mukhoti et al.20 and Kendall and Cippola17 and only insert dropout layers after the central
+encoder and decoder layers. At test time, we perform T = 8 stochastic forward passes and use the
+predictive entropy, defined in Equation (5), as a measure of uncertainty, and use the average of this
+measure over all training instances as a threshold to mask out high uncertainty regions in the image.
+Data Generation
+High-fidelity DTMs (i.e., 5 cm ground sample distance) of the four prospective Touch-And-Go
+(TAG) sample sites developed as part of the OSIRIS-REx mission to Asteroid 101955 Bennu, i.e.,
+Nightingale, Kingfisher, Osprey, and Sandpiper, were used to generate ground truth safety map
+labels for reconnaissance imagery from the mission. Specifically, we leverage monocular recon-
+naissance imagery and the corresponding camera pose labels, relative to a body fixed frame of the
+asteroid, provided through the AstroVision dataset.22 For each image, DEMs are constructed by
+transforming the DTM into a local coordinate system in which the +z-axis points opposite the
+vector corresponding to the direction of the gravitational force due to the target body at the point
+on the surface closest to the center of the image. The gravity due to body was computed using a
+global shape model of Bennu23 and assuming a constant-denity polyhedron.24 Safety mapping was
+conducted on the DEM and then projected back into the image to produce pixel-wise landing safety
+labels. Example reconnaissance images for each prospective TAG site are provided in Figure 2.
+Landing safety was computed from the DEMs using the method developed by the Autonomous
+Landing Hazard Avoidance Technology (ALHAT) project.25 The ALHAT method evaluates the
+lander contact locations for all pixels and for all orientations to assess the worst-case surface slope
+and roughness values with respect to the surface elevation data contained in the ground truth DEMs.
+Specifically, a landing plane is computed for each pixel by assessing the elevation of four evenly
+spaced contact points, emulating lander foot pads, on the perimeter of a circle specified by the di-
+ameter of the lander. Slope is defined as the largest angle between the landing plane and x-y plane
+of the ground truth DEM for all orientations, and the roughness is the largest perpendicular distance
+to the terrain above the the landing plane for all orientations. Any pixel with slope and roughness
+exceeding a given threshold is labeled as unsafe, where we chose a threshold of 30◦ for slope and 3.5
+cm for roughness. We specify a lander with a 35 cm diameter, similar to the MASCOT (Mobile As-
+teroid surface SCOuT) lander that was deployed during the Hayabusa2 mission to Asteroid 162173
+5
+
+(a) Image
+20
+40
+60
+80
+Slope ( )
+(b) Slope
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Roughness (m)
+(c) Roughness
+(d) Slope & Roughness Safety Map
+(e) Roughness-only Safety Map
+Figure 3: Ground truth safety map example. Safe and unsafe regions are drawn in green and red,
+respectively, in the safety map.
+Ryugu.26 An example safety map along with its corresponding monocular image is provided in
+Figure 3.
+Moreover, we provide the data distributions of our datasets with respect to the ground sample
+distance, imaging depth, viewing angle, and visibility ratio in Figure 4. Ground sample distance
+(GSD) measures the average distance on the surface spanned by a single pixel, which is a function
+of the distance to the surface and the camera intrinsics, as landing safety becomes increasingly
+difficult to observe as the relative size of the lander in the image decreases. The imaging depth
+measures the average distance to the surface when the image was taken, and provides context for
+the GSD values. Specifically, the MapCam of the OSIRIS-REx Camera Suite (OCAMS),15 with
+a focal length of ∼125 mm, can provide 5 cm GSD measuresments of the surface at distances
+of approximately 1 km, while the PolyCam, with a focal length of ∼620 mm, provides the same
+resolution at distances of almost 4 km. Viewing angle measures the angle between the −z-axis of
+the ground truth DEM and the camera boresight. Finally, the visibility ratio is the ratio of visible
+(i.e., not occluded by shadows) pixels to total pixels in the image and provides a measure of the
+illumination conditions in the image.
+6
+
++PolyCam
+MapCam
+SamCam
+Figure 4: Data distributions with respect to imaging depth, GSD, viewing angle, and visibility
+ratio. Our dataset features a total of 770 images annotated with per-pixel safety labels: 133 of
+Kingfisher, 342 of Nightingale, 162 of Osprey, and 91 of Sandpiper, and 42 from the TAG sample
+collection event at Nightingale.
+RESULTS
+In this section, we first present our suite of metrics used to evaluate the performance of our
+approach. We then validate our approach on two different experiments using real images from the
+OSIRIS-REx mission to Asteroid 101955 Bennu, including images captured during the actual TAG
+sample collection event.
+Metric Definitions
+We measure the quality of the predicted per-pixel safety map labels of our model with respect to
+precision, sensitivity, accuracy, and mean intersection over union (mIoU):
+precision =
+true safe
+true safe + false safe,
+(6)
+7
+
+Table 1: Overall performance for the Sandpiper landing site experiment. The values in paren-
+theses are the metrics with shadowed pixels ignored. All reported values are percentages.
+METHOD
+PRECISION
+SENSITIVITY
+ACCURACY
+MIOU
+SLOPE & ROUGHNESS
+WITHOUT UNCERTAINTY
+60.66 (62.91)
+67.21 (70.05)
+69.53 (69.41)
+52.05 (52.27)
+WITH UNCERTAINTY
+76.98 (77.67)
+20.09 (21.86)
+82.29 (82.01)
+65.76 (65.93)
+ROUGHNESS ONLY
+WITHOUT UNCERTAINTY
+77.24 (78.92)
+61.61 (63.66)
+63.53 (64.41)
+44.87 (45.31)
+WITH UNCERTAINTY
+85.71 (86.32)
+28.78 (31.63)
+73.77 (73.93)
+55.11 (54.98)
+sensitivity =
+true safe
+true safe + false unsafe,
+(7)
+accuracy = true safe + true unsafe
+valid pixels
+,
+(8)
+mIoU = 1
+2
+�
+true safe
+valid pixels − true unsafe +
+true unsafe
+valid pixels − true safe
+�
+.
+(9)
+True safe (false safe) includes pixels predicted to be safe by our models that are safe (unsafe) in
+the ground truth labels, and true unsafe (false unsafe) includes pixels predicted to be unsafe that
+are unsafe (safe) in the ground truth labels. Note that false unsafe includes safe pixels that are
+ignored and not labeled safe due to high uncertainty. For our application, we can interpret precision
+as the reliability of the pixels predicted to be safe, and sensitivity as detection rate of true safe
+sites, respectively. Accuracy and mIoU are the metrics evaluated for the valid pixels, which are
+the pixels with smaller uncertainty than the threshold. In other words, valid pixels correspond
+to the predictions that the network is most “certain” about. For the results without uncertainty
+thresholding, accuracy and mIoU are evaluated for all the pixels with valid safety labels. In the
+following analysis, we refer to the ratio of pixels that fall above our uncertainty threshold, and
+consequently marked as unsafe, to valid pixels as the screening rate.
+Experiment 1: Prospective Landing Site Sandpiper
+We train our model on images from three of the prospective landing sites from the OSIRIS-
+Rex mission, namely, Nightingale, Osprey, and Kingfisher, and test our model on images of the
+remaining sample site, Sandpiper. This emulates a scenario in which data from previously mapped
+landing sites could be used to train a network to predict landing safety in a new, unexplored region of
+the target body without requiring the construction of high-fidelity DEMs. These results are detailed
+in Table 1, and qualitative examples are provided in Figure 5, where we consider performance with
+respect to identifying both slope and roughness hazards and roughness-only hazards.
+The results illustrate that our models are able to predict safety maps from just a single monocular
+image of the propspective landing site, which is completely unseen during training, with accuracy
+over 69% for the slope and roughness hazard detection case, and over 63% for the roughness-only
+8
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+(a) Slope & roughness
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+Test Image
+Without Uncertainty
+With Uncertainty
+Uncertainty
+(b) Roughness-only
+Figure 5: Qualitative monocular safety mapping results for the Sandpiper experiment. Green,
+yellow, blue, and red labels represent true safe, true unsafe, false unsafe, and false safe, respectively.
+9
+
+hazard detection case, even without uncertainty thresholding. Moreover, our Bayesian ICNet archi-
+tecture enables uncertainty thresholding in order to further boost performance by ignoring regions in
+which the models’ prediction has high entropy. With the uncertainty threshold, accuracy increases
+to 82.29% and 73.77% for the slope and roughness and roughness-only cases, respectively, at the
+cost of decreased sensitivity. Importantly, we are able to achieve 76.98% and 85.71% precision for
+the slope and roughness and roughness-only cases, respectively, after uncertainty thresholding. We
+also have a slight increase in all the metrics by ignoring shadowed pixels, which are reported in
+parentheses in Table 1.
+Comparing the two different hazard detection tasks, i.e., slope and roughness hazards and roughness-
+only hazards, the roughness-only case has lower values of sensitivity, accuracy, and mIoU, but
+higher values of precision. This suggests that the roughness-only case is a harder task in terms of
+precisely labeling safe and unsafe pixels on average, resulting in lower accuracy and mIoU, but is
+an easier task in terms of identifying only safe pixels, thus resulting in higher precision. This is
+partially due to the higher incidence of safe pixels for the roughness-only case as compared to the
+slope and roughness case, as illustrated in Figure 3.
+Additionally, we analyzed the per-image metrics with respect to GSD, viewing angle, and visibil-
+ity ratio for the slope and roughness case, shown in Figure 6, and the roughness-only case, shown in
+Figure 7, in order to identify possible causes of uncertainty in the predictions. As a general trend, we
+can observe that a higher uncertainty results in lower performance metrics of precision, sensitivity,
+accuracy, and mIoU. Intuitively, low visibility is a common factor that results in higher uncertainty
+in our model for both the slope and roughness case and the roughness-only case. Our models as-
+sign a higher uncertainty to images with larger GSD for the slope and roughness case, and larger
+viewing angle for the roughness-only case. Note that increased uncertainty for images at higher
+GSDs may also be due to these instances being less represented in the training data as shown in
+Figure 4. In either case, we demonstrate that the uncertainty threshold serves as a powerful tool for
+detecting and accounting for difficult or out-of-distribution input conditions, allowing our models
+to predict precise and accurate safety maps across multiple GSDs, viewing angles, and illumination
+conditions.
+Experiment 2: OSIRIS-REx TAG Sequence
+For the second experiment, we trained two models using different combinations of images from
+the OSIRIS-REx mission: one model, denoted by BICNet-NKO, is trained on Nightingale, King-
+fisher, and Osprey, and the other model, denoted by BICNet-KOS, is trained on Kingfisher, Osprey,
+and Sandpiper. These models were trained for the slope and roughness case only. We tested the two
+models on images captured during the TAG sample collection event at the Nightingale sample site.
+We present both models to illustrate the effect of the different training data distributions on the test
+results. A subset of the 42 image sequence is shown in Figure 8. Note that the 42 test images are
+not included in the training set for the Nightingale images.
+Table 2 and Figure 10 show quantitative results and qualitative examples, respectively. Compar-
+ing BICNet-NKO and BICNet-KOS in Table 2, we can see sensitivity of BICNet-KOS is signifi-
+cantly lower than that of BICNet-NKO after uncertainty thresholding. Indeed, BICNet-KOS assigns
+a high uncertainty to almost all regions of the input images and are thus overwritten as unsafe after
+uncertainty thresholding, as shown in Figure 10, which is not the case for BICNet-NKO. These
+differences are most likely explained by the difference between the training and testing data distri-
+butions for BICNet-KOS, as illustrated in Figure 4. Specifially, the TAG images have less overlap
+10
+
+0.6
+0.8
+Precision
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.0
+0.2
+0.4
+Sensitivity
+0.6
+0.8
+Accuracy
+uncertainty
+0.44
+0.48
+0.52
+0.56
+0.60
+0.4
+0.6
+0.8
+IoU
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.6
+0.7
+0.8
+0.9
+Screen Rate
+0.6
+0.8
+Precision
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.0
+0.2
+0.4
+Sensitivity
+0.6
+0.8
+Accuracy
+uncertainty
+0.44
+0.48
+0.52
+0.56
+0.60
+0.4
+0.6
+0.8
+IoU
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.6
+0.7
+0.8
+0.9
+Screen Rate
+0.6
+0.8
+Precision
+0.4
+0.6
+0.8
+Visibility Ratio
+0.0
+0.2
+0.4
+Sensitivity
+0.6
+0.8
+Accuracy
+uncertainty
+0.44
+0.48
+0.52
+0.56
+0.60
+0.4
+0.6
+0.8
+IoU
+0.4
+0.6
+0.8
+Visibility Ratio
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.6
+0.7
+0.8
+0.9
+Screen Rate
+Figure 6: Per-image metrics for slope & roughness safety on the Sandpiper experiment with
+respect to GSD, viewing angle, and visibility ratio.
+11
+
+0.7
+0.8
+0.9
+Precision
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.0
+0.2
+0.4
+0.6
+Sensitivity
+0.5
+0.6
+0.7
+0.8
+Accuracy
+uncertainty
+0.40
+0.44
+0.48
+0.52
+0.56
+0.60
+0.3
+0.4
+0.5
+0.6
+0.7
+IoU
+0.02
+0.03
+0.04
+0.05
+0.06
+GSD (m/pixel)
+0.40
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.4
+0.6
+0.8
+Screen Rate
+0.7
+0.8
+0.9
+Precision
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.0
+0.2
+0.4
+0.6
+Sensitivity
+0.5
+0.6
+0.7
+0.8
+Accuracy
+uncertainty
+0.40
+0.44
+0.48
+0.52
+0.56
+0.60
+0.3
+0.4
+0.5
+0.6
+0.7
+IoU
+10
+20
+30
+40
+50
+Viewing Angle ( )
+0.40
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.4
+0.6
+0.8
+Screen Rate
+0.7
+0.8
+0.9
+Precision
+0.4
+0.6
+0.8
+Visibility Ratio
+0.0
+0.2
+0.4
+0.6
+Sensitivity
+0.5
+0.6
+0.7
+0.8
+Accuracy
+uncertainty
+0.40
+0.44
+0.48
+0.52
+0.56
+0.60
+0.3
+0.4
+0.5
+0.6
+0.7
+IoU
+0.4
+0.6
+0.8
+Visibility Ratio
+0.40
+0.45
+0.50
+0.55
+0.60
+Uncertainty
+0.4
+0.6
+0.8
+Screen Rate
+Figure 7: Per-image metrics for roughness-only safety on the Sandpiper experiment with re-
+spect to GSD, viewing angle, and visibility ratio.
+12
+
+2020-10-20T21:30:48
+2020-10-20T21:31:48
+2020-10-20T21:32:48
+2020-10-20T21:33:48
+2020-10-20T21:34:48
+2020-10-20T21:35:48
+2020-10-20T21:36:48
+2020-10-20T21:37:48
+2020-10-20T21:38:48
+2020-10-20T21:39:48
+2020-10-20T21:40:48
+2020-10-20T21:41:48
+Figure 8: Frames from the OSIRIS-Rex TAG sequence captured by the SamCam.
+with the images of the prospective landing sites except for Nightingale with respect to viewing angle
+and visibility ratio. Therefore, the exclusion of Nightingale from training data increases the predic-
+tive uncertainty for TAG images at test time for BICNet-KOS. Note that the Nightingale images
+(excluding the TAG images) were used for validation during training for the BICNet-KOS model in
+order to rule out overfitting as a cause for the decreased prediction performance.
+This effect of training data distributions on the uncertainty level, and the accompanying predictive
+performance, is consistent with the per-image metrics with respect to GSD, as shown in Figure 9.
+Indeed, for BICNet-NKO, as the GSD of testing images gets closer to the peak of the training data,
+the predictive performance increases and uncertainty decreases. Lower precision for the higher
+GSD images is also partly due to the very low incidence of safe regions for these instances (see
+Figure 10). Conversely, for BICNet-KOS, all test images have a high screening rate and a low
+sensitivity due to the high uncertainty, purportedly due to the out-of-distribution training data with
+respect to the viewing angle and visibility ratio and the relatively small size of the training set (386
+images). We do not provide the per-image metrics with respect to the viewing angle and visibility
+ratio, as these values remain relatively constant over the entire TAG sequence at ∼7◦ and ∼71%,
+respectively, as shown in Figure 4. These results illustrate the effect of training data distributions
+and the ability of the uncertainty measure to identify out-of-distribution data for uncertainty-aware
+segmentation networks. We postulate that training our model with a more comprehensive set of
+images will decrease prediction uncertainty and increase the performance.
+CONCLUSION
+In this paper we presented a novel landing hazard detection approach for small body missions that
+predicts safety maps directly from monocular imagery. We implemented an efficient, uncertainty-
+aware segmentation network that demonstrated hazard detection performance at over 80% accuracy
+and over 85% precision on real images of unseen landing sites captured during the OSIRIS-REx
+mission to Asteroid 101955 Bennu. We believe that monocular safety mapping is a promising
+technology for reducing reliance on human-in-the-loop procedures used in current safety mapping
+methodologies. Future work will involve developing a more comprehensive distribution of training
+data and identifying and rectifying causes of uncertainty to increase the reliability of the proposed
+13
+
+0.4
+0.6
+0.8
+Precision
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.2
+0.4
+0.6
+Sensitivity
+0.7
+0.8
+0.9
+Accuracy
+uncertainty
+0.38
+0.40
+0.42
+0.44
+0.46
+0.5
+0.6
+0.7
+0.8
+IoU
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.375
+0.400
+0.425
+0.450
+0.475
+Uncertainty
+0.5
+0.6
+Screen Rate
+(a) BICNet-NKO
+0.5
+0.6
+0.7
+0.8
+Precision
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.00
+0.02
+0.04
+0.06
+0.08
+Sensitivity
+0.75
+0.80
+0.85
+Accuracy
+uncertainty
+0.49
+0.50
+0.51
+0.52
+0.53
+0.5
+0.6
+IoU
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+GSD (m/pixel)
+0.50
+0.52
+Uncertainty
+0.65
+0.70
+0.75
+0.80
+Screen Rate
+(b) BICNet-KOS
+Figure 9: Per-image metrics for the TAG experiment with respect to the GSD for slope &
+roughness safety.
+14
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+(a) BICNet-NKO
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Uncertainty
+Test Image
+Without Uncertainty
+With Uncertainty
+Uncertainty
+(b) BICNet-KOS
+Figure 10: Qualitative monocular safety mapping results for the TAG experiment. Green,
+yellow, blue, and red labels represent true safe, true unsafe, false unsafe, and false safe, respectively.
+15
+
+Table 2: Overall performance for the TAG experiment for slope & roughness safety. BICNet-
+NKO is our Bayesian ICNet model trained on Nightingale, Kingfisher, and Osprey, and BICNet-
+KOS is trained on Kingfisher, Osprey, and Sandpiper. All reported values are percentages.
+METHOD
+PRECISION
+SENSITIVITY
+ACCURACY
+MIOU
+BICNET-NKO
+WITHOUT UNCERTAINTY
+49.02 (50.21)
+61.16 (66.53)
+67.44 (67.09)
+48.65 (48.66)
+WITH UNCERTAINTY
+61.38 (61.66)
+26.60 (30.80)
+78.62 (77.09)
+60.04 (59.23)
+BICNET-KOS
+WITHOUT UNCERTAINTY
+50.08 (50.42)
+42.18 (50.37)
+67.24 (65.32)
+45.48 (45.11)
+WITH UNCERTAINTY
+65.05 (64.89)
+2.87 (4.05)
+83.11 (82.65)
+52.58 (55.87)
+approach. Our code, data, and trained models will be made available to the public at https:
+//github.com/travisdriver/deep_monocular_hd.
+ACKNOWLEDGMENTS
+This work supported by a NASA Space Technology Graduate Research Opportunity and the
+NASA Early Career Faculty Program (grant no. 80NSSC20K0064). The authors would like to
+thank Kenneth Getzandanner and Michael Shoemaker from NASA Goddard Space Flight Center
+for several helpful discussions and comments.
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+
diff --git a/9NFQT4oBgHgl3EQfJDUU/content/tmp_files/load_file.txt b/9NFQT4oBgHgl3EQfJDUU/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5a2f24b6b1f6040ee1b63d151983eddc5e3504ca
--- /dev/null
+++ b/9NFQT4oBgHgl3EQfJDUU/content/tmp_files/load_file.txt
@@ -0,0 +1,945 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf,len=944
+page_content='(Preprint) AAS 23-294  DEEP MONOCULAR HAZARD DETECTION FOR  SAFE SMALL BODY LANDING  Travis Driver⋆*, Kento Tomita⋆†, Koki Ho‡, and Panagiotis Tsiotras§  Hazard detection and avoidance is a key technology for future robotic small body  sample return and lander missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Current state-of-the-practice methods rely on  high-fidelity, a priori terrain maps, which require extensive human-in-the-loop  verification and expensive reconnaissance campaigns to resolve mapping uncer-  tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We propose a novel safety mapping paradigm that leverages deep se-  mantic segmentation techniques to predict landing safety directly from a single  monocular image, thus reducing reliance on high-fidelity, a priori data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  We demonstrate precise and accurate safety mapping performance on real in-situ  imagery of prospective sample sites from the OSIRIS-REx mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  INTRODUCTION  Hazard detection and avoidance (HD&A) is a key technology for future robotic small body sam-  ple return and lander missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Current approaches rely on high-fidelity digital elevation maps  (DEMs) derived from digital terrain models (DTMs), local topography and albedo maps, generated  on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1 However, DTM construction involves extensive human-in-the-loop verification,  carefully designed image acquisition plans, and expensive reconnaissance campaigns to resolve  mapping uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2,3 We, instead, propose a novel safety mapping paradigm that leverages  Bayesian deep learning techniques to accurately predict landing safety maps directly from monocu-  lar images in order to reduce reliance on expensive high-fidelity, a priori data products (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=', DTMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Safety mapping methodologies that leverage deep learning have demonstrated potential to im-  prove the accuracy of onboard hazard detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Previous works4,5 have leveraged deep semantic  segmentation to classify safe and unsafe landing locations from digital elevation maps (DEMs)  derived from simulated LiDAR scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' However, generating reliable DEMs from LiDAR scans is  non-trivial and requires accurate state estimates and range measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Moreover, LiDARs typ-  ically feature a relatively small effective operating range6 and increased size, weight, and power  (SWaP) requirements relative to passive sensors such as monocular cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Thus, we propose to  derive landing safety maps directly from monocular images without assuming any a priori data or  relying on the fidelity of the current state estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Deep semantic segmentation has been previously  employed for surface characterization of small bodies from simulated monocular images, primar-  ily focusing on boulder detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='7,8 Conversely, we apply our models to real images and directly  predict safety maps that conform to realistic landing parameters and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  ⋆These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  PhD Student, Institute for Robotics and Intelligent Machines, School of Aerospace Engineering, Georgia Institute of  Technology, Atlanta, GA 30332, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  †PhD Student, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  ‡Associate Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  §David & Andrew Lewis Chair, Professor, Institute for Robotics and Intelligent Machines, School of Aerospace Engineer-  ing, Georgia Institute of Technology, Atlanta, GA 30332, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='13254v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='CV]  30 Jan 2023  The contributions of this paper are as follows: first, we develop a novel safety mapping paradigm  that leverages Bayesian deep learning techniques to predict landing safety directly from monoc-  ular images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' second, we construct a dataset of real monocular images and corresponding land-  ing safety maps that conform to realistic landing parameters for training and testing our mod-  els;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' third, we demonstrate precise and accurate safety mapping performance on real imagery of  prospective sample sites from the recent OSIRIS-REx mission to Asteroid 101955 Bennu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Our  code, data, and trained models will be made available to the public at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='com/  travisdriver/deep_monocular_hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  RELATED WORK  Current hazard detection methodologies for small body missions rely on high-fidelity digital el-  evation maps (DEMs) derived from digital terrain models (DTMs), local topography and albedo  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1 However, DTM construction typically involves extensive human-in-the-loop verification  and carefully designed image acquisition plans to achieve optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2,3 Consequently, au-  tonomous hazard detection and avoidance (HD&A) has been identified as a high-priority technol-  ogy9 to promote and enable new mission concepts to near-earth asteroids, comets, the Moon, Mars,  and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  The Autonomous Landing Hazard Avoidance Technology (ALHAT)10,11 program was launched  in 2005, followed by the Safe & Precise Landing—Integrated Capabilities Evolution (SPLICE)  program12 in 2018, in order to develop autonomous landing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' These programs have fo-  cused on developing HD&A algorithms that operate on DEMs generated from range measurements  acquired by active sensors such as flash LiDARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' However, these methods are constrained by the  relatively small effective operating range and the increased size, weight, and power (SWaP) require-  ments of LiDARs relative to passive sensors such as monocular cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Indeed, the OSIRIS-REx  Guidance, Navigation, and Control (GNC) flash LiDAR had a maximum operational range of ap-  proximately 1 km,13,14 while preliminary testing of the Hazard Detection LiDAR of the SPLICE  program demonstrated a 5 cm ground sample distance at 500 meters and near-nadir pointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='12  Conversely, the OSIRIS-REx Camera Suite (OCAMS)15 was able to acquire 5 cm GSD images  at almost 4 km, providing higher-resolution measurements earlier in the mission than the active  sensors onboard and allowing for detailed surface characterization during the early phases of the  mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6 Moreover, constructing a DEM from LiDAR scans is non-trivial and requires accurate  range measurements and precise knowledge of the spacecraft’s relative pose to the landing plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Instead, we focus on estimating landing safety directly from a single monocular image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Safety mapping methodologies that leverage deep learning have demonstrated potential to im-  prove hazard detection accuracy and have also been shown to offer competitive runtimes on flight  relevant hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='16 Previous works have leveraged deep semantic segmentation to classify safe  and unsafe landing locations from high-resolution DEMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Moghe and Zanetti4 leverage a deep neu-  ral network architecture for predicting safety maps from a DEM and design a novel loss function  specifically designed to decrease the false safe rate and encourage more precise safe predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Tomita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5 employ a Bayesian SegNet architecture17 for segmentation of input DEMs into safe  and unsafe landing locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' The Bayesian architecture implemented by Tomita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5 enables  uncertainty quantification of the predicted safety map through the predictive entropy of the model,  allowing for more precise predictions through global thresholding with respect to this uncertainty  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We build upon this work and demonstrate its efficacy on monocular imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Methods based on deep learning have also been developed for surface segmentation from monoc-  2  ular imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Pugliatti and Maestrini7 employ a custom U-Net architecture for classification of  surface landmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=', boulders and crater rims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Caroselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='8 apply deep semantic segmen-  tation to boulder detection on synthetic images of a fabricated small body model and post-process  the network prediction to derive a landing safety map based on boulder density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Conversely, we  apply our models to real images and directly predict safety maps that conform to realistic landing  parameters and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  PROPOSED APPROACH  Our novel safety mapping paradigm leverages Bayesian deep learning techniques to develop an  uncertainty-aware semantic segmentation model to predict safety maps from monocular imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  We train and test our model on real in-situ imagery from the OSIRIS-REx mission to asteroid  101955 Bennu with corresponding ground truth safety maps generated using realistic landing pa-  rameters and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Bayesian Deep Learning  Given training input data X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' , xN} with corresponding labels Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' , yN},  Bayesian deep learning employs Bayesian inference to maximize the posterior distribution of the  network parameters θ given the training data X, Y:  p(θ | X, Y) = p(Y | X, θ)p(θ)  p(Y | X)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  (1)  The distribution above can then be used to predict the likelihood of an output y∗ for a new input x∗  via  p(y∗ | x∗, X, Y) = Ep(θ | X,Y)[p(y∗ | x∗, θ)],  (2)  where we may assume a softmax likelihood for p(y∗ | x∗, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' However, computing p(θ | X, Y) is  intractable and must be approximated using variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Gal and Ghahramani18,19 showed that training a deep convolutional neural network (CNN) with  dropout layers is equivalent to approximate variational inference with a variational distribution q(θ)  which imposes a Bernoulli distribution over the model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Specifically, consider a convolu-  tional layer i with ci−1 input channels, ci output channels, and kernel size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Then dropout can be  viewed as imposing a distribution over the layer weights Wi according to  Wi = Mi diag([ϵj]ci  j=1),  ϵj ∼ Bern(pj),  (3)  where ϵj are Bernoulli distributed random variables with parameter pj (which we take to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5),  and Mi ∈ Rci−1×k×k×ci are the variational weight parameters optimized during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Therefore,  Equation (2) may be approximated by  p(y∗ | x∗, X, Y) ≈ Eq(θ)[p(y∗ | x∗, θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  (4)  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' employing dropout at test time permits the use of the predictive entropy approximated  through T stochastic forward passes through the network as a measure of uncertainty:20  H[y | x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Y] ≈ −  d  �  k=1  �  1  T  T  �  t=1  p(y = k | x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' θt)  �  log  �  1  T  T  �  t=1  p(y = k | x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' θt)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  (5)  3  CFF  (1/2)  CFF  Entropy  Mean  Stochastic dropout  samples  Uncertainty  Safety map  32  32  64  128  2  64  128  256  128  1024  CFF  Conv + BN + ReLU  Dropout  Max Pool  Pyramid Pool  (1/4)  (1/8)  (1/16)  Bilinear downsampling  (1/32)  256  256  2x Bilinear Upsample  Cascade Feature Fusion  (1/2)  (1/4)  (1/4)  (1/2)  (1)  Figure 1: Bayesian ICNet architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Multiscale fusion is conducted within the cascade feature  fusion (CFF)21 modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' The ratio in parentheses denotes the relative magnitude of the spatial  dimensions with respect to the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  where θt corresponds to a realization of the network parameters, distributed according to q(θ),  sampled during a forward pass through the network, and the average over the forward passes is  taken to be the final prediction probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' This process is referred to as Monte Carlo (MC)  dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='18,19 For the task of semantic segmentation, assume x is a tuple (X, u) containing an  image tensor X ∈ Rh×w×c and an image coordinate u ∈ R2, and k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' , d} is a pixel-wise  class label for the pixel located at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We will demonstrate that leveraging this uncertainty measure  for our network predictions leads to increased precision and accuracy of safe landing locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Uncertainty-Aware Semantic Segmentation  We leverage an uncertainty-aware semantic segmentation architecture based on the image cas-  cade network (ICNet),21 shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' ICNet is a highly efficient segmentation architecture  that blends coarse prediction maps obtained from down-sampled inputs with high-resolution feature  maps obtained from high throughput networks that operate on the full-resolution image, allowing  for fast inference on high-resolution images while maintaining accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Multiscale feature map  fusion is conducted by the cascade feature fusion (CFF) modules, whereby a reduced-resolution  segmentation map is computed from the two multiscale feature map inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' The multiscale predic-  tions are used to train the network via a weighted softmax cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='21  We implement a Bayesian version of ICNet, which we denote as BICNet, where dropout layers  are added to allow for stochastic sampling with respect to the model parameters using techniques  from Bayesian deep learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=', MC dropout, as described in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Ideally, a  Bayesian NN would feature a dropout layer after every hidden layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='18,19 However,  as observed in previous works,17,20 adding dropout layers after every convolutional layer in more  complex networks is too strong of a regularizer, resulting in underfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Therefore, we follow the  4  00(a) Global shape model  Nightingale  Osprey  Kingfisher  Sandpiper  (b) DTM (left) and reconnaissance imagery (right) for each tag site  Figure 2: OSIRIS-REx TAG site datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' TAG site locations are indicated by the corresponding  color in the global shape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  work of Mukhoti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='20 and Kendall and Cippola17 and only insert dropout layers after the central  encoder and decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' At test time, we perform T = 8 stochastic forward passes and use the  predictive entropy, defined in Equation (5), as a measure of uncertainty, and use the average of this  measure over all training instances as a threshold to mask out high uncertainty regions in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Data Generation  High-fidelity DTMs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=', 5 cm ground sample distance) of the four prospective Touch-And-Go  (TAG) sample sites developed as part of the OSIRIS-REx mission to Asteroid 101955 Bennu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=',  Nightingale, Kingfisher, Osprey, and Sandpiper, were used to generate ground truth safety map  labels for reconnaissance imagery from the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Specifically, we leverage monocular recon-  naissance imagery and the corresponding camera pose labels, relative to a body fixed frame of the  asteroid, provided through the AstroVision dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='22 For each image, DEMs are constructed by  transforming the DTM into a local coordinate system in which the +z-axis points opposite the  vector corresponding to the direction of the gravitational force due to the target body at the point  on the surface closest to the center of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' The gravity due to body was computed using a  global shape model of Bennu23 and assuming a constant-denity polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='24 Safety mapping was  conducted on the DEM and then projected back into the image to produce pixel-wise landing safety  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Example reconnaissance images for each prospective TAG site are provided in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Landing safety was computed from the DEMs using the method developed by the Autonomous  Landing Hazard Avoidance Technology (ALHAT) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='25 The ALHAT method evaluates the  lander contact locations for all pixels and for all orientations to assess the worst-case surface slope  and roughness values with respect to the surface elevation data contained in the ground truth DEMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Specifically, a landing plane is computed for each pixel by assessing the elevation of four evenly  spaced contact points, emulating lander foot pads, on the perimeter of a circle specified by the di-  ameter of the lander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Slope is defined as the largest angle between the landing plane and x-y plane  of the ground truth DEM for all orientations, and the roughness is the largest perpendicular distance  to the terrain above the the landing plane for all orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Any pixel with slope and roughness  exceeding a given threshold is labeled as unsafe, where we chose a threshold of 30◦ for slope and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  cm for roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We specify a lander with a 35 cm diameter, similar to the MASCOT (Mobile As-  teroid surface SCOuT) lander that was deployed during the Hayabusa2 mission to Asteroid 162173  5  (a) Image  20  40  60  80  Slope ( )  (b) Slope  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  Roughness (m)  (c) Roughness  (d) Slope & Roughness Safety Map  (e) Roughness-only Safety Map  Figure 3: Ground truth safety map example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Safe and unsafe regions are drawn in green and red,  respectively, in the safety map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Ryugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='26 An example safety map along with its corresponding monocular image is provided in  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Moreover, we provide the data distributions of our datasets with respect to the ground sample  distance, imaging depth, viewing angle, and visibility ratio in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Ground sample distance  (GSD) measures the average distance on the surface spanned by a single pixel, which is a function  of the distance to the surface and the camera intrinsics, as landing safety becomes increasingly  difficult to observe as the relative size of the lander in the image decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' The imaging depth  measures the average distance to the surface when the image was taken, and provides context for  the GSD values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Specifically, the MapCam of the OSIRIS-REx Camera Suite (OCAMS),15 with  a focal length of ∼125 mm, can provide 5 cm GSD measuresments of the surface at distances  of approximately 1 km, while the PolyCam, with a focal length of ∼620 mm, provides the same  resolution at distances of almost 4 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Viewing angle measures the angle between the −z-axis of  the ground truth DEM and the camera boresight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Finally, the visibility ratio is the ratio of visible  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=', not occluded by shadows) pixels to total pixels in the image and provides a measure of the  illumination conditions in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  6  +PolyCam  MapCam  SamCam  Figure 4: Data distributions with respect to imaging depth, GSD, viewing angle, and visibility  ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Our dataset features a total of 770 images annotated with per-pixel safety labels: 133 of  Kingfisher, 342 of Nightingale, 162 of Osprey, and 91 of Sandpiper, and 42 from the TAG sample  collection event at Nightingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  RESULTS  In this section, we first present our suite of metrics used to evaluate the performance of our  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We then validate our approach on two different experiments using real images from the  OSIRIS-REx mission to Asteroid 101955 Bennu, including images captured during the actual TAG  sample collection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Metric Definitions  We measure the quality of the predicted per-pixel safety map labels of our model with respect to  precision, sensitivity, accuracy, and mean intersection over union (mIoU):  precision =  true safe  true safe + false safe,  (6)  7  Table 1: Overall performance for the Sandpiper landing site experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' The values in paren-  theses are the metrics with shadowed pixels ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' All reported values are percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  METHOD  PRECISION  SENSITIVITY  ACCURACY  MIOU  SLOPE & ROUGHNESS  WITHOUT UNCERTAINTY  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='66 (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='91)  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='21 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='05)  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='53 (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='41)  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='05 (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='27)  WITH UNCERTAINTY  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='98 (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='67)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='09 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='86)  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='29 (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='01)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='76 (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='93)  ROUGHNESS ONLY  WITHOUT UNCERTAINTY  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='24 (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='92)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='61 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='66)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='53 (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='41)  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='87 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='31)  WITH UNCERTAINTY  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='71 (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='32)  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='78 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='63)  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='77 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='93)  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='11 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='98)  sensitivity =  true safe  true safe + false unsafe,  (7)  accuracy = true safe + true unsafe  valid pixels  ,  (8)  mIoU = 1  2  �  true safe  valid pixels − true unsafe +  true unsafe  valid pixels − true safe  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  (9)  True safe (false safe) includes pixels predicted to be safe by our models that are safe (unsafe) in  the ground truth labels, and true unsafe (false unsafe) includes pixels predicted to be unsafe that  are unsafe (safe) in the ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Note that false unsafe includes safe pixels that are  ignored and not labeled safe due to high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' For our application, we can interpret precision  as the reliability of the pixels predicted to be safe, and sensitivity as detection rate of true safe  sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Accuracy and mIoU are the metrics evaluated for the valid pixels, which are  the pixels with smaller uncertainty than the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' In other words, valid pixels correspond  to the predictions that the network is most “certain” about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' For the results without uncertainty  thresholding, accuracy and mIoU are evaluated for all the pixels with valid safety labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' In the  following analysis, we refer to the ratio of pixels that fall above our uncertainty threshold, and  consequently marked as unsafe, to valid pixels as the screening rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Experiment 1: Prospective Landing Site Sandpiper  We train our model on images from three of the prospective landing sites from the OSIRIS-  Rex mission, namely, Nightingale, Osprey, and Kingfisher, and test our model on images of the  remaining sample site, Sandpiper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' This emulates a scenario in which data from previously mapped  landing sites could be used to train a network to predict landing safety in a new, unexplored region of  the target body without requiring the construction of high-fidelity DEMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' These results are detailed  in Table 1, and qualitative examples are provided in Figure 5, where we consider performance with  respect to identifying both slope and roughness hazards and roughness-only hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  The results illustrate that our models are able to predict safety maps from just a single monocular  image of the propspective landing site, which is completely unseen during training, with accuracy  over 69% for the slope and roughness hazard detection case, and over 63% for the roughness-only  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  Uncertainty  (a) Slope & roughness  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  Uncertainty  Test Image  Without Uncertainty  With Uncertainty  Uncertainty  (b) Roughness-only  Figure 5: Qualitative monocular safety mapping results for the Sandpiper experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Green,  yellow, blue, and red labels represent true safe, true unsafe, false unsafe, and false safe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  9  hazard detection case, even without uncertainty thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Moreover, our Bayesian ICNet archi-  tecture enables uncertainty thresholding in order to further boost performance by ignoring regions in  which the models’ prediction has high entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' With the uncertainty threshold, accuracy increases  to 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='29% and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='77% for the slope and roughness and roughness-only cases, respectively, at the  cost of decreased sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Importantly, we are able to achieve 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='98% and 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='71% precision for  the slope and roughness and roughness-only cases, respectively, after uncertainty thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We  also have a slight increase in all the metrics by ignoring shadowed pixels, which are reported in  parentheses in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Comparing the two different hazard detection tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=', slope and roughness hazards and roughness-  only hazards, the roughness-only case has lower values of sensitivity, accuracy, and mIoU, but  higher values of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' This suggests that the roughness-only case is a harder task in terms of  precisely labeling safe and unsafe pixels on average, resulting in lower accuracy and mIoU, but is  an easier task in terms of identifying only safe pixels, thus resulting in higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' This is  partially due to the higher incidence of safe pixels for the roughness-only case as compared to the  slope and roughness case, as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Additionally, we analyzed the per-image metrics with respect to GSD, viewing angle, and visibil-  ity ratio for the slope and roughness case, shown in Figure 6, and the roughness-only case, shown in  Figure 7, in order to identify possible causes of uncertainty in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' As a general trend, we  can observe that a higher uncertainty results in lower performance metrics of precision, sensitivity,  accuracy, and mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Intuitively, low visibility is a common factor that results in higher uncertainty  in our model for both the slope and roughness case and the roughness-only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Our models as-  sign a higher uncertainty to images with larger GSD for the slope and roughness case, and larger  viewing angle for the roughness-only case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Note that increased uncertainty for images at higher  GSDs may also be due to these instances being less represented in the training data as shown in  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' In either case, we demonstrate that the uncertainty threshold serves as a powerful tool for  detecting and accounting for difficult or out-of-distribution input conditions, allowing our models  to predict precise and accurate safety maps across multiple GSDs, viewing angles, and illumination  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Experiment 2: OSIRIS-REx TAG Sequence  For the second experiment, we trained two models using different combinations of images from  the OSIRIS-REx mission: one model, denoted by BICNet-NKO, is trained on Nightingale, King-  fisher, and Osprey, and the other model, denoted by BICNet-KOS, is trained on Kingfisher, Osprey,  and Sandpiper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' These models were trained for the slope and roughness case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We tested the two  models on images captured during the TAG sample collection event at the Nightingale sample site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  We present both models to illustrate the effect of the different training data distributions on the test  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' A subset of the 42 image sequence is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Note that the 42 test images are  not included in the training set for the Nightingale images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Table 2 and Figure 10 show quantitative results and qualitative examples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Compar-  ing BICNet-NKO and BICNet-KOS in Table 2, we can see sensitivity of BICNet-KOS is signifi-  cantly lower than that of BICNet-NKO after uncertainty thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Indeed, BICNet-KOS assigns  a high uncertainty to almost all regions of the input images and are thus overwritten as unsafe after  uncertainty thresholding, as shown in Figure 10, which is not the case for BICNet-NKO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' These  differences are most likely explained by the difference between the training and testing data distri-  butions for BICNet-KOS, as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Specifially, the TAG images have less overlap  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='8  Precision  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='06  GSD (m/pixel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='4  Sensitivity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='8  Accuracy  uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='8  IoU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='06  GSD (m/pixel)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='60  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='9  Screen Rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='8  Precision  10  20  30  40  50  Viewing Angle ( )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='4  Sensitivity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='8  Accuracy  uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='8  IoU  10  20  30  40  50  Viewing Angle ( )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='60  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='8  Precision  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='4  Sensitivity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='8  Accuracy  uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='8  IoU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='60  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='9  Screen Rate  Figure 6: Per-image metrics for slope & roughness safety on the Sandpiper experiment with  respect to GSD, viewing angle, and visibility ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='60  Uncertainty  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='8  Screen Rate  Figure 7: Per-image metrics for roughness-only safety on the Sandpiper experiment with re-  spect to GSD, viewing angle, and visibility ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  12  2020-10-20T21:30:48  2020-10-20T21:31:48  2020-10-20T21:32:48  2020-10-20T21:33:48  2020-10-20T21:34:48  2020-10-20T21:35:48  2020-10-20T21:36:48  2020-10-20T21:37:48  2020-10-20T21:38:48  2020-10-20T21:39:48  2020-10-20T21:40:48  2020-10-20T21:41:48  Figure 8: Frames from the OSIRIS-Rex TAG sequence captured by the SamCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  with the images of the prospective landing sites except for Nightingale with respect to viewing angle  and visibility ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Therefore, the exclusion of Nightingale from training data increases the predic-  tive uncertainty for TAG images at test time for BICNet-KOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Note that the Nightingale images  (excluding the TAG images) were used for validation during training for the BICNet-KOS model in  order to rule out overfitting as a cause for the decreased prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  This effect of training data distributions on the uncertainty level, and the accompanying predictive  performance, is consistent with the per-image metrics with respect to GSD, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  Indeed, for BICNet-NKO, as the GSD of testing images gets closer to the peak of the training data,  the predictive performance increases and uncertainty decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Lower precision for the higher  GSD images is also partly due to the very low incidence of safe regions for these instances (see  Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Conversely, for BICNet-KOS, all test images have a high screening rate and a low  sensitivity due to the high uncertainty, purportedly due to the out-of-distribution training data with  respect to the viewing angle and visibility ratio and the relatively small size of the training set (386  images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We do not provide the per-image metrics with respect to the viewing angle and visibility  ratio, as these values remain relatively constant over the entire TAG sequence at ∼7◦ and ∼71%,  respectively, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' These results illustrate the effect of training data distributions  and the ability of the uncertainty measure to identify out-of-distribution data for uncertainty-aware  segmentation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We postulate that training our model with a more comprehensive set of  images will decrease prediction uncertainty and increase the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  CONCLUSION  In this paper we presented a novel landing hazard detection approach for small body missions that  predicts safety maps directly from monocular imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We implemented an efficient, uncertainty-  aware segmentation network that demonstrated hazard detection performance at over 80% accuracy  and over 85% precision on real images of unseen landing sites captured during the OSIRIS-REx  mission to Asteroid 101955 Bennu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' We believe that monocular safety mapping is a promising  technology for reducing reliance on human-in-the-loop procedures used in current safety mapping  methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' Future work will involve developing a more comprehensive distribution of training  data and identifying and rectifying causes of uncertainty to increase the reliability of the proposed  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content=' Green,  yellow, blue, and red labels represent true safe, true unsafe, false unsafe, and false safe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content=' BICNet-  NKO is our Bayesian ICNet model trained on Nightingale, Kingfisher, and Osprey, and BICNet-  KOS is trained on Kingfisher, Osprey, and Sandpiper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content=' All reported values are percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content=' Our code, data, and trained models will be made available to the public at https:  //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content='  ACKNOWLEDGMENTS  This work supported by a NASA Space Technology Graduate Research Opportunity and the  NASA Early Career Faculty Program (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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+page_content=' The authors would like to  thank Kenneth Getzandanner and Michael Shoemaker from NASA Goddard Space Flight Center  for several helpful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
+page_content='  REFERENCES  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFQT4oBgHgl3EQfJDUU/content/2301.13254v1.pdf'}
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diff --git a/9tFLT4oBgHgl3EQfui8A/content/tmp_files/2301.12155v1.pdf.txt b/9tFLT4oBgHgl3EQfui8A/content/tmp_files/2301.12155v1.pdf.txt
new file mode 100644
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@@ -0,0 +1,1637 @@
+epl draft
+Towards a liquid-state theory for active matter
+Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E. Cates1 and ´Etienne Fodor2
+1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
+2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
+3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK
+PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics
+Abstract – In equilibrium, the collective behaviour of particles interacting via steep, short-ranged
+potentials is well captured by the virial expansion of the free energy at low density. Here, we extend
+this approach beyond equilibrium to the case of active matter with self-propelled particles. Given
+that active systems do not admit any free-energy description in general, our aim is to build the
+dynamics of the coarse-grained density from first principles without any equilibrium assumption.
+Starting from microscopic equations of motion, we obtain the hierarchy of density correlations,
+which we close with an ansatz for the two-point density valid in the dilute regime at small activity.
+This closure yields the nonlinear dynamics of the one-point density, with hydrodynamic coefficients
+depending explicitly on microscopic interactions, by analogy with the equilibrium virial expan-
+sion. This dynamics admits a spinodal instability for purely repulsive interactions, a signature
+of motility-induced phase separation. Therefore, although our approach should be restricted to
+dilute, weakly-active systems a priori, it actually captures the features of a broader class of active
+matter.
+Active matter is a wide category of systems out of equi-
+librium, where a net flow of energy takes place at the local,
+individual level [1–3]. Examples are swimming bacteria [4]
+and active emulsions [5], amongst others. The realm of
+motile active matter includes interacting, many-particle
+systems, where particles move persistently. Different the-
+oretical approaches have been proposed to describe such
+systems. They include (i) particle-based models, such as
+run-and-tumble particles (RTPs) [6], active Brownian par-
+ticles (ABPs) [7], and active Ornstein-Uhlenbeck particles
+(AOUPs) [8,9], (ii) microscopic field theories [10,11], and
+(iii) coarse-grained field theories [12–14].
+Self-propelled particles tend to slow down at high densi-
+ties, due to either biochemical reasons or steric repulsions.
+In contrast with passively diffusing particles, this slow-
+down in turn increases the local density, creating a pos-
+itive feedback loop that can result in a phase separation
+between a dense and a dilute phase, known as motility-
+induced phase separation (MIPS) [15].
+To predict an-
+alytically the emergence of MIPS, previous works have
+relied on a local mean-field approximation to replace mi-
+croscopic interaction with density-dependent motility [16],
+and also on an adiabatic elimination of orientational de-
+gree of freedom [15]. Other studies have developed some
+aspects of a liquid-state theory for active matter with pair-
+wise interactions. This is done, for instance, by expressing
+thermodynamic observables, such as pressure [17–19] and
+dissipation [20,21], in terms of correlations between hydro-
+dynamic fields, typically density and polarization. Also,
+exact results for density correlations have been obtained
+at infinite dimensions [22,23].
+The success of equilibrium thermodynamics largely
+stems from the ability to detect phase transitions by an-
+alyzing the free-energy.
+In the dilute regime, the virial
+expansion provides an approximate expression of the free-
+energy that averages the effect of steep, short-ranged pair-
+wise interactions [24].
+For passive Brownian particles
+(PBPs) interacting with an isotropic pairwise potential
+�
+i,j<i Ψ(|xi − xj|) at temperature T, the free energy per
+unit volume is given in terms of the uniform density ρ as
+fvirial(ρ)/T = ρ log ρ + B(T)ρ2 + O(ρ3),
+B = 1
+2
+�
+dr
+�
+1 − e−Ψ(r)/T �
+.
+(1)
+Here B is commonly referred to as the second virial coef-
+ficient. The major success of the virial expansion lies in
+predicting the onset of liquid-gas phase separation in parti-
+cle systems with a combination of strongly repulsive short-
+ranged forces and weakly attractive long-range forces, such
+as the Lennard-Jones potential. As temperature increases,
+p-1
+arXiv:2301.12155v1  [cond-mat.soft]  28 Jan 2023
+
+Y. I. Li et al.
+the free energy, as a function of density ρ, goes from convex
+(B > 0) to concave (B < 0): It yields a phase transition
+from homogeneous to non-homogeneous density profiles,
+with a separation between dilute and dense phases. Im-
+portantly, the virial expansion holds for a generic micro-
+scopic potential (excluding those of such long range that
+the excess free energy is not analytic in density).
+In classical thermodynamics, the virial expansion is of-
+ten derived from the equilibrium partition function [24].
+Interestingly, it is also possible to arrive at approx-
+imate expressions of free energy by truncating the
+hierarchy of density correlations,
+known as the Bo-
+goliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierar-
+chy [25]. For instance, using an ansatz for the two-particle
+density, informed by the steady-state solution of the two-
+body problem [26], the dynamics of the one-body density
+follows a gradient flow with respect to fvirial, as shown
+in the SM. This type of derivation can potentially be ex-
+tended to a large class of nonequilibrum systems, where
+the dynamics does not derive from any free energy a pri-
+ori. Such an extension requires proposing an ansatz for
+the two-particle density, which should be appropriate to
+the specific nonequilibrium system at hand.
+Interestingly, the two-body probability distribution can
+be approached perturbatively for AOUPs, where the per-
+sistence time is the small parameter in natural units [8,9].
+To first order, this solution already predicts that repul-
+sive interaction yields effective attraction. This suggests
+that the onset of MIPS can likewise already be captured
+by approximating the dynamics of density at this order.
+Therefore, inspired by the equilibrium virial expansion, it
+is tempting to explore whether the density dynamics al-
+ready contains any spinodal instability for dilute active
+systems at weak persistence. Note that, although previ-
+ous work refer to “active virial” as an expansion of pres-
+sure [27, 28], here we are instead interested in deriving
+dynamical equations of motion for the density. Indeed, at
+variance with equilibrium, phase transitions in active sys-
+tems are usually not thermodynamically controlled by any
+equation of state, but they can still be directly detected
+as instability in the dynamics.
+In this Letter, we start with the particle-based descrip-
+tion of AOUPs, and consider their steady-state probability
+distribution. Our roadmap is essentially a “small density-
+small persistence” bottom-up derivation of the density dy-
+namics. To this end, we first introduce the corresponding
+hierarchy of density correlations. Inspired by equilibrium
+thermodynamics [25, 26], we then close the hierarchy to
+arrive at a nonlinear dynamics for the one-point density.
+It allows us to identify the MIPS spinodal instability for
+any microscopic repulsive interactions. An advantage of
+our approach is that does not impose an equilibrium map-
+ping a priori (see [8,9] for a discussion of closures to the
+AOUP equations that do so), instead retaining the non-
+equilibrium structure of the microscopic theory. A disad-
+vantage is that extensions to higher order in density would
+involved increasingly complicated calculations of higher
+virial coefficients, which we do not attempt here.
+Interacting AOUPs:
+Joint distribution func-
+tion – We consider a system of N AOUPs at positions
+xi interacting with pairwise potential U.
+The particles
+are subject to stochastic self-propulsion velocities vi with
+persistence time τ and diffusivity D [8,9]:
+˙xi = vi − ∂xiU,
+U =
+N
+�
+i=1
+N
+�
+j<i
+Ψ(rij),
+τ ˙vi = −vi +
+√
+2DΛi,
+(2)
+where rij = |xi − xj|, and we have set the mobility to
+unity. Here Λi is a Gaussian white noise, with correlation
+⟨(Λi)α(t)(Λj)β(0)⟩ = δijδαβδ(t), where latin and Greek
+letters respectively denote particle index and spatial co-
+ordinates. It follows that vi is a colored Gaussian noise,
+with correlation ⟨(vi)α(t)(vj)β(0)⟩ = δijδαβ(D/τ)e−|t|/τ.
+In the limit of vanishing persistence (τ → 0), the corre-
+lation of vi becomes white, so that the system reduces to
+a set of overdamped PBPs. Note that, at finite τ, if the
+potential term −∂xiU was in the rhs of the vi-equation
+(instead of the xi-equation), the system would represent
+underdamped PBPs in equilibrium. For this reason, over-
+damped AOUPs and underdamped PBPs coincide in the
+noninteracting limit.
+Following
+standard
+procedures
+[29],
+the
+Fokker-
+Planck equation of the joint probability distribution
+pN(x1, v1, x1, v2, . . . , xN, vN, t) associated with eq. (2) is
+∂tpN =
+N
+�
+i=1
+�
+Lf,ipN + ∂xi · (pN ∂xiU)
+�
+,
+Lf,i = D
+τ 2 ∂2
+vi +
+�1
+τ ∂vi − ∂xi
+�
+· vi,
+(3)
+where Lf,i is the non-interacting Fokker-Planck operator
+governing the free-particle motion of the i-th particle. Al-
+though it cannot be solved exactly, its steady state can be
+computed to lowest orders in the persistence time τ in a
+similar manner to [8, 9]. Such a perturbative calculation
+can be performed in arbitrary dimension, though we now
+restrict ourselves to 1D for simplicity, as a proof of prin-
+ciple. After scaling units as v → v
+�
+τ/D , x → x/
+√
+τD,
+t → t/τ, and Ψ → Ψ/D, we find the many-body steady
+state probability [see SM for details],
+pN ∝ e−U−
+v2
+1+v2
+2
+2
+�
+1 + √τ
+N
+�
+i=1
+vi∂xiU
++ τ
+N
+�
+i=1
+�v2
+i
+2
+�
+(∂xiU)2 − ∂2
+xiU
+�
+− (∂xiU)2 + 3
+2∂2
+xiU
+�
++ o(τ)
+�
+.
+(4)
+Note that the steady state probability here is given in
+(x, v) space, whereas [8] gives the corresponding probabil-
+ity in (x, ˙x) space.
+p-2
+
+Towards a liquid-state theory for active matter
+Nonequilibrium density correlations: Hierarchy
+of equations – Although the joint distribution function
+pN in eq. (4) contains all information about the steady
+state density, it is generally difficult to predict the emer-
+gence of phase transitions from the perspective of the full
+phase space. Instead, our aim is to obtain a reduced de-
+scription of the systel in terms of density correlations. The
+n-particle density pn is found by marginalising the joint
+distribution function pN as
+pn(x1, v1, . . . , xn, vn, t)
+= PN,n
+�
+pN(x1, v1, . . . , xN, vN, t)
+N
+�
+j=n+1
+dxjdvj,
+(5)
+where PN,n = N!/(N − n)! is the permutation coefficient.
+Thereafter, for simplicity, we use the shorthand notation
+pn = pn(x1, v1, . . . , xn, vn, t) for arbitrary n, where the
+dependence on positions, self-propulsion velocities, and
+time will be omitted. Integrating eq. (3) over the positions
+and velocities of all particles but one yields an equation for
+the one-particle density that depends on the two-particle
+density:
+∂tp1 = Lf,1p1 + F1[p2],
+F1[p2] = ∂x1 ·
+�
+p2 ∂x1Ψ(r12) dx2dv2.
+(6)
+The first term Lf,1p1 corresponds to the free single-particle
+dynamics, whereas the second term F1[p2] represents the
+effects of pairwise interaction between one particle and all
+the others.
+Following the same marginalisation procedure for the
+n-particle density pn, we obtain a hierarchy of equations,
+each depending on the density pn+1, analoguous to the
+BBGKY hierarchy of PBPs [25]. The resulting equation
+for the two-particle density reads
+∂tp2 = (Lf,1 + Lf,2 + Lint)p2 + F2[p3],
+Lint =
+�
+i=1,2
+∂xi ·
+�
+∂xiΨ(r12)
+�
+,
+F2[p3] =
+�
+i=1,2
+∂xi ·
+�
+p3 ∂xiΨ (ri3) dx3dv3.
+(7)
+Here Lint governs the pairwise interactions between any
+two particles, and F2[p3] represents the interactions be-
+tween either of the first two particles and all the other
+particles. As we will show now, this is a higher-order term
+in the average number density ρ = N/V compared to the
+rest, and hence can be neglected in the dilute limit.
+To determine the scaling of pn with respect to ρ, one
+can start with their normalisation properties [24]
+�
+pn
+n
+�
+i=1
+dxidvi = PN,n,
+(8)
+from which follows pn ∼ ρn for small n. Physically, this
+is indeed expected as p1(x1, v1) is the probability density
+of finding n particles at a given set of coordinates. With
+this scaling in mind, we estimate the scaling of each term
+in eq. (7) as
+(∂t − Lf,1 − Lf,2 − Lint
+����
+∼1/τint
+) p2
+����
+∼ρ2
+=
+�
+i=1,2
+∂xi ·
+�
+p3
+����
+∼ρ3
+∂xiΨ(ri3)
+�
+��
+�
+∼1/τint
+dx3
+����
+∼(r0)d
+dv3,
+(9)
+where τint is the typical timescale of interaction, r0 is the
+lengthscale of the potential, and d denotes the sptial di-
+mension. One can see that the rhs of eq. (9) is small if
+rd
+0ρ ≪ 1. Since rd
+0 is effectively the volume of a particle,
+this condition is satisfied if the volume fraction of parti-
+cles in the system is low. Therefore, as expected from the
+analogy with the BBKY hierarchy [25], the interaction of
+two particles with respect to others in the bath can be
+neglected at small volume fraction.
+Closure of hierarchy:
+Insight from quasistatic
+approximation – We now propose a closure of the hierar-
+chy of equation for density correlations. At small volume
+fraction, neglecting the rhs in eq. (9) yields
+∂tp2 = (Lf,1 + Lf,2 + Lint) p2.
+(10)
+The dynamics in eq. (10) is equivalent to the Fokker-
+Planck equation of two AOUPs interacting through the
+pairwise potential Ψ(r12).
+Recall from eq. (4) the N-
+particle stationary probability, calculated order by order
+in the persistence time τ. The two-particle probability is
+hence,
+p2,ss(r12, v1, v2) ∝ e−Ψ(r12)−
+v2
+1+v2
+2
+2
+g(r12, v1 − v2),
+(11)
+where
+g(r, w) = 1 + √τwΨ′ + (τw2/2)((Ψ′)2 − Ψ′′)
++ τ(3Ψ′′ − 2(Ψ′)2) + O(τ 3/2),
+(12)
+and Ψ′ = dΨ/dr [see SM]. In the quasistatic limit, where
+the interaction timescale is much shorter than the persis-
+tence (τint ≪ τ), we assume that every pair of particles
+reaches steady state. Inspired by previous works [26, 30],
+this assumption motivates the following ansatz to close the
+hierarchy of density correlations:
+p2(x1, v1, x2, v2, t) = p1(x1, v1, t) p1(x2, v2, t)
+× e−Ψ(r12)g(r12, v1 − v2).
+(13)
+There is evidence that the above ansatz works well for
+strong short-ranged interactions in the equilibrium case
+[see SM]. We assume it would also work in the non-
+equilibrium case. Indeed, when the interparticle distance
+is larger than the interaction range (r12 > r0), the two-
+point function p2 reduces to the product of one-point func-
+tions p1, as expected from a mean-field approach for non-
+interacting systems. The second line of eq. (13) accounts
+p-3
+
+Y. I. Li et al.
+for corrections due to interactions. For repulsive poten-
+tial, the factor e−Ψ captures the reduction of the two-
+point density caused by the interaction, as expected from
+equilibrium, whereas g(r12, v1 − v2) encapsulates nonequi-
+librium effects.
+Substituting the ansatz from eq. (13) into eq. (6), we
+arrive at the following dynamics for the one-particle den-
+sity:
+(∂t−Lf) p1(x, v, t)
+= τ∂x
+�
+p1(x, v, t)
+�
+Lvp1(x, v − w, t)dw
+�
+.
+(14)
+Here, Lf is the free Fokker-Planck operator in scaled units,
+and the operator Lv effectively takes into account interac-
+tions:
+Lf = ∂2
+v +
+�
+∂v − √τ∂x
+�
+v,
+Lv =
+�
+Ψ′(r)e−Ψ(r)g(r, w)e−r∂xdr,
+(15)
+where we have introduced the translation operator e−r∂x,
+which shifts the position of the function acted upon as
+e−r∂xp1(x, v) = p1(x − r, v). Hence, Lv corresponds to an
+infinite series of the gradient ∂x, with series coefficients set
+by the microscopic potential. Equation (14) is the central
+result of this Letter: It provides the dynamics of the one-
+particle density in a closed form for small average density.
+Importantly, this closed form depends explicitly on the
+details of the pairwise potential Ψ.
+To obtain a more explicit expression of Lv, we next
+substitute our perturbative result for g [eq. (12)] into the
+definition of Lv [eq. (15)], yielding
+Lv = 2wLa + 2Lb∂x + w2Lc∂x,
+(16)
+where, after integration by parts, we get
+La = √τ
+� ∞
+0
+dr(Ψ′)2 e−Ψe−r∂x,
+Lb = −
+� ∞
+0
+drf0(r)e−r∂x,
+Lc = −
+� ∞
+0
+dr f1(r)e−r∂x,
+(17)
+and
+f0(s) =
+� ∞
+s
+dr Ψ′e−Ψ �
+1 − τ
+�
+2(Ψ′)2 − 3Ψ′′��
+,
+f1(s) = τ
+� ∞
+s
+dr Ψ′ �
+(Ψ′)2 − Ψ′′�
+e−Ψ.
+(18)
+The integrands featuring in the definition of the operators
+{La, Lb, Lc} [eq. (17)] are even with respect to r, provided
+that Ψ(r) = Ψ(−r). It follows that these operators can be
+expressed as a series of ∂2
+x.
+The functions f0 and f1 are analoguous to the Mayer
+function which appears in the equilibrium virial expan-
+sion [25]. Indeed, in the equilibrium limit (τ → 0), the
+only surviving term in Lv stems from the leading order in
+f0, in agreement with the derivation for equilibrium over-
+damped PBPs [see SM]. This term is responsible for the
+B coefficient in eq. (1) when that result is derived dynam-
+ically for equilibrium systems. In that respect, eq. (16)
+can be regarded as a direct generalization of the equilib-
+rium virial expansion to the AOUP setting. Importantly,
+eq. (16) shows that activity produces additional velocity-
+dependent terms that are absent in equilibrium. In what
+follows, we analyze in detail the corresponding dynam-
+ics, in search for the onset of instability as a signature of
+MIPS.
+Linear stability analysis: Eigenvalue problem –
+The stationary solution of eq. (14) is given by p(0)
+1 (v) =
+(ρ/
+√
+2π)e−v2/2. This solution corresponds to a uniform
+density for the position and a Gaussian distribution for
+the self-propulsion.
+In addition, as we will see shortly,
+this is also the ground state of the free operator. In the
+following, we expand perturbatively around p(0)
+1
+to find its
+instability regions in parameter space: If the uniform state
+is not stable, then we argue that the system undergoes
+spinodal decomposition via a MIPS mechanism.
+We consider p1(x, v, t) = p(0)
+1 (v) + ε(x, v, t), with ε a
+small perturbation about the ground state p(0)
+1 . Expand-
+ing eq. (14) to linear order in ε, we get
+(∂t− ¯Lf) ε(x, v, t) = τp(0)
+1 (v)
+�
+Lv∂xε(x, v−w, t)dw, (19)
+where Lv is defined in eq. (16), and
+¯Lf = ∂2
+v + (∂v − α∂x)v,
+α = √τ − 2ρτ 3/2
+� ∞
+0
+(Ψ′)2e−Ψdr.
+(20)
+To analyze the time evolution of the perturbation given in
+eq. (20), the difficulty lies in treating the effect of the op-
+erator Lv, which contains information about microscopic
+interactions.
+Then, it is convenient to expand ε in the
+eigenfunctions of the bare theory, namely in the absence
+of Lv. Interestingly, as stated previously, the bare theory
+maps into underdamped passive Brownian motion, and
+one can readily find the solution in the literature [29].
+The corresponding eigenfunctions, which we here call the
+Fourier-Hermite basis, are
+ψnk(x, v) = e−ik(x+αv)Un(v − 2iαk),
+(21)
+where Un is the Hermite function
+Un(z) =
+1
+√
+2π Hn(z) e−z2/2 = (−1)n
+√
+2π ∂n
+z e−z2/2,
+(22)
+with the property that (∂2
+z + ∂zz)Un(z) = −nUn(z). In-
+deed, acting on the Fourier-Hermite basis with the modi-
+fied free operator ¯Lf yields
+¯Lfψnk = −λknψnk,
+λkn = (αk)2 + n.
+(23)
+p-4
+
+Towards a liquid-state theory for active matter
+As the operator ¯Lf is not Hermitian, the conjugate basis is
+not simply the complex conjugate. Instead, we introduce
+the following conjugate basis
+¯ψnk(x, v) = eik(x+αv) ¯Un(v − 2iαk),
+¯Un(z) = 1
+n!Hn(z),
+(24)
+yielding
+�
+dxdv ¯ψmk′(x, v)ψnk(x, v) = 2πδ(k − k′)δmn,
+(25)
+so that the orthogonality relation between the basis ψnk
+and its conjugate ¯ψnk indeed holds as expected.
+Having obtained the eigenvectors and eigenvalues of the
+bare theory, we decompose the perturbation ε in eigen-
+functions of ¯Lf as
+ε(x, v, t) =
+�
+n
+�
+εkn(t)ψnk(x, v)dk.
+(26)
+Multiplying eq. (19) with the conjugate basis ¯ψkn and inte-
+grating over {x, v} yields the dynamics of the perturbation
+ε, in the Fourier-Hermite basis, as
+˙εkm = −λkmεkm+
+�
+n
+�
+M (1)
+kmn+M (2)
+kmn+M (3)
+kmn
+�
+εkn, (27)
+where
+M (1)
+kmn = 2τρLa,k
+(−iαk)n+m
+αm!
+e(αk)2(m − n),
+M (2)
+kmn = −2τρLb,kk2 (−iαk)n+m
+m!
+e(αk)2,
+M (3)
+kmn = τρLc,k
+(−iαk)n+m
+α2m!
+e(αk)2
+×
+�
+m(m − 1) + n(n − 1) − 2mn − 2(αk)2�
+.
+(28)
+Here, La,k =
+�
+eikxLadk, with similar definitions for Lb,k
+and Lc,k, where the operators {La, Lb, Lc} are defined in
+eq. (17).
+If the matrix Mknm = −λkmδmn + M (1)
+kmn +
+M (2)
+kmn +M (3)
+kmn, which controls the growth rate of the per-
+turbation ε, has no positive eigenvalue, the uniform so-
+lution is stable; otherwise the system undergoes spinodal
+decomposition. Hence we only need the largest eigenvalue
+of M. As shown in the SM, the matrix M can be diagnon-
+alized exactly. This calculation actually only amounts to
+finding the maximum eigenvalue of a 3 × 3 matrix, which
+can be done straightforwardly, while also perturbatively
+keeping track of orders of τ.
+Spinodal instability – We now illustrate how our ap-
+proach, valid for an arbitrary pairwise potential, can be
+deployed to detect the onset of instability. We consider the
+following short-ranged, weakly repusive interaction poten-
+tial:
+Ψ(r) = ν exp
+�
+−
+1
+1 − (r/r0)2
+�
+,
+(29)
+0.00
+0.25
+0.50
+0.75
+1.00
+k
+−0.10
+−0.05
+0.00
+0.05
+0.10
+λmax
+τ=0.20
+0.00
+0.25
+0.50
+0.75
+1.00
+k
+τ=0.50
+Fig. 1: The value of the largest eigenvalue λmax as a function of
+the wavevector k for ν = 10, r0 = 2, varying τ and φ (blue solid
+line). The orange dotted line represents the short length-scale
+cut-off as we are only concerned with length-scales beyond the
+particle size r0.
+characterised by its strength ν and range r0, whose deriva-
+tives are continuous at any order for r within [0, r0]. The
+maximum eigenvalue of the corresponding growth rate ma-
+trix M, expanded analytically to first order in the persis-
+tence time τ, is plotted against wavevector k in Fig. 1
+for various values of τ and volume fraction φ = 2r0N/V ,
+where 2r0 is taken as the size of the particle in 1D and
+V = L is the system volume. As expected [15], the sys-
+tem exhibits a spinodal instability at high τ and packing
+fraction φ, for which the linear perturbation ε is unstable,
+namely limk→0 λmax(k) = 0+.
+Fig. 2 shows the stabil-
+ity diagram based on the sign of the longest wavelength
+perturbation, λmax(2π/L), as a function of τ and φ, show-
+ing the transition between a uniform stable and a phase-
+separated state. (Note that we could equally have chosen
+the fastest growing mode to locate the spinodal.)
+It is notable that our theory of interacting AOUPs
+shows a spinodal instability. As in equilibrium systems
+with attractions, it does this even at lowest order in the
+virial-type expansion in powers of density. Just as holds
+there, finding the instability requires using a low-density
+theory at finite densities. However the purpose of this ap-
+proach is not to gain quantitative predictions about the ex-
+act location of the spinodal curve, but rather to establish
+that the macroscopic conditions for phase separation can
+be satisfied, by considering microscopic interaction laws
+and dynamics. Notably, unlike in equilibrium, the spin-
+odal instability happens here for purely repulsive interac-
+tions – a key feature of MIPS [15]. Our microscopic calcu-
+lation complements previous viewpoints based on effective
+quasi-equilibrium attractions [31], and/or collisional slow-
+ing down [16,32].
+Nonetheless, there are several caveats and limitations to
+our method. Firstly, the chosen potential is not strictly
+hard-core, but depends on the strength parameter ν. In
+fact it is not possible to implement a true hard-core po-
+tential due to the nature of the small τ expansion for the
+two-particle density, as there are terms directly propor-
+tional to Ψ or its derivatives.
+Unsurprisingly then, the
+p-5
+
+Y. I. Li et al.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+φ
+10−1
+100
+τ
+Phase separation
+Uniform
+0
+λmax
+Fig. 2: Stability diagram in τ − φ space showing λmax(2π/L)
+for φ = 0.6, L = 200, ν = 10, r0 = 2. The white region in the
+top-right corner corresponds to when α < 0, where the method
+breaks down, as it is no longer in the small τ limit. The solid
+black line is the spinodal instability.
+results presented here do depend on ν. While overly large
+ν will break the small-τ expansion, we have checked that a
+range of moderate values produce stability diagrams qual-
+itatively similar to Fig. 2. Secondly, for some parameters,
+we found that the largest eigenvalue stays negative at low
+wavenumber but turns positive (unstable) at large ones,
+resembling an upside-down version of the right frame of
+Fig. 1. At first sight this might be taken as evidence of
+new physics in the form of microphase separation at finite
+wavenumber, an outcome generically predicted by some
+field-theoretic models [14]. However, when this happens
+the wavenumbers in question (at or around the orange line
+in Fig. 1) are comparable to the inverse particle separation
+1/r0. As the theory proposed here is a macroscopic one,
+it may not operate reliably in this large k region, so we
+do not take this as evidence of microphase separation in
+repulsive AOUPs.
+Conclusion – Inspired by liquid-state theory of equi-
+librium systems, we have started with the hierarchy of
+density correlations for AOUPs, and closed it at second
+order using a quasistatic approximation. This approach is
+closely related to the virial expansion in the equilibrium
+case. It yields a mean field theory for the one-point den-
+sity from first principles. We have analysed the stability
+of the uniform solution by looking at the largest eigen-
+value of the growth matrix, and determined the onset of
+a spinodal instability.
+Our method deals directly with particle velocities as well
+as positions. Perhaps more importantly, it is able to cap-
+ture (to lowest order in density) the physics of steep, short-
+ranged interactions.
+In that respect, it is distinct from
+standard coarse-graining methods for non-equilibrium sys-
+tems, such as those based on Dean’s equation [33], which
+starts as an exact representation but upon assuming a
+smooth (coarse-grained) density becomes an expansion in
+weak or slowly varying interaction forces.
+The Dean’s
+approach neglects strong, short-range interactions that
+change the statistics of close encounters even when the
+density, and hence the average interaction force, is small.
+In equilibrium statistical mechanics, the virial expansion
+is designed to handle exactly this situation. We have pre-
+sented the leading order counterpart of this, for an active
+system comprising interacting AOUPs.
+There are some limitations to our method. Firstly, the
+two-particle stationary solution used in the quasistatic
+ansatz is not exact, but rather a small-persistence approx-
+imation, unlike the equilibrium case. Secondly, in truncat-
+ing the density-correlation hierarchy at the second order,
+we assumed that the number density is low. Hence, even
+assuming small persistence, our approach is quantitatively
+accurate only in the dilute limit. Nonetheless, our method
+gives bottom-up confirmation of how phase separation can
+occur in purely repulsive active systems, and it can be used
+to study how the spinodal density varies with interaction
+parameters and with the persistence time.
+∗ ∗ ∗
+The authors acknowledge insightful discussions with
+Alexander Grosberg, Jean-Francois Joanny and Marius
+Bothe.
+YIL acknowledges support from Royal Society
+grant (RP\R1\180165).
+´EF acknowledges support from
+the Luxembourg National Research Fund (FNR), grant
+reference 14389168.
+Work funded in part by the Euro-
+pean Research Council under the EU’s Horizon 2020 Pro-
+gramme (Grant number 740269).
+RG-M acknowledges
+support from a St John’s College Research Fellowship,
+University of Cambridge.
+REFERENCES
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+erpool T. B., Prost J., Rao M. and Simha R. A., Rev.
+Mod. Phys., 85 (2013) 1143.
+[2] O’Byrne J., Kafri Y., Tailleur J. and van Wijland
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+F., Phys. Rev. Lett., 123 (2019) 260602.
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+[24] Kardar M., Statistical physics of particles (Cambridge
+University Press) 2007.
+[25] Hansen J.-P. and McDonald I. R., Theory of simple
+liquids: with applications to soft matter (Academic press)
+2013.
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+(2015) 032118.
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+matter, 11 (2015) 6680.
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+(2015) 032117.
+[33] Dean D. S., J. Phys. A, 29 (1996) L613.
+p-7
+
+epl draft
+Supplementary material:
+“Towards a liquid-state theory for active matter”
+Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E. Cates1 and ´Etienne
+Fodor2
+1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,
+Cambridge CB3 0WA, UK
+2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxem-
+bourg, Luxembourg
+3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK
+PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics
+Abstract – Supplementary material to “Towards a liquid-state theory for active matter”.
+Overdamped passive Brownian particles: Quasistatic ansatz for density cor-
+relations. –
+In this section, we demonstrate that, for overdamped passive Brownian par-
+ticles (PBPs), one recovers the virial expansion of the free energy when using the quasistatic
+ansatz p2(x1, x2, t) = p1(x1, t)p1(x2, t)pss(x1, x2). We consider the following equations of
+motion for N PBPs with pair-interaction potential Ψ:
+˙xi = −∂xiU +
+√
+2TΛi,
+U =
+N
+�
+i=1
+N
+�
+j<i
+Ψ(|xi − xj|),
+(S1)
+where we have set the mobility to unity, so that T is the diffusion constant, and Λi is a
+Gaussian white noise. Similar to the main text, we write down the corresponding hierarchy
+for one- and two-particle densities, assuming that the density is sufficiently low to ignore
+contributions from three-particle density:
+∂tp1(x1, t) = T∇2
+x1p1(x1, t) + ∇x1 ·
+�
+p2(x1, x2, t)∇x1Ψ(r)dx2,
+∂tp2(x1, x2, t) = T
+�
+∇2
+x1 + ∇2
+x2
+�
+p2(x1, x2, t) +
+�
+i=1,2
+∇xi · (p2(x1, x2, t)∇xiΨ(r)),
+(S2)
+where r = |x1 −x2|. In the quasistatic limit, where the interaction time is much faster than
+the free particle travel time, it is reasonable to suppose that every pair of particles reach
+steady state. This inspires the following ansatz to close the density hierarchy [1,2]:
+p2(x1, x2, t) = p1(x1, t) p1(x2, t) exp(−Ψ(r)/T),
+(S3)
+where we have used that the Boltzmann distribution is the the steady-state distribution of
+two particles. Substituting this ansatz into the p1 equation, we have
+∂tp1(x1, t) = T∇2
+x1p1(x1, t) + ∇x1 ·
+�
+p1(x1, t)
+�
+p1(x2, t)e−Ψ(r)/T ∇x1Ψ(r)dx2
+�
+.
+(S4)
+p-1
+arXiv:2301.12155v1  [cond-mat.soft]  28 Jan 2023
+
+Y. I. Li et al.
+Notice that e−Ψ(r)/T ∇x1Ψ(r) = −T∇x1f(r), where f(r) = exp(−Ψ(r)/T) − 1 is usually
+called the Mayer-f function [3]. Changing variables from x2 to r = x1 −x2 in the integrand
+of eq. (S4), and integrating by parts, we get
+1
+T ∂tp1(x1, t) = ∇2
+x1p1(x1, t) − ∇x1 ·
+�
+p1(x1, t)
+�
+f(r)∇x1p1(x1 − r, t)dr
+�
+.
+(S5)
+Thus far, we have closed the density hierarchy and obtained a mean-field equation for the
+one-particle density. In the following, we seek to connect eq. (S5) with existing knowledge
+of equilibrium physics of interacting gases, in particular, the virial expansion.
+Virial expansion.
+We assume that the density distribution ρ does not vary dramatically
+over the length-scale of the pair-potential. Therefore, expanding p1(x1−r) in eq. (S5) around
+x1 yields
+∂tp1 = ∇ · (p1∇µ),
+µ = T(log p1 + 2Bp1),
+B = −1
+2
+�
+f(r)dr.
+(S6)
+Here B(T) is the second virial coefficient in equilibrium statistical physics [3]. The stationary
+solution minimises the following free energy:
+F = T
+� �
+p1 log p1 + Bp2
+1
+�
+dx.
+(S7)
+The lack of gradient terms in the free energy is a result of neglecting higher-order terms in
+the expansion of p1(x1 − r). Assuming that the stationary state is uniform with density
+ρ = N/V , we obtain
+F(N, V, T) = TN log(N/V )
+�
+��
+�
+ideal gas
++
+TBN 2/V
+�
+��
+�
+virial correction
+.
+(S8)
+This is exactly the equilibrium virial expansion to second order [3].
+All-gradient expansion.
+To explore the effects of higher-gradient terms, we expand
+p1(x − r) in eq. (S5) to all orders in gradient:
+p1(x − r) = e−r·∇xp1(x),
+(S9)
+where e−r·∇x = �
+n
+(−r·∇x)n
+n!
+. Substituting into eq. (S5), we obtain
+1
+T ∂tp1 = ∇2p1 + 2∇ ·
+�
+p1∇
+� ¯Bp1
+��
+,
+¯B = −1
+2
+�
+f(r)e−r·∇xdr,
+(S10)
+where the second virial coefficient has the same form as in eq. (S6) but ¯B is now an operator.
+Proceeding as in the previous section, the chemical potential µ has the same form as eq. (S6).
+In Fourier space, ¯Bk =
+�
+eik·x ¯Bdx is the Fourier transform of the Meyer-f function. Note
+that ¯Bk=0 = B, since we only captured the zeroth mode when truncating the gradient
+expansion at the lowest order. The corresponding free energy is
+¯F = T
+�
+p1 log p1dx
+�
+��
+�
+ideal gas
++ T
+�
+p1,k ¯Bk p1,−k
+dk
+(2π)d
+�
+��
+�
+interactions
+.
+(S11)
+Overall, the gradient expansion is a natural extension of the equilibrium virial expansion
+for non-uniform densities. We note that one cannot trust the high-k behaviour of B(k), as
+it is sensitive to the details of the potential at close range. However, since we are always
+interested in length-scales much larger than the range of the potential (i.e.
+size of the
+individual particles), we do not need to be concerned with such a high-k behaviour.
+p-2
+
+Supplementary material: “Towards a liquid-state theory for active matter”
+Steady state of AOUPs: Small-τ expansion. –
+In this section, we derive the sta-
+tionary probability as an expansion at small τ . As opposed to [4], where the expansion was
+performed in the (x, ˙x) space, we consider here expansion in (x, v) space. This is necessary
+as we need the steady-state two-particle distribution in (x, v) space in the quasistatic ansatz.
+We start from the Fokker-Planck equation for two particles in one spatial dimension:
+∂tP =
+�
+i=1,2
+LiP,
+Li = D
+τ 2 ∂2
+vi +
+�1
+τ ∂vi − ∂xi
+�
+vi + ∂xi(∂xiΨ).
+(S12)
+Scaling units as v → v
+�
+τ/D, x → x/
+√
+τD, t → t/τ, and Ψ → Ψ/D, we obtain
+Li = Li,0 + √τLi,1 + τLi,2,
+(S13)
+where
+Li,0 = ∂2
+vi + ∂vivi,
+Li,1 = −∂xivi,
+Li,2 = ∂xi(∂xiΨ).
+(S14)
+We assume the following τ-expansion for the stationary distribution:
+p2,ss = e− 1
+2(v2
+1+v2
+2)−Ψ
+�
+1 +
+�
+n
+(√τ)nAi({xi, vi})
+�
+,
+(S15)
+where Ai is the i-th order term. Solving order by order, we get
+p2,ss ∝ exp
+�
+−Ψ(r) − 1
+2
+�
+v2
+1 + v2
+2
+��
+×
+�
+1 + √τ(v1 − v2)Ψ′ + τ
+�(v1 − v2)2
+2
+((Ψ′)2 − Ψ′′) − 2(Ψ′)2 + 3Ψ′′
+�
++ o(τ)
+�
+,
+(S16)
+where Ψ′ = dΨ/dr. This result is used to derive the quasistatic approximation in eq. (12)
+of the main text.
+The interaction matrix in Fourier-Hermite basis. –
+In this section, we show the
+details of the change of basis to the Fourier-Hermite basis. Recall that the perturbation
+density field ε can be decomposed in the Fourier-Hermite basis as
+ε(x, v) =
+�
+n
+�
+εkn(t)ψnk(x, v)dk,
+ψnk = e−ik(x+αv)Un(v − 2iαk),
+(S17)
+where the Hermite function Un is defined in eq. (22) of the main text. Substituting the
+decomposition into eq. (19) in the main text, we get
+�
+n
+�
+( ˙εkn + εknλnk)ψnk(x, v)dk = −τρ
+�
+n
+�
+ikεknU0(v)dk
+�
+ψnk(x, v − w)dw
+×
+�
+2La,kw + 2Lb,k(−ik) + Lc,k(−ik)w2�
+,
+(S18)
+where
+La,k = √τ
+� ∞
+0
+dr (Ψ′)2 e−Ψeirk,
+Lb,k = −
+� ∞
+0
+drf0(r)eirk,
+f0(s) =
+� ∞
+s
+dr Ψ′e−Ψ �
+1 − τ2(Ψ′)2 + τ3Ψ′′�
+,
+Lc,k = −
+� ∞
+0
+dr f1(r)eirk,
+f1(s) = τ
+� ∞
+s
+drΨ′((Ψ′)2 − Ψ′′)e−Ψ.
+(S19)
+p-3
+
+Y. I. Li et al.
+We then multiply both sides of eq. (S18) by the dual basis ¯ψmk′, as defined in eq. (24) of
+the main text, and integrate over the (x, v). By orthogonality of the basis, as outlined in
+eq. (25) of the main text, we have
+˙εkm = −λkmεkm +
+�
+n
+�
+M (1)
+kmn + M (2)
+kmn + M (3)
+kmn
+�
+εkn,
+(S20)
+where
+M (1)
+kmn = 2τρLa,k(−ik)
+�
+dv U0(v) ¯Um(v − 2iαk)
+�
+dw w eiαkwUn(v − w − 2iαk),
+M (2)
+kmn = 2τρLb,k(−ik)2
+�
+dv U0(v) ¯Um(v − 2iαk)
+�
+dw eiαkwUn(v − w − 2iαk),
+M (3)
+kmn = τρLc,k(−ik)2
+�
+dv U0(v) ¯Um(v − 2iαk)
+�
+dw w2eiαkwUn(v − w − 2iαk).
+(S21)
+We first perform all the w-integrals:
+In
+0 (v, k) ≡
+�
+dw eikαwUn(v − w − 2iαk) = e
+3
+2 α2k2+iαkv(−iαk)n
+In
+1 (v, k) ≡
+�
+dw
+√
+2π w eiαkwUn(v − w − 2iαk) = −e
+3
+2 α2k2+iαkv(−iαk)n−1 �
+iαkv + n + α2k2�
+In
+2 (v, k) ≡
+�
+dw
+√
+2π w2eiαkwUn(v − w − 2iαk)
+= e
+3
+2 α2k2+iαkv(−iαk)n−2 �
+n(n − 1) − α2k2(1 + (v − iαk)2) + 2inαk(v − iαk)
+�
+.
+(S22)
+Performing next the v-integrals, we find:
+M (1)
+kmn = 2τρLa,k(−ik)
+�
+dv
+√
+2π e− 1
+2 v2Hm(v − 2iαk)In
+1 (v, k)
+= 2τρLa,k
+(−iαk)n+m
+αm!
+eα2k2(m − n),
+M (2)
+kmn = −2τρLb,kk2
+�
+dv
+√
+2π e− 1
+2 v2Hm(v − 2iαk)In
+0 (v, k)
+= −2τρLb,kk2 (−iαk)n+m
+m!
+eα2k2,
+M (3)
+kmn = τρLc,k(−ik)2
+�
+dv
+√
+2π e− 1
+2 v2Hm(v − 2iαk)In
+2 (v, k)
+= τρLc,k
+(−iαk)n+m
+α2m!
+eα2k2 �
+m(m − 1) + n(n − 1) − 2mn − 2α2k2�
+.
+(S23)
+If the overall growth rate matrix Mkmn = −λkmδmn + M (1)
+kmn + M (2)
+kmn + M (3)
+kmn has no
+positive eigenvalue, the uniform solution is stable; otherwise the system undergoes spinodal
+decomposition. Hence, all we need is the sign of the largest eigenvalue of the M matrix.
+Despite the appearance of M as an infinite-dimensional matrix, it is in fact possible to
+find another set of basis where M is block diagonal. Indeed, {M (1), M (2), M (3)} are all
+linear combinations of the following matrices:
+(−iαk)m+n
+m!
+, (−iαk)m+n
+m!
+m, (−iαk)m+n
+m!
+n, (−iαk)m+n
+m!
+m(m − 1), (−iαk)m+n
+m!
+n(n − 1),
+(S24)
+p-4
+
+Supplementary material: “Towards a liquid-state theory for active matter”
+so that they can be written as
+M (1) = 2τρLa,k
+α
+eα2k2 (u1v⊺
+0 − u0v⊺
+1) ,
+M (2) = −2τρLb,kk2eα2k2u0v⊺
+0,
+M (3) = τρLc,k
+α2
+eα2k2 �
+u2v⊺
+0 + u0v⊺
+2 − 2u1v⊺
+1 − 2α2k2u0v⊺
+0
+�
+.
+(S25)
+where ⊺ denotes matrix transpose, and
+(u0)m = (−iαk)m
+m!
+,
+(u1)m = (−iαk)m
+m!
+m,
+(u2)m = (−iαk)m
+m!
+m(m − 1),
+(v0)n = (−iαk)n,
+(v1)n = (−iαk)nn,
+(v2)n = (−iαk)nn(n − 1),
+(S26)
+Observe that when acting on an arbitrary vector y, uv⊺y = (v⊺y)u, meaning that a matrix of
+this form projects all vectors onto the direction of u. We introduce ∆ = M (1) +M (2) +M (3)
+in this section, and, grouping the terms, we have
+e−(αk)2∆ = h0u0v⊺
+0 + h10(u1v⊺
+0 − u0v⊺
+1) + h11u1v⊺
+1 + h2(u2v⊺
+0 + u0v⊺
+2),
+(S27)
+where
+h0 = −2k2τρ (Lb,k + Lc,k) ,
+h10 = 2τρLa,k/α,
+h11 = 2τρLc,k/α2,
+h2 = τρLc,k/α2.
+(S28)
+We now consider the following spanning set: (v0, v1, v2, ...), where (vj)n = (−iαk)nPn,j
+(recall Pn,j = n!/(n − j)! is the permutation coefficient). One can easily check that it is
+a spanning set by spotting that the first j elements of vj are zeros, and hence no vector
+in this set can be written as a linear combination of others. Furthermore, let us introduce
+(q0, q1, q2, ..) as the dual basis of {vj}, such that v⊺
+i qj = δij. We proceed to write ∆ in the
+new basis: ˜∆ij ≡ v⊺
+i ∆ qj, where
+˜∆ = eα2k2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+h0v⊺
+0u0 + h10v⊺
+0u1 + h2v⊺
+0u2
+−h10v⊺
+0u0 + h11v⊺
+0u1
+h2v⊺
+0u0
+0
+. . .
+h0v⊺
+1u0 + h10v⊺
+1u1 + h2v⊺
+1u2
+−h10v⊺
+1u0 + h11v⊺
+1u1
+h2v⊺
+1u0
+0
+. . .
+h0v⊺
+2u0 + h10v⊺
+2u1 + h2v⊺
+2u2
+−h10v⊺
+2u0 + h11v⊺
+2u1
+h2v⊺
+2u0
+0
+. . .
+0
+0
+0
+0
+. . .
+...
+...
+...
+...
+...
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(S29)
+which is nonzero only in the top 3 × 3 due to the orthogonality relation between v’s and q’s.
+p-5
+
+Y. I. Li et al.
+Computing the inner products and setting ˜k = αk for convenience, we have
+v⊺
+0u0 =
+�
+n
+(−˜k2)n
+n!
+= e−˜k2,
+v⊺
+1u0 = v⊺
+0u1 =
+�
+n
+(−˜k2)n
+n!
+n = −˜k2 �
+n
+(−˜k2)n
+n!
+= −˜k2e−˜k2,
+v⊺
+2u0 = v⊺
+0u2 =
+�
+n
+(−˜k2)n
+n!
+n(n − 1) = ˜k4e−˜k2,
+v⊺
+1u1 =
+�
+n
+(−˜k2)n
+n!
+[n(n − 1) + n] = (˜k4 − ˜k2)e−˜k2,
+v⊺
+1u2 = v⊺
+2u1 =
+�
+n
+(−˜k2)n
+n!
+[n(n − 1)(n − 2) + 2n(n − 1)] = (−˜k6 + 2˜k4)e−˜k2,
+v⊺
+2u2 =
+�
+n
+(−˜k2)n
+n!
+[n(n − 1)(n − 2)(n − 3) + 4n(n − 1)(n − 2) + 2n(n − 1)]
+= (˜k8 − 4˜k6 + 2˜k4)e−˜k2.
+(S30)
+The resulting 3 × 3 matrix appearing in eq. (S29) can then be written as
+�
+�
+�
+�
+�
+h0 − h10˜k2 + h2˜k4
+−h10 − h11˜k2
+h2
+−h0˜k2 + h10
+�
+˜k4 − ˜k2�
++ h2
+�
+−˜k6 + 2˜k4�
+h10˜k2 + h11
+�
+˜k4 − ˜k2�
+−h2˜k2
+h0˜k4 + h10
+�
+−˜k6 + 2˜k4�
++ h2
+�
+˜k8 − 4˜k6 + 2˜k4�
+−h10˜k4 + h11
+�
+−˜k6 + 2˜k4�
+h2˜k4
+�
+�
+�
+�
+� .
+(S31)
+To compute the eigenvalues of the full matrix M = D + ∆, where Dkmn = (−n − ˜k2)δmn,
+we also need to write D in the new basis:
+˜Djl = v⊺
+j D ql = −δj+1,l − (j − ˜k2)δjl,
+(S32)
+where we have used n(vj)n = (vj+1 + jvj)n, or explicitly
+˜D =
+�
+�
+�
+�
+�
+�
+�
+�
+−˜k2
+−1
+0
+. . .
+0
+−1 − ˜k2
+−1
+0
+. . .
+0
+0
+−2 − ˜k2
+−1
+0
+. . .
+...
+...
+...
+...
+...
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(S33)
+Notice that ˜D is upper triangular and therefore the eigenvalues are simply the diagonal
+elements −n − ˜k2, which are the same as for D.
+Having written both the diagonal and non-diagonal parts in the new basis, we proceed
+to solve for the eigenvalues λ of �
+M = ˜D + ˜∆, satisfying det
+�
+�
+M − λI
+�
+= 0, where I is the
+identity matrix. Observe that �
+M is of the following form,
+�
+M =
+�
+�X
+Y
+0
+Z
+�
+� ,
+(S34)
+where X is a 3 × 3 matrix. One can prove that det
+�
+�
+M − λI
+�
+= det(X − λI) det(Z − λI).
+So we only need to solve for the eigenvalues of X and Z separately, but we already know
+p-6
+
+Supplementary material: “Towards a liquid-state theory for active matter”
+the eigenvalues of Z as it is upper-triangular with diagonal elements (−n − ˜k2) for n ≥ 3.
+Hence what remains are the eigenvalues of X, which can be solved numerically easily as it
+is only 3 × 3. Overall, we have reduced the problem of finding the maximum eigenvalue of
+an infinite-dimensional matrix to diagonalising the 3 × 3 matrix ˜∆ + ˜D. We then compute
+the largest eigenvalue as
+λmax = −k2
+�
+1 + τ 1 + 2(B2(k) − 2A1(k)2 + C2(k)) − 4k2B0(k)(B2(k) + C2(k))
+1 − 2B0(k)k2
+�
++ o(τ),
+(S35)
+where
+B0(k) = Lb,k
+��
+τ=0,
+B2(k) = (d/dτ)Lb,k,
+A1(k) = (1/√τ)La,k,
+C2(k) = (1/τ)Lc,k.
+(S36)
+REFERENCES
+[1] Grosberg A. Y. and Joanny J.-F., Phys. Rev. E, 92 (2015) 032118.
+[2] Ilker E. and Joanny J.-F., Phys. Rev. Research, 2 (2020) 023200.
+[3] Kardar M., Statistical physics of particles (Cambridge University Press) 2007.
+[4] Fodor ´E., Nardini C., Cates M. E., Tailleur J., Visco P. and van Wijland F., Phys.
+Rev. Lett., 117 (2016) 038103.
+p-7
+
diff --git a/9tFLT4oBgHgl3EQfui8A/content/tmp_files/load_file.txt b/9tFLT4oBgHgl3EQfui8A/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7d08b30878d133ee65c6e54a644d11585f026a50
--- /dev/null
+++ b/9tFLT4oBgHgl3EQfui8A/content/tmp_files/load_file.txt
@@ -0,0 +1,685 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf,len=684
+page_content='epl draft  Towards a liquid-state theory for active matter  Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Cates1 and ´Etienne Fodor2  1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK  2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg  3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK  PACS 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Ln – Nonequilibrium and irreversible thermodynamics  Abstract – In equilibrium, the collective behaviour of particles interacting via steep, short-ranged  potentials is well captured by the virial expansion of the free energy at low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Here, we extend  this approach beyond equilibrium to the case of active matter with self-propelled particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Given  that active systems do not admit any free-energy description in general, our aim is to build the  dynamics of the coarse-grained density from first principles without any equilibrium assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Starting from microscopic equations of motion, we obtain the hierarchy of density correlations,  which we close with an ansatz for the two-point density valid in the dilute regime at small activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  This closure yields the nonlinear dynamics of the one-point density, with hydrodynamic coefficients  depending explicitly on microscopic interactions, by analogy with the equilibrium virial expan-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This dynamics admits a spinodal instability for purely repulsive interactions, a signature  of motility-induced phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Therefore, although our approach should be restricted to  dilute, weakly-active systems a priori, it actually captures the features of a broader class of active  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Active matter is a wide category of systems out of equi-  librium, where a net flow of energy takes place at the local,  individual level [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Examples are swimming bacteria [4]  and active emulsions [5], amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The realm of  motile active matter includes interacting, many-particle  systems, where particles move persistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Different the-  oretical approaches have been proposed to describe such  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' They include (i) particle-based models, such as  run-and-tumble particles (RTPs) [6], active Brownian par-  ticles (ABPs) [7], and active Ornstein-Uhlenbeck particles  (AOUPs) [8,9], (ii) microscopic field theories [10,11], and  (iii) coarse-grained field theories [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Self-propelled particles tend to slow down at high densi-  ties, due to either biochemical reasons or steric repulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In contrast with passively diffusing particles, this slow-  down in turn increases the local density, creating a pos-  itive feedback loop that can result in a phase separation  between a dense and a dilute phase, known as motility-  induced phase separation (MIPS) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To predict an-  alytically the emergence of MIPS, previous works have  relied on a local mean-field approximation to replace mi-  croscopic interaction with density-dependent motility [16],  and also on an adiabatic elimination of orientational de-  gree of freedom [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Other studies have developed some  aspects of a liquid-state theory for active matter with pair-  wise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This is done, for instance, by expressing  thermodynamic observables, such as pressure [17–19] and  dissipation [20,21], in terms of correlations between hydro-  dynamic fields, typically density and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Also,  exact results for density correlations have been obtained  at infinite dimensions [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The success of equilibrium thermodynamics largely  stems from the ability to detect phase transitions by an-  alyzing the free-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In the dilute regime, the virial  expansion provides an approximate expression of the free-  energy that averages the effect of steep, short-ranged pair-  wise interactions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  For passive Brownian particles  (PBPs) interacting with an isotropic pairwise potential  �  i,j<i Ψ(|xi − xj|) at temperature T, the free energy per  unit volume is given in terms of the uniform density ρ as  fvirial(ρ)/T = ρ log ρ + B(T)ρ2 + O(ρ3),  B = 1  2  �  dr  �  1 − e−Ψ(r)/T �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (1)  Here B is commonly referred to as the second virial coef-  ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The major success of the virial expansion lies in  predicting the onset of liquid-gas phase separation in parti-  cle systems with a combination of strongly repulsive short-  ranged forces and weakly attractive long-range forces, such  as the Lennard-Jones potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As temperature increases,  p-1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='12155v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='soft]  28 Jan 2023  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  the free energy, as a function of density ρ, goes from convex  (B > 0) to concave (B < 0): It yields a phase transition  from homogeneous to non-homogeneous density profiles,  with a separation between dilute and dense phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Im-  portantly, the virial expansion holds for a generic micro-  scopic potential (excluding those of such long range that  the excess free energy is not analytic in density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In classical thermodynamics, the virial expansion is of-  ten derived from the equilibrium partition function [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Interestingly, it is also possible to arrive at approx-  imate expressions of free energy by truncating the  hierarchy of density correlations,  known as the Bo-  goliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierar-  chy [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' For instance, using an ansatz for the two-particle  density, informed by the steady-state solution of the two-  body problem [26], the dynamics of the one-body density  follows a gradient flow with respect to fvirial, as shown  in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This type of derivation can potentially be ex-  tended to a large class of nonequilibrum systems, where  the dynamics does not derive from any free energy a pri-  ori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Such an extension requires proposing an ansatz for  the two-particle density, which should be appropriate to  the specific nonequilibrium system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Interestingly, the two-body probability distribution can  be approached perturbatively for AOUPs, where the per-  sistence time is the small parameter in natural units [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To first order, this solution already predicts that repul-  sive interaction yields effective attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This suggests  that the onset of MIPS can likewise already be captured  by approximating the dynamics of density at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Therefore, inspired by the equilibrium virial expansion, it  is tempting to explore whether the density dynamics al-  ready contains any spinodal instability for dilute active  systems at weak persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Note that, although previ-  ous work refer to “active virial” as an expansion of pres-  sure [27, 28], here we are instead interested in deriving  dynamical equations of motion for the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, at  variance with equilibrium, phase transitions in active sys-  tems are usually not thermodynamically controlled by any  equation of state, but they can still be directly detected  as instability in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In this Letter, we start with the particle-based descrip-  tion of AOUPs, and consider their steady-state probability  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Our roadmap is essentially a “small density-  small persistence” bottom-up derivation of the density dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' To this end, we first introduce the corresponding  hierarchy of density correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Inspired by equilibrium  thermodynamics [25, 26], we then close the hierarchy to  arrive at a nonlinear dynamics for the one-point density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  It allows us to identify the MIPS spinodal instability for  any microscopic repulsive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' An advantage of  our approach is that does not impose an equilibrium map-  ping a priori (see [8,9] for a discussion of closures to the  AOUP equations that do so), instead retaining the non-  equilibrium structure of the microscopic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' A disad-  vantage is that extensions to higher order in density would  involved increasingly complicated calculations of higher  virial coefficients, which we do not attempt here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Interacting AOUPs:  Joint distribution func-  tion – We consider a system of N AOUPs at positions  xi interacting with pairwise potential U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The particles  are subject to stochastic self-propulsion velocities vi with  persistence time τ and diffusivity D [8,9]:  ˙xi = vi − ∂xiU,  U =  N  �  i=1  N  �  j<i  Ψ(rij),  τ ˙vi = −vi +  √  2DΛi,  (2)  where rij = |xi − xj|, and we have set the mobility to  unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Here Λi is a Gaussian white noise, with correlation  ⟨(Λi)α(t)(Λj)β(0)⟩ = δijδαβδ(t), where latin and Greek  letters respectively denote particle index and spatial co-  ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' It follows that vi is a colored Gaussian noise,  with correlation ⟨(vi)α(t)(vj)β(0)⟩ = δijδαβ(D/τ)e−|t|/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In the limit of vanishing persistence (τ → 0), the corre-  lation of vi becomes white, so that the system reduces to  a set of overdamped PBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Note that, at finite τ, if the  potential term −∂xiU was in the rhs of the vi-equation  (instead of the xi-equation), the system would represent  underdamped PBPs in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' For this reason, over-  damped AOUPs and underdamped PBPs coincide in the  noninteracting limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Following  standard  procedures  [29],  the  Fokker-  Planck equation of the joint probability distribution  pN(x1, v1, x1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xN, vN, t) associated with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (2) is  ∂tpN =  N  �  i=1  �  Lf,ipN + ∂xi · (pN ∂xiU)  �  ,  Lf,i = D  τ 2 ∂2  vi +  �1  τ ∂vi − ∂xi  �  vi,  (3)  where Lf,i is the non-interacting Fokker-Planck operator  governing the free-particle motion of the i-th particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Al-  though it cannot be solved exactly, its steady state can be  computed to lowest orders in the persistence time τ in a  similar manner to [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Such a perturbative calculation  can be performed in arbitrary dimension, though we now  restrict ourselves to 1D for simplicity, as a proof of prin-  ciple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' After scaling units as v → v  �  τ/D , x → x/  √  τD,  t → t/τ, and Ψ → Ψ/D, we find the many-body steady  state probability [see SM for details],  pN ∝ e−U−  v2  1+v2  2  2  �  1 + √τ  N  �  i=1  vi∂xiU  + τ  N  �  i=1  �v2  i  2  �  (∂xiU)2 − ∂2  xiU  �  − (∂xiU)2 + 3  2∂2  xiU  �  + o(τ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (4)  Note that the steady state probability here is given in  (x, v) space, whereas [8] gives the corresponding probabil-  ity in (x, ˙x) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-2  Towards a liquid-state theory for active matter  Nonequilibrium density correlations: Hierarchy  of equations – Although the joint distribution function  pN in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (4) contains all information about the steady  state density, it is generally difficult to predict the emer-  gence of phase transitions from the perspective of the full  phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Instead, our aim is to obtain a reduced de-  scription of the systel in terms of density correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The  n-particle density pn is found by marginalising the joint  distribution function pN as  pn(x1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xn, vn, t)  = PN,n  �  pN(x1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xN, vN, t)  N  �  j=n+1  dxjdvj,  (5)  where PN,n = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='/(N − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' is the permutation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Thereafter, for simplicity, we use the shorthand notation  pn = pn(x1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xn, vn, t) for arbitrary n, where the  dependence on positions, self-propulsion velocities, and  time will be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (3) over the positions  and velocities of all particles but one yields an equation for  the one-particle density that depends on the two-particle  density:  ∂tp1 = Lf,1p1 + F1[p2],  F1[p2] = ∂x1 ·  �  p2 ∂x1Ψ(r12) dx2dv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (6)  The first term Lf,1p1 corresponds to the free single-particle  dynamics, whereas the second term F1[p2] represents the  effects of pairwise interaction between one particle and all  the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Following the same marginalisation procedure for the  n-particle density pn, we obtain a hierarchy of equations,  each depending on the density pn+1, analoguous to the  BBGKY hierarchy of PBPs [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The resulting equation  for the two-particle density reads  ∂tp2 = (Lf,1 + Lf,2 + Lint)p2 + F2[p3],  Lint =  �  i=1,2  ∂xi ·  �  ∂xiΨ(r12)  �  ,  F2[p3] =  �  i=1,2  ∂xi ·  �  p3 ∂xiΨ (ri3) dx3dv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (7)  Here Lint governs the pairwise interactions between any  two particles, and F2[p3] represents the interactions be-  tween either of the first two particles and all the other  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As we will show now, this is a higher-order term  in the average number density ρ = N/V compared to the  rest, and hence can be neglected in the dilute limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To determine the scaling of pn with respect to ρ, one  can start with their normalisation properties [24]  �  pn  n  �  i=1  dxidvi = PN,n,  (8)  from which follows pn ∼ ρn for small n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Physically, this  is indeed expected as p1(x1, v1) is the probability density  of finding n particles at a given set of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' With  this scaling in mind, we estimate the scaling of each term  in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (7) as  (∂t − Lf,1 − Lf,2 − Lint  ����  ∼1/τint  ) p2  ����  ∼ρ2  =  �  i=1,2  ∂xi ·  �  p3  ����  ∼ρ3  ∂xiΨ(ri3)  �  ��  �  ∼1/τint  dx3  ����  ∼(r0)d  dv3,  (9)  where τint is the typical timescale of interaction, r0 is the  lengthscale of the potential, and d denotes the sptial di-  mension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' One can see that the rhs of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (9) is small if  rd  0ρ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Since rd  0 is effectively the volume of a particle,  this condition is satisfied if the volume fraction of parti-  cles in the system is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Therefore, as expected from the  analogy with the BBKY hierarchy [25], the interaction of  two particles with respect to others in the bath can be  neglected at small volume fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Closure of hierarchy:  Insight from quasistatic  approximation – We now propose a closure of the hierar-  chy of equation for density correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' At small volume  fraction, neglecting the rhs in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (9) yields  ∂tp2 = (Lf,1 + Lf,2 + Lint) p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (10)  The dynamics in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (10) is equivalent to the Fokker-  Planck equation of two AOUPs interacting through the  pairwise potential Ψ(r12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Recall from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (4) the N-  particle stationary probability, calculated order by order  in the persistence time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The two-particle probability is  hence,  p2,ss(r12, v1, v2) ∝ e−Ψ(r12)−  v2  1+v2  2  2  g(r12, v1 − v2),  (11)  where  g(r, w) = 1 + √τwΨ′ + (τw2/2)((Ψ′)2 − Ψ′′)  + τ(3Ψ′′ − 2(Ψ′)2) + O(τ 3/2),  (12)  and Ψ′ = dΨ/dr [see SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the quasistatic limit, where  the interaction timescale is much shorter than the persis-  tence (τint ≪ τ), we assume that every pair of particles  reaches steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Inspired by previous works [26, 30],  this assumption motivates the following ansatz to close the  hierarchy of density correlations:  p2(x1, v1, x2, v2, t) = p1(x1, v1, t) p1(x2, v2, t)  × e−Ψ(r12)g(r12, v1 − v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (13)  There is evidence that the above ansatz works well for  strong short-ranged interactions in the equilibrium case  [see SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We assume it would also work in the non-  equilibrium case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, when the interparticle distance  is larger than the interaction range (r12 > r0), the two-  point function p2 reduces to the product of one-point func-  tions p1, as expected from a mean-field approach for non-  interacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The second line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (13) accounts  p-3  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  for corrections due to interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' For repulsive poten-  tial, the factor e−Ψ captures the reduction of the two-  point density caused by the interaction, as expected from  equilibrium, whereas g(r12, v1 − v2) encapsulates nonequi-  librium effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Substituting the ansatz from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (13) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (6), we  arrive at the following dynamics for the one-particle den-  sity:  (∂t−Lf) p1(x, v, t)  = τ∂x  �  p1(x, v, t)  �  Lvp1(x, v − w, t)dw  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (14)  Here, Lf is the free Fokker-Planck operator in scaled units,  and the operator Lv effectively takes into account interac-  tions:  Lf = ∂2  v +  �  ∂v − √τ∂x  �  v,  Lv =  �  Ψ′(r)e−Ψ(r)g(r, w)e−r∂xdr,  (15)  where we have introduced the translation operator e−r∂x,  which shifts the position of the function acted upon as  e−r∂xp1(x, v) = p1(x − r, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence, Lv corresponds to an  infinite series of the gradient ∂x, with series coefficients set  by the microscopic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Equation (14) is the central  result of this Letter: It provides the dynamics of the one-  particle density in a closed form for small average density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Importantly, this closed form depends explicitly on the  details of the pairwise potential Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To obtain a more explicit expression of Lv, we next  substitute our perturbative result for g [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (12)] into the  definition of Lv [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (15)], yielding  Lv = 2wLa + 2Lb∂x + w2Lc∂x,  (16)  where, after integration by parts, we get  La = √τ  � ∞  0  dr(Ψ′)2 e−Ψe−r∂x,  Lb = −  � ∞  0  drf0(r)e−r∂x,  Lc = −  � ∞  0  dr f1(r)e−r∂x,  (17)  and  f0(s) =  � ∞  s  dr Ψ′e−Ψ �  1 − τ  �  2(Ψ′)2 − 3Ψ′′��  ,  f1(s) = τ  � ∞  s  dr Ψ′ �  (Ψ′)2 − Ψ′′�  e−Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (18)  The integrands featuring in the definition of the operators  {La, Lb, Lc} [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (17)] are even with respect to r, provided  that Ψ(r) = Ψ(−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' It follows that these operators can be  expressed as a series of ∂2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The functions f0 and f1 are analoguous to the Mayer  function which appears in the equilibrium virial expan-  sion [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, in the equilibrium limit (τ → 0), the  only surviving term in Lv stems from the leading order in  f0, in agreement with the derivation for equilibrium over-  damped PBPs [see SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This term is responsible for the  B coefficient in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (1) when that result is derived dynam-  ically for equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In that respect, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (16)  can be regarded as a direct generalization of the equilib-  rium virial expansion to the AOUP setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Importantly,  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (16) shows that activity produces additional velocity-  dependent terms that are absent in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In what  follows, we analyze in detail the corresponding dynam-  ics, in search for the onset of instability as a signature of  MIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Linear stability analysis: Eigenvalue problem –  The stationary solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (14) is given by p(0)  1 (v) =  (ρ/  √  2π)e−v2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This solution corresponds to a uniform  density for the position and a Gaussian distribution for  the self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In addition, as we will see shortly,  this is also the ground state of the free operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the  following, we expand perturbatively around p(0)  1  to find its  instability regions in parameter space: If the uniform state  is not stable, then we argue that the system undergoes  spinodal decomposition via a MIPS mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We consider p1(x, v, t) = p(0)  1 (v) + ε(x, v, t), with ε a  small perturbation about the ground state p(0)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Expand-  ing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (14) to linear order in ε, we get  (∂t− ¯Lf) ε(x, v, t) = τp(0)  1 (v)  �  Lv∂xε(x, v−w, t)dw, (19)  where Lv is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (16), and  ¯Lf = ∂2  v + (∂v − α∂x)v,  α = √τ − 2ρτ 3/2  � ∞  0  (Ψ′)2e−Ψdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (20)  To analyze the time evolution of the perturbation given in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (20), the difficulty lies in treating the effect of the op-  erator Lv, which contains information about microscopic  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Then, it is convenient to expand ε in the  eigenfunctions of the bare theory, namely in the absence  of Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Interestingly, as stated previously, the bare theory  maps into underdamped passive Brownian motion, and  one can readily find the solution in the literature [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The corresponding eigenfunctions, which we here call the  Fourier-Hermite basis, are  ψnk(x, v) = e−ik(x+αv)Un(v − 2iαk),  (21)  where Un is the Hermite function  Un(z) =  1  √  2π Hn(z) e−z2/2 = (−1)n  √  2π ∂n  z e−z2/2,  (22)  with the property that (∂2  z + ∂zz)Un(z) = −nUn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In-  deed, acting on the Fourier-Hermite basis with the modi-  fied free operator ¯Lf yields  ¯Lfψnk = −λknψnk,  λkn = (αk)2 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (23)  p-4  Towards a liquid-state theory for active matter  As the operator ¯Lf is not Hermitian, the conjugate basis is  not simply the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Instead, we introduce  the following conjugate basis  ¯ψnk(x, v) = eik(x+αv) ¯Un(v − 2iαk),  ¯Un(z) = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Hn(z),  (24)  yielding  �  dxdv ¯ψmk′(x, v)ψnk(x, v) = 2πδ(k − k′)δmn,  (25)  so that the orthogonality relation between the basis ψnk  and its conjugate ¯ψnk indeed holds as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Having obtained the eigenvectors and eigenvalues of the  bare theory, we decompose the perturbation ε in eigen-  functions of ¯Lf as  ε(x, v, t) =  �  n  �  εkn(t)ψnk(x, v)dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (26)  Multiplying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (19) with the conjugate basis ¯ψkn and inte-  grating over {x, v} yields the dynamics of the perturbation  ε, in the Fourier-Hermite basis, as  ˙εkm = −λkmεkm+  �  n  �  M (1)  kmn+M (2)  kmn+M (3)  kmn  �  εkn, (27)  where  M (1)  kmn = 2τρLa,k  (−iαk)n+m  αm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  e(αk)2(m − n),  M (2)  kmn = −2τρLb,kk2 (−iαk)n+m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  e(αk)2,  M (3)  kmn = τρLc,k  (−iαk)n+m  α2m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  e(αk)2  ×  �  m(m − 1) + n(n − 1) − 2mn − 2(αk)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (28)  Here, La,k =  �  eikxLadk, with similar definitions for Lb,k  and Lc,k, where the operators {La, Lb, Lc} are defined in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  If the matrix Mknm = −λkmδmn + M (1)  kmn +  M (2)  kmn +M (3)  kmn, which controls the growth rate of the per-  turbation ε, has no positive eigenvalue, the uniform so-  lution is stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' otherwise the system undergoes spinodal  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence we only need the largest eigenvalue  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As shown in the SM, the matrix M can be diagnon-  alized exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This calculation actually only amounts to  finding the maximum eigenvalue of a 3 × 3 matrix, which  can be done straightforwardly, while also perturbatively  keeping track of orders of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Spinodal instability – We now illustrate how our ap-  proach, valid for an arbitrary pairwise potential, can be  deployed to detect the onset of instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We consider the  following short-ranged, weakly repusive interaction poten-  tial:  Ψ(r) = ν exp  �  −  1  1 − (r/r0)2  �  ,  (29)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  k  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='10  λmax  τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  k  τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='50  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1: The value of the largest eigenvalue λmax as a function of  the wavevector k for ν = 10, r0 = 2, varying τ and φ (blue solid  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The orange dotted line represents the short length-scale  cut-off as we are only concerned with length-scales beyond the  particle size r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  characterised by its strength ν and range r0, whose deriva-  tives are continuous at any order for r within [0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The  maximum eigenvalue of the corresponding growth rate ma-  trix M, expanded analytically to first order in the persis-  tence time τ, is plotted against wavevector k in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1  for various values of τ and volume fraction φ = 2r0N/V ,  where 2r0 is taken as the size of the particle in 1D and  V = L is the system volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As expected [15], the sys-  tem exhibits a spinodal instability at high τ and packing  fraction φ, for which the linear perturbation ε is unstable,  namely limk→0 λmax(k) = 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 2 shows the stabil-  ity diagram based on the sign of the longest wavelength  perturbation, λmax(2π/L), as a function of τ and φ, show-  ing the transition between a uniform stable and a phase-  separated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (Note that we could equally have chosen  the fastest growing mode to locate the spinodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=')  It is notable that our theory of interacting AOUPs  shows a spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As in equilibrium systems  with attractions, it does this even at lowest order in the  virial-type expansion in powers of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Just as holds  there, finding the instability requires using a low-density  theory at finite densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' However the purpose of this ap-  proach is not to gain quantitative predictions about the ex-  act location of the spinodal curve, but rather to establish  that the macroscopic conditions for phase separation can  be satisfied, by considering microscopic interaction laws  and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Notably, unlike in equilibrium, the spin-  odal instability happens here for purely repulsive interac-  tions – a key feature of MIPS [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Our microscopic calcu-  lation complements previous viewpoints based on effective  quasi-equilibrium attractions [31], and/or collisional slow-  ing down [16,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Nonetheless, there are several caveats and limitations to  our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Firstly, the chosen potential is not strictly  hard-core, but depends on the strength parameter ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In  fact it is not possible to implement a true hard-core po-  tential due to the nature of the small τ expansion for the  two-particle density, as there are terms directly propor-  tional to Ψ or its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Unsurprisingly then, the  p-5  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='0  φ  10−1  100  τ  Phase separation  Uniform  0  λmax  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 2: Stability diagram in τ − φ space showing λmax(2π/L)  for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='6, L = 200, ν = 10, r0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The white region in the  top-right corner corresponds to when α < 0, where the method  breaks down, as it is no longer in the small τ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The solid  black line is the spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  results presented here do depend on ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' While overly large  ν will break the small-τ expansion, we have checked that a  range of moderate values produce stability diagrams qual-  itatively similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Secondly, for some parameters,  we found that the largest eigenvalue stays negative at low  wavenumber but turns positive (unstable) at large ones,  resembling an upside-down version of the right frame of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' At first sight this might be taken as evidence of  new physics in the form of microphase separation at finite  wavenumber, an outcome generically predicted by some  field-theoretic models [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' However, when this happens  the wavenumbers in question (at or around the orange line  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1) are comparable to the inverse particle separation  1/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As the theory proposed here is a macroscopic one,  it may not operate reliably in this large k region, so we  do not take this as evidence of microphase separation in  repulsive AOUPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Conclusion – Inspired by liquid-state theory of equi-  librium systems, we have started with the hierarchy of  density correlations for AOUPs, and closed it at second  order using a quasistatic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This approach is  closely related to the virial expansion in the equilibrium  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' It yields a mean field theory for the one-point den-  sity from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We have analysed the stability  of the uniform solution by looking at the largest eigen-  value of the growth matrix, and determined the onset of  a spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Our method deals directly with particle velocities as well  as positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Perhaps more importantly, it is able to cap-  ture (to lowest order in density) the physics of steep, short-  ranged interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In that respect, it is distinct from  standard coarse-graining methods for non-equilibrium sys-  tems, such as those based on Dean’s equation [33], which  starts as an exact representation but upon assuming a  smooth (coarse-grained) density becomes an expansion in  weak or slowly varying interaction forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The Dean’s  approach neglects strong, short-range interactions that  change the statistics of close encounters even when the  density, and hence the average interaction force, is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In equilibrium statistical mechanics, the virial expansion  is designed to handle exactly this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We have pre-  sented the leading order counterpart of this, for an active  system comprising interacting AOUPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  There are some limitations to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Firstly, the  two-particle stationary solution used in the quasistatic  ansatz is not exact, but rather a small-persistence approx-  imation, unlike the equilibrium case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Secondly, in truncat-  ing the density-correlation hierarchy at the second order,  we assumed that the number density is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence, even  assuming small persistence, our approach is quantitatively  accurate only in the dilute limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Nonetheless, our method  gives bottom-up confirmation of how phase separation can  occur in purely repulsive active systems, and it can be used  to study how the spinodal density varies with interaction  parameters and with the persistence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  ∗ ∗ ∗  The authors acknowledge insightful discussions with  Alexander Grosberg, Jean-Francois Joanny and Marius  Bothe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  YIL acknowledges support from Royal Society  grant (RP\\R1\\180165).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  ´EF acknowledges support from  the Luxembourg National Research Fund (FNR), grant  reference 14389168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Work funded in part by the Euro-  pean Research Council under the EU’s Horizon 2020 Pro-  gramme (Grant number 740269).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  RG-M acknowledges  support from a St John’s College Research Fellowship,  University of Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
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+page_content='  [29] Risken H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Fokker-planck equation (Springer) 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [30] Ilker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
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+page_content='  [31] Farage T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Krinninger P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Brader J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 91 (2015) 042310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [32] Takatori S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Brady J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 91  (2015) 032117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [33] Dean D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' A, 29 (1996) L613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-7  epl draft  Supplementary material:  “Towards a liquid-state theory for active matter”  Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Cates1 and ´Etienne  Fodor2  1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,  Cambridge CB3 0WA, UK  2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxem-  bourg, Luxembourg  3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK  PACS 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Ln – Nonequilibrium and irreversible thermodynamics  Abstract – Supplementary material to “Towards a liquid-state theory for active matter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Overdamped passive Brownian particles: Quasistatic ansatz for density cor-  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' –  In this section, we demonstrate that, for overdamped passive Brownian par-  ticles (PBPs), one recovers the virial expansion of the free energy when using the quasistatic  ansatz p2(x1, x2, t) = p1(x1, t)p1(x2, t)pss(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We consider the following equations of  motion for N PBPs with pair-interaction potential Ψ:  ˙xi = −∂xiU +  √  2TΛi,  U =  N  �  i=1  N  �  j<i  Ψ(|xi − xj|),  (S1)  where we have set the mobility to unity, so that T is the diffusion constant, and Λi is a  Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Similar to the main text, we write down the corresponding hierarchy  for one- and two-particle densities, assuming that the density is sufficiently low to ignore  contributions from three-particle density:  ∂tp1(x1, t) = T∇2  x1p1(x1, t) + ∇x1 ·  �  p2(x1, x2, t)∇x1Ψ(r)dx2,  ∂tp2(x1, x2, t) = T  �  ∇2  x1 + ∇2  x2  �  p2(x1, x2, t) +  �  i=1,2  ∇xi · (p2(x1, x2, t)∇xiΨ(r)),  (S2)  where r = |x1 −x2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the quasistatic limit, where the interaction time is much faster than  the free particle travel time, it is reasonable to suppose that every pair of particles reach  steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This inspires the following ansatz to close the density hierarchy [1,2]:  p2(x1, x2, t) = p1(x1, t) p1(x2, t) exp(−Ψ(r)/T),  (S3)  where we have used that the Boltzmann distribution is the the steady-state distribution of  two particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Substituting this ansatz into the p1 equation, we have  ∂tp1(x1, t) = T∇2  x1p1(x1, t) + ∇x1 ·  �  p1(x1, t)  �  p1(x2, t)e−Ψ(r)/T ∇x1Ψ(r)dx2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S4)  p-1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='12155v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='soft]  28 Jan 2023  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Notice that e−Ψ(r)/T ∇x1Ψ(r) = −T∇x1f(r), where f(r) = exp(−Ψ(r)/T) − 1 is usually  called the Mayer-f function [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Changing variables from x2 to r = x1 −x2 in the integrand  of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S4), and integrating by parts, we get  1  T ∂tp1(x1, t) = ∇2  x1p1(x1, t) − ∇x1 ·  �  p1(x1, t)  �  f(r)∇x1p1(x1 − r, t)dr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S5)  Thus far, we have closed the density hierarchy and obtained a mean-field equation for the  one-particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the following, we seek to connect eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5) with existing knowledge  of equilibrium physics of interacting gases, in particular, the virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We assume that the density distribution ρ does not vary dramatically  over the length-scale of the pair-potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Therefore, expanding p1(x1−r) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5) around  x1 yields  ∂tp1 = ∇ · (p1∇µ),  µ = T(log p1 + 2Bp1),  B = −1  2  �  f(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S6)  Here B(T) is the second virial coefficient in equilibrium statistical physics [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The stationary  solution minimises the following free energy:  F = T  � �  p1 log p1 + Bp2  1  �  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S7)  The lack of gradient terms in the free energy is a result of neglecting higher-order terms in  the expansion of p1(x1 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Assuming that the stationary state is uniform with density  ρ = N/V , we obtain  F(N, V, T) = TN log(N/V )  �  ��  �  ideal gas  +  TBN 2/V  �  ��  �  virial correction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S8)  This is exactly the equilibrium virial expansion to second order [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  All-gradient expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To explore the effects of higher-gradient terms, we expand  p1(x − r) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5) to all orders in gradient:  p1(x − r) = e−r·∇xp1(x),  (S9)  where e−r·∇x = �  n  (−r·∇x)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Substituting into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5), we obtain  1  T ∂tp1 = ∇2p1 + 2∇ ·  �  p1∇  � ¯Bp1  ��  ,  ¯B = −1  2  �  f(r)e−r·∇xdr,  (S10)  where the second virial coefficient has the same form as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S6) but ¯B is now an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Proceeding as in the previous section, the chemical potential µ has the same form as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In Fourier space, ¯Bk =  �  eik·x ¯Bdx is the Fourier transform of the Meyer-f function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Note  that ¯Bk=0 = B, since we only captured the zeroth mode when truncating the gradient  expansion at the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The corresponding free energy is  ¯F = T  �  p1 log p1dx  �  ��  �  ideal gas  + T  �  p1,k ¯Bk p1,−k  dk  (2π)d  �  ��  �  interactions  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S11)  Overall, the gradient expansion is a natural extension of the equilibrium virial expansion  for non-uniform densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We note that one cannot trust the high-k behaviour of B(k), as  it is sensitive to the details of the potential at close range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' However, since we are always  interested in length-scales much larger than the range of the potential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  size of the  individual particles), we do not need to be concerned with such a high-k behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-2  Supplementary material: “Towards a liquid-state theory for active matter”  Steady state of AOUPs: Small-τ expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' –  In this section, we derive the sta-  tionary probability as an expansion at small τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As opposed to [4], where the expansion was  performed in the (x, ˙x) space, we consider here expansion in (x, v) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This is necessary  as we need the steady-state two-particle distribution in (x, v) space in the quasistatic ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We start from the Fokker-Planck equation for two particles in one spatial dimension:  ∂tP =  �  i=1,2  LiP,  Li = D  τ 2 ∂2  vi +  �1  τ ∂vi − ∂xi  �  vi + ∂xi(∂xiΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S12)  Scaling units as v → v  �  τ/D, x → x/  √  τD, t → t/τ, and Ψ → Ψ/D, we obtain  Li = Li,0 + √τLi,1 + τLi,2,  (S13)  where  Li,0 = ∂2  vi + ∂vivi,  Li,1 = −∂xivi,  Li,2 = ∂xi(∂xiΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S14)  We assume the following τ-expansion for the stationary distribution:  p2,ss = e− 1  2(v2  1+v2  2)−Ψ  �  1 +  �  n  (√τ)nAi({xi, vi})  �  ,  (S15)  where Ai is the i-th order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Solving order by order, we get  p2,ss ∝ exp  �  −Ψ(r) − 1  2  �  v2  1 + v2  2  ��  ×  �  1 + √τ(v1 − v2)Ψ′ + τ  �(v1 − v2)2  2  ((Ψ′)2 − Ψ′′) − 2(Ψ′)2 + 3Ψ′′  �  + o(τ)  �  ,  (S16)  where Ψ′ = dΨ/dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This result is used to derive the quasistatic approximation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (12)  of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The interaction matrix in Fourier-Hermite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' –  In this section, we show the  details of the change of basis to the Fourier-Hermite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Recall that the perturbation  density field ε can be decomposed in the Fourier-Hermite basis as  ε(x, v) =  �  n  �  εkn(t)ψnk(x, v)dk,  ψnk = e−ik(x+αv)Un(v − 2iαk),  (S17)  where the Hermite function Un is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (22) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Substituting the  decomposition into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (19) in the main text, we get  �  n  �  ( ˙εkn + εknλnk)ψnk(x, v)dk = −τρ  �  n  �  ikεknU0(v)dk  �  ψnk(x, v − w)dw  ×  �  2La,kw + 2Lb,k(−ik) + Lc,k(−ik)w2�  ,  (S18)  where  La,k = √τ  � ∞  0  dr (Ψ′)2 e−Ψeirk,  Lb,k = −  � ∞  0  drf0(r)eirk,  f0(s) =  � ∞  s  dr Ψ′e−Ψ �  1 − τ2(Ψ′)2 + τ3Ψ′′�  ,  Lc,k = −  � ∞  0  dr f1(r)eirk,  f1(s) = τ  � ∞  s  drΨ′((Ψ′)2 − Ψ′′)e−Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S19)  p-3  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We then multiply both sides of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S18) by the dual basis ¯ψmk′, as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (24) of  the main text, and integrate over the (x, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' By orthogonality of the basis, as outlined in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (25) of the main text, we have  ˙εkm = −λkmεkm +  �  n  �  M (1)  kmn + M (2)  kmn + M (3)  kmn  �  εkn,  (S20)  where  M (1)  kmn = 2τρLa,k(−ik)  �  dv U0(v) ¯Um(v − 2iαk)  �  dw w eiαkwUn(v − w − 2iαk),  M (2)  kmn = 2τρLb,k(−ik)2  �  dv U0(v) ¯Um(v − 2iαk)  �  dw eiαkwUn(v − w − 2iαk),  M (3)  kmn = τρLc,k(−ik)2  �  dv U0(v) ¯Um(v − 2iαk)  �  dw w2eiαkwUn(v − w − 2iαk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S21)  We first perform all the w-integrals:  In  0 (v, k) ≡  �  dw eikαwUn(v − w − 2iαk) = e  3  2 α2k2+iαkv(−iαk)n  In  1 (v, k) ≡  �  dw  √  2π w eiαkwUn(v − w − 2iαk) = −e  3  2 α2k2+iαkv(−iαk)n−1 �  iαkv + n + α2k2�  In  2 (v, k) ≡  �  dw  √  2π w2eiαkwUn(v − w − 2iαk)  = e  3  2 α2k2+iαkv(−iαk)n−2 �  n(n − 1) − α2k2(1 + (v − iαk)2) + 2inαk(v − iαk)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S22)  Performing next the v-integrals, we find:  M (1)  kmn = 2τρLa,k(−ik)  �  dv  √  2π e− 1  2 v2Hm(v − 2iαk)In  1 (v, k)  = 2τρLa,k  (−iαk)n+m  αm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  eα2k2(m − n),  M (2)  kmn = −2τρLb,kk2  �  dv  √  2π e− 1  2 v2Hm(v − 2iαk)In  0 (v, k)  = −2τρLb,kk2 (−iαk)n+m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  eα2k2,  M (3)  kmn = τρLc,k(−ik)2  �  dv  √  2π e− 1  2 v2Hm(v − 2iαk)In  2 (v, k)  = τρLc,k  (−iαk)n+m  α2m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  eα2k2 �  m(m − 1) + n(n − 1) − 2mn − 2α2k2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S23)  If the overall growth rate matrix Mkmn = −λkmδmn + M (1)  kmn + M (2)  kmn + M (3)  kmn has no  positive eigenvalue, the uniform solution is stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' otherwise the system undergoes spinodal  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence, all we need is the sign of the largest eigenvalue of the M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Despite the appearance of M as an infinite-dimensional matrix, it is in fact possible to  find another set of basis where M is block diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, {M (1), M (2), M (3)} are all  linear combinations of the following matrices:  (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  , (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m, (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n, (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m(m − 1), (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n(n − 1),  (S24)  p-4  Supplementary material: “Towards a liquid-state theory for active matter”  so that they can be written as  M (1) = 2τρLa,k  α  eα2k2 (u1v⊺  0 − u0v⊺  1) ,  M (2) = −2τρLb,kk2eα2k2u0v⊺  0,  M (3) = τρLc,k  α2  eα2k2 �  u2v⊺  0 + u0v⊺  2 − 2u1v⊺  1 − 2α2k2u0v⊺  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S25)  where ⊺ denotes matrix transpose, and  (u0)m = (−iαk)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  ,  (u1)m = (−iαk)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m,  (u2)m = (−iαk)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m(m − 1),  (v0)n = (−iαk)n,  (v1)n = (−iαk)nn,  (v2)n = (−iαk)nn(n − 1),  (S26)  Observe that when acting on an arbitrary vector y, uv⊺y = (v⊺y)u, meaning that a matrix of  this form projects all vectors onto the direction of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We introduce ∆ = M (1) +M (2) +M (3)  in this section, and, grouping the terms, we have  e−(αk)2∆ = h0u0v⊺  0 + h10(u1v⊺  0 − u0v⊺  1) + h11u1v⊺  1 + h2(u2v⊺  0 + u0v⊺  2),  (S27)  where  h0 = −2k2τρ (Lb,k + Lc,k) ,  h10 = 2τρLa,k/α,  h11 = 2τρLc,k/α2,  h2 = τρLc,k/α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S28)  We now consider the following spanning set: (v0, v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='), where (vj)n = (−iαk)nPn,j  (recall Pn,j = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='/(n − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' is the permutation coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' One can easily check that it is  a spanning set by spotting that the first j elements of vj are zeros, and hence no vector  in this set can be written as a linear combination of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Furthermore, let us introduce  (q0, q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='.) as the dual basis of {vj}, such that v⊺  i qj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We proceed to write ∆ in the  new basis: ˜∆ij ≡ v⊺  i ∆ qj, where  ˜∆ = eα2k2  �  �  �  �  �  �  �  �  �  �  �  h0v⊺  0u0 + h10v⊺  0u1 + h2v⊺  0u2  −h10v⊺  0u0 + h11v⊺  0u1  h2v⊺  0u0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  h0v⊺  1u0 + h10v⊺  1u1 + h2v⊺  1u2  −h10v⊺  1u0 + h11v⊺  1u1  h2v⊺  1u0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  h0v⊺  2u0 + h10v⊺  2u1 + h2v⊺  2u2  −h10v⊺  2u0 + h11v⊺  2u1  h2v⊺  2u0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  �  �  ,  (S29)  which is nonzero only in the top 3 × 3 due to the orthogonality relation between v’s and q’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-5  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Computing the inner products and setting ˜k = αk for convenience, we have  v⊺  0u0 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  = e−˜k2,  v⊺  1u0 = v⊺  0u1 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n = −˜k2 �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  = −˜k2e−˜k2,  v⊺  2u0 = v⊺  0u2 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n(n − 1) = ˜k4e−˜k2,  v⊺  1u1 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [n(n − 1) + n] = (˜k4 − ˜k2)e−˜k2,  v⊺  1u2 = v⊺  2u1 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [n(n − 1)(n − 2) + 2n(n − 1)] = (−˜k6 + 2˜k4)e−˜k2,  v⊺  2u2 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [n(n − 1)(n − 2)(n − 3) + 4n(n − 1)(n − 2) + 2n(n − 1)]  = (˜k8 − 4˜k6 + 2˜k4)e−˜k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S30)  The resulting 3 × 3 matrix appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S29) can then be written as  �  �  �  �  �  h0 − h10˜k2 + h2˜k4  −h10 − h11˜k2  h2  −h0˜k2 + h10  �  ˜k4 − ˜k2�  + h2  �  −˜k6 + 2˜k4�  h10˜k2 + h11  �  ˜k4 − ˜k2�  −h2˜k2  h0˜k4 + h10  �  −˜k6 + 2˜k4�  + h2  �  ˜k8 − 4˜k6 + 2˜k4�  −h10˜k4 + h11  �  −˜k6 + 2˜k4�  h2˜k4  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S31)  To compute the eigenvalues of the full matrix M = D + ∆, where Dkmn = (−n − ˜k2)δmn,  we also need to write D in the new basis:  ˜Djl = v⊺  j D ql = −δj+1,l − (j − ˜k2)δjl,  (S32)  where we have used n(vj)n = (vj+1 + jvj)n, or explicitly  ˜D =  �  �  �  �  �  �  �  �  −˜k2  −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0  −1 − ˜k2  −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0  0  −2 − ˜k2  −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S33)  Notice that ˜D is upper triangular and therefore the eigenvalues are simply the diagonal  elements −n − ˜k2, which are the same as for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Having written both the diagonal and non-diagonal parts in the new basis, we proceed  to solve for the eigenvalues λ of �  M = ˜D + ˜∆, satisfying det  �  �  M − λI  �  = 0, where I is the  identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Observe that �  M is of the following form,  �  M =  �  �X  Y  0  Z  �  � ,  (S34)  where X is a 3 × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' One can prove that det  �  �  M − λI  �  = det(X − λI) det(Z − λI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  So we only need to solve for the eigenvalues of X and Z separately, but we already know  p-6  Supplementary material: “Towards a liquid-state theory for active matter”  the eigenvalues of Z as it is upper-triangular with diagonal elements (−n − ˜k2) for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Hence what remains are the eigenvalues of X, which can be solved numerically easily as it  is only 3 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Overall, we have reduced the problem of finding the maximum eigenvalue of  an infinite-dimensional matrix to diagonalising the 3 × 3 matrix ˜∆ + ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We then compute  the largest eigenvalue as  λmax = −k2  �  1 + τ 1 + 2(B2(k) − 2A1(k)2 + C2(k)) − 4k2B0(k)(B2(k) + C2(k))  1 − 2B0(k)k2  �  + o(τ),  (S35)  where  B0(k) = Lb,k  ��  τ=0,  B2(k) = (d/dτ)Lb,k,  A1(k) = (1/√τ)La,k,  C2(k) = (1/τ)Lc,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S36)  REFERENCES  [1] Grosberg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 92 (2015) 032118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [2] Ilker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Research, 2 (2020) 023200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [3] Kardar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Statistical physics of particles (Cambridge University Press) 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [4] Fodor ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nardini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Visco P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and van Wijland F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 117 (2016) 038103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
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+An Efficient Solution to s-Rectangular Robust Markov
+Decision Processes
+Navdeep Kumar1, Kfir Levy1, Kaixin Wang2, and Shie Mannor1
+1Technion
+2National University of Singapore
+February 1, 2023
+Abstract
+We present an efficient robust value iteration for s-rectangular robust Markov
+Decision Processes (MDPs) with a time complexity comparable to standard (non-
+robust) MDPs which is significantly faster than any existing method. We do so by
+deriving the optimal robust Bellman operator in concrete forms using our Lp water
+filling lemma. We unveil the exact form of the optimal policies, which turn out to be
+novel threshold policies with the probability of playing an action proportional to its
+advantage.
+1
+Introduction
+In Markov Decision Processes (MDPs), an agent interacts with the environment and learns
+to optimally behave in it [28]. However, the MDP solution may be very sensitive to little
+changes in the model parameters [23]. Hence we should be cautious applying the solution of
+the MDP, when the model is changing or when there is uncertainty in the model parameters.
+Robust MDPs provide a way to address this issue, where an agent can learn to optimally
+behave even when the model parameters are uncertain [15, 29, 18]. Another motivation to
+study robust MDPs is that they can lead to better generalization [33, 34, 25] compared to
+non-robust solutions.
+Unfortunately, solving robust MDPs is proven to be NP-hard for general uncertainty
+sets [32]. As a result, the uncertainty set is often assumed to be rectangular, which enables
+the existence of a contractive robust Bellman operators to obtain the optimal robust value
+function [24, 18, 22, 12, 32]. Recently, there has been progress in solving robust MDPs
+for some sa-rectangular uncertainty sets via both value-based and policy-based methods
+[30, 31]. An uncertainty set is said to be sa-rectangular if it can be expressed as a Cartesian
+product of the uncertainty in all states and actions. It can be further generalized to a
+s-rectangular uncertainty set if it can be expressed as a Cartesian product of the uncertainty
+in all states only. Compared to sa-rectangular robust MDPs, s-rectangular robust MDPs
+are less conservative and hence more desirable; however, they are also much more difficult
+and poorly understood [32]. Currently, there are few works that consider s-rectangular Lp
+robust MDPs where uncertainty set is further constrained by Lp norm, but they rely on
+1
+arXiv:2301.13642v1  [cs.LG]  31 Jan 2023
+
+black box methods which limits its applicability and offers little insights [7, 16, 9, 32]. No
+effective value or policy based methods exist for solving any s-robust MDPs. Moreover, it is
+known that optimal policies in s-rectangular robust MDPs can be stochastic, in contrast to
+sa-rectangular robust MDPs and non-robust MDPs [32]. However, so far, nothing is known
+about the stochastic nature of the optimal policies in s-rectangular MDPs.
+In this work, we mainly focus on s-rectangular Lp robust MDPs. We first revise the
+unrealistic assumptions made in the noise transition kernel in [9] and introduce forbidden
+transitions, which leads to novel regularizers. Then we derive robust Bellman operator
+(policy evaluation) for a s-rectangular robust MDPs in closed form which is equivalent to
+reward-value-policy-regularized non-robust Bellman operator without radius assumption 5.1
+in [9]. We exploit this equivalence to derive an optimal robust Bellman operator in concrete
+forms using our Lp-water pouring lemma which generalizes existing water pouring lemma for
+L2 case [1]. We can compute these operators in closed form for p = 1, ∞ and exactly by a
+simple algorithm for p = 2, and approximately by binary search for general p. We show that
+the time complexity of robust value iteration for p = 1, 2 is the same as that of non-robust
+value iteration. For general p, the complexity includes some additional log-factors due to
+binary searches.
+In addition, we derive a complete characterization of the stochastic nature of optimal
+policies in s-rectangular robust MDPs. The optimal policies in this case, are threshold
+policies, that plays only actions with positive advantage with probability proportional to
+(p − 1)-th power to its advantage.
+Related Work. For sa-rectangular R-contamination robust MDPs, [30] derived robust
+Bellman operators which are equivalent to value-regularized-non-robust Bellman operators,
+enabling efficient robust value iteration. Building upon this work, [31] derived robust policy
+gradient which is equivalvent to non-robust policy gradient with regularizer and correction
+terms. Unfortunately, these methods can’t be naturally generalized to s-rectangular robust
+MDPs.
+For s-rectangular robust MDPs, methods such as robust value iteration [6, 32], robust
+modified policy iteration [19], partial robust policy iteration [16] etc tries to approximately
+evaluate robust Bellman operators using variety of tools to estimate optimal robust value
+function. The scalability of these methods has been limited due to their reliance on an
+external black-box solver such as Linear Programming.
+Previous works have explored robust MDPs from a regularization perspective [9, 10, 17, 11].
+Specifically, [9] showed that s-rectangular robust MDPs is equivalent to reward-value-policy
+regularized MDPs, and proposed a gradient based policy iteration for s-rectangular Lp
+robust MDPs ( where uncertainty set is s-rectangular and constrained by Lp norm). But
+this gradient based policy improvement relies on black box simplex projection, hence very
+slow and not scalable.
+The detailed discussion of the above works can be found in the appendix.
+2
+Preliminary
+2.1
+Notations
+For a set S, |S| denotes its cardinality. ⟨u, v⟩ := �
+s∈S u(s)v(s) denotes the dot product
+between functions u, v : S → R.
+∥v∥q
+p := (�
+s|v(s)|p)
+q
+p denotes the q-th power of Lp
+norm of function v, and we use ∥v∥p := ∥v∥1
+p and ∥v∥ := ∥v∥2 as shorthand.
+For a
+2
+
+set C, ∆C := {a : C → R|a(c) ≥ 0, ∀c, �
+c∈C ac = 1} is the probability simplex over
+C. 0, 1 denotes all zero vector and all ones vector/function respectively of appropriate
+dimension/domain. 1(a = b) := 1 if a = b, 0 otherwise, is the indicator function. For
+vectors u, v, 1(u ≥ v) is component wise indicator vector, i.e. 1(u ≥ v)(x) = 1(u(x) ≥ v(x)).
+A × B = {(a, b) | a ∈ A, b ∈ B} is cartesain product between set A and B.
+2.2
+Markov Decision Processes
+A Markov Decision Process (MDP) can be described as a tuple (S, A, P, R, γ, µ), where S is
+the state space, A is the action space, P is a transition kernel mapping S × A to ∆S, R is a
+reward function mapping S × A to R, µ is an initial distribution over states in S, and γ is a
+discount factor in [0, 1). The expected discounted cumulative reward (return) is defined as
+ρπ
+(P,R) :=E
+� ∞
+�
+n=0
+γnR(sn, an)
+��� s0 ∼ µ, π, P
+�
+.
+The return can be written compactly as
+ρπ
+(P,R) = ⟨µ, vπ
+(P,R)⟩,
+(1)
+[26] where vπ
+(P,R) is the value function , defined as
+vπ
+(P,R)(s) := E
+� ∞
+�
+n=0
+γnR(sn, an)
+��� s0 = s, π, P
+�
+.
+(2)
+Our objective is to find an optimal policy π∗
+(P,R) that maximizes the performance ρπ
+(P,R).
+This performance can be written as :
+ρ∗
+(P,R) := max
+π
+ρπ
+(P,R) = ⟨µ, v∗
+(P,R)⟩,
+(3)
+where v∗
+(P,R) := maxπ vπ
+(P,R) is the optimal value function [26].
+The value function vπ
+(P,R) and the optimal value function v∗
+(P,R) are the fixed points of the
+Bellman operator T π
+(P,R) and the robust Bellman operator T ∗
+(P,R), respectively [28]. These
+γ-contraction operators are defined as follows: For any vector v, and state s ∈ S,
+(T π
+(P,R)v)(s) :=
+�
+a
+π(a|s)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+,
+and
+T ∗
+(P,R)v := max
+π
+T π
+(P,R)v.
+Therefore, the value iteration vn+1 := T ∗
+(P,R)vn converges linearly to the optimal value
+function v∗
+(P,R). Given this optimal value function, the optimal policy can be computed as:
+π∗
+(P,R) ∈ arg maxπ T π
+(P,R)v∗
+(P,R).
+Remark 1. The vector minimum of a set U of vectors is defined component wise, i.e.
+(minu∈U u)(i) := minu∈U u(i).
+This operation is well-defined only when there exists a
+minimal vector u∗ ∈ U such that u∗ ⪯ u, ∀u ∈ U. The same holds for other operations such
+as maximum, argmin, argmax, etc.
+3
+
+2.3
+Robust Markov Decision Processes
+A robust Markov Decision Process (MDP) is a tuple (S, A, P, R, γ, µ) which generalizes the
+standard MDP by containing a set of transition kernels P and set of reward functions R.
+Let uncertainty set U = P × R be set of tuples of transition kernels and reward functions
+[18, 24]. The robust performance ρπ
+U of a policy π is defined to be its worst performance on
+the entire uncertainty set U as
+ρπ
+U :=
+min
+(P,R)∈U ρπ
+(P,R).
+(4)
+Our objective is to find an optimal robust policy π∗
+U that maximizes the robust performance
+ρπ
+U, defined as
+ρ∗
+U := max
+π
+ρπ
+U.
+(5)
+Solving the above robust objectives 4 and 5 are strongly NP-hard for general uncertainty
+sets, even if they are convex [32]. Hence, the uncertainty set U = P × R is commonly
+assumed to be s-rectangular, meaning that R and P can be decomposed state-wise as
+R = ×s∈SRs and P = ×s∈SPs. For further simplification, U = P × R is assumed to
+decompose state-action-wise as R = ×(s,a)∈S×ARs,a and P = ×(s,a)∈S×APs,a, known as
+sa-rectangular uncertainty set. Throughout the paper, the uncertainty set is assumed to
+be s-rectangular (or sa-rectangular) unless stated otherwise. Under the s-rectangularity
+assumption, for every policy π, there exists a robust value function vπ
+U which is the minimum
+of vπ
+(P,R) for all (P, R) ∈ U, and the optimal robust value function v∗
+U which is the maximum
+of vπ
+U for all policies π [32], that is
+vπ
+U :=
+min
+(P,R)∈U vπ
+(P,R),
+and
+v∗
+U := max
+π
+vπ
+U.
+This implies, robust policy performance can be rewritten as
+ρπ
+U = ⟨µ, vπ
+U⟩,
+and
+ρ∗
+U = ⟨µ, v∗
+U⟩.
+Furthermore, the robust value function vπ
+U is the fixed point of the robust Bellmen operator
+T π
+U [32, 18], defined as
+(T π
+U v)(s) :=
+min
+(P,R)∈U
+�
+a
+π(a|s)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+,
+and the optimal robust value function v∗
+U is the fixed point of the optimal robust Bellman
+operator T ∗
+U [18, 32], defined as
+T ∗
+U v := max
+π
+T π
+U v.
+The optimal robust Bellman operator T ∗
+U and robust Bellman operators T π
+U are γ contraction
+maps for all policy π [32], that is
+∥T ∗
+U v − T ∗
+U u∥∞ ≤ γ∥u − v∥∞,
+∥T π
+U v − T π
+U u∥∞ ≤ γ∥u − v∥∞,
+∀π, u, v.
+So for all initial values vπ
+0 , v∗
+0, sequences defined as
+vπ
+n+1 := T π
+U vπ
+n,
+v∗
+n+1 := T ∗
+U v∗
+n
+(6)
+converges linearly to their respective fixed points, that is vπ
+n → vπ
+U and v∗
+n → v∗
+U. Given
+this optimal robust value function, the optimal robust policy can be computed as: π∗
+U ∈
+arg maxπ T π
+U v∗
+U [32]. This makes the robust value iteration an attractive method for solving
+s-rectangular robust MDPs.
+4
+
+Table 1: p-variance
+x
+κx(v)
+Remark
+p
+minω∈R∥v − ω1∥p
+Binary search
+∞
+maxs v(s)−mins v(s)
+2
+Semi-norm
+2
+��
+s
+�
+v(s) −
+�
+s v(s)
+S
+�2
+Variance
+1
+�⌊(S+1)/2⌋
+i=1
+v(si)
+Top half - lower half
+− �S
+i=⌈(S+1)/2⌉ v(si)
+where v is sorted, i.e. v(si) ≥ v(si+1)
+∀i.
+3
+Method
+In this section, we consider constraining the uncertainty set around nominal values by the Lp
+norm, which is a natural way of limiting the broad class of s (or sa)-rectangular uncertainty
+sets [9, 16, 3]. We will then derive robust Bellman operators for these uncertainty sets, which
+can be used to obtain robust value functions. This will be done separately for sa-rectangular
+in Subsection 3.1 and s-rectangular case in Section 3.2.
+We begin by making a few useful definitions. We reserve q for Holder conjugate of p, i.e.
+1
+p + 1
+q = 1. Let p-variance function κp : S → R be defined as
+κp(v) := min
+ω∈R∥v − ω1∥p.
+(7)
+For p = 1, 2, ∞, the p-variance function κp has intuitive closed forms as summarized in Table
+1. For general p, it can be calculated by binary search in the range [mins v(s), maxs v(s)] (
+see appendix I for proofs).
+3.1
+(Sa)-rectangular Lp robust Markov Decision Processes
+In accordance with [9], we define sa-rectangular Lp constrained uncertainty set Usa
+p
+as
+Usa
+p := (P0 + P) × (R0 + R)
+where P, R are noise sets around nominal kernel P0 and nominal reward R0 respectively.
+Furthermore, noise sets are sa-rectangular, that is
+P = ×s∈S,a∈APs,a,
+and
+R = ×s∈S,a∈ARs,a,
+and each component are bounded by Lp norm that is
+Rs,a =
+�
+Rs,a ∈ R
+��� |Rs,a| ≤ αs,a
+�
+,
+and
+Ps,a = {Ps,a : S → R
+���
+�
+s′
+Ps,a(s′) = 0
+�
+��
+�
+simplex condition
+, ∥Ps,a∥p ≤ βs,a}
+5
+
+with radius vector α and β. Radius vector β is chosen small enough so that all the transition
+kernels in (P0 + P) are well defined. Further, all transition kernels in (P0 + P) must have
+the sum of each row equal to one, with P0 being a valid transition kernel satisfying this
+requirement. This implies that the elements of P must have a sum of zero across each row
+as ensured by simplex condition above.
+Our setting differs from [9] as they didn’t impose this simplex condition on the kernel
+noise, which renders their setting unrealistic as not all transition kernels in their uncertainty
+set satisfy the properties of transition kernels. This makes our reward regularizer dependent
+on the q-variance of the value function κq(v), instead of the q-th norm of value function ∥v∥q
+in [9].
+The main result of this subsection below states that robust Bellman operators can be
+evaluated using only nominal values and regularizers.
+Theorem 1. sa-rectangular Lp robust Bellman operators are equivalent to reward-value
+regularized (non-robust) Bellman operators:
+(T π
+Usa
+p v)(s) =
+�
+a
+π(a|s)
+�
+−αs,a − γβs,aκq(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+,
+and
+(T ∗
+Usa
+p v)(s) = max
+a∈A
+�
+−αs,a − γβs,aκq(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+.
+Proof. The proof in appendix, it mainly consists of two parts: a) Separating the noise from
+nominal values. b) The reward noise to yields the term −αs,a and noise in kernel yields
+−γβs,aκq(v).
+Note, the reward penalty is proportional to both the uncertainty radiuses and a novel
+variance function κp(v).
+We recover non-robust value iteration by putting uncertainty radiuses (i.e. αs,a, βs,a) to
+zero, in the above results. Furthermore, the same is true for all subsequent robust results in
+this paper.
+Q-Learning
+The above result immediately implies the robust value iteration, and also suggests the Q-value
+iteration of the following form
+Qn+1(s, a) = max
+a
+�
+R0(s, a) − αs,a − γβs,aκq(vn) +
+�
+s′
+P0(s′|s, a) max
+a
+Qn(s′, a′)
+�
+,
+where vn(s) = maxa Qn(s, a), which is further discussed in appendix E.
+Observe that value-variance κp(v) can be estimated online, using batches or other more
+sophisticated methods. This paves the path for generalizing to a model-free setting similar
+to [30].
+Forbidden Transitions
+Now, we focus on the cases where P0(s′|s, a) = 0 for some states s′, that is, forbidden
+transitions. In many practical situations, for a given state, many transitions are impossible.
+For example, consider a grid world example where only a single-step jumps (left, right, up,
+down) are allowed, so in this case, the probability of making a multi-step jump is impossible.
+6
+
+Table 2: Optimal robust Bellman operator evaluation
+U
+(T ∗
+U v)(s)
+remark
+Us
+p
+min x
+s.t.
+���
+�
+Qs − x1
+�
+◦1
+�
+Qs ≥ x
+����
+p= σq(v, s)
+Solve by binary search
+Us
+1
+maxk
+�k
+i=1 Q(s,ai)−σ∞(v,s)
+k
+Highest penalized average
+Us
+2
+By algorithm 1
+High mean and variance
+Us
+∞
+maxa∈A Q(s, a) − σ1(v, s)
+Best action
+Usa
+p
+maxa∈A
+�
+Q(s, a) − αsa − γβsaκq(v)
+�
+Best penalized action
+nr
+maxa Q(s, a)
+Best action
+where nr stands for Non-Robust MDP,
+Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′),
+sorted Q-value: Q(s, a1) ≥ · · · ≥ Q(s, aA)
+, σq(v, s) = αs + γβsκq(v), Qs = Q(s, ·),
+and ◦ is Hadamard product.
+So upon adding noise to the kernel, the system should not start making impossible transitions.
+Therefore, noise set P must satisfy additional constraint: For any (s, a) if P0(s′|s, a) = 0
+then
+P(s′|s, a) = 0,
+∀P ∈ P.
+Incorporating this constraint without much change in the theory is one of our novel contri-
+bution, and is discussed in the appendix C.
+3.2
+S-rectangular Lp robust Markov Decision Processes
+In this subsection, we discuss the core contribution of this paper: the evaluation of robust
+Bellman operators for the s-rectangular uncertainty set.
+We begin by defining s-rectangular Lp constrained uncertainty set Us
+p as
+Us
+p := (P0 + P) × (R0 + R)
+where noise sets are s-rectangular,
+P = ×s∈SPs,
+and
+R = ×s∈SRs,
+and each component are bounded by Lp norm,
+Rs =
+�
+Rs : A → R
+��� ∥Rs∥p ≤ αs
+�
+,
+and
+Ps =
+�
+Ps : S × A → R
+��� ∥Ps∥p ≤ βs,
+�
+s′
+Ps(s′, a) = 0, ∀a
+�
+,
+with radius vectors α and small enough β.
+The result below shows that, compared to the sa-rectangular case, the policy evaluation
+for the s-rectangular case has an extra dependence on the policy.
+7
+
+Theorem 2. (Policy Evaluation) S-rectangular Lp robust Bellman operator is equivalent to
+reward-value-policy regularized (non-robust) Bellman operator:
+(T π
+Uspv)(s) = −
+�
+αs + γβsκq(v)
+�
+∥πs∥q +
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+,
+where ∥πs∥q is q-norm of the vector π(·|s) ∈ ∆A.
+Proof. The proof in the appendix: the techniques are similar to as its sa-rectangular
+counterpart.
+The reward penalty in this case has an additional dependence on the norm of the policy
+(∥πs∥q). This norm is conceptually similar to entropy regularization �
+a π(a|s) ln(π(a|s)),
+which is widely studied in the literature [21, 13, 20, 14, 27], and other regularizers such as
+�
+a π(a|s)tsallis( 1−π(a|s)
+2
+), �
+a π(a|s)cos(cos( π(a|s)
+2
+)), etc.
+Note: These regularizers, which are convex functions, are often used to promote stochas-
+ticity in the policy and thus improve exploration during learning. However, the above result
+shows another benefit of these regularizers: they can improve robustness, which in turn can
+lead to better generalization.
+In literature, the above regularizers are scaled with arbitrary chosen constant, here we
+have the different constant αs + γβsκq(v) for different states.
+This extra dependence makes the policy improvement a more challenging task and thus,
+presents a richer theory.
+Theorem 3. (Policy improvement) For any vector v and state s, (T ∗
+Uspv)(s) is the minimum
+value of x that satisfies
+� �
+a
+�
+Q(s, a) − x
+�p
+1
+�
+Q(s, a) ≥ x
+�� 1
+p = σ,
+(8)
+where Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′), and σ = αs + γβsκq(v).
+Proof. The proof is in the appendix; the main steps are:
+(T ∗
+Uspv)(s) = max
+π (T π
+Uspv)(s),
+(from definition)
+( Using policy evaluation Theorem 2)
+= max
+π
+�
+(T π
+(P0,R0)v)(s)−
+�
+αs + γβsκq(v)
+�
+∥πs∥q
+�
+= max
+πs∈∆A⟨πs, Qs⟩ − σ∥πs∥q
+( where Qs = Q(·|s)).
+The solution to the above optimization problem is technically complex. Specifically, for
+p = 2, the solution is known as the water filling/pouring lemma [1], we generalize it to the
+Lp case, in the appendix.
+To better understand the nature of (8), lets look at the ’sub-optimality distance’ function
+g,
+g(x) :=
+� �
+a
+�
+Q(s, a) − x
+�p
+1
+�
+Q(s, a) ≥ x
+�� 1
+p .
+8
+
+Algorithm 1 s-rectangular L2 robust Bellman operator
+(see algorithm 1 of [1]
+Input: v, s, x = Q(s, ·), and σ = αs + γβsκq(v)
+Output: (T ∗
+Uspv)(s)
+1: Sort x such that x1 ≥ x2, · · · ≥ xA.
+2: Set k = 0 and λ = x1 − σ
+3: while k ≤ A − 1 and λ ≤ xk do
+4:
+k = k + 1
+5:
+λ = 1
+k
+�
+k
+�
+i=1
+xi −
+�
+�
+�
+�kσ2 + (
+k
+�
+i=1
+x2
+i − k
+k
+�
+i=1
+xi)2
+�
+6: end while
+7: return λ
+The g(x) is the cumulative difference between x and the Q-values of actions whose
+Q-value is greater than x. The function is monotonically decreasing, with a lower bound
+of σ at x = maxa Q(s, a) − σ and a value of zero for all x ≥ maxa Q(s, a). Since, (T ∗
+Uspv)(s)
+is the value of x at which the "sub-optimality distance" g(x) is equal to the "uncertainty
+penalty" σ. Hence, (8) can be approximately solved using a binary search between the
+interval [maxa Q(s, a) − σ,
+maxa Q(s, a)].
+We invite the readers to consider the dependence of (T ∗
+Uspv)(s) on p, αs, and βs, specifically:
+1. If αs = βs = 0 then σ = 0 which implies (T ∗
+Uspv)(s) = maxa Q(s, a), same as non-robust
+case.
+2. If p = ∞ then (T ∗
+Uspv)(s) = maxa Q(s, a) − σ, as in the sa-rectangular case.
+3. For p = 1, 2, (8) becomes linear and quadratic equation respectively, hence can be
+solved exactly.
+4. As αs and βs increase, σ increases, resulting in a decrease in (T ∗
+Uspv)(s) at a rate that
+becomes smaller as σ increases. When σ is sufficiently small, (T ∗
+Uspv)(s) = maxa Q(s, a)−
+σ.
+Solution to (8) can be obtained in closed form for the cases of p = 1, ∞, exactly by
+algorithm 1 for p = 2, and approximately by binary search for general p, as summarized in
+table 2.
+In this section, we have demonstrated that robust Bellman operators can be efficiently
+evaluated for both sa and s rectangular Lp robust MDPs, thus enabling efficient robust
+value iteration. In the following sections, we discuss the nature of optimal policies and the
+time complexity of robust value iteration. Finally, we present experiments validating the
+time complexity of robust value iteration.
+4
+Optimal Policies
+In the previous sections, we discussed how to efficiently obtain the optimal robust value
+functions. This section focuses on utilizing these optimal robust value functions to derive
+9
+
+Table 3: Optimal Policy
+U
+π∗
+U(a|s) ∝
+Remark
+Us
+p
+A(s, a)p−11(A(s, a) ≥ 0)
+Top actions proportional to
+(p − 1)-th power of advantage
+Us
+1
+1(A(s, a) ≥ 0)
+Top actions with uniform probability
+Us
+2
+A(s, a)1(A(s, a) ≥ 0)
+Top actions proportion to advantage
+Us
+∞
+1(A(s, a) = 0)
+Best action
+Usa
+p
+1(A(s, a) = maxa A(s, a))
+Best regularized action
+(P0, R0)
+1(A(s, a) = 0)
+Non-robust MDP: Best action
+where Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v∗
+U(s′), and A(s, a) = Q(s, a) − v∗
+U(s).
+the optimal robust policy using
+π∗
+U ∈ arg max
+π
+T π
+U v∗
+U.
+This implies, the robust optimal policy π∗
+U(·|s) at state s, is the policy π that maximizes
+�
+a
+π(a|s)
+min
+(P,R)∈U
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v∗
+U(s′)
+�
+.
+Non-robust MDP admits a deterministic optimal policy that maximizes the optimal
+Q-value Q(s, a) := R(s, a) + γ �
+s′ P(s′|s, a)v∗
+(P,R)(s′).
+sa-rectangular robust MDPs are known to admit a deterministic optimal robust
+policy [18, 24]. Moreover, from Theorem 1, it clear that a sa-rectangular Lp robust MDP
+has a deterministic optimal robust policy that maximizes the regularized Q-value Q(s, a) =
+−αs,a − γβs,aκq(v) + R0(s, a) + γ �
+s′ P0(s′|s, a)v∗
+Usa
+p (s′).
+s-rectangular robust MDPs: For this case, it is known that all optimal robust
+policies can be stochastic [32], however, it was not previously known what the nature of
+this stochasticity was. The result below provides the first explicit characterization of robust
+optimal policies.
+Theorem 4. The optimal robust policy π∗
+Usp can be computed using optimal robust value
+function as:
+π∗
+Usp(a|s) ∝ [Q(s, a) − v∗
+Usp(s)]p−11
+�
+Q(s, a) ≥ v∗
+Usp(s)
+�
+where Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v∗
+Us
+p(s).
+The above policy is a threshold policy that takes actions with a positive advantage, which
+is proportional to the advantage function, while giving more weight to actions with higher
+advantages and avoiding playing actions that are not useful. This policy is different from
+the optimal policy in soft-Q learning with entropy regularization, which is a softmax policy
+10
+
+Algorithm 2 Online s-rectangular Lp robust value iteration
+Input: Initialize Q, v randomly, s0 ∼ µ, and n = 0.
+Output: v = v∗
+Usp.
+1: while not converged; n = n + 1 do
+2:
+Estimate κp(v) using table 1.
+3:
+Approximate (T ∗
+Uspv)(sn) using table 2 and update
+v(sn) = v(sn) + ηn[(T ∗
+Uspv)(sn) − v(sn)].
+4:
+Play action an = a with probability proportional to
+[Q(sn, a) − v(sn)]p−11(Q(sn, a) ≥ v(sn)),
+and get next state sn+1 from the environment.
+5:
+Update Q-value:
+Q(sn, an) =Q(sn, an) + η′
+n[R(sn, an) + γv(sn+1) − Q(sn, an)].
+6: end while
+Table 4: Relative running cost (time) for value iteration
+S
+A
+nr
+Usa
+1
+LP
+Us
+1 LP
+Usa
+1
+Usa
+2
+Usa
+∞
+Us
+1
+Us
+2
+Us
+∞
+Usa
+10
+Us
+10
+10
+10
+1
+1438
+72625
+1.7
+1.5
+1.5
+1.4
+2.6
+1.4
+5.5
+33
+30
+10
+1
+6616
+629890
+1.3
+1.4
+1.4
+1.5
+2.8
+3.0
+5.2
+78
+50
+10
+1
+6622
+4904004
+1.5
+1.9
+1.3
+1.2
+2.4
+2.2
+4.1
+41
+100
+20
+1
+16714
+NA
+1.4
+1.5
+1.5
+1.1
+2.1
+1.5
+3.2
+41
+nr stands for Non-robust MDP
+of the form π(a|s) ∝ eη(Q(a|s)−v(s)) [14, 21, 27]. To the best of our knowledge, this type of
+policy has not been presented in literature before.
+The special cases of the above theorem for p = 1, 2, ∞ along with others are summarized
+in table 3.
+5
+Time complexity
+In this section, we examine the time complexity of robust value iteration:
+vn+1 := T ∗
+U vn
+for different Lp robust MDPs assuming the knowledge of nominal values (P0, R0). Since, the
+optimal robust Bellman operator T ∗
+U is γ-contraction operator [32], meaning that it requires
+only O(log( 1
+ϵ )) iterations to obtain an ϵ-close approximation of the optimal robust value.
+The main challenge is to calculate the cost of one iteration.
+The evaluation of the optimal robust Bellman operators in Theorem 1 and Theorem 3
+has three main components. A) Computing κp(v), which can be done differently depending
+11
+
+Table 5: Time complexity
+Total cost O
+Non-Robust MDP
+log(1/ϵ)S2A
+Usa
+1
+log(1/ϵ)S2A
+Usa
+2
+log(1/ϵ)S2A
+Usa
+∞
+log(1/ϵ)S2A
+Us
+1
+log(1/ϵ)(S2A + SA log(A))
+Us
+2
+log(1/ϵ)(S2A + SA log(A))
+Us
+∞
+log(1/ϵ)S2A
+Usa
+p
+log(1/ϵ)
+�
+S2A + S log(S/ϵ)
+�
+Us
+p
+log(1/ϵ)
+�
+S2A + SA log(A/ϵ)
+�
+Convex U
+Strongly NP Hard
+on the value of p, as shown in table 1. B) Computing the Q-value from v, which requires
+O(S2A) in all cases. And finally, C) Evaluating optimal robust Bellman operators from
+Q-values, which requires different operations such as sorting of the Q-value, calculating the
+best action, and performing a binary search, etc., as shown in table 2. The overall complexity
+of the evaluation is presented in table 5, with the proofs provided in appendix L.
+We can observe that when the state space S is large, the complexity of the robust MDPs
+is the same as that of the non-robust MDPs, as the complexity of all robust MDPs is the
+same as non-robust MDPs at the limit S → ∞ (keeping action space A and tolerance ϵ
+constant). This is verified by our experiments, thus concluding that the Lp robust MDPs
+are as easy as non-robust MDPs.
+6
+Experiments
+In this section, we present numerical results that demonstrate the effectiveness of our methods,
+verifying our theoretical claims.
+Table 4 and Figure 1 demonstrate the relative cost (time) of robust value iteration
+compared to non-robust MDP, for randomly generated kernel and reward functions with
+varying numbers of states S and actions A. The results show that s and sa-rectangular
+MDPs are indeed costly to solve using numerical methods such as Linear Programming (LP).
+Our methods perform similarly to non-robust MDPs, especially for p = 1, 2, ∞. For general
+p, binary search is required for acceptable tolerance, which requires 30−50 iterations, leading
+to a little longer computation time.
+As our complexity analysis shows, value iteration’s relative cost converges to 1 as the
+number of states increases while keeping the number of actions fixed. This is confirmed by
+Figure 1.
+The rate of convergence for all the settings tested was the same as that of the non-robust
+setting, as predicted by the theory. The experiments ran a few times, resulting in some
+stochasticity in the results, but the trend is clear. Further details can be found in section G.
+12
+
+Figure 1: Relative cost of value iteration w.r.t. non-robust MDP at different S with fixed
+A = 10.
+13
+
+Relative cost of value iteration
+U^sa 1
+1.5
+U^s 1
+relative cost to non-robust MDPs
+non-robust
+1.4
+L.3
+1.0
+0
+200
+600
+800
+400
+1000
+1200
+1400
+number of states7
+Conclusion and future work
+We present an efficient robust value iteration for s-rectangular Lp-robust MDPs. Our method
+can be easily adapted to an online setting, as shown in Algorithm 2 for s-rectangular Lp-robust
+MDPs. Algorithm 2 is a two-time-scale algorithm, where the Q-values are approximated at
+a faster time scale and the value function is approximated from the Q-values at a slower
+time scale. The p-variance function κp can be estimated in an online fashion using batches
+or other sophisticated methods. The convergence of the algorithm can be guaranteed from
+[8]; however, its analysis is left for future work.
+Additionally, we introduce a novel value regularizer (κp) and a novel threshold policy
+which may help to obtain more robust and generalizable policies.
+Further research could focus on other types of uncertainty sets, potentially resulting in
+different kinds of regularizers and optimal policies.
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+How to read appendix
+1. Section A contains related work.
+2. Section B contains additional properties and results that couldn’t be included in the
+main section for the sake of clarity and space. Many of the results in the main paper
+is special cases of the results in this section.
+3. Section C contains the discussion on zero transition kernel (forbidden transitions).
+4. Section D contains a possible connection this work to UCRL.
+5. Section G contains additional experimental results and a detailed discussion.
+6. All the proofs of the main body of the paper is presented in the section K and L.
+7. Section I contains helper results for section K. Particularly, it discusses p-mean function
+ωp and p-variance function κp.
+8. Section J contains helper results for section K. Particularly, it discusses Lp water
+pouring lemma, necessary to evaluate robust optimal Bellman operator (learning) for
+s-rectangular Lp robust MDPs.
+9. Section L contains time complexity proof for model based algorithms.
+16
+
+10. Section E develops Q-learning machinery for (sa)-rectangular Lp robust MDPs based
+on the results in the main section. It is not used in the main body or anywhere
+else, but this provides a good understanding for algorithms proposed in section F for
+(sa)-rectangular case.
+11. Section F contains model-based algorithms for s and (sa)-rectangular Lp robust MDPs.
+It also contains, remarks for special cases for p = 1, 2, ∞.
+A
+Related Work
+R-Contamination Uncertainty Robust MDPs
+The paper [30] considers the following uncertainty set for some fixed constant 0 ≤ R ≤ 1,
+Psa = {(1 − R)(P0)(·|s, a) + RP | P ∈ ∆S},
+s ∈ S, a ∈ A,
+(9)
+and P = ⊗s,aPs,a,
+U = {R0} × P. The robust value function vπ
+U is the fixed point of
+the robust Bellman operator defined as
+(T π
+U v)(s) := min
+P ∈P
+�
+a
+π(a|s)[R0(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)],
+(10)
+=
+�
+a
+π(a|s)[R0(s, a) − γR max
+s
+v(s) + (1 − R)γ
+�
+s′
+P0(s′|s, a)v(s′)].
+(11)
+And the optimal robust value function v∗
+U⊣ is the fixed point of the optimal robust Bellman
+operator defined as
+(T ∗
+U v)(s) := max
+π
+min
+P ∈P
+�
+a
+π(a|s)[R0(s, a) + γ(1 − R)
+�
+s′
+P(s′|s, a)v(s′)],
+(12)
+= max
+a [R0(s, a) − γR max
+s
+v(s) + γ(1 − R)
+�
+s′
+P0(s′|s, a)v(s′)].
+(13)
+Since, the uncertainty set is sa-rectangular, hence the map is a contraction [24], so the
+robust value iteration here, will also converge linearly similar to non-robust MDPs. It is also
+possible to obtain Q-learning as following
+Qn+1(s, a) = R0(s, a) − γR max
+s,a Qn(s, a) + γ(1 − R)
+�
+s′
+P0(s′|s, a) max
+s′
+Qn(s′, a′).
+(14)
+Convergence of the above Q-learning follows from the contraction of robust value iteration.
+Further, it is easy to see that model-free Q-learning can be obtained from the above.
+A follow-up work [31] proposes a policy gradient method for the same.
+Proposition 1. (Theorem 3.3 of [31]) Consider a class of policies Π satisfying Assumption
+3.2 of [31]. The gradient of the robust return is given by
+∇ρπθ =
+γR
+(1 − γ)(1 − γ + γR)
+�
+s,a
+dπθ
+µ (s, a)∇πθ(a|s)Qπθ
+U (s, a)
++
+1
+1 − γ + γR
+�
+s,a
+dπθ
+sθ (s, a)∇πθ(a|s)Qπθ
+U (s, a),
+where sθ ∈ arg max vπθ
+U (s), and Qπ
+U(s, a) = �
+a π(a|s)
+�
+R0(s, a) − γR maxs vπ
+U(s) + γ(1 −
+R) �
+s′ P0(s′|s, a)vπ
+U(s′)
+�
+.
+17
+
+The work shows that the proposed robust policy gradient method converges to the global
+optimum asymptotically under direct policy parameterization.
+The uncertainty set considered here, is sa-rectangular, as uncertainty in each state-action
+is independent, hence the regularizer term (γR maxs v(s)) is independent of policy, and the
+optimal (and greedy) policy is deterministic. It is unclear, how the uncertainty set can be
+generalized to the s-rectangular case. Observe that the above results resemble very closely
+our sa-rectangular L1 robust MDPs results.
+Twice Regularized MDPs
+The paper [9] converts robust MDPs to twice regularized MDPs, and proposes a gradient
+based policy iteration method for solving them.
+Proposition 2. (corollary 3.1 of [9]) (s-rectangular reward robust policy evaluation) Let the
+uncertainty set be U = (R0 + R) × {P0}, where Rs = {rs ∈ RA | ∥rs∥ ≤ αs} for all s ∈ S.
+Then the robust value function vπ
+U is the optimal solution to the convex optimization problem:
+max
+v∈RA⟨µ, v⟩
+s.t.
+v(s) ≤ (T π
+R0,P0v)(s) − αs∥πs∥,
+∀s ∈ S.
+It derives the policy gradient for reward robust MDPs to obtain the optimal robust policy
+π∗
+U.
+Proposition 3. (Proposition 3.2 of [9]) (s-rectangular reward robust policy gradient) Let
+the uncertainty set be U = (R0 + R) × {P0}, where Rs = {rs ∈ RA | ∥rs∥ ≤ αs} for all
+s ∈ S. Then the gradient of the reward robust objective ρπ
+U := ⟨µ, vπ
+U⟩ is given by
+∇ρπ
+U = E(s,a)∼dπ
+P0
+�
+∇ ln(π(a|s))
+�
+Qπ
+U(s, a) − αs
+π(a|s)
+∥πs∥
+��
+,
+where Qπ
+U(s, a) := min(R,P )∈U[R(s, a) + γ �
+s′ P(s′|s, a)vπ
+U(s′)].
+Proposition 4. (Corollary 4.1 of [9]) (s-rectangular general robust policy evaluation) Let
+the uncertainty set be U = (R0 + R) × {P0 + P}, where Rs = {rs ∈ RA | ∥rs∥ ≤ αs} and
+Ps = {Ps ∈ RS×A | ∥Ps∥ ≤ βs} for all s ∈ S. Then the robust value function vπ
+U is the
+optimal solution to the convex optimization problem:
+max
+v∈RA⟨µ, v⟩
+s.t.
+v(s) ≤ (T π
+R0,P0v)(s) − αs∥πs∥ − γβs∥v∥∥πs∥,
+∀s ∈ S.
+Same as the reward robust case, the paper tries to find a policy gradient method to
+obtain the optimal robust policy. Unfortunately, the dependence of regularizer terms on
+value makes it a very difficult task. Hence it proposes the R2MPI algorithm (algorithm 1 of
+[9]) for the purpose that optimizing the greedy step via projection onto the simplex using
+a black box solver. Note that the above proposition is not same as our policy evaluation
+(although it looks similar), it requires some extra assumptions (assumption 5.1 [9]) and lot
+of work ensure R2 Bellman operator is contraction etc. In our case, we directly evaluate
+robust Bellman operator that has already proven to be a contraction, hence we don’t require
+any extra assumption nor any other work as [9].
+Our work makes improvements over this work by explicitly solving both policy evaluation
+and policy improvement in general robust MDPs. It also makes more realistic assumptions
+on the transition kernel uncertainty set.
+18
+
+Regularizer solves Robust MDPs
+The work [11] looks in the opposite direction than we do. It investigates the impact of the
+popularly used entropy regularizer on robustness. It finds that MaxEnt can be used to
+maximize a lower bound on a certain robust RL objective (reward robust).
+As we noticed that ∥πs∥q behaves like entropy in our regularization. Further, our work
+also deals with uncertainty in transition kernel in addition to the uncertainty in reward
+function.
+Upper Confidence RL
+The upper confidence setting in [4, 3] is very similar to our Lp robust setting. We refer to
+this discussion in section D.
+B
+S-rectangular: More Properties
+Definition 1. We begin with the following notational definitions.
+1. Q-value at value function v is defined as
+Qv(s, a) := R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′).
+2. Optimal Q-value is defined as
+Q∗
+U(s, a) = R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v∗
+U(s′)
+3. With little abuse of notation, Q(s, ai) shall denote the ith best value in state s, that is
+Q(s, a1) ≥ Q(s, a2) ≥, · · · , ≥ Q(s, aA).
+4. πv
+U denotes the greedy policy at value function v, that is
+T ∗
+U v = T πv
+U
+U
+v.
+5. χp(s) denotes the number of active actions in state s in s-rectangular Lp robust MDPs,
+defined as
+χp(s) :=
+�� {a | π∗
+Us
+p(a|s) ≥ 0}
+�� .
+6. χp(p, s) denotes the number of active actions in state s at value function v in s-
+rectangular Lp robust MDPs, defined as
+χp(v, s) :=
+�� {a | πv
+Us
+p(a|s) ≥ 0}
+�� .
+We saw above that optimal policy in s-rectangular robust MDPs may be stochastic. The
+action that has a positive advantage is active and the rest are inactive. Let χp(s) be the
+number of active actions in state s, defined as
+χp(s) :=
+�� {a | π∗
+Us
+p(a|s) ≥ 0}
+��=
+�� {a | Q∗
+Us
+p(s, a) ≥ v∗
+Us
+p(s)}
+�� .
+(15)
+19
+
+Last equality comes from Theorem 4. One direct relation between Q-value and value function
+is given by
+v∗
+Us
+p(s) =
+�
+a
+π∗
+Us
+p(a|s)
+�
+−
+�
+αs + γβsκq(v)
+�
+∥π∗
+Us
+p(·|s)∥q + Q∗
+Us
+p(s, a)
+�
+.
+(16)
+The above relation is very convoluted compared to non-robust and sa-rectangular robust
+cases. The property below illuminates an interesting relation.
+Property 1. (Optimal Value vs Q-value) v∗
+Us
+p(s) is bounded by the Q-value of χp(s)th and
+(χp(s) + 1)th actions, that is ,
+Q∗
+Us
+p(s, aχp(s)+1) < v∗
+Us
+p(s) ≤ Q∗
+Us
+p(s, aχp(s)).
+This special case of the property 2, similarly table 6 is special case of table 8.
+Table 6: Optimal value function and Q-value
+v∗(s) = maxa Q∗(s, a)
+Best value
+v∗
+Usa
+p (s) = maxa[αs,a − γβs,aκq(v∗
+Usa
+p ) − Q∗
+Usa
+p (s, a)]
+Best regularized value
+Q∗
+Us
+p(s, aχp(s)+1) < v∗
+Us
+p(s) ≤ Q∗
+Us
+p(s, aχp(s))
+Sandwich!
+where v∗, Q∗ is the optimal value function and Q-value respectively
+of non-robust MDP.
+The same is true for the non-optimal Q-value and value function.
+Theorem 5. (Greedy policy) The greedy policy πv
+Us
+p is a threshold policy, that is proportional
+to the advantage function, that is
+πv
+Us
+p(a|s) ∝
+�
+Qv(s, a) − (T ∗
+Us
+pv)(s)
+�p−1 1
+�
+Qv(s, a) ≥ (T ∗
+Us
+pv)(s)
+�
+.
+The above theorem is proved in the appendix, and Theorem 4 is its special case. So is
+table 3 special case of table 7.
+Table 7: Greedy policy at value function v
+U
+πv
+U(a|s) ∝
+remark
+Us
+p
+(Qv(s, a) − (T ∗
+U v)(s))p−11(Av
+U(s, a) ≥ 0)
+top actions proportional to
+(p − 1)th power of its advantage
+Us
+1
+1(Av
+U(s,a)≥0)
+�
+a 1(Av
+U(s,a)≥0)
+top actions with uniform probability
+Us
+2
+Av
+U(s,a)1Av
+U(s,a)≥0)
+�
+a Av
+U(s,a)1(Av
+U(s,a)≥0)
+top actions proportion to advantage
+Us
+∞
+arg maxa∈A Qv(s, a)
+best action
+Usa
+p
+arg maxa[−αsa − γβsaκq(v) + Qv(s, a)]
+best action
+where Av
+U(s, a) = Qv(s, a) − (T ∗
+U v)(s) and Qv(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′).
+20
+
+The above result states that the greedy policy takes actions that have a positive advantage,
+so we have.
+χp(v, s) :=
+�� {a | πv
+Us
+p(a|s) ≥ 0}
+��=
+�� {a | Qv(s, a) ≥ (T ∗
+Us
+p)v(s)}
+�� .
+(17)
+Property 2. (Greedy Value vs Q-value) (T ∗
+Us
+pv)(s) is bounded by the Q-value of χp(v, s)th
+and (χp(v, s) + 1)th actions, that is ,
+Qv(s, aχp(v,s)+1) < (T ∗
+Us
+pv)(s) ≤ Qv(s, aχp(v,s)).
+Table 8:
+Greedy value function and Q-value
+(T ∗v)(s) = maxa Qv(s, a)
+Best value
+(T ∗
+Usa
+p )v(s) = maxa[αs,a − γβs,aκq(v) − Qv(s, a)]
+Best regularized value
+Qv(s, aχp(v,s)+1) < (T ∗
+Us
+p)v(s) ≤ Qv(s, aχp(v,s))
+Sandwich!
+where Qv(s, a1) ≥, · · · , ≥ Qv(s, aA).
+The property below states that we can compute the number of active actions χp(v, s)
+(and χp(s)) directly without computing greedy (optimal) policy.
+Property 3. χp(v, s) is number of actions that has positive advantage, that is
+χp(v, s) := max{k |
+k
+�
+i=1
+�
+Qv(s, ai) − Qv(s, ak)
+�p≤ σp},
+where σ = αs + γβsκq(v), and Qv(s, a1) ≥ Qv(s, a2), ≥ · · · ≥ Q(s, aA).
+When uncertainty radiuses (αs, βs) are zero (essentially σ = 0 ), then χp(v, s) = 1, ∀v, s,
+that means, greedy policy taking the best action. In other words, all the robust results
+reduce to non-robust results as discussed in section 2.2 as the uncertainty radius becomes
+zero.
+Algorithm 3 Algorithm to compute s-rectangular Lp robust optimal Bellman Operator
+1: Input: σ = αs + γβsκq(v),
+Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′).
+2: Output (T ∗
+Us
+pv)(s), χp(v, s)
+3: Sort Q(s, ·) and label actions such that Q(s, a1) ≥ Q(s, a2), · · · .
+4: Set initial value guess λ1 = Q(s, a1) − σ and counter k = 1.
+5: while k ≤ A − 1 and λk ≤ Q(s, ak) do
+6:
+Increment counter: k = k + 1
+7:
+Take λk to be a solution of the following
+k
+�
+i=1
+�
+Q(s, ai) − x
+�p= σp,
+and
+x ≤ Q(s, ak).
+(18)
+8: end while
+9: Return: λk, k
+21
+
+C
+Revisiting kernel noise assumption
+Sa-Rectangular Uncertainty
+Suppose at state s, we know that it is impossible to have transition (next) to some states
+(forbidden states Fs,a) under some action. That is, we have the transition uncertainty set P
+and nominal kernel P0 such that
+P0(s′|s, a) = P(s′|s, a) = 0,
+∀P ∈ P, ∀s′ ∈ Fs,a.
+(19)
+Then we define, the kernel noise as
+Ps,a = {P | ∥P∥p = βs,a,
+�
+s′
+P(s′) = 0,
+P(s”) = 0, ∀s” ∈ Fs,a}.
+(20)
+In this case, our p-variance function is redefined as
+κp(v, s, a) =
+min
+∥P ∥p=βs,a,
+�
+s′ P (s′)=0,
+P (s”)=0,
+∀s”∈Fs,a⟨P, v⟩
+(21)
+= min
+ω∈R∥u − ω1∥p,
+where u(s) = v(s)1(s /∈ Fs,a).
+(22)
+=κp(u)
+(23)
+This basically says, we consider value of only those states that is allowed (not forbidden) in
+calculation of p-variance. For example, we have
+κ∞(v, s, a) = maxs/∈Fs,a v(s) − mins/∈Fs,a v(s)
+2
+.
+(24)
+(25)
+So theorem 1 of the main paper can be re-stated as
+Theorem 6. (Restated) (Sa)-rectangular Lp robust Bellman operator is equivalent to reward
+regularized (non-robust) Bellman operator. That is, using κp above, we have
+(T π
+Usa
+p v)(s) =
+�
+a
+π(a|s)[−αs,a − γβs,aκq(v, s, a) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)],
+(T ∗
+Usa
+p v)(s) = max
+a∈A[−αs,a − γβs,aκq(v, s, a) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)].
+S-Rectangular Uncertainty
+This notion can also be applied to s-rectanular uncertainty, but with little caution. Here, we
+define forbidden states in state s to be Fs (state dependent) instead of state-action dependent
+in sa-rectangular case. Here, we define p-variance as
+κp(v, s) = κp(u),
+where u(s) = v(s)1(s /∈ Fs).
+(26)
+So the theorem 2 can be restated as
+Theorem 7. (restated) (Policy Evaluation) S-rectangular Lp robust Bellman operator is
+equivalent to reward regularized (non-robust) Bellman operator, that is
+(T π
+Us
+pv)(s) = −
+�
+αs +γβsκq(v, s)
+�
+∥π(·|s)∥q +
+�
+a
+π(a|s)
+�
+R0(s, a)+γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+where κp is defined above and ∥π(·|s)∥q is q-norm of the vector π(·|s) ∈ ∆A.
+22
+
+All the other results (including theorem 4), we just need to replace the old p-variance
+function with new p-variance function appropriately.
+D
+Application to UCRL
+In robust MDPs, we consider the minimization over uncertainty set to avoid risk. When we
+want to discover the underlying kernel by exploration, then we seek optimistic policy, then
+we consider the maximization over uncertainty set [4, 3, 2]. We refer the reader to the step 3
+of the UCRL algorithm [4], which seeks to find
+arg max
+π
+max
+R,P ∈U⟨µ, vπ
+P,R⟩,
+(27)
+where
+U = {(R, P) | |R(s, a) − R0(s, a)| ≤ αs,a, |P(s′|s, a) − P0(s′|s, a)| ≤ βs,a,s′, P ∈ (∆S)S×A}
+for current estimated kernel P0 and reward function R0. We refer section 3.1.1 and step 4 of
+the UCRL 2 algorithm of [3], which seeks to find
+arg max
+π
+max
+R,P ∈U⟨µ, vπ
+P,R⟩,
+(28)
+where
+U ={(R, P) | |R(s, a) − R0(s, a)| ≤ αs,a,
+∥P(·|s, a) − P0(·|s, a)∥1 ≤ βs,a, P ∈ (∆S)S×A}
+The uncertainty radius α, β depends on the number of samples of different transitions and
+observations of the reward. The paper [4] doesn’t explain any method to solve the above
+problem. UCRL 2 algorithm [3], suggests to solve it by linear programming that can be very
+slow. We show that it can be solved by our methods.
+The above problem can be tackled as following
+max
+π
+max
+R,P ∈Usa
+p
+⟨µ, vπ
+P,R⟩.
+(29)
+We can define, optimistic Bellman operators as
+ˆT π
+U v := max
+R,P ∈U vπ
+P,R,
+ˆT ∗
+U v := max
+π
+max
+R,P ∈U vπ
+P,R.
+(30)
+The well definition and contraction of the above optimistic operators may follow directly
+from their pessimistic (robust) counterparts. We can evaluate above optimistic operators as
+( ˆT π
+Usa
+p v)(s) =
+�
+a
+π(a|s)
+�
+R0(s, a) + αs,a + βs,aγκq(v) +
+�
+s′
+P0(s′|s, a)v(s′)
+�
+,
+(31)
+( ˆT ∗
+Usa
+p v)(s) = max
+a
+�
+R0(s, a) + αs,a + βs,aγκq(v) +
+�
+s′
+P0(s′|s, a)v(s′)
+�
+.
+(32)
+The uncertainty radiuses α, β and nominal values P0, R0 can be found by similar analysis by
+[4, 3]. We can get the Q-learning from the above results as
+Q(s, a) → R0(s, a) − αs,a − γβs,aκq(v) + γ
+�
+s′
+P0(s′|s, a) max
+a′ Q(s′, a′),
+(33)
+23
+
+where v(s) = maxa Q(s, a). From law of large numbers, we know that uncertainty radiuses
+αs,a, βs,a behaves as O( 1
+√n) asymptotically with number of iteration n. This resembles very
+closely to UCB VI algorithm [5]. We emphasize that similar optimistic operators can be
+defined and evaluated for s-rectangular uncertainty sets too.
+E
+Q-Learning for sa-rectangular MDPs
+In view of Theorem 1, we can define Qπ
+Usa
+p , the robust Q-values under policy π for (sa)-
+rectangular Lp constrained uncertainty set Usa
+p
+as
+Qπ
+Usa
+p (s, a) := −αs,a − γβs,aκq(vπ
+Usa
+p ) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)vπ
+Usa
+p (s′).
+(34)
+This implies that we have the following relation between robust Q-values and robust value
+function, same as its non-robust counterparts,
+vπ
+Usa
+p (s) =
+�
+a
+π(a|s)Qπ
+Usa
+p (s, a).
+(35)
+Let Q∗
+Usa
+p denote the optimal robust Q-values associated with optimal robust value v∗
+Usa
+p , given
+as
+Q∗
+Usa
+p (s, a) := −αs,a − γβs,aκq(v∗
+Usa
+p ) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v∗
+Usa
+p (s′).
+(36)
+It is evident from Theorem 1 that optimal robust value and optimal robust Q-values satisfies
+the following relation, same as its non-robust counterparts,
+v∗
+Usa
+p (s′) = max
+a∈A Q∗
+Usa
+p (s, a).
+(37)
+Combining 37 and 36, we have optimal robust Q-value recursion as follows
+Q∗
+Usa
+p (s, a) = −αs,a − γβs,aκq(v∗
+Usa
+p ) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a) max
+a∈A Q∗
+Usa
+p (s, a).
+(38)
+The above robust Q-value recursion enjoys similar properties as its non-robust counterparts.
+Corollary 1. ((sa)-rectangular Lp regularized Q-learning) Let
+Qn+1(s, a) = R0(s, a) − αsa − γβsaκq(vn) + γ
+�
+s′
+P0(s′|s, a) max
+a∈A Qn(s′, a),
+where vn(s) = maxa∈A Qn(s, a), then Qn converges to Q∗
+Usa
+p linearly.
+Observe that the above Q-learning equation is exactly the same as non-robust MDP
+except the reward penalty. Recall that κ1(v) = 0.5(maxs v(s) − mins v(s)) is difference
+between peak to peak values and κ2(v) is variance of v, that can be easily estimated. Hence,
+model free algorithms for (sa)-rectangular Lp robust MDPs for p = 1, 2, can be derived
+easily from the above results. This implies that (sa)-rectangular L1 and L2 robust MDPs
+are as easy as non-robust MDPs.
+24
+
+F
+Model Based Algorithms
+In this section, we assume that we know the nominal transitional kernel and nominal reward
+function. Algorithm 4, algorithm 5 is model based algorithm for (sa)-rectangular and s
+rectangular Lp robust MDPs respectively. It is explained in the algorithms, how to get deal
+with specail cases (p = 1, 2, ∞) in a easy way.
+Algorithm 4 Model Based Q-Learning Algorithm for SA Rectangular Lp Robust MDP
+1: Input: αs,a, βs,a are uncertainty radius in reward and transition kernel respectively in
+state s and action a. Transition kernel P and reward vector R. Take initial Q-values Q0
+randomly and v0(s) = maxa Q0(s, a).
+2: while not converged do
+3:
+Do binary search in [mins vn(s), maxs vn(s)] to get q-mean ωn, such that
+�
+s
+(vn(s) − ωn)
+|vn(s) − ωn| |vn(s) − ωn|
+1
+p−1 = 0.
+(39)
+4:
+Compute q-variance:
+κn = ∥v − ωn∥q.
+5:
+Note: For p = 1, 2, ∞, we can compute κn exactly in closed from, see table 1.
+6:
+for s ∈ S do
+7:
+for a ∈ A do
+8:
+Update Q-value as
+Qn+1(s, a) = R0(s, a) − αsa − γβsaκn + γ
+�
+s′
+P0(s′|s, a) max
+a
+Qn(s′, a).
+9:
+end for
+10:
+Update value as
+vn+1(s) = max
+a
+Qn+1(s, a).
+11:
+end for
+n → n + 1
+12: end while
+25
+
+Algorithm 5 Model Based Algorithm for S Rectangular Lp Robust MDP
+1: Take initial Q-values Q0 and value function v0 randomly.
+2: Input: αs, βs are uncertainty radius in reward and transition kernel respectively in state
+s.
+3: while not converged do
+4:
+Do binary search in [mins vn(s), maxs vn(s)] to get q-mean ωn, such that
+�
+s
+(vn(s) − ωn)
+|vn(s) − ωn| |vn(s) − ωn|
+1
+p−1 = 0.
+(40)
+5:
+Compute q-variance:
+κn = ∥v − ωn∥q.
+6:
+Note: For p = 1, 2, ∞, we can compute κn exactly in closed from, see table 1.
+7:
+for s ∈ S do
+8:
+for a ∈ A do
+9:
+Update Q-value as
+Qn+1(s, a) = R0(s, a) + γ
+�
+s′
+P0(s′|s, a)vn+1(s′).
+(41)
+10:
+end for
+11:
+Sort actions in decreasing order of the Q-value, that is
+Qn+1(s, ai) ≥ Qn+1(s, ai+1).
+(42)
+12:
+Value evaluation:
+vn+1(s) = x
+such that
+(αs + γβsκn)p =
+�
+Qn+1(s,ai)≥x
+|Qn+1(s, ai) − x|p.
+(43)
+13:
+Note: We can compute vn+1(s) exactly in closed from for p = ∞ and for p = 1, 2,
+we can do the same using algorithm 8,7 respectively, see table 2.
+14:
+end for
+n → n + 1
+15: end while
+26
+
+Algorithm 6 Model based algorithm for s-recantangular L1 robust MDPs
+1: Take initial value function v0 randomly and start the counter n = 0.
+2: while not converged do
+3:
+Calculate q-variance:
+κn = 1
+2
+�
+maxs vn(s) − mins vn(s)
+�
+4:
+for s ∈ S do
+5:
+for a ∈ A do
+6:
+Update Q-value as
+Qn(s, a) = R0(s, a) + γ
+�
+s′
+P0(s′|s, a)vn(s′).
+(44)
+7:
+end for
+8:
+Sort actions in state s, in decreasing order of the Q-value, that is
+Qn(s, a1) ≥ Qn(s, a2), · · · ≥ Qn(s, aA).
+(45)
+9:
+Value evaluation:
+vn+1(s) = max
+m
+�m
+i=1 Qn(s, ai) − αs − βsγκn
+m
+.
+(46)
+10:
+Value evaluation can also be done using algorithm 8.
+11:
+end for
+n → n + 1
+12: end while
+G
+Experiments
+The table 4 contains relative cost (time) of robust value iteration w.r.t. non-robust MDP,
+for randomly generated kernel and reward function with the number of states S and the
+number of action A.
+Notations
+S : number of state, A: number of actions, Usa
+p
+LP: Sa rectangular Lp robust MPDs by Linear
+Programming, Us
+p LP: S rectangular Lp robust MPDs by Linear Programming and other
+numerical methods, Usa/s
+p=1,2,∞ : Sa/S rectangular L1/L2/L∞ robust MDPs by closed form
+method (see table 2, theorem 3) Usa/s
+p=5,10 : Sa/S rectangular L5/L10 robust MDPs by binary
+search (see table 2, theorem 3 of the paper)
+Observations
+1. Our method for s/sa rectangular L1/L2/L∞ robust MDPs takes almost same (1-3 times)
+the time as non-robust MDP for one iteration of value iteration. This confirms our complexity
+analysis (see table 4 of the paper) 2. Our binary search method for sa rectangular L5/L10
+robust MDPs takes around 4 − 6 times more time than non-robust counterpart. This is
+27
+
+Table 9: Relative running cost (time) for value iteration
+U
+S=10 A=10
+S=30 A=10
+S=50 A=10
+S=100 A=20
+remark
+non-robust
+1
+1
+1
+1
+Usa
+∞ by LP
+1374
+2282
+2848
+6930
+lp
+Usa
+1
+by LP
+1438
+6616
+6622
+16714
+lp
+Us
+1 by LP
+72625
+629890
+4904004
+NA
+lp/minimize
+Usa
+1
+1.77
+1.38
+1.54
+1.45
+closed form
+Usa
+2
+1.51
+1.43
+1.91
+1.59
+closed form
+Usa
+∞
+1.58
+1.48
+1.37
+1.58
+closed form
+Us
+1
+1.41
+1.58
+1.20
+1.16
+closed form
+Us
+2
+2.63
+2.82
+2.49
+2.18
+closed form
+Us
+∞
+1.41
+3.04
+2.25
+1.50
+closed form
+Usa
+5
+5.4
+4.91
+4.14
+4.06
+binary search
+Usa
+10
+5.56
+5.29
+4.15
+3.26
+binary search
+Us
+5
+33.30
+89.23
+40.22
+41.22
+binary search
+Us
+10
+33.59
+78.17
+41.07
+41.10
+binary search
+lp stands for scipy.optimize.linearprog
+28
+
+Table 10: Relative running cost (time) for value iteration
+U
+S=10 A=10
+S=100 A=20
+remark
+non-robust
+1
+1
+Usa
+1
+0.999
+0.999
+closed form
+Usa
+2
+0.999
+0.999
+closed form
+Usa
+∞
+1.000
+0.998
+closed form
+Us
+1
+0.999
+0.999
+closed form
+Us
+2
+0.999
+0.999
+closed form
+Us
+∞
+1.000
+0.998
+closed form
+Usa
+5
+0.999
+0.995
+binary search
+Usa
+10
+1.000
+0.999
+binary search
+Us
+5
+1.000
+0.999
+binary search
+Us
+10
+1.000
+0.995
+binary search
+due to extra iterations required to find p-variance function κp(v) through binary search. 3.
+Our binary search method for s rectangular L5/L10 robust MDPs takes around 30 − 100
+times more time than non-robust counterpart. This is due to extra iterations required to
+find p-variance function κp(v) through binary search and Bellman operator. 4. One common
+feature of our method is that time complexity scales moderately as guranteed through our
+complexity analysis. 5. Linear programming methods for sa-rectangualr L1/L∞ robsust
+MDPs take atleast 1000 times more than our methods for small state-action space, and
+it scales up very fast. 6. Numerical methods (Linear programming for minimization over
+uncertianty and ’scipy.optimize.minimize’ for maximization over policy) for s-rectangular L1
+robust MDPs take 4-5 order more time than our mehtods (and non-robust MDPs) for very
+small state-action space, and scales up too fast. The reason is obvious, as it has to solve
+two optimization, one minimization over uncertainty and other maximization over policy,
+whereas in the sa-rectangular case, only minimization over uncertainty is required. This
+confirms that s-rectangular uncertainty set is much more challenging.
+Rate of convergence
+The rate of convergence for all were approximately the same as 0.9 = γ, as predicted by
+theory. And it is well illustrated by the relative rate of convergence w.r.t. non-robust by the
+table G.
+In the above experiments, Bellman updates for sa/s rectangular L1/L2/L∞ were done in
+29
+
+closed form, and for L5/L10 were done by binary search as suggested by table 2 and theorem
+3.
+Note: Above experiments’ results are for few runs, hence containing some stochas-
+ticity but the general trend is clear. In the final version, we will do averaging of many
+runs to minimize the stochastic nature. Results for many different runs can be found at
+https://github.com/******.
+Note that the above experiments were done without using too much parallelization.
+There is ample scope to fine-tune and improve the performance of robust MDPs. The above
+experiments confirm the theoretical complexity provided in Table 4 of the paper. The codes
+and results can be found at https://github.com/******.
+Experiments parameters
+Number of states S (variable), number of actions A (variable), transition kernel and reward
+function generated randomly, discount factor 0.9, uncertainty radiuses =0.1 (for all states
+and action, just for convenience ), number of iterations = 100, tolerance for binary search =
+0.00001
+Hardware
+The experiments are done on the following hardware: Intel(R) Core(TM) i5-4300U CPU @
+1.90GHz 64 bits, memory 7862MiB Software: Experiments were done in python, using numpy,
+scipy.optimize.linprog for Linear programmig for policy evalution in s/sa rectangular robust
+MDPs, scipy.optimize.minize and scipy.optimize.LinearConstraints for policy improvement
+in s-rectangular L1 robust MDPs.
+H
+Extension to Model Free Settings
+Extension of Q-learning (in section E ) for sa-rectangular MDPs to model free setting can
+easily done similar to [30], also policy gradient method can be obtained as [31]. The only
+thing, we need to do, is to be able to compute/estimate κq online. It can be estimated
+using an ensemble (samples). Further, κ2 can be estimated by the estimated mean and the
+estimated second moment. κ∞ can be estimated by tracking maximum and minimum values.
+For s-rectangular case too, we can obtain model-free algorithms easily, by estimating κq
+online and keeping track of Q-values and value function. The convergence analysis may be
+similar to [30], especially for sa-rectangular case, and for the other, it would be two time
+scale, which can be dealt with techniques in [8]. We leave this for future work. It would be
+interesting to obtain policy gradient methods for this case, which we believe can be obtained
+from the policy evaluation theorem.
+I
+p-variance
+Recall that κp is defined as follows
+κp(v) = min
+w ∥v − ω1∥p = ∥v − ωp∥p.
+30
+
+Now, observe that
+∂∥v − ω∥p
+∂ω
+= 0
+=⇒
+�
+s
+sign(v(s) − ω)|v(s) − ω|p−1 = 0,
+=⇒
+�
+s
+sign(v(s) − ωp(v))|v(s) − ωq(v)|p−1 = 0.
+(47)
+For p = ∞, we have
+lim
+p→∞
+���
+�
+s
+sign
+�
+v(s) − ω∞(v)
+��� v(s) − ω∞(v)
+��p���
+1
+p = 0
+=
+�
+max
+s |v(s) − ω∞(v)|
+�
+lim
+p→∞
+���
+�
+s
+sign
+�
+v(s) − ω∞(v)
+��
+|v(s) − ω∞(v)
+��
+maxs|v(s) − ω∞(v)|
+�p���
+1
+p
+Assuming max
+s |v(s) − ω∞(v)| ̸= 0 otherwise ω∞ = v(s) = v(s′),
+∀s, s′
+=⇒ lim
+p→∞
+���
+�
+s
+sign
+�
+v(s) − ω∞(v)
+��
+|v(s) − ω∞(v)
+��
+maxs|v(s) − ω∞(v)|
+�p���
+1
+p = 0
+To avoid technical complication, we assume max
+s
+v(s) > v(s) < min
+s
+v(s),
+∀s
+=⇒ lim
+p→∞|max
+s
+v(s) − ω∞(v)| = lim
+p→∞|min
+s
+v(s) − ω∞(v)|
+=⇒ max
+s
+v(s) − lim
+q→∞ ω∞(v) = −(min
+s
+v(s) − lim
+p→∞ ω∞(v)),
+(managing signs)
+=⇒ lim
+p→∞ ω∞(v) = maxs v(s) + mins v(s)
+2
+.
+(48)
+κ∞(v) =∥v − ω∞1∥∞
+=∥v − maxs v(s) + mins v(s)
+2
+1∥∞,
+(putting in value of ω∞)
+=maxs v(s) − mins v(s)
+2
+(49)
+For p = 2, we have
+κ2(v) =∥v − ω21∥2
+=∥v −
+�
+s v(s)
+S
+1∥2,
+=
+��
+s
+(v(s) −
+�
+s v(s)
+S
+)2
+(50)
+For p = 1, we have
+�
+s∈S
+sign
+�
+v(s) − ω1(v)
+�
+= 0
+(51)
+Note that there may be more than one values of ω1(v) that satisfies the above equation and
+each solution does equally good job (as we will see later). So we will pick one ( is median of
+31
+
+Table 11: p-mean, where v(si) ≥ v(si+1)
+∀i.
+x
+ωx(v)
+remark
+p
+�
+s sign(v(s) − ωp(v))|v(s) − ωp(v)|
+1
+p−1 = 0
+Solve by binary search
+1
+v(s⌊(S+1)/2⌋)+v(s⌈(S+1)/2⌉)
+2
+Median
+2
+�
+s v(s)
+S
+Mean
+∞
+maxs v(s)+mins v(s)
+2
+Average of peaks
+v) according to our convenience as
+ω1(v) = v(s⌊(S+1)/2⌋) + v(s⌈(S+1)/2⌉)
+2
+where
+v(si) ≥ v(si+1)
+∀i.
+κ1(v) =∥v − ω11∥1
+=∥v − med(v)1∥1,
+(putting in value of ω0, see table 11)
+=
+�
+s
+|v(s) − med(v)|
+=
+⌊(S+1)/2⌋
+�
+i=1
+(v(s) − med(v)) +
+S
+�
+⌈(S+1)/2⌉
+(med(v) − v(s))
+=
+⌊(S+1)/2⌋
+�
+i=1
+v(s) −
+S
+�
+⌈(S+1)/2⌉
+v(s)
+(52)
+where med(v) :=
+v(s⌊(S+1)/2⌋)+v(s⌈(S+1)/2⌉)
+2
+where
+v(si) ≥ v(si+1)
+∀i is median of v. The
+results are summarized in table 1 and 11.
+I.1
+p-variance function and kernel noise
+Lemma 1. q-variance function κq is the solution of the following optimization problem
+(kernel noise),
+κq(v) = −1
+ϵ min
+c ⟨c, v⟩,
+∥c∥p ≤ ϵ,
+�
+s
+c(s) = 0.
+Proof. Writing Lagrangian L, as
+L :=
+�
+s
+c(s)v(s) + λ
+�
+s
+c(s) + µ(
+�
+s
+|c(s)|p − ϵp),
+where λ ∈ R is the multiplier for the constraint �
+s c(s) = 0 and µ ≥ 0 is the multiplier for
+the inequality constraint ∥c∥q≤ ϵ. Taking its derivative, we have
+∂L
+∂c(s) = v(s) + λ + µp|c(s)|p−1 c(s)
+|c(s)|
+(53)
+32
+
+From the KKT (stationarity) condition, the solution c∗ has zero derivative, that is
+v(s) + λ + µp|c∗(s)|p−1 c∗(s)
+|c∗(s)| = 0,
+∀s ∈ S.
+(54)
+Using Lagrangian derivative equation (54), we have
+v(s) + λ + µp|c∗(s)|p−1 c∗(s)
+|c∗(s)| = 0
+=⇒
+�
+s
+c∗(s)[v(s) + λ + µp|c∗(s)|p−1 c∗(s)
+|c∗(s)|] = 0,
+(multiply with c∗(s) and summing )
+=⇒
+�
+s
+c∗(s)v(s) + λ
+�
+s
+c∗(s) + µp
+�
+s
+|c∗(s)|p−1 (c∗(s))2
+|c∗(s)| = 0
+=⇒ ⟨c∗, v⟩ + µp
+�
+s
+|c∗(s)|p = 0
+(using
+�
+s
+c∗(s) = 0 and (c∗(s))2 = |c∗(s)|2 )
+=⇒ ⟨c∗, v⟩ = −µpϵp,
+(using
+�
+s
+|c∗(s)|p = ϵp ).
+(55)
+It is easy to see that µ ≥ 0, as minimum value of the objective must not be positive ( at
+c = 0, the objective value is zero). Again we use Lagrangian derivative (54) and try to get
+the objective value (−µpϵp) in terms of λ, as
+v(s) + λ + µp|c∗(s)|p−1 c∗(s)
+|c∗(s)| = 0
+=⇒ |c∗(s)|p−2c∗(s) = −v(s) + λ
+µp
+,
+(re-arranging terms)
+=⇒
+�
+s
+|(|c∗(s)|p−2c∗(s))|
+p
+p−1 =
+�
+s
+| − v(s) + λ
+µp
+|
+p
+p−1 ,
+(doing
+�
+s
+|·|
+p
+p−1 )
+=⇒ ∥c∗∥p
+p =
+�
+s
+| − v(s) + λ
+µp
+|
+p
+p−1 =
+�
+s
+|v(s) + λ
+µp
+|q = ∥v + λ∥q
+q
+|µp|q
+=⇒ |µp|q∥c∗∥p
+p = ∥v + λ∥q
+q,
+(re-arranging terms)
+=⇒ |µp|qϵp = ∥v + λ∥q
+q,
+(using
+�
+s
+|c∗(s)|p = ϵp )
+=⇒ ϵ(µpϵp/q) = ϵ∥v + λ∥q
+(taking 1
+q the power then multiplying with ϵ)
+=⇒ µpϵp = ϵ∥v + λ∥q.
+(56)
+33
+
+Again, using Lagrangian derivative (54) to solve for λ, we have
+v(s) + λ + µp|c∗(s)|p−1 c∗(s)
+|c∗(s)| = 0
+=⇒ |c∗(s)|p−2c∗(s) = −v(s) + λ
+µp
+,
+(re-arranging terms)
+=⇒ |c∗(s)| = |v(s) + λ
+µp
+|
+1
+p−1 ,
+(looking at absolute value)
+and
+c∗(s)
+|c∗(s)| = − v(s) + λ
+|v(s) + λ|,
+(looking at sign: and note µ, p ≥ 0)
+=⇒
+�
+s
+c∗(s)
+|c∗(s)||c∗(s)| = −
+�
+s
+v(s) + λ
+|v(s) + λ||v(s) + λ
+µp
+|
+1
+p−1 ,
+(putting back)
+=⇒
+�
+s
+c∗(s) = −
+�
+s
+v(s) + λ
+|v(s) + λ||v(s) + λ
+µp
+|
+1
+p−1 ,
+=⇒
+�
+s
+v(s) + λ
+|v(s) + λ||v(s) + λ|
+1
+p−1 = 0,
+( using
+�
+i
+c∗(s) = 0)
+(57)
+Combining everything, we have
+− 1
+ϵ min
+c ⟨c, v⟩,
+∥c∥p ≤ ϵ,
+�
+s
+c(s) = 0
+=∥v − λ∥q,
+such that
+�
+s
+sign(v(s) − λ)|v(s) − λ|
+1
+p−1 = 0.
+(58)
+Now, observe that
+∂∥v − λ∥q
+∂λ
+= 0
+=⇒
+�
+s
+sign(v(s) − λ)|v(s) − λ|
+1
+p−1 = 0,
+=⇒ κq(v) = ∥v − λ∥q,
+such that
+�
+s
+sign(v(s) − λ)|v(s) − λ|
+1
+p−1 = 0.
+(59)
+The last equality follows from the convexity of p-norm ∥·∥q, where every local minima is
+global minima.
+For the sanity check, we re-derive things for p = 1 from scratch. For p = 1, we have
+− 1
+ϵ min
+c ⟨c, v⟩,
+∥c∥1 ≤ ϵ,
+�
+s
+c(s) = 0.
+= − 1
+2(min
+s
+v(s) − max
+s
+v(s))
+=κ1(v).
+(60)
+It is easy to see the above result, just by inspection.
+I.2
+Binary search for p-mean and estimation of p-variance
+If the function f : [−B/2, B/2] → R, B ∈ R is monotonic (WLOG let it be monotonically
+decreasing) in a bounded domain, and it has a unique root x∗ s.t. f(x∗) = 0. Then we can
+34
+
+find x that is an ϵ-approximation x∗ (i.e. ∥x − x∗∥ ≤ ϵ ) in O(B/ϵ) iterations. Why? Let
+x0 = 0 and
+xn+1 :=
+�
+�
+�
+�
+�
+−B+xn
+2
+if
+f(xn) > 0
+B+xn
+2
+if
+f(xn) < 0
+xn
+if
+f(xn) = 0
+.
+It is easy to observe that ∥xn − x∗∥ ≤ B(1/2)n. This is proves the above claim. This
+observation will be referred to many times.
+Now, we move to the main claims of the section.
+Proposition 5. The function
+hp(λ) :=
+�
+s
+sign
+�
+v(s) − λ
+��� v(s) − λ
+��p
+is monotonically strictly decreasing and also has a root in the range [mins v(s), maxs v(s)].
+Proof.
+hp(λ) =
+�
+s
+v(s) − λ
+|v(s) − λ||v(s) − λ|p
+dhp
+dλ (λ) = −p
+�
+s
+|v(s) − λ|p−1 ≤ 0,
+∀p ≥ 0.
+(61)
+Now, observe that hp(maxs v(s)) ≤ 0 and hp(mins v(s)) ≥ 0, hence by hp must have a root
+in the range [mins v(s), maxs v(s)] as the function is continuous.
+The above proposition ensures that a root ωp(v) can be easily found by binary search
+between [mins v(s), maxs v(s)].
+Precisely, ϵ approximation of ωp(v) can be found in O(log( maxs v(s)−mins v(s)
+ϵ
+)) number of
+iterations of binary search. And one evaluation of the function hp requires O(S) iterations.
+And we have finite state-action space and bounded reward hence WLOG we can assume
+|maxs v(s)|, |mins v(s)| are bounded by a constant. Hence, the complexity to approximate
+ωp is O(S log( 1
+ϵ )).
+Let ˆωp(v) be an ϵ-approximation of ωp(v), that is
+�� ωp(v) − ˆωp(v)
+��≤ ϵ.
+And let ˆκp(v) be approximation of κp(v) using approximated mean, that is,
+ˆκp(v) := ∥v − ˆωp(v)1∥p.
+Now we will show that ϵ error in calculation of p-mean ωp, induces O(ϵ) error in estimation
+of p-variance κp. Precisely,
+��� κp(v) − ˆκp(v)
+���=
+���
+�� v − ωp(v)1
+��
+p −
+�� v − ˆωp(v)1
+��
+p
+���
+≤
+�� ωp(v)1 − ˆωp(v)1
+��
+p,
+(reverse triangle inequality)
+=
+�� 1
+��
+p
+�� ωp(v) − ˆωp(v)
+��
+≤
+�� 1
+��
+p ϵ
+=S
+1
+p ϵ ≤ Sϵ.
+(62)
+35
+
+For general p, an ϵ approximation of κp(v) can be calculated in O(S log( S
+ϵ ) iterations. Why?
+We will estimate mean ωp to an ϵ/S tolerance (with cost O(S log( S
+ϵ ) ) and then approximate
+the κp with this approximated mean (cost O(S)).
+J
+Lp Water Filling/Pouring lemma
+In this section, we are going to discuss the following optimization problem,
+max
+c
+−α∥c∥q + ⟨c, b⟩
+such that
+A
+�
+i=1
+ci = 1,
+ci ≥ 0,
+∀i
+where α ≥ 0, referred as Lp-water pouring problem. We are going to assume WLOG that b
+is sorted component wise, that is b1 ≥ b2, · · · ≥ bA. The above problem for p = 2, is studied
+in [1]. The approach we are going to solve the problem is as follows: a) Write Lagrangian b)
+Since the problem is convex, any solutions of KKT condition is global maximum. c) Obtain
+conditions using KKT conditions.
+Lemma 2. Let b ∈ RA be such that its components are in decreasing order (i,e bi ≥ bi+1),
+α ≥ 0 be any non-negative constant, and
+ζp := max
+c
+−α∥c∥q + ⟨c, b⟩
+such that
+A
+�
+i=1
+ci = 1,
+ci ≥ 0,
+∀i,
+(63)
+and let c∗ be a solution to the above problem. Then
+1. Higher components of b, gets higher weight in c∗. In other words, c∗ is also sorted
+component wise in descending order, that is
+c∗
+1 ≥ c∗
+2, · · · , ≥ c∗
+A.
+2. The value ζp satisfies the following equation
+αp =
+�
+bi≥ζp
+(bi − ζp)p
+3. The solution c of (63), is related to ζp as
+ci =
+(bi − ζp)p−11(bi ≥ ζp)
+�
+s(bi − ζp)p−11(bi ≥ ζp)
+4. Observe that the top χp := max{i|bi ≥ ζp} actions are active and rest are passive. The
+number of active actions can be calculated as
+{k|αp ≥
+k
+�
+i=1
+(bi − bk)p} = {1, 2, · · · , χp}.
+5. Things can be re-written as
+ci ∝
+�
+(bi − ζp)p−1
+if
+i ≤ χp
+0
+else
+and
+αp =
+χp
+�
+i=1
+(bi − ζp)p
+36
+
+6. The function �
+bi≥x(bi − x)p is monotonically decreasing in x, hence the root ζp can
+be calculated efficiently by binary search between [b1 − α, b1].
+7. Solution is sandwiched as follows
+bχp+1 ≤ ζp ≤ bχp
+8. k ≤ χp if and only if there exist the solution of the following,
+k
+�
+i=1
+(bi − x)p = αp
+and
+x ≤ bk.
+9. If action k is active and there is greedy increment hope then action k + 1 is also active.
+That is
+k ≤ χp
+and
+λk ≤ bk+1 =⇒ k + 1 ≤ χp,
+where
+k
+�
+i=1
+(bi − λk)p = αp
+and
+λk ≤ bk.
+10. If action k is active, and there is no greedy hope and then action k + 1 is not active.
+That is,
+k ≤ χp
+and
+λk > bk+1 =⇒ k + 1 > χp,
+where
+k
+�
+i=1
+(bi − λk)p = αp
+and
+λk ≤ bk.
+And this implies k = χp.
+Proof.
+1. Let
+f(c) := −α∥c∥q + ⟨b, c⟩.
+Let c be any vector, and c′ be rearrangement c in descending order. Precisely,
+c′
+k := cik,
+where
+ci1 ≥ ci2, · · · , ≥ ciA.
+Then it is easy to see that f(c′) ≥ f(c). And the claim follows.
+2. Writting Lagrangian of the optimization problem, and its derivative,
+L = −α∥c∥q + ⟨c, b⟩ + λ(
+�
+i
+ci − 1) + θici
+∂L
+∂ci
+= −α∥c∥1−q
+q
+|ci|q−2ci + bi + λ + θi,
+(64)
+λ ∈ R is multiplier for equality constraint �
+i ci = 1 and θ1, · · · , θA ≥ 0 are multipliers
+for inequality constraints ci ≥ 0,
+∀i ∈ [A]. Using KKT (stationarity) condition, we
+have
+−α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i + bi + λ + θi = 0
+(65)
+37
+
+Let B := {i|c∗
+i > 0}, then
+�
+i∈B
+c∗
+i [−α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i + bi + λ] = 0
+=⇒ − α∥c∗∥1−q
+q
+∥c∗∥q
+q + ⟨c∗, b⟩ + λ = 0,
+(using
+�
+i
+c∗
+i = 1 and (c∗
+i )2 = |c∗
+i |2)
+=⇒ − α∥c∗∥q + ⟨c∗, b⟩ + λ = 0
+=⇒ − α∥c∗∥q + ⟨c∗, b⟩ = −λ,
+(re-arranging)
+(66)
+Now again using (65), we have
+− α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i + bi + λ + θi = 0
+=⇒ α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i = bi + λ + θi,
+∀i,
+(re-arranging)
+(67)
+Now, if i ∈ B then θi = 0 from complimentry slackness, so we have
+α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i = bi + λ > 0,
+∀i ∈ B
+by definition of B. Now, if for some i, bi + λ > 0 then bi + λ + θi > 0 as θi ≥ 0, that
+implies
+α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i = bi + λ + θi > 0
+=⇒ c∗
+i > 0 =⇒ i ∈ B.
+So, we have,
+i ∈ B ⇐⇒ bi + λ > 0.
+To summarize, we have
+α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i = (bi + λ)1(bi ≥ −λ),
+∀i,
+(68)
+=⇒
+�
+i
+α
+q
+q−1 ∥c∗∥−q
+q (c∗
+i )q =
+�
+i
+(bi + λ)
+q
+q−1 1(bi ≥ −λ),
+(taking q/(q − 1)th power and summing)
+=⇒ αp =
+A
+�
+i=1
+(bi + λ)p1(bi ≥ −λ).
+(69)
+So, we have,
+ζp = −λ
+such that
+αp =
+�
+bi≥λ
+(bi + λ)p.
+=⇒ αp =
+�
+bi≥ζp
+(bi − ζp)p
+(70)
+3. Furthermore, using (68), we have
+α∥c∗∥1−q
+q
+|c∗
+i |q−2c∗
+i = (bi + λ)1(bi ≥ −λ) = (bi − ζp)1(bi ≥ ζp)
+∀i,
+=⇒ c∗
+i ∝ (bi − ζp)
+1
+q−1 1(bi ≥ ζp) =
+(bi − ζp)p−11(bi ≥ ζp)
+�
+i(bi − ζp)p−11(bi ≥ ζp),
+(using
+�
+i
+c∗
+i = 1).
+(71)
+38
+
+4. Now, we move on to calculate the number of active actions χp. Observe that the
+function
+f(λ) :=
+A
+�
+i=1
+(bi − λ)p1(bi ≥ λ) − αp
+(72)
+is monotonically decreasing in λ and ζp is a root of f. This implies
+f(x) ≤ 0 ⇐⇒ x ≥ ζp
+=⇒ f(bi) ≤ 0 ⇐⇒ bi ≥ ζp
+=⇒ {i|bi ≥ ζp} = {i|f(bi) ≤ 0}
+=⇒ χp = max{i|bi ≥ ζp} = max{i|f(bi) ≤ 0}.
+(73)
+Hence, things follows by putting back in the definition of f.
+5. We have,
+αp =
+A
+�
+i=1
+(bi − ζp)p1(bi ≥ ζp),
+and
+χp = max{i|bi ≥ ζp}.
+Combining both we have
+αp =
+χp
+�
+i=1
+(bi − ζp)p.
+And the other part follows directly.
+6. Continuity and montonocity of the function �
+bi≥x(bi − x)p is trivial. Now observe
+that �
+bi≥b1(bi − b1)p = 0 and �
+bi≥b1−α(bi − (b1 − α))p ≥ αp, so it implies that it is
+equal to αp in the range [b1 − α, b1].
+7. Recall that the ζp is the solution to the following equation
+αp =
+�
+bi≥x
+(bi − x)p.
+And from the definition of χp, we have
+αp <
+χp+1
+�
+i=1
+(bi − bχp+1)p =
+�
+bi≥bχp+1
+(bi − bχp+1)p,
+and
+αp ≥
+χp
+�
+i=1
+(bi − bχp)p =
+�
+bi≥bχp
+(bi − bχp)p.
+So from continuity, we infer the root ζp must lie between [bχp+1, bχ].
+8. We prove the first direction, and assume we have
+k ≤ χp
+=⇒
+k
+�
+i=1
+(bi − bk)p ≤ αp
+(from definition of χp).
+(74)
+39
+
+Observe the function f(x) := �k
+i=1(bi − x)p is monotically decreasing in the range
+(−∞, bk].
+Further, f(bk) ≤ αp and limx→−∞ f(x) = ∞, so from the continuity
+argument there must exist a value y ∈ (−∞, bk] such that f(y) = αp. This implies that
+k
+�
+i=1
+(bi − y)p ≤ αp,
+and
+y ≤ bk.
+Hence, explicitly showed the existence of the solution. Now, we move on to the second
+direction, and assume there exist x such that
+k
+�
+i=1
+(bi − x)p = αp,
+and
+x ≤ bk.
+=⇒
+k
+�
+i=1
+(bi − bk)p ≤ αp,
+(as x ≤ bk ≤ bk−1 · · · ≤ b1)
+=⇒ k ≤ χp.
+9. We have k ≤ χp and λk such that
+αp =
+k
+�
+i=1
+(bi − λk)p,
+and
+λk ≤ bk,
+(from above item)
+≥
+k
+�
+i=1
+(bi − bk+1)p,
+(as λk ≤ bk+1 ≤ bk)
+≥
+k+1
+�
+i=1
+(bi − bk+1)p,
+(addition of 0).
+(75)
+From the definition of χp, we get k + 1 ≤ χp.
+10. We are given
+k
+�
+i=1
+(bi − λk)p = αp
+=⇒
+k
+�
+i=1
+(bi − bk+1)p > αp,
+(as λk > bk+1)
+=⇒
+k+1
+�
+i=1
+(bi − bk+1)p > αp,
+(addition of zero)
+=⇒ k + 1 > χp.
+40
+
+J.0.1
+Special case: L1
+For p = 1, by definition, we have
+ζ1 = max
+c
+−α∥c∥∞ + ⟨c, b⟩
+such that
+�
+a∈A
+ca = 1,
+c ⪰ 0.
+(76)
+And χ1 is the optimal number of actions, that is
+α =
+χ1
+�
+i=1
+(bi − ζ1)
+=⇒ ζ1 =
+�χ1
+i=1 bi − α
+χ1
+.
+Let λk be the such that
+α =
+k
+�
+i=1
+(bi − λk)
+=⇒ λk =
+�k
+i=1 bi − α
+k
+.
+Proposition 6.
+ζ1 = max
+k
+λk
+Proof. From lemma 2, we have
+λ1 ≤ λ2 · · · ≤ λχ1.
+Now, we have
+λk − λk+m =
+�k
+i=1 bi − α
+k
+−
+�k+m
+i=1 bi − α
+k + m
+=
+�k
+i=1 bi − α
+k
+−
+�k
+i=1 bi − α
+k + m
+−
+�m
+i=1 bk+i
+k + m
+= m(�k
+i=1 bi − α
+k(k + m))
+−
+�m
+i=1 bk+i
+k + m
+=
+m
+k + m(
+�k
+i=1 bi − α
+k
+−
+�m
+i=1 bk+i
+m
+)
+=
+m
+k + m(λk −
+�m
+i=1 bk+i
+m
+)
+(77)
+From lemma 2, we also know the stopping criteria for χ1, that is
+λχ1 > bχ1+1
+=⇒ λχ1 > bχ1+i,
+i ≥ 1,
+(as bi are in descending order)
+=⇒ λχ1 >
+�m
+i=1 bχ1+i
+m
+,
+∀m ≥ 1.
+41
+
+Combining it with the (77), for all m ≥ 0 , we get
+λχ1 − λχ1+m =
+m
+χ1 + m(λχ1 −
+�m
+i=1 bχ1+i
+m
+)
+≥ 0
+=⇒ λχ1 ≥ λχ1+m
+(78)
+Hence, we get the desired result,
+ζ1 = λχ1 = max
+k
+λk.
+J.0.2
+Special case: max norm
+For p = ∞, by definition, we have
+ζ∞(b) = max
+c
+−α∥c∥1 + ⟨c, b⟩
+such that
+�
+a∈A
+ca = 1,
+c ⪰ 0.
+= max
+c
+−α + ⟨c, b⟩
+such that
+�
+a∈A
+ca = 1,
+c ⪰ 0.
+= − α + max
+i
+bi
+(79)
+J.0.3
+Special case: L2
+The problem is discussed in great details in [1], here we outline the proof. For p = 2, we
+have
+ζ2 = max
+c
+−α∥c∥2 + ⟨c, b⟩
+such that
+�
+a∈A
+ca = 1,
+c ⪰ 0.
+(80)
+Let λk be the solution of the following equation
+α2 =
+k
+�
+i=1
+(bi − λ)2,
+λ ≤ bk
+= kλ2 − 2
+k
+�
+i=1
+λbi. +
+k
+�
+i=1
+(bi)2,
+λ ≤ bk
+=⇒ λk =
+�k
+i=1 bi ±
+�
+(�k
+i=1 bi)2 − k(�k
+i=1(bi)2 − α2)
+k
+,
+and
+λk ≤ bk
+=
+�k
+i=1 bi −
+�
+(�k
+i=1 bi)2 − k(�k
+i=1(bi)2 − α2)
+k
+=
+�k
+i=1 bi
+k
+−
+�
+�
+�
+�α2 −
+k
+�
+i=1
+(bi −
+�k
+i=1 bi
+k
+)2
+(81)
+From lemma 2, we know
+λ1 ≤ λ2 · · · ≤ λχ2 = ζ2
+42
+
+where χ2 calculated in two ways: a)
+χ2 = max
+m {m|
+m
+�
+i=1
+(bi − bm)2 ≤ α2}
+b)
+χ2 = min
+m {m|λm ≤ bm+1}
+We proceed greedily until stopping condition is met in lemma 2. Concretely, it is illustrated
+in algorithm 7.
+J.1
+L1 Water Pouring lemma
+In this section, we re-derive the above water pouring lemma for p = 1 from scratch, just for
+sanity check. As in the above proof, there is a possibility of some breakdown, as we had take
+limits q → ∞. We will see that all the above results for p = 1 too.
+Let b ∈ RA be such that its components are in decreasing order, i,e bi ≥ bi+1 and
+ζ1 := max
+c
+−α∥c∥∞ + ⟨c, b⟩
+such that
+A
+�
+i=1
+ci = 1,
+ci ≥ 0,
+∀i.
+(82)
+Lets fix any vector c ∈ RA, and let k1 := ⌊
+1
+maxi ci ⌋ and let
+c1
+i =
+�
+�
+�
+�
+�
+maxi ci
+if
+i ≤ k1
+1 − k1 maxi ci
+if
+i = k1 + 1
+0
+else
+Then we have,
+−α∥c∥∞ + ⟨c, b⟩ = − α max
+i
+ci +
+A
+�
+i=1
+cibi
+≤ − α max
+i
+ci +
+A
+�
+i=1
+c1
+i bi,
+(recall bi is in decreasing order)
+= − α∥c1∥∞ + ⟨c1, b⟩
+(83)
+Now, lets define c2 ∈ RA. Let
+k2 =
+�
+k1 + 1
+if
+�k1
+i=1 bi−α
+k1
+≤ bk+1
+k1
+else
+43
+
+and let c2
+i = 1(i≤k2)
+k2
+. Then we have,
+−α∥c1∥∞ + ⟨c1, b⟩ = − α max
+i
+ci +
+A
+�
+i=1
+c1
+i bi
+= − α max
+i
+ci +
+k1
+�
+i=1
+max
+i
+cibi + (1 − k1 max
+i
+ci)bk1+1,
+(definition of c1)
+=(−α + �k1
+i=1 bi
+k1
+)k1 max
+i
+ci + bk1+1(1 − k1 max
+i
+ci),
+(re-arranging)
+≤−α + �k2
+i=1 bi
+k2
+= − α∥c2∥∞ + ⟨c2, b⟩
+(84)
+The last inequality comes from the definition of k2 and c2. So we conclude that a optimal
+solution is uniform over some actions, that is
+ζ1 = max
+c∈C −α∥c∥∞ + ⟨c, b⟩
+= max
+k
+� −α + �k
+i=1 bi
+k
+�
+(85)
+where C := {ck ∈ RA|ck
+i = 1(i≤k)
+k
+} is set of uniform actions. Rest all the properties follows
+same as Lp water pouring lemma.
+K
+Robust Value Iteration (Main)
+In this section, we will discuss the main results from the paper except for time complexity
+results. It contains the proofs of the results presented in the main body and also some other
+corollaries/special cases.
+K.1
+sa-rectangular robust policy evaluation and improvement
+Theorem 8. (sa)-rectangular Lp robust Bellman operator is equivalent to reward regularized
+(non-robust) Bellman operator, that is
+(T π
+Usa
+p v)(s) =
+�
+a
+π(a|s)[−αs,a − γβs,aκq(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)],
+and
+(T ∗
+Usa
+p v)(s) = max
+a∈A[−αs,a − γβs,aκq(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)],
+where κp is defined in (7).
+44
+
+Proof. From definition robust Bellman operator and Usa
+p = (R0 + R) × (P0 + P), we have,
+(T π
+Usa
+p v)(s) =
+min
+R,P ∈Usa
+p
+�
+a
+π(a|s)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+=
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
++
+min
+p∈P,r∈R
+�
+a
+π(a|s)
+�
+r(s, a) + γ
+�
+s′
+p(s′|s, a)v(s′)
+�
+,
+(from (sa)-rectangularity, we get)
+=
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
++
+�
+a
+π(a|s)
+min
+ps,a∈Psa,rs,a∈Rs,a
+�
+rs,a + γ
+�
+s′
+ps,a(s′)v(s′)
+�
+�
+��
+�
+:=Ωsa(v)
+(86)
+Now we focus on regularizer function Ω, as follows
+Ωsa(v) =
+min
+ps,a∈Ps,a,rs,a∈Rs,a
+�
+rs,a + γ
+�
+s′
+ps,a(s′)v(s′)
+�
+=
+min
+rs,a∈Rs,a rs,a + γ
+min
+ps,a∈Psa
+�
+s′
+ps,a(s′)v(s′)
+= −αs,a + γ
+min
+∥psa∥p≤βs,a,�
+s′ psa(s′)=0⟨ps,a, v⟩,
+= − αs,a − γβs,aκq(v),
+(from lemma 1).
+(87)
+Putting back, we have
+(T π
+Usa
+p v)(s) =
+�
+a
+π(a|s)
+�
+−αs,a − γβs,aκq(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+Again, reusing above results in optimal robust operator, we have
+(T ∗
+Usa
+p v)(s) = max
+πs∈∆A
+min
+R,P ∈Usa
+p
+�
+a
+πs(a)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+= max
+πs∈∆A
+�
+a
+πs(a)
+�
+−αs,a − γβs,aκp(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+= max
+a∈A
+�
+−αs,a − γβs,aκq(v) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+(88)
+The claim is proved.
+K.2
+S-rectangular robust policy evaluation
+Theorem 9. S-rectangular Lp robust Bellman operator is equivalent to reward regularized
+(non-robust) Bellman operator, that is
+(T π
+Us
+pv)(s) = −
+�
+αs + γβsκq(v)
+�
+∥π(·|s)∥q +
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+45
+
+where κp is defined in (7) and ∥π(·|s)∥q is q-norm of the vector π(·|s) ∈ ∆A.
+Proof. From definition of robust Bellman operator and Us
+p = (R0 + R) × (P0 + P), we have
+(T π
+Us
+pv)(s) =
+min
+R,P ∈Us
+p
+�
+a
+π(a|s)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+=
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+��
+�
+nominal values
+�
++
+min
+p∈P,r∈R
+�
+a
+π(a|s)
+�
+r(s, a) + γ
+�
+s′
+p(s′|s, a)v(s′)
+�
+(from s-rectangularity we have)
+=
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
++
+min
+ps∈Ps,rs∈Rs
+�
+a
+π(a|s)
+�
+rs(a) + γ
+�
+s′
+ps(s′|a)v(s′)
+�
+�
+��
+�
+:=Ωs(πs,v)
+(89)
+where we denote πs(a) = π(a|s) as a shorthand. Now we calculate the regularizer function
+as follows
+Ωs(πs, v) :=
+min
+rs∈Rs,ps∈Ps⟨rs + γvT ps, πs⟩ = min
+rs∈Rs⟨rs, πs⟩ + γ min
+ps∈Ps vT psπs
+= −αs∥πs∥q + γ min
+ps∈Ps vT psπs,
+(using 1
+p + 1
+q = 1 )
+= − αs∥πs∥q + γ min
+ps∈Ps
+�
+a
+πs(a)⟨ps,a, v⟩
+= − αs∥πs∥q + γ
+min
+�
+a(βs,a)p≤(βs)p
+min
+∥psa∥p≤βs,a,�
+s′ psa(s′)=0
+�
+a
+πs(a)⟨ps,a, v⟩
+= − αs∥πs∥q + γ
+min
+�
+a(βs,a)p≤(βs)p
+�
+a
+πs(a)
+min
+∥psa∥p≤βs,a,�
+s′ psa(s′)=0
+⟨ps,a, v⟩
+= − αs∥πs∥q + γ
+min
+�
+a(βsa)p≤(βs)p
+�
+a
+πs(a)(−βsaκp(v))
+( from lemma 1)
+= − αs∥πs∥q − γκq(v)
+max
+�
+a(βsa)p≤(βs)p
+�
+a
+πs(a)βsa
+= − αs∥πs∥q − γκp(v)∥πs∥qβs
+(using Holders)
+= − (αs + γβsκq(v))∥πs∥q.
+(90)
+Now putting above values in robust operator, we have
+(T π
+Us
+pv)(s) = −
+�
+αs + γβsκq(v)
+�
+∥π(·|s)∥q+
+�
+a
+π(a|s)
+�
+R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+�
+.
+46
+
+K.3
+s-rectangular robust policy improvement
+Reusing robust policy evaluation results in section K.2, we have
+(T ∗
+Uspv)(s) = max
+πs∈∆A
+min
+R,P ∈Usa
+p
+�
+a
+πs(a)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+= max
+πs∈∆A
+�
+−(αs + γβsκq(v))∥πs∥q +
+�
+a
+πs(a)(R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′))
+�
+.
+(91)
+Observe that, we have the following form
+(T ∗
+Uspv)(s) = max
+c
+−α∥c∥q + ⟨c, b⟩
+such that
+A
+�
+i=1
+ci = 1,
+c ⪰ 0,
+(92)
+where α = αs +γβsκq(v) and bi = R(s, ai)+γ �
+s′ P(s′|s, ai)v(s′). Now all the results below,
+follows from water pouring lemma ( lemma 2).
+Theorem 10. (Policy improvement) The optimal robust Bellman operator can be evaluated
+in following ways.
+1. (T ∗
+Us
+pv)(s) is the solution of the following equation that can be found using binary search
+between
+�
+maxa Q(s, a) − σ, maxa Q(s, a)
+�
+,
+�
+a
+�
+Q(s, a) − x
+�p 1
+�
+Q(s, a) ≥ x
+�
+= σp.
+(93)
+2. (T ∗
+Us
+pv)(s) and χp(v, s) can also be computed through algorithm 3.
+where σ = αs + γβsκq(v), and Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′).
+Proof. The first part follows from lemma 2, point 2. The second part follows from lemma 2,
+point 9 (greedy inclusion ) and point 10 (stopping condition).
+Theorem 11. (Go To Policy) The greedy policy π w.r.t. value function v, defined as
+T ∗
+Us
+pv = T π
+Us
+pv is a threshold policy. It takes only those actions that has positive advantage,
+with probability proportional to (p − 1)th power of its advantage. That is
+π(a|s) ∝ (A(s, a))p−11(A(s, a) ≥ 0),
+where A(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′) − (T ∗
+Us
+pv)(s).
+Proof. Follows from lemma 2, point 3.
+Property 4. χp(v, s) is number of actions that has positive advantage, that is
+χp(v, s) =
+���
+�
+a | (T ∗
+Us
+pv)(s) ≤ R0(s, a) + γ
+�
+s′
+P0(s′|s, a)v(s′)
+���� .
+Proof. Follows from lemma 2, point 4.
+47
+
+Property 5. ( Value vs Q-value) (T ∗
+Us
+pv)(s) is bounded by the Q-value of χth and (χ + 1)th
+actions. That is
+Q(s, aχ+1) < (T ∗
+Us
+pv)(s) ≤ Q(s, aχ),
+where
+χ = χp(v, s),
+Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′), and Q(s, a1) ≥ Q(s, a2), · · · Q(s, aA).
+Proof. Follows from lemma 2, point 7.
+Corollary 2. For p = 1, the optimal policy π1 w.r.t. value function v and uncertainty set
+Us
+1, can be computed directly using χ1(s) without calculating advantage function. That is
+π1(as
+i|s) = 1(i ≤ χ1(s))
+χ1(s)
+.
+Proof. Follows from Theorem 11 by putting p = 1. Note that it can be directly obtained
+using L1 water pouring lemma (see section J.1)
+Corollary 3. (For p = ∞) The optimal policy π w.r.t. value function v and uncertainty set
+Us
+∞ (precisely T ∗
+Us
+∞v = T π
+Us
+∞v), is to play the best response, that is
+π(a|s) = 1(a ∈ arg maxa Q(s, a))
+�� arg maxa Q(s, a)
+��
+.
+In case of tie in the best response, it is optimal to play any of the best responses with any
+probability.
+Proof. Follows from Theorem 11 by taking limit p → ∞.
+Corollary 4. For p = ∞, T ∗
+Us
+pv, the robust optimal Bellman operator evaluation can be
+obtained in closed form. That is
+(T ∗
+Us
+∞v)(s) = max
+a
+Q(s, a) − σ,
+where σ = αs + γβsκ1(v), Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′).
+Proof. Let π be such that
+T ∗
+Us
+∞v = T π
+Us
+∞v.
+This implies
+(T ∗
+Uspv)(s) =
+min
+R,P ∈Usa
+p
+�
+a
+π(a|s)
+�
+R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+= −(αs + γβsκp(v))∥π(·|s)∥q +
+�
+a
+π(a|s)(R(s, a) + γ
+�
+s′
+P(s′|s, a)v(s′)).
+(94)
+From corollary 3, we know the that π is deterministic best response policy. Putting this we
+get the desired result.
+There is a another way of proving this, using Theorem 3 by taking limit p → ∞ carefully as
+lim
+p→∞
+�
+a
+�
+Q(s, a) − T ∗
+Uspv)(s)
+�p
+1
+�
+Q(s, a) ≥ T ∗
+Uspv)(s)
+�
+)
+1
+p = σ,
+(95)
+where σ = αs + γβsκ1(v).
+48
+
+Corollary 5. For p = 1, the robust optimal Bellman operator T ∗
+Us
+p, can be computed in
+closed form. That is
+(T ∗
+Us
+pv)(s) = max
+k
+�k
+i=1 Q(s, ai) − σ
+k
+,
+where σ = αs + γβsκ∞(v), Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′), and Q(s, a1) ≥
+Q(s, a2), ≥ · · · ≥ Q(s, aA).
+Proof. Follows from section J.0.1.
+Corollary 6. The s rectangular Lp robust Bellman operator can be evaluated for p = 1, 2
+by algorithm 8 and algorithm 7 respectively.
+Proof. It follows from the algorithm 3, where we solve the linear equation and quadratic
+equation for p = 1, 2 respectively. For p = 2, it can be found in [1].
+Algorithm 7 Algorithm to compute S-rectangular L2 robust optimal Bellman Operator
+1: Input: σ = αs + γβsκ2(v),
+Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′).
+2: Output (T ∗
+Us
+2 v)(s), χ2(v, s)
+3: Sort Q(s, ·) and label actions such that Q(s, a1) ≥ Q(s, a2), · · · .
+4: Set initial value guess λ1 = Q(s, a1) − σ and counter k = 1.
+5: while k ≤ A − 1 and λk ≤ Q(s, ak) do
+6:
+Increment counter: k = k + 1
+7:
+Update value estimate:
+λk = 1
+k
+�
+k
+�
+i=1
+Q(s, ai) −
+�
+�
+�
+�kσ2 + (
+k
+�
+i=1
+Q(s, ai))2 − k
+k
+�
+i=1
+(Q(s, ai))2
+�
+8: end while
+9: Return: λk, k
+Algorithm 8 Algorithm to compute S-rectangular L1 robust optimal Bellman Operator
+1: Input: σ = αs + γβsκ∞(v),
+Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)v(s′).
+2: Output (T ∗
+Us
+1 v)(s), χ1(v, s)
+3: Sort Q(s, ·) and label actions such that Q(s, a1) ≥ Q(s, a2), · · · .
+4: Set initial value guess λ1 = Q(s, a1) − σ and counter k = 1.
+5: while k ≤ A − 1 and λk ≤ Q(s, ak) do
+6:
+Increment counter: k = k + 1
+7:
+Update value estimate:
+λk = 1
+k
+�
+k
+�
+i=1
+Q(s, ai) − σ
+�
+8: end while
+9: Return: λk, k
+49
+
+L
+Time Complexity
+In this section, we will discuss time complexity of various robust MDPs and compare it with
+time complexity of non-robust MDPs. We assume that we have the knowledge of nominal
+transition kernel and nominal reward function for robust MDPs, and in case of non-robust
+MDPs, we assume the knowledge of the transition kernel and reward function. We divide
+the discussion into various parts depending upon their similarity.
+L.1
+Exact Value Iteration: Best Response
+In this section, we will discuss non-robust MDPs, (sa)-rectangular L1/L2/L∞ robust MDPs
+and s-rectangular L∞ robust MDPs. They all have a common theme for value iteration as
+follows, for the value function v, their Bellman operator ( T ) evaluation is done as
+(T v)(s) =
+max
+a
+����
+action cost
+�
+R(s, a) + αs,a
+κ(v)
+����
+reward penalty/cost
++γ
+�
+s′
+P(s′|s, a)v(s′)
+�
+��
+�
+sweep
+�
+.
+(96)
+’Sweep’ requires O(S) iterations and ’action cost’ requires O(A) iterations. Note that the
+reward penalty κ(v) doesn’t depend on state and action. It is calculated only once for value
+iteration for all states. The above value update has to be done for each states , so one full
+update requires
+O
+�
+S(action cost)(sweep cost
+�
++reward cost
+�
+= O
+�
+S2A + reward cost
+�
+Since the value iteration is a contraction map, so to get ϵ-close to the optimal value, it
+requires O(log( 1
+ϵ )) full value update, so the complexity is
+O
+�
+log(1
+ϵ )
+�
+S2A + reward cost
+��
+.
+1. Non-robust MDPs: The cost of ’reward is zero as there is no regularizer to compute.
+The total complexity is
+O
+�
+log(1
+ϵ )
+�
+S2A + 0
+��
+= O
+�
+log(1
+ϵ )S2A
+�
+.
+2. (sa)-rectangular L1/L2/L∞ and s-rectangular L∞ robust MDPs: We need
+to calculate the reward penalty (κ1(v)/κ2(v)/κ∞) that takes O(S) iterations. As
+calculation of mean, variance and median, all are linear time compute. Hence the
+complexity is
+O
+�
+log(1
+ϵ )
+�
+S2A + S
+��
+= O
+�
+log(1
+ϵ )S2A
+�
+.
+L.2
+Exact Value iteration: Top k response
+In this section, we discuss the time complexity of s-rectangular L1/L2 robust MDPs as in
+algorithm 5. We need to calculate the reward penalty (κ∞(v)/κ2(v) in (40)) that takes O(S)
+iterations. Then for each state we do: sorting of Q-values in (45), value evaluation in (46),
+50
+
+update Q-value in (44) that takes O(A log(A)), O(A), O(SA) iterations respectively. Hence
+the complexity is
+total iteration(reward cost (40) + S( sorting (45) + value evaluation (46) +Q-value(44))
+= log(1
+ϵ )(S + S(A log(A) + A + SA)
+O
+�
+log(1
+ϵ )
+�
+S2A + SA log(A)
+��
+.
+For general p, we need little caution as kp(v) can’t be calculated exactly but approximately
+by binary search. And it is the subject of discussion for the next sections.
+L.3
+Inexact Value Iteration: sa-rectangular Lp robust MDPs (U sa
+p )
+In this section, we will study the time complexity for robust value iteration for (sa)-rectangular
+Lp robust MDPs for general p. Recall, that value iteration takes best penalized action, that
+is easy to compute. But reward penalization depends on p-variance measure κp(v), that we
+will estimate by ˆκp(v) through binary search. We have inexact value iterations as
+vn+1(s) := max
+a∈A[αsa − γβsaˆκq(vn) + R0(s, a) + γ
+�
+s′
+P0(s′|s, a)vn(s′)]
+where ˆκq(vn) is a ϵ1 approximation of κq(vn), that is |ˆκq(vn) − κq(vn)| ≤ ϵ1. Then it is easy
+to see that we have bounded error in robust value iteration, that is
+∥vn+1 − T ∗
+Usa
+p vn∥∞ ≤ γβmaxϵ1
+where βmax := maxs,a βs,a
+Proposition 7. Let T ∗
+U be a γ contraction map, and v∗ be its fixed point. And let {vn, n ≥ 0}
+be approximate value iteration, that is
+∥vn+1 − T ∗
+U vn∥∞ ≤ ϵ
+then
+lim
+n→∞∥vn − v∗∥∞ ≤
+ϵ
+1 − γ
+moreover, it converges to the
+ϵ
+1−γ radius ball linearly, that is
+∥vn − v∗∥∞ −
+ϵ
+1 − γ ≤ cγn
+where c =
+1
+1−γ ϵ + ∥v0 − v∗∥∞.
+51
+
+Proof.
+∥vn+1 − v∗∥∞ =∥vn+1 − T ∗
+U v∗∥∞
+=∥vn+1 − T ∗
+U vn + T ∗
+U vn − T ∗
+U v∗∥∞
+≤∥vn+1 − T ∗
+U vn∥∞ + ∥T ∗
+U vn − T ∗
+U v∗∥∞
+≤∥vn+1 − T ∗
+U vn∥∞ + γ∥vn − v∗∥∞,
+(contraction)
+≤ϵ + γ∥vn − v∗∥∞,
+(approximate value iteration)
+=⇒ ∥vn − v∗∥∞ =
+n−1
+�
+k=0
+γkϵ + γn∥v0 − v∗∥∞,
+(unrolling above recursion)
+=1 − γn
+1 − γ ϵ + γn∥v0 − v∗∥∞
+=γn[
+1
+1 − γ ϵ + ∥v0 − v∗∥∞] +
+ϵ
+1 − γ
+(97)
+Taking limit n → ∞ both sides, we get
+lim
+n→∞∥vn − v∗∥∞ ≤
+ϵ
+1 − γ .
+Lemma 3. For Usa
+p , the total iteration cost is log( 1
+ϵ )S2A + (log( 1
+ϵ ))2 to get ϵ close to the
+optimal robust value function.
+Proof. We calculate κq(v) with ϵ1 = (1−γ)ϵ
+3
+tolerance that takes O(S log( S
+ϵ1 )) using binary
+search (see section I.2). Now, we do approximate value iteration for n = log( 3∥v0−v∗∥∞
+ϵ
+).
+Using the above lemma, we have
+∥vn − v∗
+Usa
+p ∥∞ =γn[
+1
+1 − γ ϵ1 + ∥v0 − v∗
+Usa
+p ∥∞] +
+ϵ1
+1 − γ
+≤γn[ ϵ
+3 + ∥v0 − v∗
+Usa
+p ∥∞] + ϵ
+3
+≤γn ϵ
+3 + ϵ
+3 + ϵ
+3 ≤ ϵ.
+(98)
+In summary, we have action cost O(A), reward cost O(S log( S
+ϵ )), sweep cost O(S) and total
+number of iterations O(log( 1
+ϵ )). So the complexity is
+(number of iterations)
+�
+S(actions cost) (sweep cost) + reward cost
+�
+= log(1
+ϵ )
+�
+S2A + S log(S
+ϵ )
+�
+= log(1
+ϵ )(S2A + S log(1
+ϵ ) + S log(S))
+= log(1
+ϵ )S2A + S(log(1
+ϵ ))2
+52
+
+L.4
+Inexact Value Iteration: s-rectangular Lp robust MDPs
+In this section, we study the time complexity for robust value iteration for s-rectangular
+Lp robust MDPs for general p ( algorithm 4). Recall, that value iteration takes regularized
+actions and penalized reward. And reward penalization depends on q-variance measure κq(v),
+that we will estimate by ˆκq(v) through binary search, then again we will calculate T ∗
+Usa
+p by
+binary search with approximated κq(v). Here, we have two error sources ((40), (46)) as
+contrast to (sa)-rectangular cases, where there was only one error source from the estimation
+of κq.
+First, we account for the error caused by the first source (κq). Here we do value iteration
+with approximated q-variance ˆκq, and exact action regularizer. We have
+vn+1(s) := λ
+s.t.
+αs + γβsˆκq(v) = (
+�
+Q(s,a)≥λ
+(Q(s, a) − λ)p)
+1
+p
+where Q(s, a) = R0(s, a) + γ �
+s′ P0(s′|s, a)vn(s′), and |ˆκq(vn) − κq(vn)| ≤ ϵ1. Then from
+the next result (proposition 8), we get
+∥vn+1 − T ∗
+Usa
+p vn∥∞ ≤ γβmaxϵ1
+where βmax := maxs,a βs,a
+Proposition 8. Let ˆκ be an an ϵ-approximation of κ, that is |ˆκ − κ| ≤ ϵ, and let b ∈ RA
+be sorted component wise, that is, b1 ≥, · · · , ≥ bA. Let λ be the solution to the following
+equation with exact parameter κ,
+α + γβκ = (
+�
+bi≥λ
+|bi − λ|p)
+1
+p
+and let ˆλ be the solution of the following equation with approximated parameter ˆκ,
+α + γβˆκ = (
+�
+bi≥ˆλ
+|bi − ˆλ|p)
+1
+p ,
+then ˆλ is an O(ϵ)-approximation of λ, that is
+|λ − ˆλ| ≤ γβϵ.
+Proof. Let the function f : [bA, b1] → R be defined as
+f(x) := (
+�
+bi≥x
+|bi − x|p)
+1
+p .
+We will show that derivative of f is bounded, implying its inverse is bounded and hence
+Lipschitz, that will prove the claim. Let proceed
+df(x)
+dx
+= −(
+�
+bi≥x
+|bi − x|p)
+1
+p −1 �
+bi≥x
+|bi − x|p−1
+= −
+�
+bi≥x |bi − x|p−1
+(�
+bi≥x |bi − x|p)
+p−1
+p
+= −
+� (�
+bi≥x |bi − x|p−1)
+1
+p−1
+(�
+bi≥x |bi − x|p)
+1
+p
+�p−1
+≤ −1.
+(99)
+53
+
+The inequality follows from the following relation between Lp norm,
+∥x∥a ≥ ∥x∥b,
+∀0 ≤ a ≤ b.
+It is easy to see that the function f is strictly monotone in the range bA, b1], so its inverse is
+well defined in the same range. Then derivative of the inverse of the function f is bounded as
+0 ≥ d
+dxf −(x) ≥ −1.
+Now, observe that λ = f −(α + γβκ) and ˆλ = f −(α + γβˆκ), then by Lipschitzcity, we have
+|λ − ˆλ| = |f −(α + γβκ) − f −(α + γβˆκ)| ≤ γβ| − κ − ˆκ)| ≤ γβϵ.
+Lemma 4. For Us
+p, the total iteration cost is O
+�
+log( 1
+ϵ )
+�
+S2A + SA log( A
+ϵ )
+��
+to get ϵ
+close to the optimal robust value function.
+Proof. We calculate κq(v) in (40) with ϵ1 = (1−γ)ϵ
+6
+tolerance that takes O(S log( S
+ϵ1 )) iterations
+using binary search (see section I.2). Then for every state, we sort the Q values (as in (45))
+that costs O(A log(A)) iterations. In each state, to update value, we do again binary search
+with approximate κq(v) upto ϵ2 := (1−γ)ϵ
+6
+tolerance, that takes O(log( 1
+ϵ2 )) search iterations
+and each iteration cost O(A), altogether it costs O(A log( 1
+ϵ2 )) iterations. Sorting of actions
+and binary search adds upto O(A log( A
+ϵ )) iterations (action cost). So we have (doubly)
+approximated value iteration as following,
+|vn+1(s) − ˆλ| ≤ ϵ1
+(100)
+where
+(αs + γβsˆκq(vn))p =
+�
+Qn(s,a)≥ˆλ
+(Qn(s, a) − ˆλ)p
+and
+Qn(s, a) = R0(s, a) + γ
+�
+s′
+P0(s′|s, a)vn(s′),
+|ˆκq(vn) − κq(vn)| ≤ ϵ1.
+And we do this approximate value iteration for n = log( 3∥v0−v∗∥∞
+ϵ
+). Now, we do error
+analysis. By accumulating error, we have
+|vn+1(s) − (T ∗
+Us
+pvn)(s)| ≤|vn+1(s) − ˆλ| + |ˆλ − (T ∗
+Us
+pvn)(s)|
+≤ϵ1 + |ˆλ − (T ∗
+Us
+pvn)(s)|,
+(by definition)
+≤ϵ1 + γβmaxϵ1,
+(from proposition 8)
+≤2ϵ1.
+(101)
+where βmax := maxs βs, γ ≤ 1.
+Now, we do approximate value iteration, and from proposition 7, we get
+∥vn − v∗
+Us
+p∥ ≤ 2ϵ1
+1 − γ + γn[
+1
+1 − γ 2ϵ1 + ∥v0 − v∗
+Us
+p∥∞]
+(102)
+54
+
+Now, putting the value of n, we have
+∥vn − v∗
+Us
+p∥∞ =γn[ 2ϵ1
+1 − γ + ∥v0 − v∗
+Us
+p∥∞] +
+2ϵ1
+1 − γ
+≤γn[ ϵ
+3 + ∥v0 − v∗
+Us
+p∥∞] + ϵ
+3
+≤γn ϵ
+3 + ϵ
+3 + ϵ
+3 ≤ ϵ.
+(103)
+To summarize, we do O(log( 1
+ϵ )) full value iterations. Cost of evaluating reward penalty
+is O(S log( S
+ϵ )). For each state: evaluation of Q-value from value function requires O(SA)
+iterations, sorting the actions according Q-values requires O(A log(A)) iterations, and binary
+search for evaluation of value requires O(A log(1/ϵ). So the complexity is
+O((total iterations)(reward cost + S(Q-value + sorting + binary search for value )))
+= O
+�
+log(1
+ϵ )
+�
+S log(S
+ϵ ) + S(SA + A log(A) + A log(1
+ϵ ))
+��
+= O
+�
+log(1
+ϵ )
+�
+S log(1
+ϵ ) + S log(S) + S2A + SA log(A) + SA log(1
+ϵ )
+��
+= O
+�
+log(1
+ϵ )
+�
+S2A + SA log(A) + SA log(1
+ϵ )
+��
+= O
+�
+log(1
+ϵ )
+�
+S2A + SA log(A
+ϵ )
+��
+55
+
diff --git a/DtFRT4oBgHgl3EQfxjhb/content/tmp_files/load_file.txt b/DtFRT4oBgHgl3EQfxjhb/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c22741dd0c92d42647512249235614cb5fb036d5
--- /dev/null
+++ b/DtFRT4oBgHgl3EQfxjhb/content/tmp_files/load_file.txt
@@ -0,0 +1,1229 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf,len=1228
+page_content='An Efficient Solution to s-Rectangular Robust Markov  Decision Processes  Navdeep Kumar1, Kfir Levy1, Kaixin Wang2, and Shie Mannor1  1Technion  2National University of Singapore  February 1, 2023  Abstract  We present an efficient robust value iteration for s-rectangular robust Markov  Decision Processes (MDPs) with a time complexity comparable to standard (non-  robust) MDPs which is significantly faster than any existing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We do so by  deriving the optimal robust Bellman operator in concrete forms using our Lp water  filling lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We unveil the exact form of the optimal policies, which turn out to be  novel threshold policies with the probability of playing an action proportional to its  advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1  Introduction  In Markov Decision Processes (MDPs), an agent interacts with the environment and learns  to optimally behave in it [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' However, the MDP solution may be very sensitive to little  changes in the model parameters [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence we should be cautious applying the solution of  the MDP, when the model is changing or when there is uncertainty in the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Robust MDPs provide a way to address this issue, where an agent can learn to optimally  behave even when the model parameters are uncertain [15, 29, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Another motivation to  study robust MDPs is that they can lead to better generalization [33, 34, 25] compared to  non-robust solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Unfortunately, solving robust MDPs is proven to be NP-hard for general uncertainty  sets [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' As a result, the uncertainty set is often assumed to be rectangular, which enables  the existence of a contractive robust Bellman operators to obtain the optimal robust value  function [24, 18, 22, 12, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Recently, there has been progress in solving robust MDPs  for some sa-rectangular uncertainty sets via both value-based and policy-based methods  [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' An uncertainty set is said to be sa-rectangular if it can be expressed as a Cartesian  product of the uncertainty in all states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It can be further generalized to a  s-rectangular uncertainty set if it can be expressed as a Cartesian product of the uncertainty  in all states only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Compared to sa-rectangular robust MDPs, s-rectangular robust MDPs  are less conservative and hence more desirable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' however, they are also much more difficult  and poorly understood [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Currently, there are few works that consider s-rectangular Lp  robust MDPs where uncertainty set is further constrained by Lp norm, but they rely on  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='13642v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='LG]  31 Jan 2023  black box methods which limits its applicability and offers little insights [7, 16, 9, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' No  effective value or policy based methods exist for solving any s-robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Moreover, it is  known that optimal policies in s-rectangular robust MDPs can be stochastic, in contrast to  sa-rectangular robust MDPs and non-robust MDPs [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' However, so far, nothing is known  about the stochastic nature of the optimal policies in s-rectangular MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  In this work, we mainly focus on s-rectangular Lp robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We first revise the  unrealistic assumptions made in the noise transition kernel in [9] and introduce forbidden  transitions, which leads to novel regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then we derive robust Bellman operator  (policy evaluation) for a s-rectangular robust MDPs in closed form which is equivalent to  reward-value-policy-regularized non-robust Bellman operator without radius assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We exploit this equivalence to derive an optimal robust Bellman operator in concrete  forms using our Lp-water pouring lemma which generalizes existing water pouring lemma for  L2 case [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We can compute these operators in closed form for p = 1, ∞ and exactly by a  simple algorithm for p = 2, and approximately by binary search for general p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We show that  the time complexity of robust value iteration for p = 1, 2 is the same as that of non-robust  value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For general p, the complexity includes some additional log-factors due to  binary searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  In addition, we derive a complete characterization of the stochastic nature of optimal  policies in s-rectangular robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The optimal policies in this case, are threshold  policies, that plays only actions with positive advantage with probability proportional to  (p − 1)-th power to its advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For sa-rectangular R-contamination robust MDPs, [30] derived robust  Bellman operators which are equivalent to value-regularized-non-robust Bellman operators,  enabling efficient robust value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Building upon this work, [31] derived robust policy  gradient which is equivalvent to non-robust policy gradient with regularizer and correction  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Unfortunately, these methods can’t be naturally generalized to s-rectangular robust  MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  For s-rectangular robust MDPs, methods such as robust value iteration [6, 32], robust  modified policy iteration [19], partial robust policy iteration [16] etc tries to approximately  evaluate robust Bellman operators using variety of tools to estimate optimal robust value  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The scalability of these methods has been limited due to their reliance on an  external black-box solver such as Linear Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Previous works have explored robust MDPs from a regularization perspective [9, 10, 17, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Specifically, [9] showed that s-rectangular robust MDPs is equivalent to reward-value-policy  regularized MDPs, and proposed a gradient based policy iteration for s-rectangular Lp  robust MDPs ( where uncertainty set is s-rectangular and constrained by Lp norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' But  this gradient based policy improvement relies on black box simplex projection, hence very  slow and not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The detailed discussion of the above works can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2  Preliminary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  Notations  For a set S, |S| denotes its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' ⟨u, v⟩ := �  s∈S u(s)v(s) denotes the dot product  between functions u, v : S → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∥v∥q  p := (�  s|v(s)|p)  q  p denotes the q-th power of Lp  norm of function v, and we use ∥v∥p := ∥v∥1  p and ∥v∥ := ∥v∥2 as shorthand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  For a  2  set C, ∆C := {a : C → R|a(c) ≥ 0, ∀c, �  c∈C ac = 1} is the probability simplex over  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 0, 1 denotes all zero vector and all ones vector/function respectively of appropriate  dimension/domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 1(a = b) := 1 if a = b, 0 otherwise, is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For  vectors u, v, 1(u ≥ v) is component wise indicator vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 1(u ≥ v)(x) = 1(u(x) ≥ v(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  A × B = {(a, b) | a ∈ A, b ∈ B} is cartesain product between set A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  Markov Decision Processes  A Markov Decision Process (MDP) can be described as a tuple (S, A, P, R, γ, µ), where S is  the state space, A is the action space, P is a transition kernel mapping S × A to ∆S, R is a  reward function mapping S × A to R, µ is an initial distribution over states in S, and γ is a  discount factor in [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The expected discounted cumulative reward (return) is defined as  ρπ  (P,R) :=E  � ∞  �  n=0  γnR(sn, an)  ��� s0 ∼ µ, π, P  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The return can be written compactly as  ρπ  (P,R) = ⟨µ, vπ  (P,R)⟩,  (1)  [26] where vπ  (P,R) is the value function , defined as  vπ  (P,R)(s) := E  � ∞  �  n=0  γnR(sn, an)  ��� s0 = s, π, P  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (2)  Our objective is to find an optimal policy π∗  (P,R) that maximizes the performance ρπ  (P,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This performance can be written as :  ρ∗  (P,R) := max  π  ρπ  (P,R) = ⟨µ, v∗  (P,R)⟩,  (3)  where v∗  (P,R) := maxπ vπ  (P,R) is the optimal value function [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The value function vπ  (P,R) and the optimal value function v∗  (P,R) are the fixed points of the  Bellman operator T π  (P,R) and the robust Bellman operator T ∗  (P,R), respectively [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' These  γ-contraction operators are defined as follows: For any vector v, and state s ∈ S,  (T π  (P,R)v)(s) :=  �  a  π(a|s)  �  R(s, a) + γ  �  s′  P(s′|s, a)v(s′)  �  ,  and  T ∗  (P,R)v := max  π  T π  (P,R)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Therefore, the value iteration vn+1 := T ∗  (P,R)vn converges linearly to the optimal value  function v∗  (P,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Given this optimal value function, the optimal policy can be computed as:  π∗  (P,R) ∈ arg maxπ T π  (P,R)v∗  (P,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The vector minimum of a set U of vectors is defined component wise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (minu∈U u)(i) := minu∈U u(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This operation is well-defined only when there exists a  minimal vector u∗ ∈ U such that u∗ ⪯ u, ∀u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The same holds for other operations such  as maximum, argmin, argmax, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  Robust Markov Decision Processes  A robust Markov Decision Process (MDP) is a tuple (S, A, P, R, γ, µ) which generalizes the  standard MDP by containing a set of transition kernels P and set of reward functions R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Let uncertainty set U = P × R be set of tuples of transition kernels and reward functions  [18, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The robust performance ρπ  U of a policy π is defined to be its worst performance on  the entire uncertainty set U as  ρπ  U :=  min  (P,R)∈U ρπ  (P,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (4)  Our objective is to find an optimal robust policy π∗  U that maximizes the robust performance  ρπ  U, defined as  ρ∗  U := max  π  ρπ  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (5)  Solving the above robust objectives 4 and 5 are strongly NP-hard for general uncertainty  sets, even if they are convex [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence, the uncertainty set U = P × R is commonly  assumed to be s-rectangular, meaning that R and P can be decomposed state-wise as  R = ×s∈SRs and P = ×s∈SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For further simplification, U = P × R is assumed to  decompose state-action-wise as R = ×(s,a)∈S×ARs,a and P = ×(s,a)∈S×APs,a, known as  sa-rectangular uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Throughout the paper, the uncertainty set is assumed to  be s-rectangular (or sa-rectangular) unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Under the s-rectangularity  assumption, for every policy π, there exists a robust value function vπ  U which is the minimum  of vπ  (P,R) for all (P, R) ∈ U, and the optimal robust value function v∗  U which is the maximum  of vπ  U for all policies π [32], that is  vπ  U :=  min  (P,R)∈U vπ  (P,R),  and  v∗  U := max  π  vπ  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This implies, robust policy performance can be rewritten as  ρπ  U = ⟨µ, vπ  U⟩,  and  ρ∗  U = ⟨µ, v∗  U⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Furthermore, the robust value function vπ  U is the fixed point of the robust Bellmen operator  T π  U [32, 18], defined as  (T π  U v)(s) :=  min  (P,R)∈U  �  a  π(a|s)  �  R(s, a) + γ  �  s′  P(s′|s, a)v(s′)  �  ,  and the optimal robust value function v∗  U is the fixed point of the optimal robust Bellman  operator T ∗  U [18, 32], defined as  T ∗  U v := max  π  T π  U v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The optimal robust Bellman operator T ∗  U and robust Bellman operators T π  U are γ contraction  maps for all policy π [32], that is  ∥T ∗  U v − T ∗  U u∥∞ ≤ γ∥u − v∥∞,  ∥T π  U v − T π  U u∥∞ ≤ γ∥u − v∥∞,  ∀π, u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  So for all initial values vπ  0 , v∗  0, sequences defined as  vπ  n+1 := T π  U vπ  n,  v∗  n+1 := T ∗  U v∗  n  (6)  converges linearly to their respective fixed points, that is vπ  n → vπ  U and v∗  n → v∗  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Given  this optimal robust value function, the optimal robust policy can be computed as: π∗  U ∈  arg maxπ T π  U v∗  U [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This makes the robust value iteration an attractive method for solving  s-rectangular robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4  Table 1: p-variance  x  κx(v)  Remark  p  minω∈R∥v − ω1∥p  Binary search  ∞  maxs v(s)−mins v(s)  2  Semi-norm  2  ��  s  �  v(s) −  �  s v(s)  S  �2  Variance  1  �⌊(S+1)/2⌋  i=1  v(si)  Top half - lower half  − �S  i=⌈(S+1)/2⌉ v(si)  where v is sorted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v(si) ≥ v(si+1)  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3  Method  In this section, we consider constraining the uncertainty set around nominal values by the Lp  norm, which is a natural way of limiting the broad class of s (or sa)-rectangular uncertainty  sets [9, 16, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We will then derive robust Bellman operators for these uncertainty sets, which  can be used to obtain robust value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This will be done separately for sa-rectangular  in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1 and s-rectangular case in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We begin by making a few useful definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We reserve q for Holder conjugate of p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1  p + 1  q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let p-variance function κp : S → R be defined as  κp(v) := min  ω∈R∥v − ω1∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (7)  For p = 1, 2, ∞, the p-variance function κp has intuitive closed forms as summarized in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For general p, it can be calculated by binary search in the range [mins v(s), maxs v(s)] (  see appendix I for proofs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  (Sa)-rectangular Lp robust Markov Decision Processes  In accordance with [9], we define sa-rectangular Lp constrained uncertainty set Usa  p  as  Usa  p := (P0 + P) × (R0 + R)  where P, R are noise sets around nominal kernel P0 and nominal reward R0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Furthermore, noise sets are sa-rectangular, that is  P = ×s∈S,a∈APs,a,  and  R = ×s∈S,a∈ARs,a,  and each component are bounded by Lp norm that is  Rs,a =  �  Rs,a ∈ R  ��� |Rs,a| ≤ αs,a  �  ,  and  Ps,a = {Ps,a : S → R  ���  �  s′  Ps,a(s′) = 0  �  ��  �  simplex condition  , ∥Ps,a∥p ≤ βs,a}  5  with radius vector α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Radius vector β is chosen small enough so that all the transition  kernels in (P0 + P) are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Further, all transition kernels in (P0 + P) must have  the sum of each row equal to one, with P0 being a valid transition kernel satisfying this  requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This implies that the elements of P must have a sum of zero across each row  as ensured by simplex condition above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Our setting differs from [9] as they didn’t impose this simplex condition on the kernel  noise, which renders their setting unrealistic as not all transition kernels in their uncertainty  set satisfy the properties of transition kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This makes our reward regularizer dependent  on the q-variance of the value function κq(v), instead of the q-th norm of value function ∥v∥q  in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The main result of this subsection below states that robust Bellman operators can be  evaluated using only nominal values and regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' sa-rectangular Lp robust Bellman operators are equivalent to reward-value  regularized (non-robust) Bellman operators:  (T π  Usa  p v)(s) =  �  a  π(a|s)  �  −αs,a − γβs,aκq(v) + R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)  �  ,  and  (T ∗  Usa  p v)(s) = max  a∈A  �  −αs,a − γβs,aκq(v) + R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The proof in appendix, it mainly consists of two parts: a) Separating the noise from  nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' b) The reward noise to yields the term −αs,a and noise in kernel yields  −γβs,aκq(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Note, the reward penalty is proportional to both the uncertainty radiuses and a novel  variance function κp(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We recover non-robust value iteration by putting uncertainty radiuses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' αs,a, βs,a) to  zero, in the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Furthermore, the same is true for all subsequent robust results in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Q-Learning  The above result immediately implies the robust value iteration, and also suggests the Q-value  iteration of the following form  Qn+1(s, a) = max  a  �  R0(s, a) − αs,a − γβs,aκq(vn) +  �  s′  P0(s′|s, a) max  a  Qn(s′, a′)  �  ,  where vn(s) = maxa Qn(s, a), which is further discussed in appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Observe that value-variance κp(v) can be estimated online, using batches or other more  sophisticated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This paves the path for generalizing to a model-free setting similar  to [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Forbidden Transitions  Now, we focus on the cases where P0(s′|s, a) = 0 for some states s′, that is, forbidden  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In many practical situations, for a given state, many transitions are impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  For example, consider a grid world example where only a single-step jumps (left, right, up,  down) are allowed, so in this case, the probability of making a multi-step jump is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6  Table 2: Optimal robust Bellman operator evaluation  U  (T ∗  U v)(s)  remark  Us  p  min x  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ���  �  Qs − x1  �  1  �  Qs ≥ x  ����  p= σq(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' s)  Solve by binary search  Us  1  maxk  �k  i=1 Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='ai)−σ∞(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='s)  k  Highest penalized average  Us  2  By algorithm 1  High mean and variance  Us  ∞  maxa∈A Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) − σ1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' s)  Best action  Usa  p  maxa∈A  �  Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) − αsa − γβsaκq(v)  �  Best penalized action  nr  maxa Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)  Best action  where nr stands for Non-Robust MDP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ �  s′ P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  sorted Q-value: Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a1) ≥ · · · ≥ Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' aA)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' σq(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' s) = αs + γβsκq(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Qs = Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' ·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  and ◦ is Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  So upon adding noise to the kernel, the system should not start making impossible transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Therefore, noise set P must satisfy additional constraint: For any (s, a) if P0(s′|s, a) = 0  then  P(s′|s, a) = 0,  ∀P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Incorporating this constraint without much change in the theory is one of our novel contri-  bution, and is discussed in the appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  S-rectangular Lp robust Markov Decision Processes  In this subsection, we discuss the core contribution of this paper: the evaluation of robust  Bellman operators for the s-rectangular uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We begin by defining s-rectangular Lp constrained uncertainty set Us  p as  Us  p := (P0 + P) × (R0 + R)  where noise sets are s-rectangular,  P = ×s∈SPs,  and  R = ×s∈SRs,  and each component are bounded by Lp norm,  Rs =  �  Rs : A → R  ��� ∥Rs∥p ≤ αs  �  ,  and  Ps =  �  Ps : S × A → R  ��� ∥Ps∥p ≤ βs,  �  s′  Ps(s′, a) = 0, ∀a  �  ,  with radius vectors α and small enough β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The result below shows that, compared to the sa-rectangular case, the policy evaluation  for the s-rectangular case has an extra dependence on the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  7  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Policy Evaluation) S-rectangular Lp robust Bellman operator is equivalent to  reward-value-policy regularized (non-robust) Bellman operator:  (T π  Uspv)(s) = −  �  αs + γβsκq(v)  �  ∥πs∥q +  �  a  π(a|s)  �  R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)  �  ,  where ∥πs∥q is q-norm of the vector π(·|s) ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The proof in the appendix: the techniques are similar to as its sa-rectangular  counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The reward penalty in this case has an additional dependence on the norm of the policy  (∥πs∥q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This norm is conceptually similar to entropy regularization �  a π(a|s) ln(π(a|s)),  which is widely studied in the literature [21, 13, 20, 14, 27], and other regularizers such as  �  a π(a|s)tsallis( 1−π(a|s)  2  ), �  a π(a|s)cos(cos( π(a|s)  2  )), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Note: These regularizers, which are convex functions, are often used to promote stochas-  ticity in the policy and thus improve exploration during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' However, the above result  shows another benefit of these regularizers: they can improve robustness, which in turn can  lead to better generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  In literature, the above regularizers are scaled with arbitrary chosen constant, here we  have the different constant αs + γβsκq(v) for different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This extra dependence makes the policy improvement a more challenging task and thus,  presents a richer theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Policy improvement) For any vector v and state s, (T ∗  Uspv)(s) is the minimum  value of x that satisfies  � �  a  �  Q(s, a) − x  �p  1  �  Q(s, a) ≥ x  �� 1  p = σ,  (8)  where Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′), and σ = αs + γβsκq(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The proof is in the appendix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' the main steps are:  (T ∗  Uspv)(s) = max  π (T π  Uspv)(s),  (from definition)  ( Using policy evaluation Theorem 2)  = max  π  �  (T π  (P0,R0)v)(s)−  �  αs + γβsκq(v)  �  ∥πs∥q  �  = max  πs∈∆A⟨πs, Qs⟩ − σ∥πs∥q  ( where Qs = Q(·|s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The solution to the above optimization problem is technically complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Specifically, for  p = 2, the solution is known as the water filling/pouring lemma [1], we generalize it to the  Lp case, in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  To better understand the nature of (8), lets look at the ’sub-optimality distance’ function  g,  g(x) :=  � �  a  �  Q(s, a) − x  �p  1  �  Q(s, a) ≥ x  �� 1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  8  Algorithm 1 s-rectangular L2 robust Bellman operator  (see algorithm 1 of [1]  Input: v, s, x = Q(s, ·), and σ = αs + γβsκq(v)  Output: (T ∗  Uspv)(s)  1: Sort x such that x1 ≥ x2, · · · ≥ xA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: Set k = 0 and λ = x1 − σ  3: while k ≤ A − 1 and λ ≤ xk do  4:  k = k + 1  5:  λ = 1  k  �  k  �  i=1  xi −  �  �  �  �kσ2 + (  k  �  i=1  x2  i − k  k  �  i=1  xi)2  �  6: end while  7: return λ  The g(x) is the cumulative difference between x and the Q-values of actions whose  Q-value is greater than x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The function is monotonically decreasing, with a lower bound  of σ at x = maxa Q(s, a) − σ and a value of zero for all x ≥ maxa Q(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Since, (T ∗  Uspv)(s)  is the value of x at which the "sub-optimality distance" g(x) is equal to the "uncertainty  penalty" σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence, (8) can be approximately solved using a binary search between the  interval [maxa Q(s, a) − σ,  maxa Q(s, a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We invite the readers to consider the dependence of (T ∗  Uspv)(s) on p, αs, and βs, specifically:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' If αs = βs = 0 then σ = 0 which implies (T ∗  Uspv)(s) = maxa Q(s, a), same as non-robust  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' If p = ∞ then (T ∗  Uspv)(s) = maxa Q(s, a) − σ, as in the sa-rectangular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = 1, 2, (8) becomes linear and quadratic equation respectively, hence can be  solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' As αs and βs increase, σ increases, resulting in a decrease in (T ∗  Uspv)(s) at a rate that  becomes smaller as σ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' When σ is sufficiently small, (T ∗  Uspv)(s) = maxa Q(s, a)−  σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Solution to (8) can be obtained in closed form for the cases of p = 1, ∞, exactly by  algorithm 1 for p = 2, and approximately by binary search for general p, as summarized in  table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  In this section, we have demonstrated that robust Bellman operators can be efficiently  evaluated for both sa and s rectangular Lp robust MDPs, thus enabling efficient robust  value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In the following sections, we discuss the nature of optimal policies and the  time complexity of robust value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Finally, we present experiments validating the  time complexity of robust value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4  Optimal Policies  In the previous sections, we discussed how to efficiently obtain the optimal robust value  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This section focuses on utilizing these optimal robust value functions to derive  9  Table 3: Optimal Policy  U  π∗  U(a|s) ∝  Remark  Us  p  A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)p−11(A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) ≥ 0)  Top actions proportional to  (p − 1)-th power of advantage  Us  1  1(A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) ≥ 0)  Top actions with uniform probability  Us  2  A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)1(A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) ≥ 0)  Top actions proportion to advantage  Us  ∞  1(A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = 0)  Best action  Usa  p  1(A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = maxa A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a))  Best regularized action  (P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' R0)  1(A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = 0)  Non-robust MDP: Best action  where Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ �  s′ P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v∗  U(s′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' and A(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = Q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) − v∗  U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  the optimal robust policy using  π∗  U ∈ arg max  π  T π  U v∗  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This implies, the robust optimal policy π∗  U(·|s) at state s, is the policy π that maximizes  �  a  π(a|s)  min  (P,R)∈U  �  R(s, a) + γ  �  s′  P(s′|s, a)v∗  U(s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Non-robust MDP admits a deterministic optimal policy that maximizes the optimal  Q-value Q(s, a) := R(s, a) + γ �  s′ P(s′|s, a)v∗  (P,R)(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  sa-rectangular robust MDPs are known to admit a deterministic optimal robust  policy [18, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Moreover, from Theorem 1, it clear that a sa-rectangular Lp robust MDP  has a deterministic optimal robust policy that maximizes the regularized Q-value Q(s, a) =  −αs,a − γβs,aκq(v) + R0(s, a) + γ �  s′ P0(s′|s, a)v∗  Usa  p (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  s-rectangular robust MDPs: For this case, it is known that all optimal robust  policies can be stochastic [32], however, it was not previously known what the nature of  this stochasticity was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The result below provides the first explicit characterization of robust  optimal policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The optimal robust policy π∗  Usp can be computed using optimal robust value  function as:  π∗  Usp(a|s) ∝ [Q(s, a) − v∗  Usp(s)]p−11  �  Q(s, a) ≥ v∗  Usp(s)  �  where Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v∗  Us  p(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The above policy is a threshold policy that takes actions with a positive advantage, which  is proportional to the advantage function, while giving more weight to actions with higher  advantages and avoiding playing actions that are not useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This policy is different from  the optimal policy in soft-Q learning with entropy regularization, which is a softmax policy  10  Algorithm 2 Online s-rectangular Lp robust value iteration  Input: Initialize Q, v randomly, s0 ∼ µ, and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Output: v = v∗  Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1: while not converged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' n = n + 1 do  2:  Estimate κp(v) using table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3:  Approximate (T ∗  Uspv)(sn) using table 2 and update  v(sn) = v(sn) + ηn[(T ∗  Uspv)(sn) − v(sn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4:  Play action an = a with probability proportional to  [Q(sn, a) − v(sn)]p−11(Q(sn, a) ≥ v(sn)),  and get next state sn+1 from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5:  Update Q-value:  Q(sn, an) =Q(sn, an) + η′  n[R(sn, an) + γv(sn+1) − Q(sn, an)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6: end while  Table 4: Relative running cost (time) for value iteration  S  A  nr  Usa  1  LP  Us  1 LP  Usa  1  Usa  2  Usa  ∞  Us  1  Us  2  Us  ∞  Usa  10  Us  10  10  10  1  1438  72625  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  33  30  10  1  6616  629890  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  78  50  10  1  6622  4904004  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  41  100  20  1  16714  NA  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  41  nr stands for Non-robust MDP  of the form π(a|s) ∝ eη(Q(a|s)−v(s)) [14, 21, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' To the best of our knowledge, this type of  policy has not been presented in literature before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The special cases of the above theorem for p = 1, 2, ∞ along with others are summarized  in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5  Time complexity  In this section, we examine the time complexity of robust value iteration:  vn+1 := T ∗  U vn  for different Lp robust MDPs assuming the knowledge of nominal values (P0, R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Since, the  optimal robust Bellman operator T ∗  U is γ-contraction operator [32], meaning that it requires  only O(log( 1  ϵ )) iterations to obtain an ϵ-close approximation of the optimal robust value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The main challenge is to calculate the cost of one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The evaluation of the optimal robust Bellman operators in Theorem 1 and Theorem 3  has three main components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' A) Computing κp(v), which can be done differently depending  11  Table 5: Time complexity  Total cost O  Non-Robust MDP  log(1/ϵ)S2A  Usa  1  log(1/ϵ)S2A  Usa  2  log(1/ϵ)S2A  Usa  ∞  log(1/ϵ)S2A  Us  1  log(1/ϵ)(S2A + SA log(A))  Us  2  log(1/ϵ)(S2A + SA log(A))  Us  ∞  log(1/ϵ)S2A  Usa  p  log(1/ϵ)  �  S2A + S log(S/ϵ)  �  Us  p  log(1/ϵ)  �  S2A + SA log(A/ϵ)  �  Convex U  Strongly NP Hard  on the value of p, as shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' B) Computing the Q-value from v, which requires  O(S2A) in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And finally, C) Evaluating optimal robust Bellman operators from  Q-values, which requires different operations such as sorting of the Q-value, calculating the  best action, and performing a binary search, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=', as shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The overall complexity  of the evaluation is presented in table 5, with the proofs provided in appendix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We can observe that when the state space S is large, the complexity of the robust MDPs  is the same as that of the non-robust MDPs, as the complexity of all robust MDPs is the  same as non-robust MDPs at the limit S → ∞ (keeping action space A and tolerance ϵ  constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This is verified by our experiments, thus concluding that the Lp robust MDPs  are as easy as non-robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6  Experiments  In this section, we present numerical results that demonstrate the effectiveness of our methods,  verifying our theoretical claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Table 4 and Figure 1 demonstrate the relative cost (time) of robust value iteration  compared to non-robust MDP, for randomly generated kernel and reward functions with  varying numbers of states S and actions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The results show that s and sa-rectangular  MDPs are indeed costly to solve using numerical methods such as Linear Programming (LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Our methods perform similarly to non-robust MDPs, especially for p = 1, 2, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For general  p, binary search is required for acceptable tolerance, which requires 30−50 iterations, leading  to a little longer computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  As our complexity analysis shows, value iteration’s relative cost converges to 1 as the  number of states increases while keeping the number of actions fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This is confirmed by  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The rate of convergence for all the settings tested was the same as that of the non-robust  setting, as predicted by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The experiments ran a few times, resulting in some  stochasticity in the results, but the trend is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Further details can be found in section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  12  Figure 1: Relative cost of value iteration w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' non-robust MDP at different S with fixed  A = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  13  Relative cost of value iteration  U^sa 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5  U^s 1  relative cost to non-robust MDPs  non-robust  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='0  0  200  600  800  400  1000  1200  1400  number of states7  Conclusion and future work  We present an efficient robust value iteration for s-rectangular Lp-robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Our method  can be easily adapted to an online setting, as shown in Algorithm 2 for s-rectangular Lp-robust  MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Algorithm 2 is a two-time-scale algorithm, where the Q-values are approximated at  a faster time scale and the value function is approximated from the Q-values at a slower  time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The p-variance function κp can be estimated in an online fashion using batches  or other sophisticated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The convergence of the algorithm can be guaranteed from  [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' however, its analysis is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Additionally, we introduce a novel value regularizer (κp) and a novel threshold policy  which may help to obtain more robust and generalizable policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Further research could focus on other types of uncertainty sets, potentially resulting in  different kinds of regularizers and optimal policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  References  [1] Oren Anava and Kfir Levy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' k*-nearest neighbors: From global to local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Advances in  neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [2] Mahsa Asadi, Mohammad Sadegh Talebi, Hippolyte Bourel, and Odalric-Ambrym  Maillard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Model-based reinforcement learning exploiting state-action equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' CoRR,  abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='04077, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [3] Peter Auer, Thomas Jaksch, and Ronald Ortner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
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+page_content='  [27] John Schulman, Xi Chen, and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Equivalence between policy gradients and  soft q-learning, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [28] Richard S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Sutton and Andrew G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Reinforcement Learning: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The MIT Press, second edition, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [29] Aviv Tamar, Shie Mannor, and Huan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Scaling up robust mdps using function  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In Eric P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Xing and Tony Jebara, editors, Proceedings of the 31st  International Conference on Machine Learning, volume 32 of Proceedings of Machine  Learning Research, pages 181–189, Bejing, China, 22–24 Jun 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [30] Yue Wang and Shaofeng Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Online robust reinforcement learning with model uncer-  tainty, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [31] Yue Wang and Shaofeng Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Policy gradient method for robust reinforcement learning,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [32] Wolfram Wiesemann, Daniel Kuhn, and Breç Rustem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Robust markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Mathematics of Operations Research, 38(1):153–183, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [33] Huan Xu and Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Robustness and generalization, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  [34] Chenyang Zhao, Olivier Sigaud, Freek Stulp, and Timothy M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hospedales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Investigating  generalisation in continuous deep reinforcement learning, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  How to read appendix  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section A contains related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section B contains additional properties and results that couldn’t be included in the  main section for the sake of clarity and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Many of the results in the main paper  is special cases of the results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section C contains the discussion on zero transition kernel (forbidden transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section D contains a possible connection this work to UCRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section G contains additional experimental results and a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' All the proofs of the main body of the paper is presented in the section K and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section I contains helper results for section K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Particularly, it discusses p-mean function  ωp and p-variance function κp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section J contains helper results for section K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Particularly, it discusses Lp water  pouring lemma, necessary to evaluate robust optimal Bellman operator (learning) for  s-rectangular Lp robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section L contains time complexity proof for model based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  16  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section E develops Q-learning machinery for (sa)-rectangular Lp robust MDPs based  on the results in the main section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It is not used in the main body or anywhere  else, but this provides a good understanding for algorithms proposed in section F for  (sa)-rectangular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Section F contains model-based algorithms for s and (sa)-rectangular Lp robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  It also contains, remarks for special cases for p = 1, 2, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  A  Related Work  R-Contamination Uncertainty Robust MDPs  The paper [30] considers the following uncertainty set for some fixed constant 0 ≤ R ≤ 1,  Psa = {(1 − R)(P0)(·|s, a) + RP | P ∈ ∆S},  s ∈ S, a ∈ A,  (9)  and P = ⊗s,aPs,a,  U = {R0} × P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The robust value function vπ  U is the fixed point of  the robust Bellman operator defined as  (T π  U v)(s) := min  P ∈P  �  a  π(a|s)[R0(s, a) + γ  �  s′  P(s′|s, a)v(s′)],  (10)  =  �  a  π(a|s)[R0(s, a) − γR max  s  v(s) + (1 − R)γ  �  s′  P0(s′|s, a)v(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (11)  And the optimal robust value function v∗  U⊣ is the fixed point of the optimal robust Bellman  operator defined as  (T ∗  U v)(s) := max  π  min  P ∈P  �  a  π(a|s)[R0(s, a) + γ(1 − R)  �  s′  P(s′|s, a)v(s′)],  (12)  = max  a [R0(s, a) − γR max  s  v(s) + γ(1 − R)  �  s′  P0(s′|s, a)v(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (13)  Since, the uncertainty set is sa-rectangular, hence the map is a contraction [24], so the  robust value iteration here, will also converge linearly similar to non-robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It is also  possible to obtain Q-learning as following  Qn+1(s, a) = R0(s, a) − γR max  s,a Qn(s, a) + γ(1 − R)  �  s′  P0(s′|s, a) max  s′  Qn(s′, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (14)  Convergence of the above Q-learning follows from the contraction of robust value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Further, it is easy to see that model-free Q-learning can be obtained from the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  A follow-up work [31] proposes a policy gradient method for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3 of [31]) Consider a class of policies Π satisfying Assumption  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2 of [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The gradient of the robust return is given by  ∇ρπθ =  γR  (1 − γ)(1 − γ + γR)  �  s,a  dπθ  µ (s, a)∇πθ(a|s)Qπθ  U (s, a)  +  1  1 − γ + γR  �  s,a  dπθ  sθ (s, a)∇πθ(a|s)Qπθ  U (s, a),  where sθ ∈ arg max vπθ  U (s), and Qπ  U(s, a) = �  a π(a|s)  �  R0(s, a) − γR maxs vπ  U(s) + γ(1 −  R) �  s′ P0(s′|s, a)vπ  U(s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  17  The work shows that the proposed robust policy gradient method converges to the global  optimum asymptotically under direct policy parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The uncertainty set considered here, is sa-rectangular, as uncertainty in each state-action  is independent, hence the regularizer term (γR maxs v(s)) is independent of policy, and the  optimal (and greedy) policy is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It is unclear, how the uncertainty set can be  generalized to the s-rectangular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Observe that the above results resemble very closely  our sa-rectangular L1 robust MDPs results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Twice Regularized MDPs  The paper [9] converts robust MDPs to twice regularized MDPs, and proposes a gradient  based policy iteration method for solving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1 of [9]) (s-rectangular reward robust policy evaluation) Let the  uncertainty set be U = (R0 + R) × {P0}, where Rs = {rs ∈ RA | ∥rs∥ ≤ αs} for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Then the robust value function vπ  U is the optimal solution to the convex optimization problem:  max  v∈RA⟨µ, v⟩  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  v(s) ≤ (T π  R0,P0v)(s) − αs∥πs∥,  ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  It derives the policy gradient for reward robust MDPs to obtain the optimal robust policy  π∗  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2 of [9]) (s-rectangular reward robust policy gradient) Let  the uncertainty set be U = (R0 + R) × {P0}, where Rs = {rs ∈ RA | ∥rs∥ ≤ αs} for all  s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then the gradient of the reward robust objective ρπ  U := ⟨µ, vπ  U⟩ is given by  ∇ρπ  U = E(s,a)∼dπ  P0  �  ∇ ln(π(a|s))  �  Qπ  U(s, a) − αs  π(a|s)  ∥πs∥  ��  ,  where Qπ  U(s, a) := min(R,P )∈U[R(s, a) + γ �  s′ P(s′|s, a)vπ  U(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1 of [9]) (s-rectangular general robust policy evaluation) Let  the uncertainty set be U = (R0 + R) × {P0 + P}, where Rs = {rs ∈ RA | ∥rs∥ ≤ αs} and  Ps = {Ps ∈ RS×A | ∥Ps∥ ≤ βs} for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then the robust value function vπ  U is the  optimal solution to the convex optimization problem:  max  v∈RA⟨µ, v⟩  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  v(s) ≤ (T π  R0,P0v)(s) − αs∥πs∥ − γβs∥v∥∥πs∥,  ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Same as the reward robust case, the paper tries to find a policy gradient method to  obtain the optimal robust policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Unfortunately, the dependence of regularizer terms on  value makes it a very difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence it proposes the R2MPI algorithm (algorithm 1 of  [9]) for the purpose that optimizing the greedy step via projection onto the simplex using  a black box solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Note that the above proposition is not same as our policy evaluation  (although it looks similar), it requires some extra assumptions (assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1 [9]) and lot  of work ensure R2 Bellman operator is contraction etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In our case, we directly evaluate  robust Bellman operator that has already proven to be a contraction, hence we don’t require  any extra assumption nor any other work as [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Our work makes improvements over this work by explicitly solving both policy evaluation  and policy improvement in general robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It also makes more realistic assumptions  on the transition kernel uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  18  Regularizer solves Robust MDPs  The work [11] looks in the opposite direction than we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It investigates the impact of the  popularly used entropy regularizer on robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It finds that MaxEnt can be used to  maximize a lower bound on a certain robust RL objective (reward robust).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  As we noticed that ∥πs∥q behaves like entropy in our regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Further, our work  also deals with uncertainty in transition kernel in addition to the uncertainty in reward  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Upper Confidence RL  The upper confidence setting in [4, 3] is very similar to our Lp robust setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We refer to  this discussion in section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  B  S-rectangular: More Properties  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We begin with the following notational definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Q-value at value function v is defined as  Qv(s, a) := R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Optimal Q-value is defined as  Q∗  U(s, a) = R0(s, a) + γ  �  s′  P0(s′|s, a)v∗  U(s′)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' With little abuse of notation, Q(s, ai) shall denote the ith best value in state s, that is  Q(s, a1) ≥ Q(s, a2) ≥, · · · , ≥ Q(s, aA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' πv  U denotes the greedy policy at value function v, that is  T ∗  U v = T πv  U  U  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' χp(s) denotes the number of active actions in state s in s-rectangular Lp robust MDPs,  defined as  χp(s) :=  �� {a | π∗  Us  p(a|s) ≥ 0}  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' χp(p, s) denotes the number of active actions in state s at value function v in s-  rectangular Lp robust MDPs, defined as  χp(v, s) :=  �� {a | πv  Us  p(a|s) ≥ 0}  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We saw above that optimal policy in s-rectangular robust MDPs may be stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The  action that has a positive advantage is active and the rest are inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let χp(s) be the  number of active actions in state s, defined as  χp(s) :=  �� {a | π∗  Us  p(a|s) ≥ 0}  ��=  �� {a | Q∗  Us  p(s, a) ≥ v∗  Us  p(s)}  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (15)  19  Last equality comes from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' One direct relation between Q-value and value function  is given by  v∗  Us  p(s) =  �  a  π∗  Us  p(a|s)  �  −  �  αs + γβsκq(v)  �  ∥π∗  Us  p(·|s)∥q + Q∗  Us  p(s, a)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (16)  The above relation is very convoluted compared to non-robust and sa-rectangular robust  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The property below illuminates an interesting relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Optimal Value vs Q-value) v∗  Us  p(s) is bounded by the Q-value of χp(s)th and  (χp(s) + 1)th actions, that is ,  Q∗  Us  p(s, aχp(s)+1) < v∗  Us  p(s) ≤ Q∗  Us  p(s, aχp(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This special case of the property 2, similarly table 6 is special case of table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Table 6: Optimal value function and Q-value  v∗(s) = maxa Q∗(s, a)  Best value  v∗  Usa  p (s) = maxa[αs,a − γβs,aκq(v∗  Usa  p ) − Q∗  Usa  p (s, a)]  Best regularized value  Q∗  Us  p(s, aχp(s)+1) < v∗  Us  p(s) ≤ Q∗  Us  p(s, aχp(s))  Sandwich!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  where v∗, Q∗ is the optimal value function and Q-value respectively  of non-robust MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The same is true for the non-optimal Q-value and value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Greedy policy) The greedy policy πv  Us  p is a threshold policy, that is proportional  to the advantage function, that is  πv  Us  p(a|s) ∝  �  Qv(s, a) − (T ∗  Us  pv)(s)  �p−1 1  �  Qv(s, a) ≥ (T ∗  Us  pv)(s)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The above theorem is proved in the appendix, and Theorem 4 is its special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' So is  table 3 special case of table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Table 7: Greedy policy at value function v  U  πv  U(a|s) ∝  remark  Us  p  (Qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) − (T ∗  U v)(s))p−11(Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) ≥ 0)  top actions proportional to  (p − 1)th power of its advantage  Us  1  1(Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)≥0)  �  a 1(Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)≥0)  top actions with uniform probability  Us  2  Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)1Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)≥0)  �  a Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)1(Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)≥0)  top actions proportion to advantage  Us  ∞  arg maxa∈A Qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)  best action  Usa  p  arg maxa[−αsa − γβsaκq(v) + Qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)]  best action  where Av  U(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = Qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) − (T ∗  U v)(s) and Qv(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) = R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ �  s′ P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  20  The above result states that the greedy policy takes actions that have a positive advantage,  so we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  χp(v, s) :=  �� {a | πv  Us  p(a|s) ≥ 0}  ��=  �� {a | Qv(s, a) ≥ (T ∗  Us  p)v(s)}  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (17)  Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Greedy Value vs Q-value) (T ∗  Us  pv)(s) is bounded by the Q-value of χp(v, s)th  and (χp(v, s) + 1)th actions, that is ,  Qv(s, aχp(v,s)+1) < (T ∗  Us  pv)(s) ≤ Qv(s, aχp(v,s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Table 8:  Greedy value function and Q-value  (T ∗v)(s) = maxa Qv(s, a)  Best value  (T ∗  Usa  p )v(s) = maxa[αs,a − γβs,aκq(v) − Qv(s, a)]  Best regularized value  Qv(s, aχp(v,s)+1) < (T ∗  Us  p)v(s) ≤ Qv(s, aχp(v,s))  Sandwich!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  where Qv(s, a1) ≥, · · · , ≥ Qv(s, aA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The property below states that we can compute the number of active actions χp(v, s)  (and χp(s)) directly without computing greedy (optimal) policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' χp(v, s) is number of actions that has positive advantage, that is  χp(v, s) := max{k |  k  �  i=1  �  Qv(s, ai) − Qv(s, ak)  �p≤ σp},  where σ = αs + γβsκq(v), and Qv(s, a1) ≥ Qv(s, a2), ≥ · · · ≥ Q(s, aA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  When uncertainty radiuses (αs, βs) are zero (essentially σ = 0 ), then χp(v, s) = 1, ∀v, s,  that means, greedy policy taking the best action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In other words, all the robust results  reduce to non-robust results as discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2 as the uncertainty radius becomes  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Algorithm 3 Algorithm to compute s-rectangular Lp robust optimal Bellman Operator  1: Input: σ = αs + γβsκq(v),  Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: Output (T ∗  Us  pv)(s), χp(v, s)  3: Sort Q(s, ·) and label actions such that Q(s, a1) ≥ Q(s, a2), · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4: Set initial value guess λ1 = Q(s, a1) − σ and counter k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5: while k ≤ A − 1 and λk ≤ Q(s, ak) do  6:  Increment counter: k = k + 1  7:  Take λk to be a solution of the following  k  �  i=1  �  Q(s, ai) − x  �p= σp,  and  x ≤ Q(s, ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (18)  8: end while  9: Return: λk, k  21  C  Revisiting kernel noise assumption  Sa-Rectangular Uncertainty  Suppose at state s, we know that it is impossible to have transition (next) to some states  (forbidden states Fs,a) under some action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is, we have the transition uncertainty set P  and nominal kernel P0 such that  P0(s′|s, a) = P(s′|s, a) = 0,  ∀P ∈ P, ∀s′ ∈ Fs,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (19)  Then we define, the kernel noise as  Ps,a = {P | ∥P∥p = βs,a,  �  s′  P(s′) = 0,  P(s”) = 0, ∀s” ∈ Fs,a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (20)  In this case, our p-variance function is redefined as  κp(v, s, a) =  min  ∥P ∥p=βs,a,  �  s′ P (s′)=0,  P (s”)=0,  ∀s”∈Fs,a⟨P, v⟩  (21)  = min  ω∈R∥u − ω1∥p,  where u(s) = v(s)1(s /∈ Fs,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (22)  =κp(u)  (23)  This basically says, we consider value of only those states that is allowed (not forbidden) in  calculation of p-variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For example, we have  κ∞(v, s, a) = maxs/∈Fs,a v(s) − mins/∈Fs,a v(s)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (24)  (25)  So theorem 1 of the main paper can be re-stated as  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Restated) (Sa)-rectangular Lp robust Bellman operator is equivalent to reward  regularized (non-robust) Bellman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is, using κp above, we have  (T π  Usa  p v)(s) =  �  a  π(a|s)[−αs,a − γβs,aκq(v, s, a) + R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)],  (T ∗  Usa  p v)(s) = max  a∈A[−αs,a − γβs,aκq(v, s, a) + R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  S-Rectangular Uncertainty  This notion can also be applied to s-rectanular uncertainty, but with little caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Here, we  define forbidden states in state s to be Fs (state dependent) instead of state-action dependent  in sa-rectangular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Here, we define p-variance as  κp(v, s) = κp(u),  where u(s) = v(s)1(s /∈ Fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (26)  So the theorem 2 can be restated as  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (restated) (Policy Evaluation) S-rectangular Lp robust Bellman operator is  equivalent to reward regularized (non-robust) Bellman operator, that is  (T π  Us  pv)(s) = −  �  αs +γβsκq(v, s)  �  ∥π(·|s)∥q +  �  a  π(a|s)  �  R0(s, a)+γ  �  s′  P0(s′|s, a)v(s′)  �  where κp is defined above and ∥π(·|s)∥q is q-norm of the vector π(·|s) ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  22  All the other results (including theorem 4), we just need to replace the old p-variance  function with new p-variance function appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  D  Application to UCRL  In robust MDPs, we consider the minimization over uncertainty set to avoid risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' When we  want to discover the underlying kernel by exploration, then we seek optimistic policy, then  we consider the maximization over uncertainty set [4, 3, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We refer the reader to the step 3  of the UCRL algorithm [4], which seeks to find  arg max  π  max  R,P ∈U⟨µ, vπ  P,R⟩,  (27)  where  U = {(R, P) | |R(s, a) − R0(s, a)| ≤ αs,a, |P(s′|s, a) − P0(s′|s, a)| ≤ βs,a,s′, P ∈ (∆S)S×A}  for current estimated kernel P0 and reward function R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We refer section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1 and step 4 of  the UCRL 2 algorithm of [3], which seeks to find  arg max  π  max  R,P ∈U⟨µ, vπ  P,R⟩,  (28)  where  U ={(R, P) | |R(s, a) − R0(s, a)| ≤ αs,a,  ∥P(·|s, a) − P0(·|s, a)∥1 ≤ βs,a, P ∈ (∆S)S×A}  The uncertainty radius α, β depends on the number of samples of different transitions and  observations of the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The paper [4] doesn’t explain any method to solve the above  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' UCRL 2 algorithm [3], suggests to solve it by linear programming that can be very  slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We show that it can be solved by our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The above problem can be tackled as following  max  π  max  R,P ∈Usa  p  ⟨µ, vπ  P,R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (29)  We can define, optimistic Bellman operators as  ˆT π  U v := max  R,P ∈U vπ  P,R,  ˆT ∗  U v := max  π  max  R,P ∈U vπ  P,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (30)  The well definition and contraction of the above optimistic operators may follow directly  from their pessimistic (robust) counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We can evaluate above optimistic operators as  ( ˆT π  Usa  p v)(s) =  �  a  π(a|s)  �  R0(s, a) + αs,a + βs,aγκq(v) +  �  s′  P0(s′|s, a)v(s′)  �  ,  (31)  ( ˆT ∗  Usa  p v)(s) = max  a  �  R0(s, a) + αs,a + βs,aγκq(v) +  �  s′  P0(s′|s, a)v(s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (32)  The uncertainty radiuses α, β and nominal values P0, R0 can be found by similar analysis by  [4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We can get the Q-learning from the above results as  Q(s, a) → R0(s, a) − αs,a − γβs,aκq(v) + γ  �  s′  P0(s′|s, a) max  a′ Q(s′, a′),  (33)  23  where v(s) = maxa Q(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' From law of large numbers, we know that uncertainty radiuses  αs,a, βs,a behaves as O( 1  √n) asymptotically with number of iteration n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This resembles very  closely to UCB VI algorithm [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We emphasize that similar optimistic operators can be  defined and evaluated for s-rectangular uncertainty sets too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  E  Q-Learning for sa-rectangular MDPs  In view of Theorem 1, we can define Qπ  Usa  p , the robust Q-values under policy π for (sa)-  rectangular Lp constrained uncertainty set Usa  p  as  Qπ  Usa  p (s, a) := −αs,a − γβs,aκq(vπ  Usa  p ) + R0(s, a) + γ  �  s′  P0(s′|s, a)vπ  Usa  p (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (34)  This implies that we have the following relation between robust Q-values and robust value  function, same as its non-robust counterparts,  vπ  Usa  p (s) =  �  a  π(a|s)Qπ  Usa  p (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (35)  Let Q∗  Usa  p denote the optimal robust Q-values associated with optimal robust value v∗  Usa  p , given  as  Q∗  Usa  p (s, a) := −αs,a − γβs,aκq(v∗  Usa  p ) + R0(s, a) + γ  �  s′  P0(s′|s, a)v∗  Usa  p (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (36)  It is evident from Theorem 1 that optimal robust value and optimal robust Q-values satisfies  the following relation, same as its non-robust counterparts,  v∗  Usa  p (s′) = max  a∈A Q∗  Usa  p (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (37)  Combining 37 and 36, we have optimal robust Q-value recursion as follows  Q∗  Usa  p (s, a) = −αs,a − γβs,aκq(v∗  Usa  p ) + R0(s, a) + γ  �  s′  P0(s′|s, a) max  a∈A Q∗  Usa  p (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (38)  The above robust Q-value recursion enjoys similar properties as its non-robust counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' ((sa)-rectangular Lp regularized Q-learning) Let  Qn+1(s, a) = R0(s, a) − αsa − γβsaκq(vn) + γ  �  s′  P0(s′|s, a) max  a∈A Qn(s′, a),  where vn(s) = maxa∈A Qn(s, a), then Qn converges to Q∗  Usa  p linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Observe that the above Q-learning equation is exactly the same as non-robust MDP  except the reward penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Recall that κ1(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='5(maxs v(s) − mins v(s)) is difference  between peak to peak values and κ2(v) is variance of v, that can be easily estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence,  model free algorithms for (sa)-rectangular Lp robust MDPs for p = 1, 2, can be derived  easily from the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This implies that (sa)-rectangular L1 and L2 robust MDPs  are as easy as non-robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  24  F  Model Based Algorithms  In this section, we assume that we know the nominal transitional kernel and nominal reward  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Algorithm 4, algorithm 5 is model based algorithm for (sa)-rectangular and s  rectangular Lp robust MDPs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It is explained in the algorithms, how to get deal  with specail cases (p = 1, 2, ∞) in a easy way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Algorithm 4 Model Based Q-Learning Algorithm for SA Rectangular Lp Robust MDP  1: Input: αs,a, βs,a are uncertainty radius in reward and transition kernel respectively in  state s and action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Transition kernel P and reward vector R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Take initial Q-values Q0  randomly and v0(s) = maxa Q0(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: while not converged do  3:  Do binary search in [mins vn(s), maxs vn(s)] to get q-mean ωn, such that  �  s  (vn(s) − ωn)  |vn(s) − ωn| |vn(s) − ωn|  1  p−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (39)  4:  Compute q-variance:  κn = ∥v − ωn∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5:  Note: For p = 1, 2, ∞, we can compute κn exactly in closed from, see table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6:  for s ∈ S do  7:  for a ∈ A do  8:  Update Q-value as  Qn+1(s, a) = R0(s, a) − αsa − γβsaκn + γ  �  s′  P0(s′|s, a) max  a  Qn(s′, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  9:  end for  10:  Update value as  vn+1(s) = max  a  Qn+1(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  11:  end for  n → n + 1  12: end while  25  Algorithm 5 Model Based Algorithm for S Rectangular Lp Robust MDP  1: Take initial Q-values Q0 and value function v0 randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: Input: αs, βs are uncertainty radius in reward and transition kernel respectively in state  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  3: while not converged do  4:  Do binary search in [mins vn(s), maxs vn(s)] to get q-mean ωn, such that  �  s  (vn(s) − ωn)  |vn(s) − ωn| |vn(s) − ωn|  1  p−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (40)  5:  Compute q-variance:  κn = ∥v − ωn∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6:  Note: For p = 1, 2, ∞, we can compute κn exactly in closed from, see table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  7:  for s ∈ S do  8:  for a ∈ A do  9:  Update Q-value as  Qn+1(s, a) = R0(s, a) + γ  �  s′  P0(s′|s, a)vn+1(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (41)  10:  end for  11:  Sort actions in decreasing order of the Q-value, that is  Qn+1(s, ai) ≥ Qn+1(s, ai+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (42)  12:  Value evaluation:  vn+1(s) = x  such that  (αs + γβsκn)p =  �  Qn+1(s,ai)≥x  |Qn+1(s, ai) − x|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (43)  13:  Note: We can compute vn+1(s) exactly in closed from for p = ∞ and for p = 1, 2,  we can do the same using algorithm 8,7 respectively, see table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  14:  end for  n → n + 1  15: end while  26  Algorithm 6 Model based algorithm for s-recantangular L1 robust MDPs  1: Take initial value function v0 randomly and start the counter n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: while not converged do  3:  Calculate q-variance:  κn = 1  2  �  maxs vn(s) − mins vn(s)  �  4:  for s ∈ S do  5:  for a ∈ A do  6:  Update Q-value as  Qn(s, a) = R0(s, a) + γ  �  s′  P0(s′|s, a)vn(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (44)  7:  end for  8:  Sort actions in state s, in decreasing order of the Q-value, that is  Qn(s, a1) ≥ Qn(s, a2), · · · ≥ Qn(s, aA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (45)  9:  Value evaluation:  vn+1(s) = max  m  �m  i=1 Qn(s, ai) − αs − βsγκn  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (46)  10:  Value evaluation can also be done using algorithm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  11:  end for  n → n + 1  12: end while  G  Experiments  The table 4 contains relative cost (time) of robust value iteration w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' non-robust MDP,  for randomly generated kernel and reward function with the number of states S and the  number of action A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Notations  S : number of state, A: number of actions, Usa  p  LP: Sa rectangular Lp robust MPDs by Linear  Programming, Us  p LP: S rectangular Lp robust MPDs by Linear Programming and other  numerical methods, Usa/s  p=1,2,∞ : Sa/S rectangular L1/L2/L∞ robust MDPs by closed form  method (see table 2, theorem 3) Usa/s  p=5,10 : Sa/S rectangular L5/L10 robust MDPs by binary  search (see table 2, theorem 3 of the paper)  Observations  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Our method for s/sa rectangular L1/L2/L∞ robust MDPs takes almost same (1-3 times)  the time as non-robust MDP for one iteration of value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This confirms our complexity  analysis (see table 4 of the paper) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Our binary search method for sa rectangular L5/L10  robust MDPs takes around 4 − 6 times more time than non-robust counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This is  27  Table 9: Relative running cost (time) for value iteration  U  S=10 A=10  S=30 A=10  S=50 A=10  S=100 A=20  remark  non-robust  1  1  1  1  Usa  ∞ by LP  1374  2282  2848  6930  lp  Usa  1  by LP  1438  6616  6622  16714  lp  Us  1 by LP  72625  629890  4904004  NA  lp/minimize  Usa  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='77  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='45  closed form  Usa  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='59  closed form  Usa  ∞  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
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+page_content='999  binary search  Us  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='999  binary search  Us  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='995  binary search  due to extra iterations required to find p-variance function κp(v) through binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Our binary search method for s rectangular L5/L10 robust MDPs takes around 30 − 100  times more time than non-robust counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This is due to extra iterations required to  find p-variance function κp(v) through binary search and Bellman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' One common  feature of our method is that time complexity scales moderately as guranteed through our  complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Linear programming methods for sa-rectangualr L1/L∞ robsust  MDPs take atleast 1000 times more than our methods for small state-action space, and  it scales up very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Numerical methods (Linear programming for minimization over  uncertianty and ’scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='minimize’ for maximization over policy) for s-rectangular L1  robust MDPs take 4-5 order more time than our mehtods (and non-robust MDPs) for very  small state-action space, and scales up too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The reason is obvious, as it has to solve  two optimization, one minimization over uncertainty and other maximization over policy,  whereas in the sa-rectangular case, only minimization over uncertainty is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This  confirms that s-rectangular uncertainty set is much more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Rate of convergence  The rate of convergence for all were approximately the same as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='9 = γ, as predicted by  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And it is well illustrated by the relative rate of convergence w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' non-robust by the  table G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  In the above experiments, Bellman updates for sa/s rectangular L1/L2/L∞ were done in  29  closed form, and for L5/L10 were done by binary search as suggested by table 2 and theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Note: Above experiments’ results are for few runs, hence containing some stochas-  ticity but the general trend is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In the final version, we will do averaging of many  runs to minimize the stochastic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Results for many different runs can be found at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='com/******.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Note that the above experiments were done without using too much parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  There is ample scope to fine-tune and improve the performance of robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The above  experiments confirm the theoretical complexity provided in Table 4 of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The codes  and results can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='com/******.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Experiments parameters  Number of states S (variable), number of actions A (variable), transition kernel and reward  function generated randomly, discount factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='9, uncertainty radiuses =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1 (for all states  and action, just for convenience ), number of iterations = 100, tolerance for binary search =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='00001  Hardware  The experiments are done on the following hardware: Intel(R) Core(TM) i5-4300U CPU @  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='90GHz 64 bits, memory 7862MiB Software: Experiments were done in python, using numpy,  scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='linprog for Linear programmig for policy evalution in s/sa rectangular robust  MDPs, scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='minize and scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='LinearConstraints for policy improvement  in s-rectangular L1 robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  H  Extension to Model Free Settings  Extension of Q-learning (in section E ) for sa-rectangular MDPs to model free setting can  easily done similar to [30], also policy gradient method can be obtained as [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The only  thing, we need to do, is to be able to compute/estimate κq online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It can be estimated  using an ensemble (samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Further, κ2 can be estimated by the estimated mean and the  estimated second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' κ∞ can be estimated by tracking maximum and minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  For s-rectangular case too, we can obtain model-free algorithms easily, by estimating κq  online and keeping track of Q-values and value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The convergence analysis may be  similar to [30], especially for sa-rectangular case, and for the other, it would be two time  scale, which can be dealt with techniques in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It would be  interesting to obtain policy gradient methods for this case, which we believe can be obtained  from the policy evaluation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  I  p-variance  Recall that κp is defined as follows  κp(v) = min  w ∥v − ω1∥p = ∥v − ωp∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  30  Now, observe that  ∂∥v − ω∥p  ∂ω  = 0  =⇒  �  s  sign(v(s) − ω)|v(s) − ω|p−1 = 0,  =⇒  �  s  sign(v(s) − ωp(v))|v(s) − ωq(v)|p−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (47)  For p = ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  lim  p→∞  ���  �  s  sign  �  v(s) − ω∞(v)  ��� v(s) − ω∞(v)  ��p���  1  p = 0  =  �  max  s |v(s) − ω∞(v)|  �  lim  p→∞  ���  �  s  sign  �  v(s) − ω∞(v)  ��  |v(s) − ω∞(v)  ��  maxs|v(s) − ω∞(v)|  �p���  1  p  Assuming max  s |v(s) − ω∞(v)| ̸= 0 otherwise ω∞ = v(s) = v(s′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∀s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' s′  =⇒ lim  p→∞  ���  �  s  sign  �  v(s) − ω∞(v)  ��  |v(s) − ω∞(v)  ��  maxs|v(s) − ω∞(v)|  �p���  1  p = 0  To avoid technical complication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we assume max  s  v(s) > v(s) < min  s  v(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∀s  =⇒ lim  p→∞|max  s  v(s) − ω∞(v)| = lim  p→∞|min  s  v(s) − ω∞(v)|  =⇒ max  s  v(s) − lim  q→∞ ω∞(v) = −(min  s  v(s) − lim  p→∞ ω∞(v)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (managing signs)  =⇒ lim  p→∞ ω∞(v) = maxs v(s) + mins v(s)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (48)  κ∞(v) =∥v − ω∞1∥∞  =∥v − maxs v(s) + mins v(s)  2  1∥∞,  (putting in value of ω∞)  =maxs v(s) − mins v(s)  2  (49)  For p = 2, we have  κ2(v) =∥v − ω21∥2  =∥v −  �  s v(s)  S  1∥2,  =  ��  s  (v(s) −  �  s v(s)  S  )2  (50)  For p = 1, we have  �  s∈S  sign  �  v(s) − ω1(v)  �  = 0  (51)  Note that there may be more than one values of ω1(v) that satisfies the above equation and  each solution does equally good job (as we will see later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' So we will pick one ( is median of  31  Table 11: p-mean, where v(si) ≥ v(si+1)  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  x  ωx(v)  remark  p  �  s sign(v(s) − ωp(v))|v(s) − ωp(v)|  1  p−1 = 0  Solve by binary search  1  v(s⌊(S+1)/2⌋)+v(s⌈(S+1)/2⌉)  2  Median  2  �  s v(s)  S  Mean  ∞  maxs v(s)+mins v(s)  2  Average of peaks  v) according to our convenience as  ω1(v) = v(s⌊(S+1)/2⌋) + v(s⌈(S+1)/2⌉)  2  where  v(si) ≥ v(si+1)  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  κ1(v) =∥v − ω11∥1  =∥v − med(v)1∥1,  (putting in value of ω0, see table 11)  =  �  s  |v(s) − med(v)|  =  ⌊(S+1)/2⌋  �  i=1  (v(s) − med(v)) +  S  �  ⌈(S+1)/2⌉  (med(v) − v(s))  =  ⌊(S+1)/2⌋  �  i=1  v(s) −  S  �  ⌈(S+1)/2⌉  v(s)  (52)  where med(v) :=  v(s⌊(S+1)/2⌋)+v(s⌈(S+1)/2⌉)  2  where  v(si) ≥ v(si+1)  ∀i is median of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The  results are summarized in table 1 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  p-variance function and kernel noise  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' q-variance function κq is the solution of the following optimization problem  (kernel noise),  κq(v) = −1  ϵ min  c ⟨c, v⟩,  ∥c∥p ≤ ϵ,  �  s  c(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Writing Lagrangian L, as  L :=  �  s  c(s)v(s) + λ  �  s  c(s) + µ(  �  s  |c(s)|p − ϵp),  where λ ∈ R is the multiplier for the constraint �  s c(s) = 0 and µ ≥ 0 is the multiplier for  the inequality constraint ∥c∥q≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Taking its derivative, we have  ∂L  ∂c(s) = v(s) + λ + µp|c(s)|p−1 c(s)  |c(s)|  (53)  32  From the KKT (stationarity) condition, the solution c∗ has zero derivative, that is  v(s) + λ + µp|c∗(s)|p−1 c∗(s)  |c∗(s)| = 0,  ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (54)  Using Lagrangian derivative equation (54), we have  v(s) + λ + µp|c∗(s)|p−1 c∗(s)  |c∗(s)| = 0  =⇒  �  s  c∗(s)[v(s) + λ + µp|c∗(s)|p−1 c∗(s)  |c∗(s)|] = 0,  (multiply with c∗(s) and summing )  =⇒  �  s  c∗(s)v(s) + λ  �  s  c∗(s) + µp  �  s  |c∗(s)|p−1 (c∗(s))2  |c∗(s)| = 0  =⇒ ⟨c∗, v⟩ + µp  �  s  |c∗(s)|p = 0  (using  �  s  c∗(s) = 0 and (c∗(s))2 = |c∗(s)|2 )  =⇒ ⟨c∗, v⟩ = −µpϵp,  (using  �  s  |c∗(s)|p = ϵp ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (55)  It is easy to see that µ ≥ 0, as minimum value of the objective must not be positive ( at  c = 0, the objective value is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Again we use Lagrangian derivative (54) and try to get  the objective value (−µpϵp) in terms of λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' as  v(s) + λ + µp|c∗(s)|p−1 c∗(s)  |c∗(s)| = 0  =⇒ |c∗(s)|p−2c∗(s) = −v(s) + λ  µp  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (re-arranging terms)  =⇒  �  s  |(|c∗(s)|p−2c∗(s))|  p  p−1 =  �  s  | − v(s) + λ  µp  |  p  p−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (doing  �  s  |·|  p  p−1 )  =⇒ ∥c∗∥p  p =  �  s  | − v(s) + λ  µp  |  p  p−1 =  �  s  |v(s) + λ  µp  |q = ∥v + λ∥q  q  |µp|q  =⇒ |µp|q∥c∗∥p  p = ∥v + λ∥q  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (re-arranging terms)  =⇒ |µp|qϵp = ∥v + λ∥q  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (using  �  s  |c∗(s)|p = ϵp )  =⇒ ϵ(µpϵp/q) = ϵ∥v + λ∥q  (taking 1  q the power then multiplying with ϵ)  =⇒ µpϵp = ϵ∥v + λ∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (56)  33  Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' using Lagrangian derivative (54) to solve for λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  v(s) + λ + µp|c∗(s)|p−1 c∗(s)  |c∗(s)| = 0  =⇒ |c∗(s)|p−2c∗(s) = −v(s) + λ  µp  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (re-arranging terms)  =⇒ |c∗(s)| = |v(s) + λ  µp  |  1  p−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (looking at absolute value)  and  c∗(s)  |c∗(s)| = − v(s) + λ  |v(s) + λ|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (looking at sign: and note µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' p ≥ 0)  =⇒  �  s  c∗(s)  |c∗(s)||c∗(s)| = −  �  s  v(s) + λ  |v(s) + λ||v(s) + λ  µp  |  1  p−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (putting back)  =⇒  �  s  c∗(s) = −  �  s  v(s) + λ  |v(s) + λ||v(s) + λ  µp  |  1  p−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  =⇒  �  s  v(s) + λ  |v(s) + λ||v(s) + λ|  1  p−1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ( using  �  i  c∗(s) = 0)  (57)  Combining everything,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  − 1  ϵ min  c ⟨c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∥c∥p ≤ ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  �  s  c(s) = 0  =∥v − λ∥q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  such that  �  s  sign(v(s) − λ)|v(s) − λ|  1  p−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (58)  Now, observe that  ∂∥v − λ∥q  ∂λ  = 0  =⇒  �  s  sign(v(s) − λ)|v(s) − λ|  1  p−1 = 0,  =⇒ κq(v) = ∥v − λ∥q,  such that  �  s  sign(v(s) − λ)|v(s) − λ|  1  p−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (59)  The last equality follows from the convexity of p-norm ∥·∥q, where every local minima is  global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  For the sanity check, we re-derive things for p = 1 from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = 1, we have  − 1  ϵ min  c ⟨c, v⟩,  ∥c∥1 ≤ ϵ,  �  s  c(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  = − 1  2(min  s  v(s) − max  s  v(s))  =κ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (60)  It is easy to see the above result, just by inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  Binary search for p-mean and estimation of p-variance  If the function f : [−B/2, B/2] → R, B ∈ R is monotonic (WLOG let it be monotonically  decreasing) in a bounded domain, and it has a unique root x∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' f(x∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then we can  34  find x that is an ϵ-approximation x∗ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' ∥x − x∗∥ ≤ ϵ ) in O(B/ϵ) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let  x0 = 0 and  xn+1 :=  �  �  �  �  �  −B+xn  2  if  f(xn) > 0  B+xn  2  if  f(xn) < 0  xn  if  f(xn) = 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  It is easy to observe that ∥xn − x∗∥ ≤ B(1/2)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This is proves the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This  observation will be referred to many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Now, we move to the main claims of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The function  hp(λ) :=  �  s  sign  �  v(s) − λ  ��� v(s) − λ  ��p  is monotonically strictly decreasing and also has a root in the range [mins v(s), maxs v(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  hp(λ) =  �  s  v(s) − λ  |v(s) − λ||v(s) − λ|p  dhp  dλ (λ) = −p  �  s  |v(s) − λ|p−1 ≤ 0,  ∀p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (61)  Now, observe that hp(maxs v(s)) ≤ 0 and hp(mins v(s)) ≥ 0, hence by hp must have a root  in the range [mins v(s), maxs v(s)] as the function is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The above proposition ensures that a root ωp(v) can be easily found by binary search  between [mins v(s), maxs v(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Precisely, ϵ approximation of ωp(v) can be found in O(log( maxs v(s)−mins v(s)  ϵ  )) number of  iterations of binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And one evaluation of the function hp requires O(S) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  And we have finite state-action space and bounded reward hence WLOG we can assume  |maxs v(s)|, |mins v(s)| are bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence, the complexity to approximate  ωp is O(S log( 1  ϵ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Let ˆωp(v) be an ϵ-approximation of ωp(v), that is  �� ωp(v) − ˆωp(v)  ��≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  And let ˆκp(v) be approximation of κp(v) using approximated mean, that is,  ˆκp(v) := ∥v − ˆωp(v)1∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Now we will show that ϵ error in calculation of p-mean ωp, induces O(ϵ) error in estimation  of p-variance κp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Precisely,  ��� κp(v) − ˆκp(v)  ���=  ���  �� v − ωp(v)1  ��  p −  �� v − ˆωp(v)1  ��  p  ���  ≤  �� ωp(v)1 − ˆωp(v)1  ��  p,  (reverse triangle inequality)  =  �� 1  ��  p  �� ωp(v) − ˆωp(v)  ��  ≤  �� 1  ��  p ϵ  =S  1  p ϵ ≤ Sϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (62)  35  For general p, an ϵ approximation of κp(v) can be calculated in O(S log( S  ϵ ) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We will estimate mean ωp to an ϵ/S tolerance (with cost O(S log( S  ϵ ) ) and then approximate  the κp with this approximated mean (cost O(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  J  Lp Water Filling/Pouring lemma  In this section, we are going to discuss the following optimization problem,  max  c  −α∥c∥q + ⟨c, b⟩  such that  A  �  i=1  ci = 1,  ci ≥ 0,  ∀i  where α ≥ 0, referred as Lp-water pouring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We are going to assume WLOG that b  is sorted component wise, that is b1 ≥ b2, · · · ≥ bA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The above problem for p = 2, is studied  in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The approach we are going to solve the problem is as follows: a) Write Lagrangian b)  Since the problem is convex, any solutions of KKT condition is global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' c) Obtain  conditions using KKT conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let b ∈ RA be such that its components are in decreasing order (i,e bi ≥ bi+1),  α ≥ 0 be any non-negative constant, and  ζp := max  c  −α∥c∥q + ⟨c, b⟩  such that  A  �  i=1  ci = 1,  ci ≥ 0,  ∀i,  (63)  and let c∗ be a solution to the above problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Higher components of b, gets higher weight in c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In other words, c∗ is also sorted  component wise in descending order, that is  c∗  1 ≥ c∗  2, · · · , ≥ c∗  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The value ζp satisfies the following equation  αp =  �  bi≥ζp  (bi − ζp)p  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The solution c of (63), is related to ζp as  ci =  (bi − ζp)p−11(bi ≥ ζp)  �  s(bi − ζp)p−11(bi ≥ ζp)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Observe that the top χp := max{i|bi ≥ ζp} actions are active and rest are passive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The  number of active actions can be calculated as  {k|αp ≥  k  �  i=1  (bi − bk)p} = {1, 2, · · · , χp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Things can be re-written as  ci ∝  �  (bi − ζp)p−1  if  i ≤ χp  0  else  and  αp =  χp  �  i=1  (bi − ζp)p  36  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The function �  bi≥x(bi − x)p is monotonically decreasing in x, hence the root ζp can  be calculated efficiently by binary search between [b1 − α, b1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Solution is sandwiched as follows  bχp+1 ≤ ζp ≤ bχp  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' k ≤ χp if and only if there exist the solution of the following,  k  �  i=1  (bi − x)p = αp  and  x ≤ bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' If action k is active and there is greedy increment hope then action k + 1 is also active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  That is  k ≤ χp  and  λk ≤ bk+1 =⇒ k + 1 ≤ χp,  where  k  �  i=1  (bi − λk)p = αp  and  λk ≤ bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' If action k is active, and there is no greedy hope and then action k + 1 is not active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  That is,  k ≤ χp  and  λk > bk+1 =⇒ k + 1 > χp,  where  k  �  i=1  (bi − λk)p = αp  and  λk ≤ bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  And this implies k = χp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let  f(c) := −α∥c∥q + ⟨b, c⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Let c be any vector, and c′ be rearrangement c in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Precisely,  c′  k := cik,  where  ci1 ≥ ci2, · · · , ≥ ciA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Then it is easy to see that f(c′) ≥ f(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Writting Lagrangian of the optimization problem, and its derivative,  L = −α∥c∥q + ⟨c, b⟩ + λ(  �  i  ci − 1) + θici  ∂L  ∂ci  = −α∥c∥1−q  q  |ci|q−2ci + bi + λ + θi,  (64)  λ ∈ R is multiplier for equality constraint �  i ci = 1 and θ1, · · · , θA ≥ 0 are multipliers  for inequality constraints ci ≥ 0,  ∀i ∈ [A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Using KKT (stationarity) condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we  have  −α∥c∗∥1−q  q  |c∗  i |q−2c∗  i + bi + λ + θi = 0  (65)  37  Let B := {i|c∗  i > 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' then  �  i∈B  c∗  i [−α∥c∗∥1−q  q  |c∗  i |q−2c∗  i + bi + λ] = 0  =⇒ − α∥c∗∥1−q  q  ∥c∗∥q  q + ⟨c∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' b⟩ + λ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (using  �  i  c∗  i = 1 and (c∗  i )2 = |c∗  i |2)  =⇒ − α∥c∗∥q + ⟨c∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' b⟩ + λ = 0  =⇒ − α∥c∗∥q + ⟨c∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' b⟩ = −λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (re-arranging)  (66)  Now again using (65),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  − α∥c∗∥1−q  q  |c∗  i |q−2c∗  i + bi + λ + θi = 0  =⇒ α∥c∗∥1−q  q  |c∗  i |q−2c∗  i = bi + λ + θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∀i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (re-arranging)  (67)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' if i ∈ B then θi = 0 from complimentry slackness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' so we have  α∥c∗∥1−q  q  |c∗  i |q−2c∗  i = bi + λ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∀i ∈ B  by definition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now, if for some i, bi + λ > 0 then bi + λ + θi > 0 as θi ≥ 0, that  implies  α∥c∗∥1−q  q  |c∗  i |q−2c∗  i = bi + λ + θi > 0  =⇒ c∗  i > 0 =⇒ i ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  So, we have,  i ∈ B ⇐⇒ bi + λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  To summarize, we have  α∥c∗∥1−q  q  |c∗  i |q−2c∗  i = (bi + λ)1(bi ≥ −λ),  ∀i,  (68)  =⇒  �  i  α  q  q−1 ∥c∗∥−q  q (c∗  i )q =  �  i  (bi + λ)  q  q−1 1(bi ≥ −λ),  (taking q/(q − 1)th power and summing)  =⇒ αp =  A  �  i=1  (bi + λ)p1(bi ≥ −λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (69)  So, we have,  ζp = −λ  such that  αp =  �  bi≥λ  (bi + λ)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  =⇒ αp =  �  bi≥ζp  (bi − ζp)p  (70)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Furthermore, using (68), we have  α∥c∗∥1−q  q  |c∗  i |q−2c∗  i = (bi + λ)1(bi ≥ −λ) = (bi − ζp)1(bi ≥ ζp)  ∀i,  =⇒ c∗  i ∝ (bi − ζp)  1  q−1 1(bi ≥ ζp) =  (bi − ζp)p−11(bi ≥ ζp)  �  i(bi − ζp)p−11(bi ≥ ζp),  (using  �  i  c∗  i = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (71)  38  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now, we move on to calculate the number of active actions χp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Observe that the  function  f(λ) :=  A  �  i=1  (bi − λ)p1(bi ≥ λ) − αp  (72)  is monotonically decreasing in λ and ζp is a root of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This implies  f(x) ≤ 0 ⇐⇒ x ≥ ζp  =⇒ f(bi) ≤ 0 ⇐⇒ bi ≥ ζp  =⇒ {i|bi ≥ ζp} = {i|f(bi) ≤ 0}  =⇒ χp = max{i|bi ≥ ζp} = max{i|f(bi) ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (73)  Hence, things follows by putting back in the definition of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We have,  αp =  A  �  i=1  (bi − ζp)p1(bi ≥ ζp),  and  χp = max{i|bi ≥ ζp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Combining both we have  αp =  χp  �  i=1  (bi − ζp)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  And the other part follows directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Continuity and montonocity of the function �  bi≥x(bi − x)p is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now observe  that �  bi≥b1(bi − b1)p = 0 and �  bi≥b1−α(bi − (b1 − α))p ≥ αp, so it implies that it is  equal to αp in the range [b1 − α, b1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Recall that the ζp is the solution to the following equation  αp =  �  bi≥x  (bi − x)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  And from the definition of χp, we have  αp <  χp+1  �  i=1  (bi − bχp+1)p =  �  bi≥bχp+1  (bi − bχp+1)p,  and  αp ≥  χp  �  i=1  (bi − bχp)p =  �  bi≥bχp  (bi − bχp)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  So from continuity, we infer the root ζp must lie between [bχp+1, bχ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We prove the first direction, and assume we have  k ≤ χp  =⇒  k  �  i=1  (bi − bk)p ≤ αp  (from definition of χp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (74)  39  Observe the function f(x) := �k  i=1(bi − x)p is monotically decreasing in the range  (−∞, bk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Further, f(bk) ≤ αp and limx→−∞ f(x) = ∞, so from the continuity  argument there must exist a value y ∈ (−∞, bk] such that f(y) = αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' This implies that  k  �  i=1  (bi − y)p ≤ αp,  and  y ≤ bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Hence, explicitly showed the existence of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now, we move on to the second  direction, and assume there exist x such that  k  �  i=1  (bi − x)p = αp,  and  x ≤ bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  =⇒  k  �  i=1  (bi − bk)p ≤ αp,  (as x ≤ bk ≤ bk−1 · · · ≤ b1)  =⇒ k ≤ χp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We have k ≤ χp and λk such that  αp =  k  �  i=1  (bi − λk)p,  and  λk ≤ bk,  (from above item)  ≥  k  �  i=1  (bi − bk+1)p,  (as λk ≤ bk+1 ≤ bk)  ≥  k+1  �  i=1  (bi − bk+1)p,  (addition of 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (75)  From the definition of χp, we get k + 1 ≤ χp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We are given  k  �  i=1  (bi − λk)p = αp  =⇒  k  �  i=1  (bi − bk+1)p > αp,  (as λk > bk+1)  =⇒  k+1  �  i=1  (bi − bk+1)p > αp,  (addition of zero)  =⇒ k + 1 > χp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  40  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  Special case: L1  For p = 1, by definition, we have  ζ1 = max  c  −α∥c∥∞ + ⟨c, b⟩  such that  �  a∈A  ca = 1,  c ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (76)  And χ1 is the optimal number of actions, that is  α =  χ1  �  i=1  (bi − ζ1)  =⇒ ζ1 =  �χ1  i=1 bi − α  χ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Let λk be the such that  α =  k  �  i=1  (bi − λk)  =⇒ λk =  �k  i=1 bi − α  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ζ1 = max  k  λk  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' From lemma 2, we have  λ1 ≤ λ2 · · · ≤ λχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Now, we have  λk − λk+m =  �k  i=1 bi − α  k  −  �k+m  i=1 bi − α  k + m  =  �k  i=1 bi − α  k  −  �k  i=1 bi − α  k + m  −  �m  i=1 bk+i  k + m  = m(�k  i=1 bi − α  k(k + m))  −  �m  i=1 bk+i  k + m  =  m  k + m(  �k  i=1 bi − α  k  −  �m  i=1 bk+i  m  )  =  m  k + m(λk −  �m  i=1 bk+i  m  )  (77)  From lemma 2, we also know the stopping criteria for χ1, that is  λχ1 > bχ1+1  =⇒ λχ1 > bχ1+i,  i ≥ 1,  (as bi are in descending order)  =⇒ λχ1 >  �m  i=1 bχ1+i  m  ,  ∀m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  41  Combining it with the (77), for all m ≥ 0 , we get  λχ1 − λχ1+m =  m  χ1 + m(λχ1 −  �m  i=1 bχ1+i  m  )  ≥ 0  =⇒ λχ1 ≥ λχ1+m  (78)  Hence, we get the desired result,  ζ1 = λχ1 = max  k  λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  Special case: max norm  For p = ∞, by definition, we have  ζ∞(b) = max  c  −α∥c∥1 + ⟨c, b⟩  such that  �  a∈A  ca = 1,  c ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  = max  c  −α + ⟨c, b⟩  such that  �  a∈A  ca = 1,  c ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  = − α + max  i  bi  (79)  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  Special case: L2  The problem is discussed in great details in [1], here we outline the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = 2, we  have  ζ2 = max  c  −α∥c∥2 + ⟨c, b⟩  such that  �  a∈A  ca = 1,  c ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (80)  Let λk be the solution of the following equation  α2 =  k  �  i=1  (bi − λ)2,  λ ≤ bk  = kλ2 − 2  k  �  i=1  λbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' +  k  �  i=1  (bi)2,  λ ≤ bk  =⇒ λk =  �k  i=1 bi ±  �  (�k  i=1 bi)2 − k(�k  i=1(bi)2 − α2)  k  ,  and  λk ≤ bk  =  �k  i=1 bi −  �  (�k  i=1 bi)2 − k(�k  i=1(bi)2 − α2)  k  =  �k  i=1 bi  k  −  �  �  �  �α2 −  k  �  i=1  (bi −  �k  i=1 bi  k  )2  (81)  From lemma 2, we know  λ1 ≤ λ2 · · · ≤ λχ2 = ζ2  42  where χ2 calculated in two ways: a)  χ2 = max  m {m|  m  �  i=1  (bi − bm)2 ≤ α2}  b)  χ2 = min  m {m|λm ≤ bm+1}  We proceed greedily until stopping condition is met in lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Concretely, it is illustrated  in algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  L1 Water Pouring lemma  In this section, we re-derive the above water pouring lemma for p = 1 from scratch, just for  sanity check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' As in the above proof, there is a possibility of some breakdown, as we had take  limits q → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We will see that all the above results for p = 1 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Let b ∈ RA be such that its components are in decreasing order, i,e bi ≥ bi+1 and  ζ1 := max  c  −α∥c∥∞ + ⟨c, b⟩  such that  A  �  i=1  ci = 1,  ci ≥ 0,  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (82)  Lets fix any vector c ∈ RA, and let k1 := ⌊  1  maxi ci ⌋ and let  c1  i =  �  �  �  �  �  maxi ci  if  i ≤ k1  1 − k1 maxi ci  if  i = k1 + 1  0  else  Then we have,  −α∥c∥∞ + ⟨c, b⟩ = − α max  i  ci +  A  �  i=1  cibi  ≤ − α max  i  ci +  A  �  i=1  c1  i bi,  (recall bi is in decreasing order)  = − α∥c1∥∞ + ⟨c1, b⟩  (83)  Now, lets define c2 ∈ RA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let  k2 =  �  k1 + 1  if  �k1  i=1 bi−α  k1  ≤ bk+1  k1  else  43  and let c2  i = 1(i≤k2)  k2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then we have,  −α∥c1∥∞ + ⟨c1, b⟩ = − α max  i  ci +  A  �  i=1  c1  i bi  = − α max  i  ci +  k1  �  i=1  max  i  cibi + (1 − k1 max  i  ci)bk1+1,  (definition of c1)  =(−α + �k1  i=1 bi  k1  )k1 max  i  ci + bk1+1(1 − k1 max  i  ci),  (re-arranging)  ≤−α + �k2  i=1 bi  k2  = − α∥c2∥∞ + ⟨c2, b⟩  (84)  The last inequality comes from the definition of k2 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' So we conclude that a optimal  solution is uniform over some actions, that is  ζ1 = max  c∈C −α∥c∥∞ + ⟨c, b⟩  = max  k  � −α + �k  i=1 bi  k  �  (85)  where C := {ck ∈ RA|ck  i = 1(i≤k)  k  } is set of uniform actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Rest all the properties follows  same as Lp water pouring lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  K  Robust Value Iteration (Main)  In this section, we will discuss the main results from the paper except for time complexity  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It contains the proofs of the results presented in the main body and also some other  corollaries/special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  sa-rectangular robust policy evaluation and improvement  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (sa)-rectangular Lp robust Bellman operator is equivalent to reward regularized  (non-robust) Bellman operator, that is  (T π  Usa  p v)(s) =  �  a  π(a|s)[−αs,a − γβs,aκq(v) + R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)],  and  (T ∗  Usa  p v)(s) = max  a∈A[−αs,a − γβs,aκq(v) + R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)],  where κp is defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  44  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' From definition robust Bellman operator and Usa  p = (R0 + R) × (P0 + P),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (T π  Usa  p v)(s) =  min  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='P ∈Usa  p  �  a  π(a|s)  �  R(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  =  �  a  π(a|s)  �  R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  +  min  p∈P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r∈R  �  a  π(a|s)  �  r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (from (sa)-rectangularity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we get)  =  �  a  π(a|s)  �  R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  +  �  a  π(a|s)  min  ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a∈Psa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a∈Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a  �  rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a + γ  �  s′  ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a(s′)v(s′)  �  �  ��  �  :=Ωsa(v)  (86)  Now we focus on regularizer function Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' as follows  Ωsa(v) =  min  ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a∈Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a∈Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a  �  rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a + γ  �  s′  ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a(s′)v(s′)  �  =  min  rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a∈Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a + γ  min  ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a∈Psa  �  s′  ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a(s′)v(s′)  = −αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a + γ  min  ∥psa∥p≤βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='�  s′ psa(s′)=0⟨ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  = − αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a − γβs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='aκq(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (from lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (87)  Putting back,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  (T π  Usa  p v)(s) =  �  a  π(a|s)  �  −αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a − γβs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='aκq(v) + R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' reusing above results in optimal robust operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  (T ∗  Usa  p v)(s) = max  πs∈∆A  min  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='P ∈Usa  p  �  a  πs(a)  �  R(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  = max  πs∈∆A  �  a  πs(a)  �  −αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a − γβs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='aκp(v) + R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  = max  a∈A  �  −αs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a − γβs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='aκq(v) + R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  (88)  The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  S-rectangular robust policy evaluation  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' S-rectangular Lp robust Bellman operator is equivalent to reward regularized  (non-robust) Bellman operator, that is  (T π  Us  pv)(s) = −  �  αs + γβsκq(v)  �  ∥π(·|s)∥q +  �  a  π(a|s)  �  R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)  �  45  where κp is defined in (7) and ∥π(·|s)∥q is q-norm of the vector π(·|s) ∈ ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' From definition of robust Bellman operator and Us  p = (R0 + R) × (P0 + P),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' we have  (T π  Us  pv)(s) =  min  R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='P ∈Us  p  �  a  π(a|s)  �  R(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  =  �  a  π(a|s)  �  R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  ��  �  nominal values  �  +  min  p∈P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r∈R  �  a  π(a|s)  �  r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  (from s-rectangularity we have)  =  �  a  π(a|s)  �  R0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a) + γ  �  s′  P0(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' a)v(s′)  �  +  min  ps∈Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='rs∈Rs  �  a  π(a|s)  �  rs(a) + γ  �  s′  ps(s′|a)v(s′)  �  �  ��  �  :=Ωs(πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='v)  (89)  where we denote πs(a) = π(a|s) as a shorthand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now we calculate the regularizer function  as follows  Ωs(πs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v) :=  min  rs∈Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='ps∈Ps⟨rs + γvT ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' πs⟩ = min  rs∈Rs⟨rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' πs⟩ + γ min  ps∈Ps vT psπs  = −αs∥πs∥q + γ min  ps∈Ps vT psπs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (using 1  p + 1  q = 1 )  = − αs∥πs∥q + γ min  ps∈Ps  �  a  πs(a)⟨ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v⟩  = − αs∥πs∥q + γ  min  �  a(βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)p≤(βs)p  min  ∥psa∥p≤βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='�  s′ psa(s′)=0  �  a  πs(a)⟨ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v⟩  = − αs∥πs∥q + γ  min  �  a(βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a)p≤(βs)p  �  a  πs(a)  min  ∥psa∥p≤βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='�  s′ psa(s′)=0  ⟨ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' v⟩  = − αs∥πs∥q + γ  min  �  a(βsa)p≤(βs)p  �  a  πs(a)(−βsaκp(v))  ( from lemma 1)  = − αs∥πs∥q − γκq(v)  max  �  a(βsa)p≤(βs)p  �  a  πs(a)βsa  = − αs∥πs∥q − γκp(v)∥πs∥qβs  (using Holders)  = − (αs + γβsκq(v))∥πs∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (90)  Now putting above values in robust operator, we have  (T π  Us  pv)(s) = −  �  αs + γβsκq(v)  �  ∥π(·|s)∥q+  �  a  π(a|s)  �  R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  46  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  s-rectangular robust policy improvement  Reusing robust policy evaluation results in section K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2, we have  (T ∗  Uspv)(s) = max  πs∈∆A  min  R,P ∈Usa  p  �  a  πs(a)  �  R(s, a) + γ  �  s′  P(s′|s, a)v(s′)  �  = max  πs∈∆A  �  −(αs + γβsκq(v))∥πs∥q +  �  a  πs(a)(R(s, a) + γ  �  s′  P(s′|s, a)v(s′))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (91)  Observe that, we have the following form  (T ∗  Uspv)(s) = max  c  −α∥c∥q + ⟨c, b⟩  such that  A  �  i=1  ci = 1,  c ⪰ 0,  (92)  where α = αs +γβsκq(v) and bi = R(s, ai)+γ �  s′ P(s′|s, ai)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now all the results below,  follows from water pouring lemma ( lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Policy improvement) The optimal robust Bellman operator can be evaluated  in following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (T ∗  Us  pv)(s) is the solution of the following equation that can be found using binary search  between  �  maxa Q(s, a) − σ, maxa Q(s, a)  �  ,  �  a  �  Q(s, a) − x  �p 1  �  Q(s, a) ≥ x  �  = σp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (93)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (T ∗  Us  pv)(s) and χp(v, s) can also be computed through algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  where σ = αs + γβsκq(v), and Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The first part follows from lemma 2, point 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The second part follows from lemma 2,  point 9 (greedy inclusion ) and point 10 (stopping condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (Go To Policy) The greedy policy π w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' value function v, defined as  T ∗  Us  pv = T π  Us  pv is a threshold policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It takes only those actions that has positive advantage,  with probability proportional to (p − 1)th power of its advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is  π(a|s) ∝ (A(s, a))p−11(A(s, a) ≥ 0),  where A(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′) − (T ∗  Us  pv)(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Follows from lemma 2, point 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' χp(v, s) is number of actions that has positive advantage, that is  χp(v, s) =  ���  �  a | (T ∗  Us  pv)(s) ≤ R0(s, a) + γ  �  s′  P0(s′|s, a)v(s′)  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Follows from lemma 2, point 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  47  Property 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' ( Value vs Q-value) (T ∗  Us  pv)(s) is bounded by the Q-value of χth and (χ + 1)th  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is  Q(s, aχ+1) < (T ∗  Us  pv)(s) ≤ Q(s, aχ),  where  χ = χp(v, s),  Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′), and Q(s, a1) ≥ Q(s, a2), · · · Q(s, aA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Follows from lemma 2, point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = 1, the optimal policy π1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' value function v and uncertainty set  Us  1, can be computed directly using χ1(s) without calculating advantage function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is  π1(as  i|s) = 1(i ≤ χ1(s))  χ1(s)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Follows from Theorem 11 by putting p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Note that it can be directly obtained  using L1 water pouring lemma (see section J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1)  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (For p = ∞) The optimal policy π w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' value function v and uncertainty set  Us  ∞ (precisely T ∗  Us  ∞v = T π  Us  ∞v), is to play the best response, that is  π(a|s) = 1(a ∈ arg maxa Q(s, a))  �� arg maxa Q(s, a)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  In case of tie in the best response, it is optimal to play any of the best responses with any  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Follows from Theorem 11 by taking limit p → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = ∞, T ∗  Us  pv, the robust optimal Bellman operator evaluation can be  obtained in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is  (T ∗  Us  ∞v)(s) = max  a  Q(s, a) − σ,  where σ = αs + γβsκ1(v), Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let π be such that  T ∗  Us  ∞v = T π  Us  ∞v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  This implies  (T ∗  Uspv)(s) =  min  R,P ∈Usa  p  �  a  π(a|s)  �  R(s, a) + γ  �  s′  P(s′|s, a)v(s′)  �  = −(αs + γβsκp(v))∥π(·|s)∥q +  �  a  π(a|s)(R(s, a) + γ  �  s′  P(s′|s, a)v(s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (94)  From corollary 3, we know the that π is deterministic best response policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Putting this we  get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  There is a another way of proving this, using Theorem 3 by taking limit p → ∞ carefully as  lim  p→∞  �  a  �  Q(s, a) − T ∗  Uspv)(s)  �p  1  �  Q(s, a) ≥ T ∗  Uspv)(s)  �  )  1  p = σ,  (95)  where σ = αs + γβsκ1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  48  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = 1, the robust optimal Bellman operator T ∗  Us  p, can be computed in  closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' That is  (T ∗  Us  pv)(s) = max  k  �k  i=1 Q(s, ai) − σ  k  ,  where σ = αs + γβsκ∞(v), Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′), and Q(s, a1) ≥  Q(s, a2), ≥ · · · ≥ Q(s, aA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Follows from section J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The s rectangular Lp robust Bellman operator can be evaluated for p = 1, 2  by algorithm 8 and algorithm 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It follows from the algorithm 3, where we solve the linear equation and quadratic  equation for p = 1, 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For p = 2, it can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Algorithm 7 Algorithm to compute S-rectangular L2 robust optimal Bellman Operator  1: Input: σ = αs + γβsκ2(v),  Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: Output (T ∗  Us  2 v)(s), χ2(v, s)  3: Sort Q(s, ·) and label actions such that Q(s, a1) ≥ Q(s, a2), · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4: Set initial value guess λ1 = Q(s, a1) − σ and counter k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5: while k ≤ A − 1 and λk ≤ Q(s, ak) do  6:  Increment counter: k = k + 1  7:  Update value estimate:  λk = 1  k  �  k  �  i=1  Q(s, ai) −  �  �  �  �kσ2 + (  k  �  i=1  Q(s, ai))2 − k  k  �  i=1  (Q(s, ai))2  �  8: end while  9: Return: λk, k  Algorithm 8 Algorithm to compute S-rectangular L1 robust optimal Bellman Operator  1: Input: σ = αs + γβsκ∞(v),  Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)v(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2: Output (T ∗  Us  1 v)(s), χ1(v, s)  3: Sort Q(s, ·) and label actions such that Q(s, a1) ≥ Q(s, a2), · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  4: Set initial value guess λ1 = Q(s, a1) − σ and counter k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  5: while k ≤ A − 1 and λk ≤ Q(s, ak) do  6:  Increment counter: k = k + 1  7:  Update value estimate:  λk = 1  k  �  k  �  i=1  Q(s, ai) − σ  �  8: end while  9: Return: λk, k  49  L  Time Complexity  In this section, we will discuss time complexity of various robust MDPs and compare it with  time complexity of non-robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We assume that we have the knowledge of nominal  transition kernel and nominal reward function for robust MDPs, and in case of non-robust  MDPs, we assume the knowledge of the transition kernel and reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We divide  the discussion into various parts depending upon their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='1  Exact Value Iteration: Best Response  In this section, we will discuss non-robust MDPs, (sa)-rectangular L1/L2/L∞ robust MDPs  and s-rectangular L∞ robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' They all have a common theme for value iteration as  follows, for the value function v, their Bellman operator ( T ) evaluation is done as  (T v)(s) =  max  a  ����  action cost  �  R(s, a) + αs,a  κ(v)  ����  reward penalty/cost  +γ  �  s′  P(s′|s, a)v(s′)  �  ��  �  sweep  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (96)  ’Sweep’ requires O(S) iterations and ’action cost’ requires O(A) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Note that the  reward penalty κ(v) doesn’t depend on state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' It is calculated only once for value  iteration for all states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' The above value update has to be done for each states , so one full  update requires  O  �  S(action cost)(sweep cost  �  +reward cost  �  = O  �  S2A + reward cost  �  Since the value iteration is a contraction map, so to get ϵ-close to the optimal value, it  requires O(log( 1  ϵ )) full value update, so the complexity is  O  �  log(1  ϵ )  �  S2A + reward cost  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Non-robust MDPs: The cost of ’reward is zero as there is no regularizer to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  The total complexity is  O  �  log(1  ϵ )  �  S2A + 0  ��  = O  �  log(1  ϵ )S2A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' (sa)-rectangular L1/L2/L∞ and s-rectangular L∞ robust MDPs: We need  to calculate the reward penalty (κ1(v)/κ2(v)/κ∞) that takes O(S) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' As  calculation of mean, variance and median, all are linear time compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence the  complexity is  O  �  log(1  ϵ )  �  S2A + S  ��  = O  �  log(1  ϵ )S2A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2  Exact Value iteration: Top k response  In this section, we discuss the time complexity of s-rectangular L1/L2 robust MDPs as in  algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We need to calculate the reward penalty (κ∞(v)/κ2(v) in (40)) that takes O(S)  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then for each state we do: sorting of Q-values in (45), value evaluation in (46),  50  update Q-value in (44) that takes O(A log(A)), O(A), O(SA) iterations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Hence  the complexity is  total iteration(reward cost (40) + S( sorting (45) + value evaluation (46) +Q-value(44))  = log(1  ϵ )(S + S(A log(A) + A + SA)  O  �  log(1  ϵ )  �  S2A + SA log(A)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  For general p, we need little caution as kp(v) can’t be calculated exactly but approximately  by binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And it is the subject of discussion for the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='3  Inexact Value Iteration: sa-rectangular Lp robust MDPs (U sa  p )  In this section, we will study the time complexity for robust value iteration for (sa)-rectangular  Lp robust MDPs for general p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Recall, that value iteration takes best penalized action, that  is easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' But reward penalization depends on p-variance measure κp(v), that we  will estimate by ˆκp(v) through binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We have inexact value iterations as  vn+1(s) := max  a∈A[αsa − γβsaˆκq(vn) + R0(s, a) + γ  �  s′  P0(s′|s, a)vn(s′)]  where ˆκq(vn) is a ϵ1 approximation of κq(vn), that is |ˆκq(vn) − κq(vn)| ≤ ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then it is easy  to see that we have bounded error in robust value iteration, that is  ∥vn+1 − T ∗  Usa  p vn∥∞ ≤ γβmaxϵ1  where βmax := maxs,a βs,a  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let T ∗  U be a γ contraction map, and v∗ be its fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And let {vn, n ≥ 0}  be approximate value iteration, that is  ∥vn+1 − T ∗  U vn∥∞ ≤ ϵ  then  lim  n→∞∥vn − v∗∥∞ ≤  ϵ  1 − γ  moreover, it converges to the  ϵ  1−γ radius ball linearly, that is  ∥vn − v∗∥∞ −  ϵ  1 − γ ≤ cγn  where c =  1  1−γ ϵ + ∥v0 − v∗∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  51  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  ∥vn+1 − v∗∥∞ =∥vn+1 − T ∗  U v∗∥∞  =∥vn+1 − T ∗  U vn + T ∗  U vn − T ∗  U v∗∥∞  ≤∥vn+1 − T ∗  U vn∥∞ + ∥T ∗  U vn − T ∗  U v∗∥∞  ≤∥vn+1 − T ∗  U vn∥∞ + γ∥vn − v∗∥∞,  (contraction)  ≤ϵ + γ∥vn − v∗∥∞,  (approximate value iteration)  =⇒ ∥vn − v∗∥∞ =  n−1  �  k=0  γkϵ + γn∥v0 − v∗∥∞,  (unrolling above recursion)  =1 − γn  1 − γ ϵ + γn∥v0 − v∗∥∞  =γn[  1  1 − γ ϵ + ∥v0 − v∗∥∞] +  ϵ  1 − γ  (97)  Taking limit n → ∞ both sides, we get  lim  n→∞∥vn − v∗∥∞ ≤  ϵ  1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For Usa  p , the total iteration cost is log( 1  ϵ )S2A + (log( 1  ϵ ))2 to get ϵ close to the  optimal robust value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We calculate κq(v) with ϵ1 = (1−γ)ϵ  3  tolerance that takes O(S log( S  ϵ1 )) using binary  search (see section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now, we do approximate value iteration for n = log( 3∥v0−v∗∥∞  ϵ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Using the above lemma, we have  ∥vn − v∗  Usa  p ∥∞ =γn[  1  1 − γ ϵ1 + ∥v0 − v∗  Usa  p ∥∞] +  ϵ1  1 − γ  ≤γn[ ϵ  3 + ∥v0 − v∗  Usa  p ∥∞] + ϵ  3  ≤γn ϵ  3 + ϵ  3 + ϵ  3 ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (98)  In summary, we have action cost O(A), reward cost O(S log( S  ϵ )), sweep cost O(S) and total  number of iterations O(log( 1  ϵ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' So the complexity is  (number of iterations)  �  S(actions cost) (sweep cost) + reward cost  �  = log(1  ϵ )  �  S2A + S log(S  ϵ )  �  = log(1  ϵ )(S2A + S log(1  ϵ ) + S log(S))  = log(1  ϵ )S2A + S(log(1  ϵ ))2  52  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='4  Inexact Value Iteration: s-rectangular Lp robust MDPs  In this section, we study the time complexity for robust value iteration for s-rectangular  Lp robust MDPs for general p ( algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Recall, that value iteration takes regularized  actions and penalized reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' And reward penalization depends on q-variance measure κq(v),  that we will estimate by ˆκq(v) through binary search, then again we will calculate T ∗  Usa  p by  binary search with approximated κq(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Here, we have two error sources ((40), (46)) as  contrast to (sa)-rectangular cases, where there was only one error source from the estimation  of κq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  First, we account for the error caused by the first source (κq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Here we do value iteration  with approximated q-variance ˆκq, and exact action regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We have  vn+1(s) := λ  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  αs + γβsˆκq(v) = (  �  Q(s,a)≥λ  (Q(s, a) − λ)p)  1  p  where Q(s, a) = R0(s, a) + γ �  s′ P0(s′|s, a)vn(s′), and |ˆκq(vn) − κq(vn)| ≤ ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then from  the next result (proposition 8), we get  ∥vn+1 − T ∗  Usa  p vn∥∞ ≤ γβmaxϵ1  where βmax := maxs,a βs,a  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let ˆκ be an an ϵ-approximation of κ, that is |ˆκ − κ| ≤ ϵ, and let b ∈ RA  be sorted component wise, that is, b1 ≥, · · · , ≥ bA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let λ be the solution to the following  equation with exact parameter κ,  α + γβκ = (  �  bi≥λ  |bi − λ|p)  1  p  and let ˆλ be the solution of the following equation with approximated parameter ˆκ,  α + γβˆκ = (  �  bi≥ˆλ  |bi − ˆλ|p)  1  p ,  then ˆλ is an O(ϵ)-approximation of λ, that is  |λ − ˆλ| ≤ γβϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let the function f : [bA, b1] → R be defined as  f(x) := (  �  bi≥x  |bi − x|p)  1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  We will show that derivative of f is bounded, implying its inverse is bounded and hence  Lipschitz, that will prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Let proceed  df(x)  dx  = −(  �  bi≥x  |bi − x|p)  1  p −1 �  bi≥x  |bi − x|p−1  = −  �  bi≥x |bi − x|p−1  (�  bi≥x |bi − x|p)  p−1  p  = −  � (�  bi≥x |bi − x|p−1)  1  p−1  (�  bi≥x |bi − x|p)  1  p  �p−1  ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (99)  53  The inequality follows from the following relation between Lp norm,  ∥x∥a ≥ ∥x∥b,  ∀0 ≤ a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  It is easy to see that the function f is strictly monotone in the range bA, b1], so its inverse is  well defined in the same range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then derivative of the inverse of the function f is bounded as  0 ≥ d  dxf −(x) ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Now, observe that λ = f −(α + γβκ) and ˆλ = f −(α + γβˆκ), then by Lipschitzcity, we have  |λ − ˆλ| = |f −(α + γβκ) − f −(α + γβˆκ)| ≤ γβ| − κ − ˆκ)| ≤ γβϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For Us  p, the total iteration cost is O  �  log( 1  ϵ )  �  S2A + SA log( A  ϵ )  ��  to get ϵ  close to the optimal robust value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' We calculate κq(v) in (40) with ϵ1 = (1−γ)ϵ  6  tolerance that takes O(S log( S  ϵ1 )) iterations  using binary search (see section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Then for every state, we sort the Q values (as in (45))  that costs O(A log(A)) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' In each state, to update value, we do again binary search  with approximate κq(v) upto ϵ2 := (1−γ)ϵ  6  tolerance, that takes O(log( 1  ϵ2 )) search iterations  and each iteration cost O(A), altogether it costs O(A log( 1  ϵ2 )) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Sorting of actions  and binary search adds upto O(A log( A  ϵ )) iterations (action cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' So we have (doubly)  approximated value iteration as following,  |vn+1(s) − ˆλ| ≤ ϵ1  (100)  where  (αs + γβsˆκq(vn))p =  �  Qn(s,a)≥ˆλ  (Qn(s, a) − ˆλ)p  and  Qn(s, a) = R0(s, a) + γ  �  s′  P0(s′|s, a)vn(s′),  |ˆκq(vn) − κq(vn)| ≤ ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  And we do this approximate value iteration for n = log( 3∥v0−v∗∥∞  ϵ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Now, we do error  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' By accumulating error, we have  |vn+1(s) − (T ∗  Us  pvn)(s)| ≤|vn+1(s) − ˆλ| + |ˆλ − (T ∗  Us  pvn)(s)|  ≤ϵ1 + |ˆλ − (T ∗  Us  pvn)(s)|,  (by definition)  ≤ϵ1 + γβmaxϵ1,  (from proposition 8)  ≤2ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (101)  where βmax := maxs βs, γ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  Now, we do approximate value iteration, and from proposition 7, we get  ∥vn − v∗  Us  p∥ ≤ 2ϵ1  1 − γ + γn[  1  1 − γ 2ϵ1 + ∥v0 − v∗  Us  p∥∞]  (102)  54  Now, putting the value of n, we have  ∥vn − v∗  Us  p∥∞ =γn[ 2ϵ1  1 − γ + ∥v0 − v∗  Us  p∥∞] +  2ϵ1  1 − γ  ≤γn[ ϵ  3 + ∥v0 − v∗  Us  p∥∞] + ϵ  3  ≤γn ϵ  3 + ϵ  3 + ϵ  3 ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content='  (103)  To summarize, we do O(log( 1  ϵ )) full value iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' Cost of evaluating reward penalty  is O(S log( S  ϵ )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' For each state: evaluation of Q-value from value function requires O(SA)  iterations, sorting the actions according Q-values requires O(A log(A)) iterations, and binary  search for evaluation of value requires O(A log(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
+page_content=' So the complexity is  O((total iterations)(reward cost + S(Q-value + sorting + binary search for value )))  = O  �  log(1  ϵ )  �  S log(S  ϵ ) + S(SA + A log(A) + A log(1  ϵ ))  ��  = O  �  log(1  ϵ )  �  S log(1  ϵ ) + S log(S) + S2A + SA log(A) + SA log(1  ϵ )  ��  = O  �  log(1  ϵ )  �  S2A + SA log(A) + SA log(1  ϵ )  ��  = O  �  log(1  ϵ )  �  S2A + SA log(A  ϵ )  ��  55' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFRT4oBgHgl3EQfxjhb/content/2301.13642v1.pdf'}
diff --git a/EtA0T4oBgHgl3EQfA__T/content/tmp_files/2301.01971v1.pdf.txt b/EtA0T4oBgHgl3EQfA__T/content/tmp_files/2301.01971v1.pdf.txt
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+Two statistical regimes in the transition to filamentation
+Alexis Gomel,1, 2 Geoffrey Gaulier,1 Debbie Eeltink,1, 2, 3 Maura Brunetti,1, 4 and Jérôme Kasparian
+1, 2, ∗
+1Group of Applied Physics, University of Geneva, 1205 Geneva, Switzerland
+2Institute for Environmental Sciences, University of Geneva, 1205 Geneva, Switzerland
+3Present address: Laboratory of Theoretical Physics of Nanosystems, EPFL, Ch-1015 Lausanne, 10 Switzerland
+4Institute for Environmental Sciences, University of Geneva, 1205 Geneva, Switzerland
+Abstract
+We experimentally investigate fluctuations in the spectrum of ultrashort laser pulses propagating in air, close to the critical
+power for filamentation. Increasing the laser peak power broadens the spectrum while the beam approaches the filamentation
+regime. We identify two regimes for this transition: In the center of the spectrum, the output spectral intensity increases
+continuously. In contrast, on the edges of the spectrum the transition implies a bimodal probability distribution function for
+intermediate incident pulse energies, where a high-intensity mode appears and grows at the expense of the original low-intensity
+mode. We argue that this dual behavior prevents the definition of a univoquial threshold for filamentation, shedding a new
+light on the long-standing lack of explicit definition of the boundary of the filamentation regime.
+∗ jerome.kasparian@unige.ch
+1
+arXiv:2301.01971v1  [physics.optics]  5 Jan 2023
+
+I.
+INTRODUCTION
+Filamentation is a self-guided propagation regime typical of ultrashort, high-power laser pulses [1–4]. Beyond a
+critical power (3 GW in air), the Kerr effect balances diffraction.
+The beam then self-focuses until higher-order
+nonlinear defocusing effects like ionization or the saturation of the Kerr effect [5] come into play.
+The resulting
+dynamical balance gives rise to self-guided light structures known as filaments. Filaments can extend over atmospheric
+scale distances [6, 7], opening the way to various applications from remote sensing [8] to lightning control [9–14],
+THz generation [15–20], fog clearing [21, 22], or the triggering of condensation in sub-saturated atmospheres [8, 23,
+24]. Filaments show very characteristic and even spectacular features, including long-distance propagation, bright
+light emission due to both plasma emission on the side and spectral broadening in the forward direction, and noise
+associated to the shockwave [25–27]. Together, these features give rise to the intuitive notion of a qualitatively distinct
+propagation regime. However, defining a clear threshold for the filamentation regime turns out to be difficult [28],
+especially in focused beams where linear propagation can be sufficient to reach the ionization threshold [29], and even
+produce a denser plasma than Kerr self-focusing does [30]. Similarly, along the propagation axis, defining the onset
+and end of filaments, and therefore their length, requires somewhat arbitrary choices.
+Here, we statistically investigate the transition to filamentation . Shot-to-shot fluctuations of the spectral amplitudes
+of a high-power laser beam, whether filamenting or not, are known to display characteristic probability distribution
+functions (PDF): A regular probability distribution is observed in the center of the spectrum, while on its edge the
+distribution is long-tailed (optical rogue wave [31]) [32]. We extend this analysis to the sub-filamenting regime and to
+the transition to filamentation, in order to characterize how the statistics evolves when the incident power progressively
+reaches the critical power for filamentation. The evolution of spectral intensity PDFs turns out to display typical
+and contrasted signatures on the edges and in the center of the spectrum, respectively, suggesting that transition
+to filamentation cannot be characterized as a whole. By doing so, we suggest that the ambiguities in the definition
+of the limits of the filamentation regime are intrinsic to the physical process rather than instrumental or conceptual
+limitations, shedding a new light on the long-standing lack of explicit definition of the boundary of the filamentation
+regime.
+II.
+EXPERIMENTAL SETUP
+Figure 1. Experimental setup
+The experimental setup (Figure 1) relied on a Coherent Astrella laser delivering ultrashort pulses at 1 kHz repetition
+rate. A half-wave plate installed in a motorized rotating stage ensureing ±1◦ precision, followed by a linear polarizer,
+allowed to adjust the pulse energy with an accuracy of ∼ 10 µJ. The angle-to-energy calibration was performed before
+each experimental run with a powermeter (Coherent PM10). The laser compressor was adjusted so as to slightly chirp
+the pulses to a duration of 55 fs (as measured with PulseCheck, APE Berlin), in order to reach the filamentation
+threshold in air within the tuning range of the energy. The beam was slightly focused from an initial diameter of 11 mm
+at 1/e2 by an f = 2 m BK7 plano-convex lens located approximately at 15 cm after the polarizer and subsequently
+through a 13.5 cm-long turbulent region generated by 48◦C to 565◦C hot air blown transversally by a hot air blower
+(Steinel HL1502 S). 15–27.5 cm downstream of the lens, the beam self-focused and filamentation started if the incident
+peak power was sufficient. The filamentation region was surrounded by a PVC tube of 18 cm diameter to shield the
+beam against air displacements in the room so as to improve its stability. After the end of filamentation region, the
+beam was imaged by a second BK7 lens (f = 20 cm) onto the entrance slit of an OceanOptics USB2000 spectrometer
+providing ∼0.5 nm spectral resolution over the visible range. A neutral density filter prevented the spectrometer from
+saturating. The spectrum was measured and recorded independently for each laser shot. For each incident pulse
+2
+
+energy, 5000 individual spectra were recorded.
+At each wavelength and incident energy, we characterized the shot to shot fluctuations by recording the PDF of
+the spectral intensity as a 100-bin histogram, using identical bins in all conditions to facilitate comparison. In the
+characterization of the PDF, we use among others Hogg’s unbiased kurtosis estimator, defined as
+Hg = U0.05 − L0.05
+U0.5 − L0.5
+− 2.59
+(1)
+where Um and Lm refer to the mean of the upper and lower m quantiles, respectively.
+This means that Um =
+1
+m
+� 1
+1−m F −1(y)dy and Lm = 1
+m
+� m
+0 F −1(y)dy, where F is the cumulative distribution function. For a Gaussian PDF,
+the Hogg estimator is 0.
+III.
+RESULTS AND DISCUSSION
+100
+102
+Intensity (arb.units)
+(a)
+(b) 0.017 mJ
+100
+102
+Intensity (arb.units)
+(c) 0.054 mJ
+(d) 0.102 mJ
+750
+800
+850
+Wavelength [nm]
+100
+102
+Intensity (arb.units)
+(e) 0.130 mJ
+750
+800
+850
+Wavelength [nm]
+(f) 0.173 mJ
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Peak energy [mJ]
+Figure 2. Evolution of the spectrum for increasing peak energy. (a) Average spectra; (b)-(f) Intensity fluctuations for incident
+pulse energies of 0.017, 0.054, 0.102, 0.130, and 0.173 mJ, respectively.
+In each panel, the solid thick line is the average
+intensity, the dark shaded area marks range between the 20th and the 80th percentiles and the light shaded area displays the
+range between the 5th and the 95th percentiles.
+Figure 2a displays the evolution of the typical spectra recorded in the center of the beam after it starts diverging,
+for incident energies ranging from 0.017 to 0.173 mJ/pulse. The dark shaded area of Panels b–f encompasses the
+20th to the 80th percentiles, while the light shadowed ranges from the 5th to the 95th ones, evidencing the typical
+range of the shot-to-shot fluctuations. As is typical for self-phase modulation (SPM) [32, 33], the spectral broadening
+3
+
+is characterized by the formation of an oscillatory plateau. For increasing incident powers, the plateau both rises
+in intensity and broadens by a progressive displacement of its edges away from the laser fundamental wavelength.
+Shot-to-shot fluctuations of the spectrum can therefore be characterized as essentially vertical (i.e., in intensity) in
+the central part of the spectrum, and essentially horizontal (i.e., in spectral range) on its edges.
+As displayed in Figure 2b–f, the shot-to-shot fluctuations can have the same order of magnitude as the mean
+spectral intensity all over the spectrum.
+In particular, above the threshold for appreciable spectral broadening,
+fluctuations reach a factor of 4, ranging from the non-filamenting to the filamenting regimes even at average incident
+pulse energies up to the critical power (0.173 mJ, P = 3.15 GW ∼ Pcr) where the filamentation visually seems to be
+well established. Such behavior randomly alternating filamenting (spectrally broadened) and non-filamenting pulses
+is consistent with the previously observed effect of turbulence on filamentation [34–36]. These fluctuations stem from
+the pre-filamentation fluctuations of the incident laser itself and from the influence of the turbulence, as well as
+the nonlinear transformation of the spectrum performed by the spectral broadening in the high-intensity region of
+propagation.
+The vertical and horizontal behaviors of the shot-to-shot fluctuations result in qualitatively different evolutions of
+the PDF as a function of incident energy (fig. 3). In the center of the spectrum (805 nm, panels g and h), increasing
+the incident pulse energy continuously shifts the peak of the PDF, from low to high spectral intensity. This shift
+becomes faster and faster when moving away from the center of the spectrum (See 785 nm, panels e and f). Reaching
+the spectrum edges, the behavior qualitatively changes, with the PDF becoming bimodal over the transition (773
+nm, panels b,c, and 860 nm, panels i,j). Over the power range where the transition occurs, the low-intensity peak
+progressively vanishes, while the high-intensity peak emerges. This behavior develops over a spectral range of ∼10 nm
+on each side of the spectrum. Even further on the edges (765 nm, panels a,b), the bimodal behavior disappears again.
+The PDF initially peaks close to zero intensity and rises when the broadening becomes sufficient.
+Note that in spite of patterns visually similar to tipping points, these transitions cannot be described as such.
+Rather, each laser pulse is essentially an independent experiment, hitting a fresh parcel of air.
+This experiment
+therefore consists in a series of random probings of the system behavior, with random initial phase profiles. The
+qualitative behaviors associated to each incident pulse energy at each wavelength can be summarized in a “phase
+diagram” (fig. 4a). We identify the regions with negligible broadening, with fully deployed broadening, and two kinds
+of transition regions: either continuous or bimodal. Remarkably, the bimodal transitions regions are characterized by
+a drop in both the skewness (fig. 4b) and Hogg’s unbiased kurtosis estimator Hg [37] (fig. 4c), while the continuous
+transitions are characterized by increasing skewness and Hg. This can be understood by considering the corresponding
+evolutions on fig. 3. In the central region of the spectrum, corresponding to continuous transitions, the PDF tail rises
+on the high-spectral intensity side, resulting in a heavier tail and growing asymmetry. In contrast, the appearance
+and growth of a secondary mode of the PDF tends to occur at the expense of the tails, and simultaneously results
+in a broadening of the central region of the PDF, which explains the smaller skewness and Hogg estimator. Further
+statistical moment like the kurtosis and coefficient of variation (i.e., the ratio between variance and mean) display
+qualitatively similar behaviors, as illustrated in fig. 5.
+From a mechanistic point of view, the bimodal transition corresponds to the wavelength range where the evolution
+of the spectrum for increasing pulse energies mostly occurs as a horizontal fluctuation of the cliff delimiting the
+broadened spectrum, i.e., the creation of new frequencies that previously were virtually absent from the spectrum.
+At these wavelengths, the sharp rise of the spectrum (large
+�� dI
+Idt
+��) followed by a relatively flat plateau of fluctuating
+width ensures the bimodal distribution, since the rising section of the spectrum has a negligible contribution to the
+PDF: The spectrum mainly features two ranges of attainable values, close to zero and on the plateau, respectively,
+while the intensity variations of the plateau itself is of second order. This interpretation also explains why the bimodal
+behavior is observed on a much narrower range of wavelengths around 850 nm (See fig. 4a), where the oscillation is
+sharper and the plateau is much less flat. In contrast, on the plateau, the fluctuations are mostly vertical. They
+consist in intensity fluctuations, i.e., an increased intensity of frequencies pre-existing in the incident pulse, so that
+the transition occurs continuously when the plateau progressively rises.
+These two qualitatively different transition modes to a broadened pulse coexist in the evolution of the same spectrum,
+driven by the same self-phase modulation process, and only some nanometers apart. The discontinuous transition to
+filamentation on the edges of the spectrum where bimodal spectral intensity distributions are observed would point
+to a system where filamenting and non-filamenting regimes are qualitatively distinct, and co-exist over some range of
+input power. In contrast, the central part of the spectrum and its smooth, continuous transition would not allow to
+define a threshold for the filamenting regime. Note that this spectral range bears most of the pulse energy.
+We argue that this dual behavior within the same spectrum during the transition to filamentation intrinsically
+prevents an unambiguous definition of the edges of a filamenting regime that would be qualitatively different from an
+essentially linear extended focus [29].
+4
+
+250
+500
+750
+1000
+1250
+Intensity (arb.units)
+0
+500
+1000
+Count
+765 [nm]
+a)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+250
+500
+750
+1000
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+765 [nm]
+b)
+10−5
+10−4
+10−3
+10−2
+Count normalized
+750
+800
+850
+100
+102
+250
+500
+750
+1000
+1250
+1500
+Intensity (arb.units)
+0
+500
+1000
+Count
+773 [nm]
+c)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+250
+500
+750
+1000
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+773 [nm]
+d)
+10−5
+10−4
+10−3
+10−2
+Count normalized
+750
+800
+850
+100
+102
+500
+1000
+1500
+2000
+Intensity (arb.units)
+0
+500
+1000
+Count
+785 [nm]
+e)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+250
+500
+750
+1000
+1250
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+785 [nm]
+f)
+10−5
+10−4
+10−3
+10−2
+Count normalized
+750
+800
+850
+100
+102
+500
+1000
+1500
+2000
+Intensity (arb.units)
+0
+500
+1000
+Count
+805 [nm]
+g)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+500
+1000
+1500
+2000
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+805 [nm]
+h)
+10−5
+10−4
+10−3
+Count normalized
+750
+800
+850
+100
+102
+50
+100
+150
+200
+250
+Intensity (arb.units)
+0
+500
+1000
+Count
+860 [nm]
+i)
+0.017
+0.054
+0.095
+0.107
+0.119
+0.130
+0.140
+0.149
+0.157
+0.169
+Pulse Energy [mJ]
+0
+50
+100
+150
+200
+Intensity (arb.units)
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+860 [nm]
+j)
+10−4
+10−3
+10−2
+10−1
+Count normalized
+750
+800
+850
+100
+102
+Figure 3. Evolution of the histogram of the spectral intensity at (a,b) 765 nm (transition through a bimodal distribution), (c,d)
+773 nm (intermediate regime), (e,f) 785 nm, (g,h) 805 nm (smooth offset of the peak), and (i,j) 860 nm (bimodal transition).
+(a,c,e,g,i): Individual histograms for increasing incident pulse energies. The color scale is the same as in fig. 2. (b,d,f,h,j):
+Colormap of the same data. Each histogram is normalised to 1. Insets display the position of the wavelength in the spectrum
+(See also Figure 2a)
+5
+
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+No broadening
+Smooth transition
+Bimodal transition
+Full broadening
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+0
+1
+2
+3
+4
+Skewness
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+−0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Hogg kurtosis
+Figure 4. (a) "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from
+non-broadened to fully-broadened, with continuous and bimodal transition modes in between.
+(b, c) Borders of the same
+diagram overlaid on (b) the skewness and (c) the Hogg’s unbiased kurtosis estimator [37].
+IV.
+CONCLUSION
+In summary, we characterized the evolution of intensity fluctuations across the spectrum of an ultrashort beam
+propagating in air, for a wide range of powers covering below and up to the critical power for filamentation. While
+on the edges of the spectrum the transition to filamentation is discontinuous and displays a bimodal distribution of
+spectral intensities around the critical power, it is continuous around the fundamental incident laser wavelength. This
+dual behavior provides an explanation for the lack of unambiguous definition of the edges of the filamentation regime
+6
+
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+0
+5
+10
+15
+20
+25
+30
+Kurtosis
+740
+760
+780
+800
+820
+840
+860
+880
+Wavelength [nm]
+0.017
+0.039
+0.062
+0.084
+0.106
+0.128
+0.151
+0.173
+Peak energy [mJ]
+0.31
+0.71
+1.12
+1.52
+1.93
+2.34
+2.74
+3.15
+Average intensity [GW]
+Smooth transition
+Full broadening
+Bimodal transition
+0
+50
+100
+150
+200
+250
+300
+Variance/Mean
+Figure 5. "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from non-
+broadened to fully-broadened, with continuous and bimodal transition modes in between overlaid on (a) the kurtosis and (b)
+the the coefficient of variation (variance/mean).
+in experiments or numerical simulations in spite of the intuitive understanding that filamentation qualitatively differs
+from a more linear propagation regime.
+Funding. Swiss National Science Foundation (SNF, grant 200020-175697)
+Acknowledgements. Experimental support was provided by Michel Moret.
+Disclosures. The authors declare no conflicts of interest.
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+
diff --git a/EtA0T4oBgHgl3EQfA__T/content/tmp_files/load_file.txt b/EtA0T4oBgHgl3EQfA__T/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf,len=719
+page_content='Two statistical regimes in the transition to filamentation  Alexis Gomel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 2 Geoffrey Gaulier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='1 Debbie Eeltink,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 3 Maura Brunetti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 4 and Jérôme Kasparian  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' ∗  1Group of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' University of Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 1205 Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Switzerland  2Institute for Environmental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' University of Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 1205 Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Switzerland  3Present address: Laboratory of Theoretical Physics of Nanosystems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' EPFL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Ch-1015 Lausanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 10 Switzerland  4Institute for Environmental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' University of Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 1205 Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Switzerland  Abstract  We experimentally investigate fluctuations in the spectrum of ultrashort laser pulses propagating in air,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' close to the critical  power for filamentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Increasing the laser peak power broadens the spectrum while the beam approaches the filamentation  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' We identify two regimes for this transition: In the center of the spectrum, the output spectral intensity increases  continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In contrast, on the edges of the spectrum the transition implies a bimodal probability distribution function for  intermediate incident pulse energies, where a high-intensity mode appears and grows at the expense of the original low-intensity  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' We argue that this dual behavior prevents the definition of a univoquial threshold for filamentation, shedding a new  light on the long-standing lack of explicit definition of the boundary of the filamentation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  ∗ jerome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='kasparian@unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='ch  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='01971v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='optics]  5 Jan 2023  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  INTRODUCTION  Filamentation is a self-guided propagation regime typical of ultrashort, high-power laser pulses [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Beyond a  critical power (3 GW in air), the Kerr effect balances diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  The beam then self-focuses until higher-order  nonlinear defocusing effects like ionization or the saturation of the Kerr effect [5] come into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  The resulting  dynamical balance gives rise to self-guided light structures known as filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Filaments can extend over atmospheric  scale distances [6, 7], opening the way to various applications from remote sensing [8] to lightning control [9–14],  THz generation [15–20], fog clearing [21, 22], or the triggering of condensation in sub-saturated atmospheres [8, 23,  24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Filaments show very characteristic and even spectacular features, including long-distance propagation, bright  light emission due to both plasma emission on the side and spectral broadening in the forward direction, and noise  associated to the shockwave [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Together, these features give rise to the intuitive notion of a qualitatively distinct  propagation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' However, defining a clear threshold for the filamentation regime turns out to be difficult [28],  especially in focused beams where linear propagation can be sufficient to reach the ionization threshold [29], and even  produce a denser plasma than Kerr self-focusing does [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Similarly, along the propagation axis, defining the onset  and end of filaments, and therefore their length, requires somewhat arbitrary choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Here, we statistically investigate the transition to filamentation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Shot-to-shot fluctuations of the spectral amplitudes  of a high-power laser beam, whether filamenting or not, are known to display characteristic probability distribution  functions (PDF): A regular probability distribution is observed in the center of the spectrum, while on its edge the  distribution is long-tailed (optical rogue wave [31]) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' We extend this analysis to the sub-filamenting regime and to  the transition to filamentation, in order to characterize how the statistics evolves when the incident power progressively  reaches the critical power for filamentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The evolution of spectral intensity PDFs turns out to display typical  and contrasted signatures on the edges and in the center of the spectrum, respectively, suggesting that transition  to filamentation cannot be characterized as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' By doing so, we suggest that the ambiguities in the definition  of the limits of the filamentation regime are intrinsic to the physical process rather than instrumental or conceptual  limitations, shedding a new light on the long-standing lack of explicit definition of the boundary of the filamentation  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  EXPERIMENTAL SETUP  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Experimental setup  The experimental setup (Figure 1) relied on a Coherent Astrella laser delivering ultrashort pulses at 1 kHz repetition  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' A half-wave plate installed in a motorized rotating stage ensureing ±1◦ precision, followed by a linear polarizer,  allowed to adjust the pulse energy with an accuracy of ∼ 10 µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The angle-to-energy calibration was performed before  each experimental run with a powermeter (Coherent PM10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The laser compressor was adjusted so as to slightly chirp  the pulses to a duration of 55 fs (as measured with PulseCheck, APE Berlin), in order to reach the filamentation  threshold in air within the tuning range of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The beam was slightly focused from an initial diameter of 11 mm  at 1/e2 by an f = 2 m BK7 plano-convex lens located approximately at 15 cm after the polarizer and subsequently  through a 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='5 cm-long turbulent region generated by 48◦C to 565◦C hot air blown transversally by a hot air blower  (Steinel HL1502 S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 15–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='5 cm downstream of the lens, the beam self-focused and filamentation started if the incident  peak power was sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The filamentation region was surrounded by a PVC tube of 18 cm diameter to shield the  beam against air displacements in the room so as to improve its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' After the end of filamentation region, the  beam was imaged by a second BK7 lens (f = 20 cm) onto the entrance slit of an OceanOptics USB2000 spectrometer  providing ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='5 nm spectral resolution over the visible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' A neutral density filter prevented the spectrometer from  saturating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The spectrum was measured and recorded independently for each laser shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' For each incident pulse  2  energy, 5000 individual spectra were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  At each wavelength and incident energy, we characterized the shot to shot fluctuations by recording the PDF of  the spectral intensity as a 100-bin histogram, using identical bins in all conditions to facilitate comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In the  characterization of the PDF, we use among others Hogg’s unbiased kurtosis estimator, defined as  Hg = U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='05 − L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='05  U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='5 − L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='5  − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='59  (1)  where Um and Lm refer to the mean of the upper and lower m quantiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  This means that Um =  1  m  � 1  1−m F −1(y)dy and Lm = 1  m  � m  0 F −1(y)dy, where F is the cumulative distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' For a Gaussian PDF,  the Hogg estimator is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  100  102  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='units)  (a)  (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='017 mJ  100  102  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='units)  (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='054 mJ  (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='102 mJ  750  800  850  Wavelength [nm]  100  102  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='units)  (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='130 mJ  750  800  850  Wavelength [nm]  (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='173 mJ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='107  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='119  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='130  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='149  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='157  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='169  Peak energy [mJ]  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Evolution of the spectrum for increasing peak energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' (a) Average spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' (b)-(f) Intensity fluctuations for incident  pulse energies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='017, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='054, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='102, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='130, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='173 mJ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  In each panel, the solid thick line is the average  intensity, the dark shaded area marks range between the 20th and the 80th percentiles and the light shaded area displays the  range between the 5th and the 95th percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Figure 2a displays the evolution of the typical spectra recorded in the center of the beam after it starts diverging,  for incident energies ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='017 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='173 mJ/pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The dark shaded area of Panels b–f encompasses the  20th to the 80th percentiles, while the light shadowed ranges from the 5th to the 95th ones, evidencing the typical  range of the shot-to-shot fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' As is typical for self-phase modulation (SPM) [32, 33], the spectral broadening  3  is characterized by the formation of an oscillatory plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' For increasing incident powers, the plateau both rises  in intensity and broadens by a progressive displacement of its edges away from the laser fundamental wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Shot-to-shot fluctuations of the spectrum can therefore be characterized as essentially vertical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=', in intensity) in  the central part of the spectrum, and essentially horizontal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=', in spectral range) on its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  As displayed in Figure 2b–f, the shot-to-shot fluctuations can have the same order of magnitude as the mean  spectral intensity all over the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  In particular, above the threshold for appreciable spectral broadening,  fluctuations reach a factor of 4, ranging from the non-filamenting to the filamenting regimes even at average incident  pulse energies up to the critical power (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='173 mJ, P = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='15 GW ∼ Pcr) where the filamentation visually seems to be  well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Such behavior randomly alternating filamenting (spectrally broadened) and non-filamenting pulses  is consistent with the previously observed effect of turbulence on filamentation [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' These fluctuations stem from  the pre-filamentation fluctuations of the incident laser itself and from the influence of the turbulence, as well as  the nonlinear transformation of the spectrum performed by the spectral broadening in the high-intensity region of  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  The vertical and horizontal behaviors of the shot-to-shot fluctuations result in qualitatively different evolutions of  the PDF as a function of incident energy (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In the center of the spectrum (805 nm, panels g and h), increasing  the incident pulse energy continuously shifts the peak of the PDF, from low to high spectral intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' This shift  becomes faster and faster when moving away from the center of the spectrum (See 785 nm, panels e and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Reaching  the spectrum edges, the behavior qualitatively changes, with the PDF becoming bimodal over the transition (773  nm, panels b,c, and 860 nm, panels i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Over the power range where the transition occurs, the low-intensity peak  progressively vanishes, while the high-intensity peak emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' This behavior develops over a spectral range of ∼10 nm  on each side of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Even further on the edges (765 nm, panels a,b), the bimodal behavior disappears again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  The PDF initially peaks close to zero intensity and rises when the broadening becomes sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Note that in spite of patterns visually similar to tipping points, these transitions cannot be described as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Rather, each laser pulse is essentially an independent experiment, hitting a fresh parcel of air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  This experiment  therefore consists in a series of random probings of the system behavior, with random initial phase profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The  qualitative behaviors associated to each incident pulse energy at each wavelength can be summarized in a “phase  diagram” (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' We identify the regions with negligible broadening, with fully deployed broadening, and two kinds  of transition regions: either continuous or bimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Remarkably, the bimodal transitions regions are characterized by  a drop in both the skewness (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 4b) and Hogg’s unbiased kurtosis estimator Hg [37] (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 4c), while the continuous  transitions are characterized by increasing skewness and Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' This can be understood by considering the corresponding  evolutions on fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In the central region of the spectrum, corresponding to continuous transitions, the PDF tail rises  on the high-spectral intensity side, resulting in a heavier tail and growing asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In contrast, the appearance  and growth of a secondary mode of the PDF tends to occur at the expense of the tails, and simultaneously results  in a broadening of the central region of the PDF, which explains the smaller skewness and Hogg estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Further  statistical moment like the kurtosis and coefficient of variation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=', the ratio between variance and mean) display  qualitatively similar behaviors, as illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  From a mechanistic point of view, the bimodal transition corresponds to the wavelength range where the evolution  of the spectrum for increasing pulse energies mostly occurs as a horizontal fluctuation of the cliff delimiting the  broadened spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=', the creation of new frequencies that previously were virtually absent from the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  At these wavelengths, the sharp rise of the spectrum (large  �� dI  Idt  ��) followed by a relatively flat plateau of fluctuating  width ensures the bimodal distribution, since the rising section of the spectrum has a negligible contribution to the  PDF: The spectrum mainly features two ranges of attainable values, close to zero and on the plateau, respectively,  while the intensity variations of the plateau itself is of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' This interpretation also explains why the bimodal  behavior is observed on a much narrower range of wavelengths around 850 nm (See fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 4a), where the oscillation is  sharper and the plateau is much less flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In contrast, on the plateau, the fluctuations are mostly vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' They  consist in intensity fluctuations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=', an increased intensity of frequencies pre-existing in the incident pulse, so that  the transition occurs continuously when the plateau progressively rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  These two qualitatively different transition modes to a broadened pulse coexist in the evolution of the same spectrum,  driven by the same self-phase modulation process, and only some nanometers apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The discontinuous transition to  filamentation on the edges of the spectrum where bimodal spectral intensity distributions are observed would point  to a system where filamenting and non-filamenting regimes are qualitatively distinct, and co-exist over some range of  input power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' In contrast, the central part of the spectrum and its smooth, continuous transition would not allow to  define a threshold for the filamenting regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Note that this spectral range bears most of the pulse energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  We argue that this dual behavior within the same spectrum during the transition to filamentation intrinsically  prevents an unambiguous definition of the edges of a filamenting regime that would be qualitatively different from an  essentially linear extended focus [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Evolution of the histogram of the spectral intensity at (a,b) 765 nm (transition through a bimodal distribution), (c,d)  773 nm (intermediate regime), (e,f) 785 nm, (g,h) 805 nm (smooth offset of the peak), and (i,j) 860 nm (bimodal transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  (a,c,e,g,i): Individual histograms for increasing incident pulse energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The color scale is the same as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Each histogram is normalised to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Insets display the position of the wavelength in the spectrum  (See also Figure 2a)  5  740  760  780  800  820  840  860  880  Wavelength [nm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content='0  Hogg kurtosis  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' (a) "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from  non-broadened to fully-broadened, with continuous and bimodal transition modes in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  (b, c) Borders of the same  diagram overlaid on (b) the skewness and (c) the Hogg’s unbiased kurtosis estimator [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  CONCLUSION  In summary, we characterized the evolution of intensity fluctuations across the spectrum of an ultrashort beam  propagating in air, for a wide range of powers covering below and up to the critical power for filamentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' While  on the edges of the spectrum the transition to filamentation is discontinuous and displays a bimodal distribution of  spectral intensities around the critical power, it is continuous around the fundamental incident laser wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' This  dual behavior provides an explanation for the lack of unambiguous definition of the edges of the filamentation regime  6  740  760  780  800  820  840  860  880  Wavelength [nm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content='173  Peak energy [mJ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='93  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='34  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='74  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='15  Average intensity [GW]  Smooth transition  Full broadening  Bimodal transition  0  50  100  150  200  250  300  Variance/Mean  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' "Phase diagram" displaying the different regimes as a function of wavelength and incident pulse energy, from non-  broadened to fully-broadened, with continuous and bimodal transition modes in between overlaid on (a) the kurtosis and (b)  the the coefficient of variation (variance/mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  in experiments or numerical simulations in spite of the intuitive understanding that filamentation qualitatively differs  from a more linear propagation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Swiss National Science Foundation (SNF, grant 200020-175697)  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Experimental support was provided by Michel Moret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Yu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Pépin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Vidal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Couture, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Mercure, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Potvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Bondiou-Clergerie,  and  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Gallimberti, Applied physics Letters 76, 819 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Ackermann, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Méjean, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Rohwetter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Salmon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Stelmaszczyk,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Yu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Sauerbrey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Wöste, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Mongin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content=' Mongin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Kasparian, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Daigle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+page_content='  [37] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' Kim and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content=' White, Finance Research Letters 1, 56 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtA0T4oBgHgl3EQfA__T/content/2301.01971v1.pdf'}
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+Prepared for submission to JINST
+Study on U/Th residual radioactivity in acrylic from
+surface treatment
+Yuanxia Li,𝑎 Xiaohui Qian,𝑎 Xiaolan Luo,𝑎 Jie Zhao,𝑎,1 Gaofeng Zhang,𝑏 Xiaoyan Ma,𝑎
+Yuekun Heng,𝑎 Liangjian Wen,𝑎 Monica Sisti,𝑐 Frédéric Perrot,𝑑 Hongqiang Tang𝑏
+𝑎Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
+𝑏Donchamp New Material Technology Co. Ltd, Taixing 225400, China
+𝑐INFN Milano Bicocca and Università di Milano-Bicocca, Milano, Italy
+𝑑Univ. Bordeaux, CNRS, CENBG, UMR 5797, F-33170 Gradignan, France
+E-mail: zhaojie@ihep.ac.cn
+Abstract: Acrylic is widely used as material for the target container in low background experiments
+due to its high light transparency and low intrinsic radioactivity. However, its surface can be easily
+contaminated during production, so careful treatment of the surface is essential to avoid direct
+contamination of the target. The Jiangmen Underground Neutrino Observatory will use about 600 t
+of acrylic to build the spherical vessel of 35.4 m in diameter for a 20 kt liquid scintillator (LS).
+Since acrylic will contact the LS directly, the cleanliness of the its surface is quite important for the
+radiopurity of the LS. A new method for measuring the radioactivity of 238U and 232Th in acrylic to
+sub-ppt (< 10−12 g/g) was developed, and it is crucial for the acrylic radioactivity screening in this
+study. We performed many background tests on different surface treatments, and the recommended
+procedure for the treatment of acrylic to achieve low radioactivity and high light transparency could
+be applicable to other low background experiments.
+1Corresponding author.
+arXiv:2301.04902v1  [physics.ins-det]  12 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Screening of surface treatment material by gamma spectrometer
+2
+3
+Method for measuring U/Th in acrylic by ICP-MS
+3
+4
+Study of the procedure for acrylic surface treatments
+5
+4.1
+Solution absorption in acrylic
+5
+4.2
+Contamination from different surface treatments
+7
+4.3
+Protection film
+8
+5
+Summary
+9
+1
+Introduction
+The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose neutrino experiment,
+which will build the world’s largest liquid scintillator (LS) detector. An acrylic vessel is used as
+the 20 kt LS container with 35.4 m in diameter and 120 mm in thickness because of its high light
+transparency and low intrinsic radioactivity [1].
+To reach the physics target, the average radio-purity of the acrylic is required to be less than
+1 ppt [2]. As shown in Ref. [3], the radio purity in the bulk has reached the requirement. However,
+the distribution of radioactivity along the thickness is not extremely uniform. The bulk measurement
+of samples with several centimeters of thickness can not reflect the surface contamination, since
+the thickness of external contamination on the acrylic surface usually reaches nanometre to micron
+level, and the surface contamination will result averaged out in the bulk measurement. If the external
+radioactive contamination stays on the surface, only gammas and few alphas/betas from radioactive
+decays can be emitted into the LS and rejected by the effective fiducial volume cut by the off-line
+analysis. However, based on the experiences at Borexino [4] and SNO+ [5], the U/Th daughters
+near the inner surface of the vessel could leak into the LS by diffusion and convection during stable
+running, thus making the acrylic vessel a stable contamination source for the LS. Borexino put
+great efforts on controlling the temperature distribution to avoid convection in LS; however, it is
+impossible to repeat the same thing at JUNO due to the large size of the experiment (70 times
+Borexino target). Therefore, the radiopurity of the inner surface of acrylic is quite important for
+JUNO, especially for the physic topics in the low energy regions, such as solar neutrino studies [1].
+The JUNO acrylic vessel is produced with 265 pieces because of its large size, as shown in
+Fig. 1. Each acrylic panel is firstly produced in a carefully cleaned flat mold and then shaped to
+bend board at high temperature at the company; finally it is bonded directly at JUNO site [1]. The
+surface of the acrylic panel is exposed to air and covered with a high-temperature resistant cloth
+– 1 –
+
+during the bending, thus the surface is eventually contaminated and not as smooth as the flat panel.
+Surface treatments are performed before shipment to the JUNO site. Typical protocols to remove
+surface contaminations include the following steps:
+• The highest contaminations can reach tens of µm depths from the surface [6]. Thus, at least
+0.1 µm thickness of the surface of acrylic is firstly removed by sanding with special paper.
+• The surface is not so smooth after sanding, so polishing is needed to improve the transparency.
+The polishing is done with a wool wheel and polishing fluid.
+• The surface should be well cleaned with detergent and deionized water after polishing.
+• Finally, a thin film is used to cover the acrylic surface to avoid external contamination during
+transportation and installation.
+All tools and detergents have risks to introduce more contamination on the surface, therefore
+each surface treatment should be well studied to minimize those risks.
+Figure 1. The acrylic vessel is produced with 265 pieces due to its very large size, as shown in the left figure.
+Each panel is produced separately, and all the panels will be finally bonded to a sphere at JUNO site. One
+JUNO panel is shown in the right figure as an example.
+The paper is organized as follows. The surface treatment material screening is described in
+Sec 2. The new method for measuring 238U and 232Th (U/Th) in acrylic to sub-ppt level is described
+in Sec 3. Various tests on radioactivity from each step of the above treatment protocol are discussed
+in Sec 4, and the recommended procedure is also shown in the same section. The concluding
+remarks are in Sec. 5.
+2
+Screening of surface treatment material by gamma spectrometer
+The main tools and detergents used for surface treatments are sanding papers, polishing fluid,
+Alconox powder [7], deionized water and protection film. We used a gamma spectrometer to screen
+the radiopurity of the tools and detergents first. We have one HPGe facility at ground level at
+IHEP [8]. The gamma spectrometer plays an important role in the primary screening of the raw
+– 2 –
+
+0
+0
+0
+0
+0
+Q
+0
+4
+0
+.materials at JUNO, and the advantage is that samples can be screened directly without any pre-
+treatment and with no damage to the sample. The principle of gamma spectroscopy is to measure
+the emitted gamma from the decay chain, and it is more sensitive to gammas with higher energy
+due to lower background. For example, the detector is more sensitive to the lower part of the
+uranium chain starting from 226Ra, and the 238U concentration can be derived by assuming secular
+equilibrium.
+The measured results for the tools and detergent used for acrylic surface treatment are sum-
+marized in Table 1. We found that special mirror wax purchased from Saint-Gobain [9] has lower
+radioactivity for 40K than one-step fast wax, due to the different chemical components. The sand-
+paper with a green base plate has lower radioactivity and better sand fastening than the one with a
+black base plate for similar mesh numbers. Regarding the Alconox powder, the first measurement
+performed at IHEP gave an upper limit of several tens to hundreds of ppb level for thorium and
+uranium concentrations, respectively, due to the limited sensitivity of HPGe. The same sample was
+further measured by HPGe at China Jinping Underground Laboratory (CJPL) with better shielding
+and sensitivity [10] down to 10 ppb for 232Th. From the obtained results, there is a higher uranium
+concentration than thorium in the Alconox powder. The polishing fluid and sandpaper with lower
+radioactivity are selected for the surface treatments on acrylic.
+After surface treatment, the acrylic surface will be protected with a thin film to avoid dust and
+radon daughters deposition on the surface during the handling operations for transportation and
+installation. The film on the inner surface will be removed by the final cleaning with high pressure
+water jet after installation, and there is no possibility for manual operation inside the acrylic vessel.
+There are two types of protection film that can be used to protect the acrylic surface, ordinary
+polyethylene (PE) film with few glue on the surface and adhesive masking paper film. The PE film
+is better in toughness, but it is not easy to be removed by the water jet. On the contrary, the paper
+film with few water soluble glue on the surface can be easily removed by water washing since the
+glue is soluble in water, but overall its mechanical strength is worse. The radioactivity screening of
+the glue that is used on the PE film showed only upper limits for U/Th at the level of hundreds of
+ppb. Concerning the adhesive masking paper film (purchased from [11]), we could only measure
+the paper film and the glue together inside the HPGe detector. The water soluble glue itself was
+further measured by ICP-MS, as it will be discussed later.
+3
+Method for measuring U/Th in acrylic by ICP-MS
+Different from gamma spectroscopy, the principle of the Inductively Coupled Plasma Mass Spec-
+troscopy (ICP-MS) is to detect the nuclei according to their exact atomic weight. ICP-MS can
+measure the isotopes 238U and 232Th directly without other information on the chain, and the sen-
+sitivity can reach the sub-ppt level. However, the sample measured by ICP-MS must be in liquid
+form, so a careful and optimized pre-treatment on the sample is needed.
+For the radio purity measurement on acrylic surface samples, we have developed a new method
+based on microwave ashing and ICP-MS equipment. The design for the pre-treatment flow is shown
+in Fig. 2. Both the acrylic sample and the tracers (the natural-non-existing nuclei 229Th and 233U)
+are ashed by the machine in a quartz vessel. The residual is collected by soaking with 35% HNO3.
+– 3 –
+
+Table 1.
+Material screening by gamma spectroscopy on tools and detergents used for the acrylic surface
+treatment. If not differently mentioned, the sample was measured at IHEP. The conversions between Bq/kg
+and mass concentration units for U/Th are given: 1 Bq/kg of 232Th activity in a material is equal to
+2.5 × 10−7 g/g (or 250 ppb) of 232Th mass concentration in that material; similarly, 1 Bq/kg of 238U means
+8.1 × 10−8 g/g (or 81 ppb) of 238U.
+238U chain [Bq/kg]
+232Th chain [Bq/kg]
+40K
+214Bi/214Pb
+226Ra
+212Bi/208Tl
+[Bq/kg]
+Polishing
+Special mirror wax
+2.8±0.2
+2.9±1.3
+5.8±0.3
+3.8±0.9
+One step fast wax
+3.1±0.3
+5.0±1.7
+5.5±0.4
+34.5±3.1
+Alconox powder
+-
+<0.37
+<4.4
+<0.24
+5.9±1.3
+Measured at CJPL
+0.23±0.02
+0.82±0.14
+0.05±0.02
+7.5±1.0
+Sand paper
+Black base-plate
+7.2±0.6
+<21
+13.6±0.6
+<3.7
+Green base-plate
+<1.7
+<10.2
+<1.6
+<2.2
+Protection film
+Glue on PE film
+<0.2
+<2.3
+<0.1
+<0.6
+Water soluble film
+<0.47
+<10.8
+0.58±0.19
+<1.56
+The eluent is collected and heated to remove the excess acid, and the solution is finally diluted for
+the ICP-MS measurement.
+Figure 2. The flow chart for the pre-treamtent of acrylic samples. The ashing of the acrylic sample is done
+with the microwave muffle furnace described in the text.
+In Ref. [3], a different approach was used: the acrylic was vaporized and burned in a two-stage
+furnace, while clean gas (mixed N2/O2) was supplied to the furnace. Monitoring of the pressure
+in the whole system is essential in this case. Compared with the pre-treatment in [3], with the new
+technique there is no need for gas input to the microwave muffle furnace, and the overall operation
+is easier and safer. All the pre-treatment process is done in a class 10,000 tent, while the ICP-MS
+measurement is performed in a class 1,000 room. The cleanliness of the quartz vessels is quite
+important for the sensitivity of the measurement, and their cleaning procedure is similar to that
+described in Ref. [3]. All the containers are soaked in the two-stage acid cylinders filled with
+6 mol/L HNO3 for at least one day at each stage to remove the U/Th on the surface. In the end,
+the containers are filled with 6 mol/L HNO3 and boiled for 10 min, then they are rinsed with fresh
+water and ready for use.
+The microwave muffle furnace (Phoenix BLACK [12]) is put in the clean tent and used for
+ashing the acrylic sample, as shown on the left of Fig. 3. The temperature for ashing is optimized
+to achieve no visible organic residual in the vessel, and the optimized temperature as a function of
+time is shown on the right of Fig. 3. The quartz vessel is used as the sample container and well
+cleaned. The 229Th and 233U standards (preserved in 2% acid) are added together with the acrylic
+– 4 –
+
+Tracer and
+Ashing in
+Soak the vessel
+acrylic
+quartz vessel
+with 35% HNO3
+Diluted for ICP-MS
+Eluent was collected and
+measurement
+heated to remove excess acid0
+20
+40
+60
+80
+Time [min]
+0
+200
+400
+600
+C]
+o
+Temperature [
+Figure 3. The microwave muffle furnace (Phoenix BLACK) used for the pre-treatment of acrylic is shown
+in the left figure. The optimized temperature as a function of time for acrylic ashing is shown in the right
+figure.
+sample to the vessel in the beginning, and the recovery efficiency can be evaluated by measuring the
+229Th/233U in the solution sent to ICP-MS. The average recovery efficiency of U/Th is calculated as
+(95±13)% based on more than one hundreds measurements (the given uncertainty is the standard
+deviation). The recovery efficiency is calibrated every time a sample is measured, and the single
+precision is better than 5%. To estimate the background for the pre-treatment, we have followed
+the same procedure without the acrylic sample (blank test). With careful background control, the
+blank test showed that the absolute background for the pre-treatment amounts to 0.24±0.07 pg for
+238U and 0.38±0.04 pg for 232Th.
+4
+Study of the procedure for acrylic surface treatments
+4.1
+Solution absorption in acrylic
+The acrylic panel is cleaned after surface treatments at the company before shipment, and the whole
+acrylic sphere is finally cleaned onsite after installation. Acrylic can absorb water, thus traces of
+radioactivity in cleaning water may diffuse into acrylic. The absorption of water in acrylic obeys
+the mathematical laws of diffusion. We have performed several water absorption tests on flat acrylic
+samples with 2 mm thickness. The mass proportion of absorbed water in acrylic as a function of
+time is shown in Fig. 4. The absorption of water in acrylic did not reach equilibrium in one month,
+and the mass of absorbed water in acrylic increased almost linearly at the beginning, which is far
+away from the equilibrium state. Therefore, there is no impact of the acrylic panel thickness when
+exposure to water is shortened to several hours. Assuming we will clean the acrylic surface during
+one hour at the company, the absorbed water in acrylic can reach about 0.08% for the sample with
+2 mm thickness, which is 1 g/m2 water absorption for one side of acrylic surface.
+For the cleaning procedure, we use Alconox solution (0.1% of water) for degreasing and
+deionized water for rinsing. Traces of radioactivity in both water and Alconox can diffuse into
+acrylic during cleaning. We use deionized water with 10−14 g/g U/Th to perform water cleaning, so
+the residual traces of radioactivity from absorbed water is negligible. Even though the Alconox is
+not as clean as the deionized water, most of Alconox can be removed by water rinsing. To validate
+how much radioactivity from Alconox can go into acrylic, we soaked the acrylic sample with 2 mm
+– 5 –
+
+Pho
+QulckTest
+方法
+CM
+OYFigure 4.
+Results of water absorption tests performed on flat acrylic samples with 2 mm thickness. The
+mass proportion of absorbed water to acrylic as a function of time is shown during 35 days, and the data
+points in the first day are magnified in the sub-figure.
+thickness in Alconox solution with different concentrations and time exposure, and the radiopurity
+of the sample for U/Th is measured by ICP-MS in the end. We have soaked the samples in Alconox
+solution for two months in order to enlarge the possible radioactivity contamination in the bulk of
+acrylic (2 mm thickness). The results of the tests for U/Th surface contamination are shown in
+Table 2.
+Table 2.
+The acrylic samples with 2 mm thickness are soaked in different solutions with different time
+exposure. The U/Th contamination from solution absorption in acrylic after absorption is measured by
+ICP-MS.
+No.
+Sample preparation
+Exposure
+Mass[g]
+238U[ppt]
+232Th[ppt]
+1
+No soaking
+-
+3.25
+0.5±0.1
+1.7±0.1
+2
+0.1% Alconox solution
+1 day
+3.21
+0.6±0.1
+2.3±0.2
+3
+0.1% Alconox solution
+2 months
+3.18
+1.9±0.1
+2.0±0.3
+4
+2% Alconox solution
+2 months
+3.74
+4.4±0.2
+2.3±0.2
+5
+0.1% Alconox solution + 30% HNO3
+2 months + 1 day
+3.11
+0.5±0.1
+1.5±0.2
+6
+Tap water after filter [739 ppt 238U, 0.02 ppt 232Th]
+1 day
+3.28
+3.0±0.1
+1.7±0.2
+7
+1% HNO3 [106.7 ppt 238U, 614.5 ppt 232Th]
+1 day
+3.18
+2.9±0.2
+32.2±1.5
+• No.1-5: we can see a clear increase on 238U in acrylic for Alconox solutions with different
+concentration and exposure. However, the increase of 232Th is not obvious due to the lower
+232Th radioactivity in Alconox powder as shown in Table 1. Even though the Alconox can
+be removed by further deionized water rinsing, the radioactivity from Alconox can diffuse
+into acrylic together with water. If the residual radioactivity stick on acrylic from Alconox
+solution is due to the active agent, the residual can not be easily removed by cleaning or
+acid. On the contrary, if the mechanism of residual consists in ionic diffusion from Alconox
+solution to acrylic, such residual can be removed by acid. We observed that the radioactivity
+absorbed in acrylic can be further removed by acid soaking, as shown in result No.5.
+• No.6-7: To further validate that the diffusion of radioactivity from Alconox to acrylic is
+– 6 –
+
+1.8
+[%]
+Water absorbtion [
+1.6
+1.4
+1.2
+0.3
+0.8
+0.2
+0.6
+0.1
+0.4
+Q
+0.2
+10
+20
+Time [h]
+0
+5
+10
+15
+20
+25
+30
+35
+Time [day]in ionic form, we have soaked another two acrylic samples in two kinds of water. One is
+tap water after 0.2 µm filter, and radioactivity exists in small particles. The other one is
+U/Th standard solution with 1% HNO3, and radioactivity is in ionic form. As shown in
+No.6 and No.7, the 238U concentration in acrylic reached equilibrium at 3 ppt, while 232Th
+can reach much higher to about 30 ppt. The reason for the difference between 238U and
+232Th is their different intrinsic physicochemical properties, which is also discussed in [13].
+Thorium is more reactive than uranium, thus it is easier for thorium to stick to acrylic. A
+similar phenomenon is observed in preparing tap water, most of the thorium is filtered and
+less residual is found in the water. From the results in No.6 and No.7, radioactivity can also
+diffuse into acrylic without the active agent, so the radioactivity in Alconox solution can
+diffuse into acrylic due to the diffusion of ion, not the active agent. The final residual on
+acrylic from the cleaning solution relies on the measurement of the surface sample.
+4.2
+Contamination from different surface treatments
+To study the surface contamination, we have treated the surface with different procedure and directly
+measured the residual of radioactivity on surface. The surface samples were taken by scraping the
+surface with well cleaned erasing knife, and the thickness of the surface sample can reach 5-10 µm.
+The radioactivity of the raw surface for one JUNO acrylic panel without any surface treatment is
+measured to be (52±1) ppt 238U and (133±5) ppt 232Th, which is 1-2 orders of magnitude higher than
+the bulk measurement. The radioactivity distribution along the depth was measured in Ref. [6], and
+has proven that U/Th contaminations can extend down to tens of µm from the surface. In addition,
+the radon daughters (210Pb, 210Po) deposited on the acrylic surface is usually non-negligible. So we
+decide to remove at least 0.1 mm depth of acrylic and to maintain a high light transparency (>96%
+at 420 nm [14] in ultrapure water environment) at the same time.
+A dedicated experiment was performed on a flat acrylic panel (not a JUNO panel) with a
+1 m2 area by applying different surface treatments including sanding, polishing, and cleaning
+successively. All of these treatments are done in a 10,000 class tent. The sanding of acrylic surfaces
+starts from 400 mesh (to remove at least 0.1 mm depth of acrylic) and is pursued by using sand
+papers with larger mesh, 800, 1200, 2000, and 3000. The higher the mesh number, the smoother
+the acrylic surface, and higher light transparency can be achieved. However, more steps increase
+the risk of contaminations. We have measured the surface sample with different mesh numbers of
+1200, 2000, and 3000, and the results for residual U/Th concentration are consistent within 20%
+among these samples.
+Polishing of acrylic is performed by a wool wheel with polishing fluid, and the typical fluid is
+purchased from Saint-Gobain [9], whose components are organic. If there is polishing with mirror
+wax after sanding, sanding to 1200 mesh number is enough to reach the required light transparency.
+Part of the polishing fluid can be further removed by degreasing with Alconox solution. To quantify
+the contamination from residual polishing fluid and Alconox solution, we have done many tests on
+the acrylic panel with sanding to 1200 mesh number, and the results are shown in Table 3. The
+water cleaning is realized by a water jet with high pressure for 20 times, and the Alconox cleaning is
+done by spraying the Alconox on surface and wiping with a clean cloth for several times. Compared
+to the sample No.1, we have done additional cleaning with Alconox solution on No.2, and there is
+obvious radioactive residual on the surface from Alconox. For the samples No. 3 and No. 4, we
+– 7 –
+
+have done polishing with the fluid for both samples, and performed cleaning with Alconox only on
+No.4. From the results, it is quite clear that a large amount of polishing fluid residual exists on the
+acrylic surface after cleaning even with Alconox solution.
+Table 3. Many cleaning tests have been done on the non-JUNO acrylic panel after sanding to the 1200 mesh
+number. The water cleaning is realized by a water jet with high pressure 20 times, and the Alconox cleaning
+is done by wiping the surface with a clean cloth several times. The polishing is done with mirror wax. After
+the treatments, surface samples are taken by scraping the surface with well cleaned erasing knife, and the
+thickness of the surface sample can reach 5-10 µm. The radioactivity in the surface samples is summarized
+in this table.
+No.
+Sample preparation
+Mass[g]
+238U[ppt]
+232Th[ppt]
+1
+Water cleaning
+0.25
+9.5±0.3
+9.3±0.5
+2
+Alconox + water cleaning
+0.34
+27±1
+22±2
+3
+Polishing + water cleaning
+0.28
+92±5
+893±27
+4
+Polishing + Alconox + water cleaning
+0.35
+123±3
+817±33
+Based on the above measurements, polishing with mirror wax and Alconox cleaning have non-
+negligible residual on the acrylic surface, and we should avoid using them for surface treatments.
+To reach light transparency greater than 96% at 420 nm in ultrapure water environment, we have
+performed sanding with 3000 mesh number, polishing with deionized water and, in the end, cleaning
+of the surface with a deionized water high pressure jet. We finally measured the surface of one JUNO
+acrylic panel following this procedure of surface treatments, with good results of (15.2±0.7) ppt
+238U and (24.3±0.7) ppt 232Th in about 10 µm depth, which is several times lower than the raw
+surface as shown in the beginning of this section. In addition, all the treatments of the inner surface
+of the acrylic panel were done in one day to avoid radon daughters deposition.
+4.3
+Protection film
+After surface treatments with sanding, polishing, and cleaning, the acrylic surface will be covered
+by a thin film to protect the acrylic from fallouts of radon and dust in the air during transportation
+and installation, which will last several months. There are two kinds of protection films that can be
+used to protect the acrylic surface, as discussed in Sec 2.
+Since the glue on the adhesive masking paper is soluble in water, we soaked the paper film
+in water for different exposure. By this way, we can measure the radiopurity of glue by ICP-MS
+with high precision. To reduce the effect of surface cleanliness in this measurement, we swab the
+surface of paper side with a little wet clean cloth. The paper is clipped out after soaking. Similar
+to pre-treatments in Figure 2, the solution is vaporized and the residual is digested by acid. The
+results of the measurements are shown in Table 4. It seems that most of the glue is dissolved in
+water within one hour. The film put on inner surface of acrylic will be removed by high pressure
+water jet during the final cleaning. The paper film with water soluble glue on it is quite easy to be
+removed with water, so the contact time is much smaller than one hour.
+The mass of glue on the paper is 5 g/m2, and the total mass of glue for the whole acrylic
+surface is 20 kg. Assume all the U/Th in glue stay on acrylic surface and further go into LS,
+the contamination to 20 kt LS is 10−16 g/g, one order higher than our requirement of U/Th in LS
+(10−17 g/g). In reality, almost all the glue can be dissolved in water when contacting with enough
+– 8 –
+
+amount of water, and the radioactivity from glue can be absorbed in acrylic together with water.
+However, the flow rate of water is about 15 liters per minute, so the glue is highly diluted in water.
+Based on the study in Sec 4.1, the water absorbed in acrylic can reach 1 g/m2 after one hour soaking.
+In reality, the absorbed U/Th in acrylic from glue is quite small.
+Table 4. The water soluble glue on paper film is digested in water solution and measured by ICP-MS.
+Exposure of soaking
+1 hour
+5 hours
+238U [ppt]
+222±11
+268±12
+232Th [ppt]
+117±5
+81±5
+Besides the direct radiopurity measurement of the glues, we have also performed the test on
+acrylic surface. We pasted the two kinds of films on acrylic surface, and took the surface sample
+after removing the film. We did not see an obvious difference between the samples with films
+and the raw panel sample, so it is safe to use both PE and paper film for acrylic protection. The
+whole acrylic sphere with 35.4 m in diameter is divided into 265 panels produced in the company
+separately. Considering the toughness of the films, we prefer to use PE film in the company, which
+is better for protection of the panels during transportation and installation. All of these panels will
+be bonded layer to layer from top to bottom at JUNO site. After bonding the panels of one layer, we
+will clean the surface of the corresponding layer and cover the inner surface with paper film. When
+the whole sphere will be finished, we will finally clean the inner surface with high pressure water
+jet, and the paper film will be easily removed without manual operation inside the sphere.
+5
+Summary
+An acrylic vessel with 35.4 m in diameter is used as the 20 kt LS container for JUNO. The cleanliness
+of the inner acrylic surface is quite important to ensure an excellent LS radiopurity. To remove
+the contamination near the acrylic surface during production, many treatments will be done on the
+acrylic surface before shipment. From this study, we found the polishing mirror wax and Alconox
+solution have non-negligible U/Th residuals on the surface. The final recommended procedure for
+the surface treatments consists of sanding up to 3000 mesh number, polishing and cleaning with
+deionized water to achieve low background and high light transparency. Finally, a thin PE film will
+cover acrylic surface during transportation, and the film will be replaced by the adhesive masking
+paper for the inner surface after onsite bonding. This operation is quite important before finally
+removing the film by water jet, since the glue on the adhesive masking paper is water soluble. This
+procedure for acrylic surface treatments is also applicable to other low background experiments. In
+addition, a new method for pre-treatment of acrylic samples based on a microwave muffle furnace
+is shown in this paper, and the sensitivity of the measurement with ICP-MS can reach sub-ppt for
+U/Th in acrylic.
+Acknowledgments
+This work is supported by the Youth Innovation Promotion Association of the Chinese Academy
+of Sciences, the National Natural Science Foundation of China (No. 11905226), and the Strategic
+Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA10010200).
+– 9 –
+
+References
+[1] A. Abusleme et al. (JUNO), Prog. Part. Nucl. Phys. 123, 103927 (2022).
+[2] A. Abusleme et al. (JUNO), JHEP 11, 102 (2021), arXiv:2107.03669 [physics.ins-det] .
+[3] C. Cao, N. Li, X. Yang, J. Zhao, Y. Li, Z. Cai, L. Wen, X. Luo, Y. Heng, and Y. Ding, Nucl. Instrum.
+Meth. A 1004, 165377 (2021), arXiv:2011.06817 [physics.ins-det] .
+[4] S. Appel et al. (Borexino), (2022), arXiv:2205.15975 [hep-ex] .
+[5] P. Khaghani, Neck sense rope system and leaching studies for SNO+, Ph.D. thesis.
+[6] F. C. F. Perrot, C. Cerna and C. P´echeyran, “Advantages and sensitivity of UV fs laser ablation
+HR-ICPMS technique for rare event experiments, talk at Low Radioactivity Techniques 2019,” .
+[7] “https://www.alconox.com/product/alconox,” .
+[8] S.-L. Niu et al., Chin. Phys. C 39, 086002 (2015), arXiv:1410.4291 [physics.ins-det] .
+[9] Saint-Gobain, “https://www.saint-gobain.com.cn/abrasives/product/non-abrasive,” .
+[10] X. Wang, X. Chen, C. Fu, X. Ji, X. Liu, Y. Mao, H. Wang, S. Wang, P. Xie, and T. Zhang, Journal of
+Instrumentation 11, T12002 (2016).
+[11] “https://www.daio-paper.co.jp/en,” .
+[12] “https://cem.com/en/phoenix-black,” .
+[13] B. D. LaFerriere, T. C. Maiti, I. J. Arnquist, and E. W. Hoppe, Nucl. Instrum. Meth. A 775, 93 (2015).
+[14] Z. Li et al., Rad. Det. Tech. Meth. 5, 356 (2021).
+– 10 –
+
diff --git a/GNE4T4oBgHgl3EQfHQyx/content/tmp_files/load_file.txt b/GNE4T4oBgHgl3EQfHQyx/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,459 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf,len=458
+page_content='Prepared for submission to JINST  Study on U/Th residual radioactivity in acrylic from  surface treatment  Yuanxia Li,𝑎 Xiaohui Qian,𝑎 Xiaolan Luo,𝑎 Jie Zhao,𝑎,1 Gaofeng Zhang,𝑏 Xiaoyan Ma,𝑎  Yuekun Heng,𝑎 Liangjian Wen,𝑎 Monica Sisti,𝑐 Frédéric Perrot,𝑑 Hongqiang Tang𝑏  𝑎Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  𝑏Donchamp New Material Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Ltd, Taixing 225400, China  𝑐INFN Milano Bicocca and Università di Milano-Bicocca, Milano, Italy  𝑑Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Bordeaux, CNRS, CENBG, UMR 5797, F-33170 Gradignan, France  E-mail: zhaojie@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='cn  Abstract: Acrylic is widely used as material for the target container in low background experiments  due to its high light transparency and low intrinsic radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, its surface can be easily  contaminated during production, so careful treatment of the surface is essential to avoid direct  contamination of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The Jiangmen Underground Neutrino Observatory will use about 600 t  of acrylic to build the spherical vessel of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter for a 20 kt liquid scintillator (LS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Since acrylic will contact the LS directly, the cleanliness of the its surface is quite important for the  radiopurity of the LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' A new method for measuring the radioactivity of 238U and 232Th in acrylic to  sub-ppt (< 10−12 g/g) was developed, and it is crucial for the acrylic radioactivity screening in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We performed many background tests on different surface treatments, and the recommended  procedure for the treatment of acrylic to achieve low radioactivity and high light transparency could  be applicable to other low background experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='04902v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='ins-det]  12 Jan 2023  Contents  1  Introduction  1  2  Screening of surface treatment material by gamma spectrometer  2  3  Method for measuring U/Th in acrylic by ICP-MS  3  4  Study of the procedure for acrylic surface treatments  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  Solution absorption in acrylic  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  Contamination from different surface treatments  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  Protection film  8  5  Summary  9  1  Introduction  The Jiangmen Underground Neutrino Observatory (JUNO) is a multi-purpose neutrino experiment,  which will build the world’s largest liquid scintillator (LS) detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' An acrylic vessel is used as  the 20 kt LS container with 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter and 120 mm in thickness because of its high light  transparency and low intrinsic radioactivity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  To reach the physics target, the average radio-purity of the acrylic is required to be less than  1 ppt [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [3], the radio purity in the bulk has reached the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However,  the distribution of radioactivity along the thickness is not extremely uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The bulk measurement  of samples with several centimeters of thickness can not reflect the surface contamination, since  the thickness of external contamination on the acrylic surface usually reaches nanometre to micron  level, and the surface contamination will result averaged out in the bulk measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If the external  radioactive contamination stays on the surface, only gammas and few alphas/betas from radioactive  decays can be emitted into the LS and rejected by the effective fiducial volume cut by the off-line  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, based on the experiences at Borexino [4] and SNO+ [5], the U/Th daughters  near the inner surface of the vessel could leak into the LS by diffusion and convection during stable  running, thus making the acrylic vessel a stable contamination source for the LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Borexino put  great efforts on controlling the temperature distribution to avoid convection in LS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' however, it is  impossible to repeat the same thing at JUNO due to the large size of the experiment (70 times  Borexino target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Therefore, the radiopurity of the inner surface of acrylic is quite important for  JUNO, especially for the physic topics in the low energy regions, such as solar neutrino studies [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The JUNO acrylic vessel is produced with 265 pieces because of its large size, as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Each acrylic panel is firstly produced in a carefully cleaned flat mold and then shaped to  bend board at high temperature at the company;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' finally it is bonded directly at JUNO site [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  surface of the acrylic panel is exposed to air and covered with a high-temperature resistant cloth  – 1 –  during the bending, thus the surface is eventually contaminated and not as smooth as the flat panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Surface treatments are performed before shipment to the JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Typical protocols to remove  surface contaminations include the following steps:  The highest contaminations can reach tens of µm depths from the surface [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Thus, at least  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 µm thickness of the surface of acrylic is firstly removed by sanding with special paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The surface is not so smooth after sanding, so polishing is needed to improve the transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The polishing is done with a wool wheel and polishing fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The surface should be well cleaned with detergent and deionized water after polishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Finally, a thin film is used to cover the acrylic surface to avoid external contamination during  transportation and installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  All tools and detergents have risks to introduce more contamination on the surface, therefore  each surface treatment should be well studied to minimize those risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The acrylic vessel is produced with 265 pieces due to its very large size, as shown in the left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Each panel is produced separately, and all the panels will be finally bonded to a sphere at JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' One  JUNO panel is shown in the right figure as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The surface treatment material screening is described in  Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The new method for measuring 238U and 232Th (U/Th) in acrylic to sub-ppt level is described  in Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Various tests on radioactivity from each step of the above treatment protocol are discussed  in Sec 4, and the recommended procedure is also shown in the same section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The concluding  remarks are in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  2  Screening of surface treatment material by gamma spectrometer  The main tools and detergents used for surface treatments are sanding papers, polishing fluid,  Alconox powder [7], deionized water and protection film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We used a gamma spectrometer to screen  the radiopurity of the tools and detergents first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have one HPGe facility at ground level at  IHEP [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The gamma spectrometer plays an important role in the primary screening of the raw  – 2 –  0  0  0  0  0  Q  0  4  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='materials at JUNO, and the advantage is that samples can be screened directly without any pre-  treatment and with no damage to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The principle of gamma spectroscopy is to measure  the emitted gamma from the decay chain, and it is more sensitive to gammas with higher energy  due to lower background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' For example, the detector is more sensitive to the lower part of the  uranium chain starting from 226Ra, and the 238U concentration can be derived by assuming secular  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The measured results for the tools and detergent used for acrylic surface treatment are sum-  marized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We found that special mirror wax purchased from Saint-Gobain [9] has lower  radioactivity for 40K than one-step fast wax, due to the different chemical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The sand-  paper with a green base plate has lower radioactivity and better sand fastening than the one with a  black base plate for similar mesh numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Regarding the Alconox powder, the first measurement  performed at IHEP gave an upper limit of several tens to hundreds of ppb level for thorium and  uranium concentrations, respectively, due to the limited sensitivity of HPGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The same sample was  further measured by HPGe at China Jinping Underground Laboratory (CJPL) with better shielding  and sensitivity [10] down to 10 ppb for 232Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From the obtained results, there is a higher uranium  concentration than thorium in the Alconox powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The polishing fluid and sandpaper with lower  radioactivity are selected for the surface treatments on acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  After surface treatment, the acrylic surface will be protected with a thin film to avoid dust and  radon daughters deposition on the surface during the handling operations for transportation and  installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The film on the inner surface will be removed by the final cleaning with high pressure  water jet after installation, and there is no possibility for manual operation inside the acrylic vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  There are two types of protection film that can be used to protect the acrylic surface, ordinary  polyethylene (PE) film with few glue on the surface and adhesive masking paper film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The PE film  is better in toughness, but it is not easy to be removed by the water jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' On the contrary, the paper  film with few water soluble glue on the surface can be easily removed by water washing since the  glue is soluble in water, but overall its mechanical strength is worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The radioactivity screening of  the glue that is used on the PE film showed only upper limits for U/Th at the level of hundreds of  ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Concerning the adhesive masking paper film (purchased from [11]), we could only measure  the paper film and the glue together inside the HPGe detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The water soluble glue itself was  further measured by ICP-MS, as it will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  3  Method for measuring U/Th in acrylic by ICP-MS  Different from gamma spectroscopy, the principle of the Inductively Coupled Plasma Mass Spec-  troscopy (ICP-MS) is to detect the nuclei according to their exact atomic weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' ICP-MS can  measure the isotopes 238U and 232Th directly without other information on the chain, and the sen-  sitivity can reach the sub-ppt level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, the sample measured by ICP-MS must be in liquid  form, so a careful and optimized pre-treatment on the sample is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  For the radio purity measurement on acrylic surface samples, we have developed a new method  based on microwave ashing and ICP-MS equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The design for the pre-treatment flow is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Both the acrylic sample and the tracers (the natural-non-existing nuclei 229Th and 233U)  are ashed by the machine in a quartz vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The residual is collected by soaking with 35% HNO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  – 3 –  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Material screening by gamma spectroscopy on tools and detergents used for the acrylic surface  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If not differently mentioned, the sample was measured at IHEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The conversions between Bq/kg  and mass concentration units for U/Th are given: 1 Bq/kg of 232Th activity in a material is equal to  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5 × 10−7 g/g (or 250 ppb) of 232Th mass concentration in that material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' similarly, 1 Bq/kg of 238U means  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 × 10−8 g/g (or 81 ppb) of 238U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  238U chain [Bq/kg]  232Th chain [Bq/kg]  40K  214Bi/214Pb  226Ra  212Bi/208Tl  [Bq/kg]  Polishing  Special mirror wax  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9  One step fast wax  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  Alconox powder <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='37  <4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  Measured at CJPL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='02  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0  Sand paper  Black base-plate  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  <21  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  <3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7  Green base-plate  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7  <10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  Protection film  Glue on PE film  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  Water soluble film  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='47  <10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='19  <1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='56  The eluent is collected and heated to remove the excess acid, and the solution is finally diluted for  the ICP-MS measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The flow chart for the pre-treamtent of acrylic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The ashing of the acrylic sample is done  with the microwave muffle furnace described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [3], a different approach was used: the acrylic was vaporized and burned in a two-stage  furnace, while clean gas (mixed N2/O2) was supplied to the furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Monitoring of the pressure  in the whole system is essential in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Compared with the pre-treatment in [3], with the new  technique there is no need for gas input to the microwave muffle furnace, and the overall operation  is easier and safer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All the pre-treatment process is done in a class 10,000 tent, while the ICP-MS  measurement is performed in a class 1,000 room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The cleanliness of the quartz vessels is quite  important for the sensitivity of the measurement, and their cleaning procedure is similar to that  described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All the containers are soaked in the two-stage acid cylinders filled with  6 mol/L HNO3 for at least one day at each stage to remove the U/Th on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In the end,  the containers are filled with 6 mol/L HNO3 and boiled for 10 min, then they are rinsed with fresh  water and ready for use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The microwave muffle furnace (Phoenix BLACK [12]) is put in the clean tent and used for  ashing the acrylic sample, as shown on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The temperature for ashing is optimized  to achieve no visible organic residual in the vessel, and the optimized temperature as a function of  time is shown on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The quartz vessel is used as the sample container and well  cleaned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The 229Th and 233U standards (preserved in 2% acid) are added together with the acrylic  – 4 –  Tracer and  Ashing in  Soak the vessel  acrylic  quartz vessel  with 35% HNO3  Diluted for ICP-MS  Eluent was collected and  measurement  heated to remove excess acid0  20  40  60  80  Time [min]  0  200  400  600  C]  o  Temperature [  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The microwave muffle furnace (Phoenix BLACK) used for the pre-treatment of acrylic is shown  in the left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The optimized temperature as a function of time for acrylic ashing is shown in the right  figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  sample to the vessel in the beginning, and the recovery efficiency can be evaluated by measuring the  229Th/233U in the solution sent to ICP-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The average recovery efficiency of U/Th is calculated as  (95±13)% based on more than one hundreds measurements (the given uncertainty is the standard  deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The recovery efficiency is calibrated every time a sample is measured, and the single  precision is better than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To estimate the background for the pre-treatment, we have followed  the same procedure without the acrylic sample (blank test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' With careful background control, the  blank test showed that the absolute background for the pre-treatment amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='07 pg for  238U and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='04 pg for 232Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  4  Study of the procedure for acrylic surface treatments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  Solution absorption in acrylic  The acrylic panel is cleaned after surface treatments at the company before shipment, and the whole  acrylic sphere is finally cleaned onsite after installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Acrylic can absorb water, thus traces of  radioactivity in cleaning water may diffuse into acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The absorption of water in acrylic obeys  the mathematical laws of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have performed several water absorption tests on flat acrylic  samples with 2 mm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The mass proportion of absorbed water in acrylic as a function of  time is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The absorption of water in acrylic did not reach equilibrium in one month,  and the mass of absorbed water in acrylic increased almost linearly at the beginning, which is far  away from the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Therefore, there is no impact of the acrylic panel thickness when  exposure to water is shortened to several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Assuming we will clean the acrylic surface during  one hour at the company, the absorbed water in acrylic can reach about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='08% for the sample with  2 mm thickness, which is 1 g/m2 water absorption for one side of acrylic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  For the cleaning procedure, we use Alconox solution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% of water) for degreasing and  deionized water for rinsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Traces of radioactivity in both water and Alconox can diffuse into  acrylic during cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We use deionized water with 10−14 g/g U/Th to perform water cleaning, so  the residual traces of radioactivity from absorbed water is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Even though the Alconox is  not as clean as the deionized water, most of Alconox can be removed by water rinsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To validate  how much radioactivity from Alconox can go into acrylic, we soaked the acrylic sample with 2 mm  – 5 –  Pho  QulckTest  方法  CM  OYFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Results of water absorption tests performed on flat acrylic samples with 2 mm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  mass proportion of absorbed water to acrylic as a function of time is shown during 35 days, and the data  points in the first day are magnified in the sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  thickness in Alconox solution with different concentrations and time exposure, and the radiopurity  of the sample for U/Th is measured by ICP-MS in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have soaked the samples in Alconox  solution for two months in order to enlarge the possible radioactivity contamination in the bulk of  acrylic (2 mm thickness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The results of the tests for U/Th surface contamination are shown in  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The acrylic samples with 2 mm thickness are soaked in different solutions with different time  exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The U/Th contamination from solution absorption in acrylic after absorption is measured by  ICP-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Sample preparation  Exposure  Mass[g]  238U[ppt]  232Th[ppt]  1  No soaking 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% Alconox solution  1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% Alconox solution  2 months  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  4  2% Alconox solution  2 months  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='74  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1% Alconox solution + 30% HNO3  2 months + 1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  6  Tap water after filter [739 ppt 238U, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='02 ppt 232Th]  1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  7  1% HNO3 [106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7 ppt 238U, 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5 ppt 232Th]  1 day  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1-5: we can see a clear increase on 238U in acrylic for Alconox solutions with different  concentration and exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, the increase of 232Th is not obvious due to the lower  232Th radioactivity in Alconox powder as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Even though the Alconox can  be removed by further deionized water rinsing, the radioactivity from Alconox can diffuse  into acrylic together with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If the residual radioactivity stick on acrylic from Alconox  solution is due to the active agent, the residual can not be easily removed by cleaning or  acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' On the contrary, if the mechanism of residual consists in ionic diffusion from Alconox  solution to acrylic, such residual can be removed by acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We observed that the radioactivity  absorbed in acrylic can be further removed by acid soaking, as shown in result No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6-7: To further validate that the diffusion of radioactivity from Alconox to acrylic is  – 6 –  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8  [%]  Water absorbtion [  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4  Q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  10  20  Time [h]  0  5  10  15  20  25  30  35  Time [day]in ionic form, we have soaked another two acrylic samples in two kinds of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' One is  tap water after 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2 µm filter, and radioactivity exists in small particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The other one is  U/Th standard solution with 1% HNO3, and radioactivity is in ionic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' As shown in  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7, the 238U concentration in acrylic reached equilibrium at 3 ppt, while 232Th  can reach much higher to about 30 ppt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The reason for the difference between 238U and  232Th is their different intrinsic physicochemical properties, which is also discussed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Thorium is more reactive than uranium, thus it is easier for thorium to stick to acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' A  similar phenomenon is observed in preparing tap water, most of the thorium is filtered and  less residual is found in the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From the results in No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='6 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7, radioactivity can also  diffuse into acrylic without the active agent, so the radioactivity in Alconox solution can  diffuse into acrylic due to the diffusion of ion, not the active agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The final residual on  acrylic from the cleaning solution relies on the measurement of the surface sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2  Contamination from different surface treatments  To study the surface contamination, we have treated the surface with different procedure and directly  measured the residual of radioactivity on surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The surface samples were taken by scraping the  surface with well cleaned erasing knife, and the thickness of the surface sample can reach 5-10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The radioactivity of the raw surface for one JUNO acrylic panel without any surface treatment is  measured to be (52±1) ppt 238U and (133±5) ppt 232Th, which is 1-2 orders of magnitude higher than  the bulk measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The radioactivity distribution along the depth was measured in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' [6], and  has proven that U/Th contaminations can extend down to tens of µm from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In addition,  the radon daughters (210Pb, 210Po) deposited on the acrylic surface is usually non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' So we  decide to remove at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 mm depth of acrylic and to maintain a high light transparency (>96%  at 420 nm [14] in ultrapure water environment) at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  A dedicated experiment was performed on a flat acrylic panel (not a JUNO panel) with a  1 m2 area by applying different surface treatments including sanding, polishing, and cleaning  successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All of these treatments are done in a 10,000 class tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The sanding of acrylic surfaces  starts from 400 mesh (to remove at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1 mm depth of acrylic) and is pursued by using sand  papers with larger mesh, 800, 1200, 2000, and 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The higher the mesh number, the smoother  the acrylic surface, and higher light transparency can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' However, more steps increase  the risk of contaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We have measured the surface sample with different mesh numbers of  1200, 2000, and 3000, and the results for residual U/Th concentration are consistent within 20%  among these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Polishing of acrylic is performed by a wool wheel with polishing fluid, and the typical fluid is  purchased from Saint-Gobain [9], whose components are organic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' If there is polishing with mirror  wax after sanding, sanding to 1200 mesh number is enough to reach the required light transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Part of the polishing fluid can be further removed by degreasing with Alconox solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To quantify  the contamination from residual polishing fluid and Alconox solution, we have done many tests on  the acrylic panel with sanding to 1200 mesh number, and the results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  water cleaning is realized by a water jet with high pressure for 20 times, and the Alconox cleaning is  done by spraying the Alconox on surface and wiping with a clean cloth for several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Compared  to the sample No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1, we have done additional cleaning with Alconox solution on No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2, and there is  obvious radioactive residual on the surface from Alconox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' For the samples No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 3 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 4, we  – 7 –  have done polishing with the fluid for both samples, and performed cleaning with Alconox only on  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From the results, it is quite clear that a large amount of polishing fluid residual exists on the  acrylic surface after cleaning even with Alconox solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Many cleaning tests have been done on the non-JUNO acrylic panel after sanding to the 1200 mesh  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The water cleaning is realized by a water jet with high pressure 20 times, and the Alconox cleaning  is done by wiping the surface with a clean cloth several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The polishing is done with mirror wax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' After  the treatments, surface samples are taken by scraping the surface with well cleaned erasing knife, and the  thickness of the surface sample can reach 5-10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The radioactivity in the surface samples is summarized  in this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Sample preparation  Mass[g]  238U[ppt]  232Th[ppt]  1  Water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='25  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='5  2  Alconox + water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='34  27±1  22±2  3  Polishing + water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='28  92±5  893±27  4  Polishing + Alconox + water cleaning  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='35  123±3  817±33  Based on the above measurements, polishing with mirror wax and Alconox cleaning have non-  negligible residual on the acrylic surface, and we should avoid using them for surface treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  To reach light transparency greater than 96% at 420 nm in ultrapure water environment, we have  performed sanding with 3000 mesh number, polishing with deionized water and, in the end, cleaning  of the surface with a deionized water high pressure jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We finally measured the surface of one JUNO  acrylic panel following this procedure of surface treatments, with good results of (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7) ppt  238U and (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='7) ppt 232Th in about 10 µm depth, which is several times lower than the raw  surface as shown in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In addition, all the treatments of the inner surface  of the acrylic panel were done in one day to avoid radon daughters deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='3  Protection film  After surface treatments with sanding, polishing, and cleaning, the acrylic surface will be covered  by a thin film to protect the acrylic from fallouts of radon and dust in the air during transportation  and installation, which will last several months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' There are two kinds of protection films that can be  used to protect the acrylic surface, as discussed in Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Since the glue on the adhesive masking paper is soluble in water, we soaked the paper film  in water for different exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' By this way, we can measure the radiopurity of glue by ICP-MS  with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To reduce the effect of surface cleanliness in this measurement, we swab the  surface of paper side with a little wet clean cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The paper is clipped out after soaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Similar  to pre-treatments in Figure 2, the solution is vaporized and the residual is digested by acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  results of the measurements are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' It seems that most of the glue is dissolved in  water within one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The film put on inner surface of acrylic will be removed by high pressure  water jet during the final cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The paper film with water soluble glue on it is quite easy to be  removed with water, so the contact time is much smaller than one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  The mass of glue on the paper is 5 g/m2, and the total mass of glue for the whole acrylic  surface is 20 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Assume all the U/Th in glue stay on acrylic surface and further go into LS,  the contamination to 20 kt LS is 10−16 g/g, one order higher than our requirement of U/Th in LS  (10−17 g/g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In reality, almost all the glue can be dissolved in water when contacting with enough  – 8 –  amount of water, and the radioactivity from glue can be absorbed in acrylic together with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  However, the flow rate of water is about 15 liters per minute, so the glue is highly diluted in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Based on the study in Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='1, the water absorbed in acrylic can reach 1 g/m2 after one hour soaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  In reality, the absorbed U/Th in acrylic from glue is quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The water soluble glue on paper film is digested in water solution and measured by ICP-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Exposure of soaking  1 hour  5 hours  238U [ppt]  222±11  268±12  232Th [ppt]  117±5  81±5  Besides the direct radiopurity measurement of the glues, we have also performed the test on  acrylic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We pasted the two kinds of films on acrylic surface, and took the surface sample  after removing the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' We did not see an obvious difference between the samples with films  and the raw panel sample, so it is safe to use both PE and paper film for acrylic protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The  whole acrylic sphere with 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter is divided into 265 panels produced in the company  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Considering the toughness of the films, we prefer to use PE film in the company, which  is better for protection of the panels during transportation and installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' All of these panels will  be bonded layer to layer from top to bottom at JUNO site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' After bonding the panels of one layer, we  will clean the surface of the corresponding layer and cover the inner surface with paper film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' When  the whole sphere will be finished, we will finally clean the inner surface with high pressure water  jet, and the paper film will be easily removed without manual operation inside the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  5  Summary  An acrylic vessel with 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='4 m in diameter is used as the 20 kt LS container for JUNO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The cleanliness  of the inner acrylic surface is quite important to ensure an excellent LS radiopurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' To remove  the contamination near the acrylic surface during production, many treatments will be done on the  acrylic surface before shipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' From this study, we found the polishing mirror wax and Alconox  solution have non-negligible U/Th residuals on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' The final recommended procedure for  the surface treatments consists of sanding up to 3000 mesh number, polishing and cleaning with  deionized water to achieve low background and high light transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Finally, a thin PE film will  cover acrylic surface during transportation, and the film will be replaced by the adhesive masking  paper for the inner surface after onsite bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' This operation is quite important before finally  removing the film by water jet, since the glue on the adhesive masking paper is water soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' This  procedure for acrylic surface treatments is also applicable to other low background experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' In  addition, a new method for pre-treatment of acrylic samples based on a microwave muffle furnace  is shown in this paper, and the sensitivity of the measurement with ICP-MS can reach sub-ppt for  U/Th in acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  Acknowledgments  This work is supported by the Youth Innovation Promotion Association of the Chinese Academy  of Sciences, the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 11905226), and the Strategic  Priority Research Program of the Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' XDA10010200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  – 9 –  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Abusleme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' (JUNO), Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' 123, 103927 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Abusleme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' (JUNO), JHEP 11, 102 (2021), arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='03669 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='ins-det] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content='  [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Cao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
+page_content=' Cai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE4T4oBgHgl3EQfHQyx/content/2301.04902v1.pdf'}
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diff --git a/GdE1T4oBgHgl3EQfXQRO/content/tmp_files/2301.03124v1.pdf.txt b/GdE1T4oBgHgl3EQfXQRO/content/tmp_files/2301.03124v1.pdf.txt
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+ 
+ 
+1 
+The COVID-19 vaccination, preventive behaviors and pro-social motivation:  
+panel data analysis from Japan 
+ 
+Short title: COVID-19 vaccination and preventive behaviors 
+ 
+Eiji Yamamura1*, Yoshiro Tsutsui2, Fumio Ohtake3 
+ 
+ 
+1 Department of Economics, Seinan Gakuin University, Fukuoka, Japan 
+ 
+*Corresponding author 
+Email: yamaei@seinan-gu.ac.jp (EY) 
+Full list of author information is available at the end of the article 
+ 
+Abstract  
+Background 
+The COVID-19 vaccine reduces infection risk: even if one contracts COVID-19, the 
+probability of complications like death or hospitalization is lower. However, vaccination may 
+prompt people to decrease preventive behaviors, such as staying indoors, handwashing, and 
+wearing a mask. Thereby, if vaccinated people pursue only their self-interest, the vaccine’s 
+effect may be lower than expected. However, if vaccinated people are pro-social (motivated 
+toward benefit for the whole society), they might maintain preventive behaviors to reduce 
+the spread of infection. 
+ 
+Methods 
+We conducted 26 surveys nearly once a month from March 2020 (the early stage of COVID-
+19) to September 2022 in Japan. By corresponding with the identical individuals, we 
+independently constructed original panel data (N = 70,979). Based on the data, we identified 
+the timing of the second vaccine shot and compared preventive behaviors before and after 
+
+ 
+ 
+2 
+vaccination. We investigated whether second-shot vaccination correlated with changes in 
+preventive behaviors. Furthermore, we explored whether the vaccination effect differs 
+between senior and younger groups. We then investigated the effect of pro-social motivation 
+on preventive behaviors. 
+ 
+Results 
+Major findings are as follows: (1) Being vaccinated led people to increase preventive 
+behaviors, such as mask-wearing by 1.04 (95% confidence intervals [Cis]: 0.96–1.11) points, 
+and handwashing by 0.34 (95% CIs: 0.30–0.38) points, on a 5-point scale. (2) vaccinated 
+people under the age of 65 years are less likely to stay indoors. (3) people with pro-social 
+motivation to be vaccinated are more likely to maintain prevention than those not so 
+motivated; the difference is 0.08 (95% CIs: 0.01–0.15) points for mask-wearing and 0.05 
+(95% CIs: 0.001–0.10) points for handwashing, on a 5-point scale. 
+ 
+Conclusion 
+After vaccination, the opportunity cost of staying indoors outweighs its benefits and people 
+are less inclined to stay home. This effect is lower for older people, who are at higher risk of 
+serious illness. The opportunity cost of mask-wearing and handwashing is lower than that of 
+staying indoors, and the benefit persists after vaccination if people have the motivation to 
+maintain these behaviors for others’ well-being.  
+Keywords: Vaccine, COVID-19, preventive behaviors, staying indoors, wearing mask, 
+washing hands, public goods, pro-social, Japan, panel data 
+ 
+
+ 
+ 
+3 
+ 
+Introduction 
+ 
+During COVID-19 pandemic, especially in the early stages, various preventive 
+behaviors were required because vaccines against COVID-19 had not been developed. 
+Preventive behaviors can be considered a kind of public good that is not sufficiently supplied 
+through market mechanisms where people pursue self-interest [1,2]. Mitigating the pandemic 
+necessitated the collective actions of citizens. However, according to the Peltzman effect, 
+people tend to increase their risky behaviors if safety measures are implemented [3].  
+Since 2021, vaccines against COVID-19 have been distributed worldwide and have 
+played a vital role in coping with COVID-19. Newly reported cases have decreased in 
+countries where vaccines have been rapidly adopted [4]. People, if they are rational, tend to 
+engage in risky behaviors when security measures are mandated [5]. In terms of economics, 
+this is considered a moral hazard. An empirical question arises as to how the spread of the 
+vaccine influences preventive behaviors [6,7]. As a result of the reduction in the risk of 
+COVID-19 infection, risk-taking behaviors increase, so that preventive behaviors such as 
+staying indoors, wearing masks, and washing hands have changed [8,9]. However, some 
+studies show no clear evidence that vaccinated people have decreased their preventive 
+behaviors in comparison with those not vaccinated [6,10].   
+The influence of vaccination on preventive behaviors may vary according to the type of 
+behavior [11]. A study found that in China, vaccination lessened the frequency of 
+handwashing but did not change mask-wearing [7]. This study aimed to explore the 
+mechanisms stopping vaccinated people from decreasing preventive behaviors. To this end, 
+
+ 
+ 
+4 
+we investigated how preventive measures can be pro-socially motivated based on altruism 
+and social solidarity [12]. 
+Using monthly individual-level panel data, we investigated whether individuals’ 
+preventive behaviors changed after vaccination. Furthermore, we looked at how the influence 
+of vaccination on preventive behaviors differs according to age and pro-social motivation.  
+ 
+Methods 
+Data collection 
+We commissioned the research company INTAGE, Inc., to conduct an online survey 
+because of their experience and reliability in academic research. The first wave of queries 
+was conducted March 13–16, 2020, and 4,359 observations were recorded. Participants 
+registered with INTAGE were recruited for this study. The participation rate was 54.7%. The 
+sampling method was designed to collect representatives of the Japanese adult population in 
+terms of educational background, gender, and residential area. For this purpose, INTAGE 
+recruited participants for a survey of pre-registered individuals. However, individuals aged 
+15 years and below were too young to be registered with INTAGE, and we considered 
+individuals over 80 years of age too old to answer pertinent questions. Inevitably, the sample 
+population was restricted to ages 16–79 years, and participants were randomly selected to fill 
+the pre-specified quotas. INTAGE provided monetary incentives to participants upon study 
+completion. 
+Internet surveys were conducted repeatedly 26 separate times (“waves”) almost every 
+month with identical individuals to construct the panel data. The exceptional period was July-
+
+ 
+ 
+5 
+September 2020, when the survey could not be conducted owing to a shortage of research 
+funds. We resumed the surveys after receiving additional funds in October 2020. Vaccination 
+was implemented in April 2021; therefore, the data cover the period before and after 
+implementation of vaccination. 
+Respondents from the first wave were targeted in subsequent waves to record how some 
+respondents changed their behaviors during the COVID-19 pandemic. In particular, the data 
+allowed us to compare identical persons’ preventive behaviors against COVID-19 before and 
+after being vaccinated. During the study period, some identical respondents were dropped 
+from the study sample because some of them stopped taking the surveys, while others did 
+not take the surveys at all. Furthermore, the sample was restricted to those who were 
+completely vaccinated by getting the second shot. In this way, we could compare their 
+behaviors before and after vaccination. Eventually, the number of identical individuals was 
+reduced from 4,359 to 3,019, and the total number of observations used in this study was 
+70,979. 
+ 
+ 
+Methods 
+Table 1 presents the key variables used in the estimation. The survey questionnaire 
+contained basic questions about demographics, such as birth year, gender, and educational 
+background. These characteristics were observed at different time points. The surveys were 
+conducted 26 times, between March 2020 and September 2022. During the study period, 
+conditions such as the spread of infection and policies against COVID-19 changed drastically. 
+Table 1 describes the key variables used in the regression estimations. As outcome variables, 
+
+ 
+ 
+6 
+the respondents were asked questions concerning preventive behaviors, such as:  
+“Within a week, to what degree have you practiced the following behaviors? Please 
+answer based on a scale of 1 (I have not practiced this behavior at all) to 5 (I have completely 
+practiced this behavior).” 
+(1) Staying indoors 
+(2) Wearing a mask 
+(3) Washing my hands thoroughly 
+The answers to these questions served as proxies for the following variables for 
+preventive behaviors: staying indoors, frequency of handwashing, and degree of wearing 
+masks. Larger values indicate that respondents are more likely to engage in preventive 
+behaviors. Further, the motivation to get a shot of COVID-19 vaccination is asked in the 
+following question: “Do you get the shot in order to decrease the spread of COVID-19 
+infection?”  
+We also asked about the subjective probability of contracting COVID-19 and their 
+perceptions of the severity of COVID-19. We asked whether they received the second shot 
+of the vaccine because the vaccination was effective only after completing the second shot. 
+The latter question was included in the questionnaire from the 12th wave conducted in May 
+2021 directly after the vaccine was introduced in Japan. The question was included until the 
+18th wave in November 2021, when most people in the sample completed the second shot. 
+The question was then excluded from the questionnaire, starting with the 19th wave in 
+January 2022. 
+ 
+
+ 
+ 
+7 
+ 
+Table 1. Definitions of key variables. 
+Variable 
+ 
+Definition 
+ 
+    Outcome variables 
+STAYING INDOORS 
+In the last week, how consistent were you at “not going out of home?” 
+Please choose among 5 choices. 
+1 (not completed at all) to 5 (completely consistent). 
+WEARING MASK 
+ 
+ 
+In the last week, how consistent were you at “wearing a mask?” Please 
+choose among 5 choices. 
+1 (not completed at all) to 5 (completely achieved). 
+HANDWASHING  
+ 
+In the last week, how consistent were you at “washing your hands?” Please 
+choose among 5 choices. 
+1 (not completed at all) to 5 (completely achieved). 
+ 
+Confounders (Independent variables) 
+VACCINE  
+ 
+Did you get the second shot? 
+1 (Yes) or 0 (No) 
+PROB_COVID19 
+ 
+What percentage do you think is the probability of your contracting the 
+novel coronavirus (COVID-19)?  
+0 to 100 (%) 
+SEVERITY COVID19 How serious do you expect your symptoms to be if you are infected with 
+the novel coronavirus? Choose from 6 choices. 
+1 (very small influence) to 6 (death) 
+AGE_25 
+Answer 1 if respondents are age 19–25, otherwise 0 
+ 
+AGE_26_64 
+Answer 1 if respondents are age 16–64, otherwise 0 
+ 
+PRO_SOCIAL 
+In deciding whether you get the shot of COVID-19 vaccine, is it important 
+in preventing the spread of COVID-19?  
+1 (strongly disagree) to 5 (strongly agree). 
+PRO-SOCIAL is 1 if respondent chooses 5, otherwise 0. 
+ 
+We pursued identical respondents from the first-wave survey to the 26th wave for 30 
+months even though some of the respondents quit the survey. The purpose of this study was 
+to explore how the preventive behaviors of identical persons changed before and after 
+vaccination. Therefore, we limited the sample to those who completed the second shot by the 
+18th wave and then pursued identical persons until the 26th wave in September 2022. We 
+used panel data containing 3,019 individuals, covering 26 time points from March 2021 to 
+September 2022.  
+Based on the panel data, we used a fixed-effects (FE) model regression. The FE model is 
+
+ 
+ 
+8 
+a type of linear regression model widely used in economics. The estimation result using an 
+FE model is equivalent to the results of a linear regression model with dummies for 
+individuals frequently included in each period [13-15]. In this study, 3,019 dummies were 
+included to control for individuals’ characteristics that do not change during the period, such 
+as gender, educational background, and childhood experience. Hence, 3,019 confounders 
+were included, reflecting the differences between individuals. Therefore, the estimated 
+results for the time-invariant confounders could not be obtained. Even if various time-variant 
+confounders are included, unobserved individual characteristics cannot be captured. This 
+inevitably results in omitted variable biases [13,15]. For instance, an increasing trend in the 
+number of newly infected persons was observed throughout Japan. This effect is common to 
+all residents in Japan and has changed over time. This can be regarded as a time-fixed effect 
+and can be controlled by including time-period dummies. In this study, 25 time-period 
+dummies were included when one base period was fixed. However, some variables changed 
+not only over time, but also between individuals. Examples are proxy variables for preventive 
+behaviors, which are outcome variables; or number of newly infected persons and deaths due 
+to COVID-19 in residential areas. Furthermore, the timing of obtaining the second vaccine 
+shot is a variable that changed over time and between individuals; therefore, the dummy for 
+vaccination is included in the estimated function as a confounder. 
+As explained, we controlled not only for unobservable individual fixed effects, but also 
+for unobservable time-fixed effects. This type of FE model is called the two-way error 
+component regression model [15]. This study focused on the correlation between vaccination 
+and preventive behaviors. The statistical software used in this study was the Stata/MP 15.0 
+
+ 
+ 
+9 
+multiprocessor from StataCorp, LLC.  
+The estimated function of an FE model takes the following form:  
+Yit = α1 VACCINEit + α6 PROB COVID19it + α7 SEVERITY COVID19it +  
+α8 EMERGENCYit + kt + mi + uit 
+ 
+where Table 1 presents the definitions and basic statistics of these variables. In this formula, 
+Yit represents the outcome variable for individual i and wave t, respectively. The time-
+invariant individual-level fixed effects are represented by mi. Furthermore, kt represents the 
+effects of different time points, controlled by 25 wave dummies, where the first wave is the 
+reference group. kt captures various shocks that occurred simultaneously throughout Japan at 
+each time point. Y includes preventive behaviors captured by three proxy variables: STAYING 
+INDOORS, HANDWASHING, and WEARING MASK. These outcome variables are discrete 
+ordered variables ranging from 1 to 5. Larger values of these variables can be interpreted as 
+indicating that the respondents are more likely to exhibit preventive behavior. In the same 
+specification, we conduct three separate estimations, and the regression parameters are 
+denoted as α. The error term is denoted by uit. A simple FE linear regression model is used in 
+this study.  
+The key confounder is the vaccination dummy; VACCINE is 1 if respondents have 
+completed the second shot of the COVID-19 vaccine, otherwise 0. People are obliged to get 
+the second shot within a month of the first shot to make the vaccine effective. Hence, in the 
+sample, there is hardly any time lag between the first and second shots because the survey 
+was conducted every month after the vaccine was approved. There are two age groups: young 
+age (AGE_25) and middle working age (AGE_26_64). The senior group is used as the 
+
+ 
+ 
+10 
+reference group. PRO_SOCIAL is a dummy variable that captures pro-social motivation to 
+get vaccinated. The mean value of PRO_SOCIAL is 0.86, which shows that 86% of people 
+have the pro-social motivation. 
+ 
+Results 
+Baseline estimations  
+The coefficient of confounders indicates marginal effects (ME). Table 2 presents the 
+estimation results for the baseline FE model. VACCINE shows a positive sign and is 
+statistically significant at the 1% level, with the exception of Column (1), where STAYING 
+INDOORS is the outcome variable. The effects of VACCINE are ME 1.036 (95% CI: 0.960-
+1.111) and ME 0.339 (95% CI: 0.300–0.378) in columns (2) and (3), respectively. Thus, 
+people after vaccination are more likely than before to wear masks by 1.036 points and to 
+wash their hands by 0.339 points, respectively, on a 5-point scale. Before vaccination, the 
+mean values of WEARING MASK and HANDWASHING are 4.39 and 4.15, respectively. 
+Hence, this indicates that the degree of wearing masks increases by 23.6% and washing hands 
+by 8.16% after vaccination, compared with before vaccination. People’s behaviors depend 
+on behaviors of others; hence; they follow the social norm [16-19]. Peer pressure is stronger 
+for wearing masks than for washing hands because the surrounding people in a public place 
+can more easily see whether one wears a mask than whether one washes one’s hands.  
+  SEVERITY COVID19 shows a significant positive value in all columns, and this means 
+that persons’ perception about seriousness of COVID-19 changes the incidence of preventive 
+behaviors. 
+
+ 
+ 
+11 
+ 
+Table 2. FE model. Dependent variables are preventive behaviors. 
+ 
+  (1) 
+STAYING 
+INDOORS 
+    (2) 
+WEARING 
+MASK 
+   (3) 
+HANDWASHING  
+VACCINE  
+−0.011 
+(−0.077-0.053) 
+1.036*** 
+(0.960-1.111) 
+0.339*** 
+(0.300-0.378) 
+PROBABILITY COVID19 
+  0.043 
+(−0.751-0.873) 
+0.487 
+(−0.094-1.070) 
+0.560 
+(0.254-0.867) 
+SEVERITY COVID19 
+ 0.026*** 
+(0.013-0.039) 
+ 0.029*** 
+(0.017-0.040) 
+ 0.019*** 
+(0.009-0.030) 
+Individual Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Time Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Controls: New deaths, 
+ Newly affected persons, 
+Household income 
+  Yes 
+Yes 
+   Yes 
+Adjusted R2 
+Observations 
+0.55 
+70,979 
+0.45 
+70,979 
+0.62 
+70,979 
+ 
+Note: Numbers within parentheses are 95% CI. For convenience, the coefficient of probability of COVID-
+19 is multiplied by 1000. The model includes the number of deaths and infected persons in the residential 
+prefectures in the surveys. However, these results have not been reported. “Yes” means that variables are 
+included. 
+*** ρ < .01 
+ 
+Estimations with cross-terms 
+The probability and seriousness of contracting COVID-19 differ by age. COVID-19 is 
+more likely to be lethal for adults aged 65 years and older than for younger people [20,21]. 
+Mask-wearing by elderly people is motivated by their self-regarding risk preferences, 
+whereas younger people are motivated by other-regarding concerns [22]. We explore how the 
+effect of COVID-19 vaccination on preventive behaviors differs between age groups. For 
+this purpose, the interaction terms between VACCINE and age groups (AGE_25 and 
+AGE_26_64) are included as key confounders. The reference age group is those over age 65. 
+The results are presented in Table 3. We find a significant negative sign in 
+
+ 
+ 
+12 
+VACCINE×AGE_25 and VACCINE×AGE_26_64 in column (1), where STAYING 
+INDOORS is the outcome variable. The effects of VACCINE×AGE_25 and 
+VACCINE×AGE_26_64 are ME −0.537 (95% CI: −0.658 - −0.416) and ME −0.295 (95% 
+CI: −0.353 - −0.238). This means that those aged below 25 are less likely to stay home by 
+0.537 points than those aged over 65, while those aged between 26 and 64 are less likely to 
+stay home by 0.297 points than those aged over 65. However, no differences in the effect of 
+the vaccination are observed when wearing masks and washing hands.  
+ 
+Table 3. FE model with cross-terms with age cohorts. Dependent variables are 
+preventive behaviors. 
+ 
+  (1) 
+STAYING 
+INDOORS 
+    (2) 
+WEARING 
+MASK 
+   (3) 
+HANDWASHING  
+VACCINE  
+0.206** 
+ 
+(0.142-0.269) 
+1.058** 
+ 
+(0.980-1.137) 
+0.319** 
+ 
+(0.281-0.356) 
+VACCINE×AGE_25 
+  −0.537*** 
+(−0.658 - −0.416) 
+−0.066 
+(−0.160-0.028) 
+−0.056 
+(−0.138-0.024) 
+VACCINE×AGE_26_64 
+−0.295*** 
+(−0.353 - −0.238) 
+ 0.030 
+(−0.077-0.015) 
+ 0.036*** 
+(0.004-0.068) 
+Individual Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Time Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Controls: New deaths, 
+ Newly affected persons, 
+Household income 
+  Yes 
+Yes 
+   Yes 
+PROBABILITY COVID19, 
+SEVERITY COVID19  
+  Yes 
+Yes 
+   Yes 
+Adjusted R2 
+Observations 
+ 
+0.57 
+70,979 
+0.47 
+70,979 
+0.64 
+70,979 
+ 
+Note: Numbers within parentheses are 95% CI. All the models include control variables, which are 
+equivalent to those in Table 2. “Yes” means that variables are included. However, these results have not 
+been reported.  
+** ρ < .01 
+*** ρ < .05 
+
+ 
+ 
+13 
+ 
+We investigated how pro-social motivation affects preventive behaviors. The interaction 
+term between VACCINE and PRO_SOCIAL was included as a key confounder. Table 4 
+shows the significant positive sign of VACCINE×PRO_SOCIAL, where WEARING MASK 
+and HANDWASHING are the outcome variables. The effects of VACCINE×PRO_SOCIAL 
+are ME 0.110 (95% CI: 0.032 - 0.187) and ME 0.076 (95% CI: 0.014 - 0.137) on WEARING 
+MASK and HANDWASHING, respectively. This suggests that pro-social persons are more 
+likely than non-pro-social persons to wear masks by 0.110 points and to wash their hands by 
+0.076 points. Effects of VACCINE are ME 0.938 (95% CI: 0.845 -1.031) and ME 0.275 (95% 
+CI: 0.209 - 0.342) on WEARING MASK and HANDWASHING, respectively. This is the 
+effect of vaccination on non-pro-social individuals’ preventive behaviors. Considering the 
+results jointly, pro-social persons are 11.8% more likely than non-pro-social persons to wear 
+masks and 27.6% more likely to wash their hands. That is, for pro-social persons, the degree 
+of the effects of vaccination on handwashing is more than twice as great as it is for wearing 
+masks. Wearing masks and washing hands are different in that the benefit of wearing a mask 
+is more likely to depend on the situation. Wearing masks in the open air is only marginally 
+effective in mitigating pandemics [23]. Pro-social vaccinated persons may consider the cost-
+benefit ratio of preventive behaviors and therefore place more importance on washing hands 
+than on wearing masks.  
+ 
+ 
+ 
+
+ 
+ 
+14 
+ 
+Table 4. FE model with cross-term with PRO_SOCIAL. Dependent variables are 
+preventive behaviors. 
+ 
+  (1) 
+STAYING 
+INDOORS 
+    (2) 
+WEARING 
+MASK 
+   (3) 
+HANDWASHING  
+VACCINE  
+−0.051 
+(−0.154-0.050) 
+0.938** 
+ 
+(0.845 -1.031) 
+0.275** 
+ 
+(0.209 - 0.342) 
+VACCINE 
+×PRO_SOCIAL 
+  0.052 
+(−0.019 – 0.122) 
+0.110** 
+ 
+(0.032 - 0.187) 
+0.076*** 
+(0.014 - 0.137) 
+Individual Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Time Fixed Effects 
+  Yes 
+Yes 
+   Yes 
+Controls: New deaths, 
+ New affected persons, 
+Household income 
+  Yes 
+Yes 
+   Yes 
+PROBABILITY COVID19, 
+SEVERITY COVID19  
+  Yes 
+Yes 
+   Yes 
+Adjusted R2 
+Observations 
+0.55 
+6,809 
+0.45 
+6,809 
+0.62 
+6,809 
+ 
+Note: Numbers within parentheses are 95% CI. All the models include control variables, which are 
+equivalent to those in Table 2. “Yes” means that variables are included. However, these results have not 
+been reported.  
+** ρ <.01 
+** ρ < .05 
+ 
+Conclusions 
+In some studies, individuals are unlikely to reduce preventive behaviors even after 
+vaccination [6,10,11]. This is contrary to rational behavior in terms of economics. However, 
+the underlying mechanisms have not yet been investigated. This study contributes to the 
+understanding of the mechanism by considering pro-social motivation. On the one hand, we 
+find that vaccinated people under 65 years of age are less likely to stay indoors than older 
+people. On the other hand, vaccinated individuals are more inclined to wash their hands and 
+
+ 
+ 
+15 
+wear masks than before being vaccinated. The motivation to be vaccinated, for 86% of 
+respondents, is to mitigate the spread of infection in society. These are considered pro-social 
+people and are more likely than others to wash their hands and wear masks after vaccination. 
+However, their staying-indoors behavior is not different from that of the others.  
+Staying indoors differs from wearing masks and washing hands when we consider the cost-
+benefit aspects of preventive behaviors. Staying home leads people to sacrifice their vacation 
+activities in the real world. Using economics terms, the sacrifice is the “opportunity cost” of 
+staying home. They would stay at home if their benefits were greater than their costs. After 
+vaccination, the opportunity cost of staying indoors is higher than its benefit. Accordingly, 
+younger people are more likely to go out.   
+Both vaccination and preventive behaviors are considered public goods for coping with the 
+pandemic. As a result of vaccination, people tend to go out, which may reduce public goods. 
+To compensate for this, vaccinated pro-social persons are more likely to be motivated to wear 
+masks and wash their hands by considering the benefit to society. Other possible 
+interpretations of the estimation results are related to the cost of getting a vaccination, 
+including the physical and psychological costs of the side effects. From an economic 
+viewpoint, the cost of vaccination can be considered the “sunk cost”—an investment already 
+incurred that cannot be recovered. Due to the sunk cost, vaccinated people continue to invest 
+in public goods by strengthening their mask-wearing and handwashing behaviors rather than 
+their staying-indoors behavior.  
+Wearing masks in the open air is only marginally effective in mitigating pandemics [23]; 
+thus, this might result in over-investment in public goods. Owing to data limitations, we 
+
+ 
+ 
+16 
+cannot analyze the situation in which vaccinated people wear masks. It is necessary to 
+determine how and to what extent the preventive behaviors of vaccinated people are effective 
+in mitigating COVID-19. 
+ 
+List of abbreviations 
+CI 
+ 
+confidence interval 
+ 
+COVID-19 
+ 
+coronavirus-19 
+FE 
+ 
+fixed effects 
+ 
+Inc. 
+ 
+Incorporated 
+LLC 
+ 
+Limited Liability Company 
+ME 
+ 
+marginal effects 
+ 
+ 
+References 
+ 
+1. Cato S, Iida T, Ishida K, Ito A, Katsumata H, McElwain KM, et al. Vaccination and altruism 
+under the COVID-19 pandemic. Public Health Pract (Oxf). 2022;3:100225. 
+doi:10.1016/j.puhip.2022.100225. 
+2. Cato S, Iida T, Ishida K, Ito A, McElwain KM, Shoji M. Social distancing as a public good under 
+the COVID-19 pandemic. Public Health. 2020;188:51-3. doi:10.1016/j.puhe.2020.08.005. 
+3. Trogen B, Caplan A. Risk compensation and COVID-19 vaccines. Ann Intern Med. 2021;174:858-
+9. doi:10.7326/M20-8251. 
+4. 
+World 
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+Organization, 
+WHO. 
+Coronavirus 
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+Dashboard; 
+2020. 
+https://covid19.who.int/. Accessed 20 Dec 2022. 
+ 
+5. Peltzman S. The effects of automobile safety regulation. J Pol Econ. 1975;83:677-725. 
+doi:10.1086/260352. 
+6. Zhang N, Lei H, Li L, Jin T, Liu X, Miao D, et al. COVID-19 vaccination did not change the 
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+2021;9:777426. doi:10.3389/fpubh.2021.777426. 
+7. Si R, Yao Y, Zhang X, Lu Q, Aziz N. Investigating the links between vaccination against COVID-
+19 and public attitudes toward protective countermeasures: Implications for public health. 
+Front Public Health. 2021;9:702699. doi:10.3389/fpubh.2021.702699. 
+8. Zhang N, Liu X, Jin T, Zhao P, Miao D, Lei H, et al. Weakening personal protective behavior by 
+Chinese university students after COVID-19 vaccination. Build Environ. 2021;206:108367. 
+doi:10.1016/j.buildenv.2021.108367. 
+9. Hossain MdE, Islam MdS, Rana MdJ, Amin MR, Rokonuzzaman M, Chakrobortty S, et al. 
+Scaling the changes in lifestyle, attitude, and behavioral patterns among COVID-19 
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+doi:10.1080/21645515.2021.2022920. 
+10. Wright L, Steptoe A, Mak HW, Fancourt D. Do people reduce compliance with COVID-19 
+guidelines following vaccination? A longitudinal analysis of matched UK adults. J Epidemiol 
+Community Health. 2022;76:109–15. doi:10.1136/jech-2021-217179. 
+11. Corea F, Folcarelli L, Napoli A, del Giudice GM, Angelillo IF. The impact of COVID-19 
+vaccination in changing the adherence to preventive measures: Evidence from Italy. 
+Vaccines (Basel). 2022;10. doi:10.3390/vaccines10050777. 
+12. Cheng KK, Lam TH, Leung CC. Wearing face masks in the community during the COVID-19 
+pandemic: Altruism and solidarity. Lancet. Published online 2020. 2022;399:e39–40. 
+doi:10.1016/S0140-6736(20)30918-1. 
+13. Wooldridge J. Introductory Econometrics. 4th ed. South-Western Cengage Learning; 2009. 
+14. Hsiao C. Analysis of Panel Data. Cambridge University Press; 1986. 
+15. Baltagi B. Econometric Analysis of Panel Data. John Wiley & Sons; 1995. 
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+18 
+16. Habersaat KB, Betsch C, Danchin M, Sunstein CR, Böhm R, Falk A, et al. Ten considerations 
+for effectively managing the COVID-19 transition. Nat Hum Behav. 2020;4:677-87. 
+doi:10.1038/s41562-020-0906-x. 
+17. Ohtake F. Can nudges save lives? Jpn Econ Rev (Oxf). 2022;73:245-68. doi:10.1007/s42973-021-
+00095-7. 
+18. Sasaki S, Saito T, Ohtake F. Nudges for COVID-19 voluntary vaccination: How to explain peer 
+information? Soc Sci Med. 2022;292:114561. doi:10.1016/j.socscimed.2021.114561. 
+19. van der Westhuizen HM, Kotze K, Tonkin-Crine S, Gobat N, Greenhalgh T. Face coverings for 
+Covid-19: From medical intervention to social practice. BMJ. 2020;370:m3021. 
+doi:10.1136/bmj.m3021. 
+20. Wu Z, McGoogan JM. Characteristics of and important lessons from the coronavirus disease 
+2019 (COVID-19) outbreak in China: Summary of a Report of 72 314 Cases From the 
+Chinese Center for Disease Control and Prevention. JAMA. 2020;323:1239–42. 
+doi:10.1001/jama.2020.2648. 
+21. Koh HK, Geller AC, Vanderweele TJ. Deaths from COVID-19. JAMA. 2021;325:133-4. 
+doi:10.1001/jama.2020.25381. 
+22. Asri A, Asri V, Renerte B, Föllmi-Heusi F, Leuppi JD, Muser J, et al. Wearing a mask-For 
+yourself or for others? Behavioral correlates of mask wearing among COVID-19 frontline 
+workers. PLOS ONE. 2021;16 (7 July):e0253621. doi:10.1371/journal.pone.0253621. 
+23. Javid B, Bassler D, Bryant MB, Cevik M, Tufekci Z, Baral S. Should masks be worn outdoors? 
+BMJ. 2021;373:n1036. doi:10.1136/bmj.n1036. 
+  
+
diff --git a/GdE1T4oBgHgl3EQfXQRO/content/tmp_files/load_file.txt b/GdE1T4oBgHgl3EQfXQRO/content/tmp_files/load_file.txt
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@@ -0,0 +1,561 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf,len=560
+page_content='1 The COVID-19 vaccination, preventive behaviors and pro-social motivation: panel data analysis from Japan  Short title: COVID-19 vaccination and preventive behaviors  Eiji Yamamura1*, Yoshiro Tsutsui2, Fumio Ohtake3  1 Department of Economics, Seinan Gakuin University, Fukuoka, Japan  Corresponding author Email: yamaei@seinan  gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='jp (EY) Full list of author information is available at the end of the article  Abstract Background The COVID-19 vaccine reduces infection risk: even if one contracts COVID-19, the probability of complications like death or hospitalization is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, vaccination may prompt people to decrease preventive behaviors, such as staying indoors, handwashing, and wearing a mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Thereby, if vaccinated people pursue only their self-interest, the vaccine’s effect may be lower than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, if vaccinated people are pro-social (motivated toward benefit for the whole society), they might maintain preventive behaviors to reduce the spread of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Methods We conducted 26 surveys nearly once a month from March 2020 (the early stage of COVID- 19) to September 2022 in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' By corresponding with the identical individuals, we independently constructed original panel data (N = 70,979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Based on the data, we identified the timing of the second vaccine shot and compared preventive behaviors before and after  2 vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We investigated whether second-shot vaccination correlated with changes in preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Furthermore, we explored whether the vaccination effect differs between senior and younger groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We then investigated the effect of pro-social motivation on preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Results Major findings are as follows: (1) Being vaccinated led people to increase preventive behaviors, such as mask-wearing by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='04 (95% confidence intervals [Cis]: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='96–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='11) points, and handwashing by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='34 (95% CIs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='30–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='38) points, on a 5-point scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' (2) vaccinated people under the age of 65 years are less likely to stay indoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' (3) people with pro-social motivation to be vaccinated are more likely to maintain prevention than those not so motivated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' the difference is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='08 (95% CIs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='01–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='15) points for mask-wearing and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='05 (95% CIs: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='001–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='10) points for handwashing, on a 5-point scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Conclusion After vaccination, the opportunity cost of staying indoors outweighs its benefits and people are less inclined to stay home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This effect is lower for older people, who are at higher risk of serious illness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The opportunity cost of mask-wearing and handwashing is lower than that of staying indoors, and the benefit persists after vaccination if people have the motivation to maintain these behaviors for others’ well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Keywords: Vaccine, COVID-19, preventive behaviors, staying indoors, wearing mask, washing hands, public goods, pro-social, Japan, panel data  3  Introduction  During COVID-19 pandemic, especially in the early stages, various preventive behaviors were required because vaccines against COVID-19 had not been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Preventive behaviors can be considered a kind of public good that is not sufficiently supplied through market mechanisms where people pursue self-interest [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Mitigating the pandemic necessitated the collective actions of citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, according to the Peltzman effect, people tend to increase their risky behaviors if safety measures are implemented [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Since 2021, vaccines against COVID-19 have been distributed worldwide and have played a vital role in coping with COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Newly reported cases have decreased in countries where vaccines have been rapidly adopted [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' People, if they are rational, tend to engage in risky behaviors when security measures are mandated [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In terms of economics, this is considered a moral hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' An empirical question arises as to how the spread of the vaccine influences preventive behaviors [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' As a result of the reduction in the risk of COVID-19 infection, risk-taking behaviors increase, so that preventive behaviors such as staying indoors, wearing masks, and washing hands have changed [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, some studies show no clear evidence that vaccinated people have decreased their preventive behaviors in comparison with those not vaccinated [6,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The influence of vaccination on preventive behaviors may vary according to the type of behavior [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  A study found that in China, vaccination lessened the frequency of handwashing but did not change mask-wearing [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This study aimed to explore the mechanisms stopping vaccinated people from decreasing preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' To this end,  4 we investigated how preventive measures can be pro-socially motivated based on altruism and social solidarity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Using monthly individual-level panel data, we investigated whether individuals’ preventive behaviors changed after vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Furthermore, we looked at how the influence of vaccination on preventive behaviors differs according to age and pro-social motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Methods Data collection We commissioned the research company INTAGE, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=', to conduct an online survey because of their experience and reliability in academic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The first wave of queries was conducted March 13–16, 2020, and 4,359 observations were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Participants registered with INTAGE were recruited for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The participation rate was 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The sampling method was designed to collect representatives of the Japanese adult population in terms of educational background, gender, and residential area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' For this purpose, INTAGE recruited participants for a survey of pre-registered individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, individuals aged 15 years and below were too young to be registered with INTAGE, and we considered individuals over 80 years of age too old to answer pertinent questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Inevitably, the sample population was restricted to ages 16–79 years, and participants were randomly selected to fill the pre-specified quotas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' INTAGE provided monetary incentives to participants upon study completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Internet surveys were conducted repeatedly 26 separate times (“waves”) almost every month with identical individuals to construct the panel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The exceptional period was July-  5 September 2020, when the survey could not be conducted owing to a shortage of research funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We resumed the surveys after receiving additional funds in October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Vaccination was implemented in April 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' therefore, the data cover the period before and after implementation of vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Respondents from the first wave were targeted in subsequent waves to record how some respondents changed their behaviors during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In particular, the data allowed us to compare identical persons’ preventive behaviors against COVID-19 before and after being vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' During the study period, some identical respondents were dropped from the study sample because some of them stopped taking the surveys, while others did not take the surveys at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Furthermore, the sample was restricted to those who were completely vaccinated by getting the second shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In this way, we could compare their behaviors before and after vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Eventually, the number of identical individuals was reduced from 4,359 to 3,019, and the total number of observations used in this study was 70,979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Methods Table 1 presents the key variables used in the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The survey questionnaire contained basic questions about demographics, such as birth year, gender, and educational background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' These characteristics were observed at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The surveys were conducted 26 times, between March 2020 and September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' During the study period, conditions such as the spread of infection and policies against COVID-19 changed drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Table 1 describes the key variables used in the regression estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' As outcome variables,  6 the respondents were asked questions concerning preventive behaviors, such as: “Within a week, to what degree have you practiced the following behaviors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Please answer based on a scale of 1 (I have not practiced this behavior at all) to 5 (I have completely practiced this behavior).” (1) Staying indoors (2) Wearing a mask (3) Washing my hands thoroughly The answers to these questions served as proxies for the following variables for preventive behaviors: staying indoors, frequency of handwashing, and degree of wearing masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Larger values indicate that respondents are more likely to engage in preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Further, the motivation to get a shot of COVID-19 vaccination is asked in the following question: “Do you get the shot in order to decrease the spread of COVID-19 infection?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We also asked about the subjective probability of contracting COVID-19 and their perceptions of the severity of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We asked whether they received the second shot of the vaccine because the vaccination was effective only after completing the second shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The latter question was included in the questionnaire from the 12th wave conducted in May 2021 directly after the vaccine was introduced in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The question was included until the 18th wave in November 2021, when most people in the sample completed the second shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The question was then excluded from the questionnaire, starting with the 19th wave in January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  7  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Definitions of key variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Variable  Definition  Outcome variables STAYING INDOORS In the last week, how consistent were you at “not going out of home?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Please choose among 5 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1 (not completed at all) to 5 (completely consistent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' WEARING MASK  In the last week, how consistent were you at “wearing a mask?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Please choose among 5 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1 (not completed at all) to 5 (completely achieved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' HANDWASHING  In the last week, how consistent were you at “washing your hands?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Please choose among 5 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1 (not completed at all) to 5 (completely achieved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Confounders (Independent variables)  VACCINE  Did you get the second shot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1 (Yes) or 0 (No) PROB_COVID19  What percentage do you think is the probability of your contracting the novel coronavirus (COVID-19)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 0 to 100 (%) SEVERITY COVID19 How serious do you expect your symptoms to be if you are infected with the novel coronavirus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Choose from 6 choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1 (very small influence) to 6 (death) AGE_25 Answer 1 if respondents are age 19–25, otherwise 0  AGE_26_64 Answer 1 if respondents are age 16–64, otherwise 0  PRO_SOCIAL In deciding whether you get the shot of COVID-19 vaccine, is it important in preventing the spread of COVID-19?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1 (strongly disagree) to 5 (strongly agree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' PRO-SOCIAL is 1 if respondent chooses 5, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  We pursued identical respondents from the first-wave survey to the 26th wave for 30 months even though some of the respondents quit the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The purpose of this study was to explore how the preventive behaviors of identical persons changed before and after vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Therefore, we limited the sample to those who completed the second shot by the 18th wave and then pursued identical persons until the 26th wave in September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We used panel data containing 3,019 individuals, covering 26 time points from March 2021 to September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Based on the panel data, we used a fixed-effects (FE) model regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The FE model is  8 a type of linear regression model widely used in economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The estimation result using an FE model is equivalent to the results of a linear regression model with dummies for individuals frequently included in each period [13-15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In this study, 3,019 dummies were included to control for individuals’ characteristics that do not change during the period, such as gender, educational background, and childhood experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Hence, 3,019 confounders were included, reflecting the differences between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Therefore, the estimated results for the time-invariant confounders could not be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Even if various time-variant confounders are included, unobserved individual characteristics cannot be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This inevitably results in omitted variable biases [13,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' For instance, an increasing trend in the number of newly infected persons was observed throughout Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This effect is common to all residents in Japan and has changed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This can be regarded as a time-fixed effect and can be controlled by including time-period dummies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In this study, 25 time-period dummies were included when one base period was fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, some variables changed not only over time, but also between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Examples are proxy variables for preventive behaviors, which are outcome variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' or number of newly infected persons and deaths due to COVID-19 in residential areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Furthermore, the timing of obtaining the second vaccine shot is a variable that changed over time and between individuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' therefore, the dummy for vaccination is included in the estimated function as a confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' As explained, we controlled not only for unobservable individual fixed effects, but also for unobservable time-fixed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This type of FE model is called the two-way error component regression model [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This study focused on the correlation between vaccination and preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The statistical software used in this study was the Stata/MP 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='0  9 multiprocessor from StataCorp, LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The estimated function of an FE model takes the following form: Yit = α1 VACCINEit + α6 PROB COVID19it + α7 SEVERITY COVID19it + α8 EMERGENCYit + kt + mi + uit  where Table 1 presents the definitions and basic statistics of these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In this formula, Yit represents the outcome variable for individual i and wave t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The time- invariant individual-level fixed effects are represented by mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Furthermore, kt represents the effects of different time points, controlled by 25 wave dummies, where the first wave is the reference group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' kt captures various shocks that occurred simultaneously throughout Japan at each time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Y includes preventive behaviors captured by three proxy variables: STAYING INDOORS, HANDWASHING, and WEARING MASK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' These outcome variables are discrete ordered variables ranging from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Larger values of these variables can be interpreted as indicating that the respondents are more likely to exhibit preventive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' In the same specification, we conduct three separate estimations, and the regression parameters are denoted as α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The error term is denoted by uit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' A simple FE linear regression model is used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The key confounder is the vaccination dummy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' VACCINE is 1 if respondents have completed the second shot of the COVID-19 vaccine, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' People are obliged to get the second shot within a month of the first shot to make the vaccine effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Hence, in the sample, there is hardly any time lag between the first and second shots because the survey was conducted every month after the vaccine was approved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  There are two age groups: young age (AGE_25) and middle working age (AGE_26_64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The senior group is used as the  10 reference group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' PRO_SOCIAL is a dummy variable that captures pro-social motivation to get vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The mean value of PRO_SOCIAL is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='86, which shows that 86% of people have the pro-social motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Results Baseline estimations The coefficient of confounders indicates marginal effects (ME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Table 2 presents the estimation results for the baseline FE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' VACCINE shows a positive sign and is statistically significant at the 1% level, with the exception of Column (1), where STAYING INDOORS is the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The effects of VACCINE are ME 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='036 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='960- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='111) and ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='339 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='300–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='378) in columns (2) and (3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Thus, people after vaccination are more likely than before to wear masks by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='036 points and to wash their hands by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='339 points, respectively, on a 5-point scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Before vaccination, the mean values of WEARING MASK and HANDWASHING are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='39 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Hence, this indicates that the degree of wearing masks increases by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='6% and washing hands by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='16% after vaccination, compared with before vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' People’s behaviors depend on behaviors of others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' hence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' they follow the social norm [16-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Peer pressure is stronger for wearing masks than for washing hands because the surrounding people in a public place can more easily see whether one wears a mask than whether one washes one’s hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' SEVERITY COVID19 shows a significant positive value in all columns, and this means that persons’ perception about seriousness of COVID-19 changes the incidence of preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  11  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' FE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Dependent variables are preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  (1)  STAYING  INDOORS  (2)  WEARING  MASK  (3)  HANDWASHING  VACCINE  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='011  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='053)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='036** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='960 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='111)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='339** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='378)  PROBABILITY COVID19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='043  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='873)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='487  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='094 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='070)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='560  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='867)  SEVERITY COVID19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='026** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='039)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='029** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='040)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='019** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='030)  Individual Fixed Effects  Yes  Yes  Yes  Time Fixed Effects  Yes  Yes  Yes  Controls: New deaths,  Newly affected persons,  Household income  Yes  Yes  Yes  Adjusted R2  Observations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='55  70,979  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='45  70,979  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='62  70,979  Note: Numbers within parentheses are 95% CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' For convenience, the coefficient of probability of COVID- 19 is multiplied by 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The model includes the number of deaths and infected persons in the residential prefectures in the surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, these results have not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' “Yes” means that variables are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' *** ρ < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='01  Estimations with cross-terms The probability and seriousness of contracting COVID-19 differ by age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' COVID-19 is more likely to be lethal for adults aged 65 years and older than for younger people [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Mask-wearing by elderly people is motivated by their self-regarding risk preferences, whereas younger people are motivated by other-regarding concerns [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We explore how the effect of COVID-19 vaccination on preventive behaviors differs between age groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' For this purpose, the interaction terms between VACCINE and age groups (AGE_25 and AGE_26_64) are included as key confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The reference age group is those over age 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The results are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' We find a significant negative sign in  12 VACCINE×AGE_25 and VACCINE×AGE_26_64 in column (1), where STAYING INDOORS is the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The effects of VACCINE×AGE_25 and VACCINE×AGE_26_64 are ME −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='537 (95% CI: −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='658 - −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='416) and ME −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='295 (95% CI: −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='353 - −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='238).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This means that those aged below 25 are less likely to stay home by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='537 points than those aged over 65, while those aged between 26 and 64 are less likely to stay home by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='297 points than those aged over 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, no differences in the effect of the vaccination are observed when wearing masks and washing hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' FE model with cross-terms with age cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Dependent variables are preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  (1)  STAYING  INDOORS  (2)  WEARING  MASK  (3)  HANDWASHING  VACCINE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='206**  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='269)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='058**  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='980 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='137)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='319**  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='356)  VACCINE×AGE_25  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='537** (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='658 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='416)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='066  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='028)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='056  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='024)  VACCINE×AGE_26_64  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='295** (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='353 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='238)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='030  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='015)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='036** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='068)  Individual Fixed Effects  Yes  Yes  Yes  Time Fixed Effects  Yes  Yes  Yes  Controls: New deaths,  Newly affected persons,  Household income  Yes  Yes  Yes  PROBABILITY COVID19,  SEVERITY COVID19  Yes  Yes  Yes  Adjusted R2  Observations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='57  70,979  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='47  70,979  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='64  70,979  Note: Numbers within parentheses are 95% CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' All the models include control variables, which are equivalent to those in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' “Yes” means that variables are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, these results have not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' ** ρ < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='01 *** ρ < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='05  13  We investigated how pro-social motivation affects preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The interaction term between VACCINE and PRO_SOCIAL was included as a key confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Table 4 shows the significant positive sign of VACCINE×PRO_SOCIAL, where WEARING MASK and HANDWASHING are the outcome variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The effects of VACCINE×PRO_SOCIAL are ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='110 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='032 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='187) and ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='076 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='014 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='137) on WEARING MASK and HANDWASHING, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This suggests that pro-social persons are more likely than non-pro-social persons to wear masks by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='110 points and to wash their hands by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='076 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Effects of VACCINE are ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='938 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='845 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='031) and ME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='275 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='209 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='342) on WEARING MASK and HANDWASHING, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This is the effect of vaccination on non-pro-social individuals’ preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Considering the results jointly, pro-social persons are 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='8% more likely than non-pro-social persons to wear masks and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='6% more likely to wash their hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' That is, for pro-social persons, the degree of the effects of vaccination on handwashing is more than twice as great as it is for wearing masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Wearing masks and washing hands are different in that the benefit of wearing a mask is more likely to depend on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Wearing masks in the open air is only marginally effective in mitigating pandemics [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Pro-social vaccinated persons may consider the cost- benefit ratio of preventive behaviors and therefore place more importance on washing hands than on wearing masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  14  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' FE model with cross-term with PRO_SOCIAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Dependent variables are preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  (1)  STAYING  INDOORS  (2)  WEARING  MASK  (3)  HANDWASHING  VACCINE  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='051  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='050)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='938**  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='845 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='031)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='275**  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='342)  VACCINE  ×PRO_SOCIAL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='052  (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='019 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='122)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='110**  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='187)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='076** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='137)  Individual Fixed Effects  Yes  Yes  Yes  Time Fixed Effects  Yes  Yes  Yes  Controls: New deaths,  New affected persons,  Household income  Yes  Yes  Yes  PROBABILITY COVID19,  SEVERITY COVID19  Yes  Yes  Yes  Adjusted R2  Observations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='55  6,809  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='45  6,809  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='62  6,809  Note: Numbers within parentheses are 95% CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' All the models include control variables, which are equivalent to those in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' “Yes” means that variables are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, these results have not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' ** ρ <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='01 ** ρ < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='05  Conclusions In some studies, individuals are unlikely to reduce preventive behaviors even after vaccination [6,10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This is contrary to rational behavior in terms of economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, the underlying mechanisms have not yet been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' This study contributes to the understanding of the mechanism by considering pro-social motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' On the one hand, we find that vaccinated people under 65 years of age are less likely to stay indoors than older people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' On the other hand, vaccinated individuals are more inclined to wash their hands and  15 wear masks than before being vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The motivation to be vaccinated, for 86% of respondents, is to mitigate the spread of infection in society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' These are considered pro-social people and are more likely than others to wash their hands and wear masks after vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' However, their staying-indoors behavior is not different from that of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Staying indoors differs from wearing masks and washing hands when we consider the cost- benefit aspects of preventive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Staying home leads people to sacrifice their vacation activities in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Using economics terms, the sacrifice is the “opportunity cost” of staying home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' They would stay at home if their benefits were greater than their costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' After vaccination, the opportunity cost of staying indoors is higher than its benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Accordingly, younger people are more likely to go out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Both vaccination and preventive behaviors are considered public goods for coping with the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' As a result of vaccination, people tend to go out, which may reduce public goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' To compensate for this, vaccinated pro-social persons are more likely to be motivated to wear masks and wash their hands by considering the benefit to society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Other possible interpretations of the estimation results are related to the cost of getting a vaccination, including the physical and psychological costs of the side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  From an economic viewpoint, the cost of vaccination can be considered the “sunk cost”—an investment already incurred that cannot be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Due to the sunk cost, vaccinated people continue to invest in public goods by strengthening their mask-wearing and handwashing behaviors rather than their staying-indoors behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Wearing masks in the open air is only marginally effective in mitigating pandemics [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' thus, this might result in over-investment in public goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Owing to data limitations, we  16 cannot analyze the situation in which vaccinated people wear masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' It is necessary to determine how and to what extent the preventive behaviors of vaccinated people are effective in mitigating COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  List of abbreviations  CI  confidence interval  COVID 19  coronavirus 19  FE  fixed effects  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  Incorporated  LLC  Limited Liability Company  ME  marginal effects  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Cato S, Iida T, Ishida K, Ito A, Katsumata H, McElwain KM, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Vaccination and altruism under the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Public Health Pract (Oxf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='3:100225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='puhip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='100225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Cato S, Iida T, Ishida K, Ito A, McElwain KM, Shoji M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Social distancing as a public good under the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Public Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='188:51-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='puhe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Trogen B, Caplan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Risk compensation and COVID-19 vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Ann Intern Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='174:858- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='7326/M20-8251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' World Health Organization, WHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Coronavirus (COVID-19) Dashboard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' https://covid19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='int/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Accessed 20 Dec 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Peltzman S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' The effects of automobile safety regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' J Pol Econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='83:677-725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
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+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Zhang N, Lei H, Li L, Jin T, Liu X, Miao D, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' COVID-19 vaccination did not change the  17 personal protective behaviors of healthcare workers in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Front Public Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
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+page_content=' Si R, Yao Y, Zhang X, Lu Q, Aziz N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Investigating the links between vaccination against COVID- 19 and public attitudes toward protective countermeasures: Implications for public health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Front Public Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
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+page_content=' Zhang N, Liu X, Jin T, Zhao P, Miao D, Lei H, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Weakening personal protective behavior by Chinese university students after COVID-19 vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Build Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
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+page_content=' Hossain MdE, Islam MdS, Rana MdJ, Amin MR, Rokonuzzaman M, Chakrobortty S, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Scaling the changes in lifestyle, attitude, and behavioral patterns among COVID-19 vaccinated people: Insights from Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Hum Vaccin Immunother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
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+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='1080/21645515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content='2022920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Wright L, Steptoe A, Mak HW, Fancourt D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' Do people reduce compliance with COVID-19 guidelines following vaccination?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' A longitudinal analysis of matched UK adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' J Epidemiol Community Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfXQRO/content/2301.03124v1.pdf'}
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diff --git a/H9AzT4oBgHgl3EQfxv52/content/tmp_files/2301.01743v1.pdf.txt b/H9AzT4oBgHgl3EQfxv52/content/tmp_files/2301.01743v1.pdf.txt
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@@ -0,0 +1,1335 @@
+CHATBOTS AS PROBLEM SOLVERS: PLAYING TWENTY 
+QUESTIONS WITH ROLE REVERSALS 
+ 
+David Noever1 and Forrest McKee2  
+PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805 
+1 david.noever@peopletec.com  2forrest.mckee@peopletec.com               
+       
+ 
+ABSTRACT 
+New chat AI applications like ChatGPT offer an advanced understanding of question context and memory 
+across multi-step tasks, such that experiments can test its deductive reasoning. This paper proposes a multi-
+role and multi-step challenge, where ChatGPT plays the classic twenty-questions game but innovatively 
+switches roles from the questioner to the answerer. The main empirical result establishes that this 
+generation of chat applications can guess random object names in fewer than twenty questions (average, 
+12) and correctly guess 94% of the time across sixteen different experimental setups. The research 
+introduces four novel cases where the chatbot fields the questions, asks the questions, both question-answer 
+roles, and finally tries to guess appropriate contextual emotions. One task that humans typically fail but 
+trained chat applications complete involves playing bilingual games of twenty questions (English answers 
+to Spanish questions). Future variations address direct problem-solving using a similar inquisitive format 
+to arrive at novel outcomes deductively, such as patentable inventions or combination thinking. Featured 
+applications of this dialogue format include complex protein designs, neuroscience metadata, and child 
+development educational materials.  
+ 
+KEYWORDS 
+Transformers, Text Generation, Malware Generation, Generative Pre-trained Transformers, GPT 
+ 
+1. INTRODUCTION 
+ 
+When large, high-quality natural language processors (NLP) surged after 2018 [1-3], the field added many 
+challenging tasks, including question-answering (QA) benchmarks that recently approached fifty 
+challenges [4-5]. Stanford's SQuAD benchmark [6] represents an example of knowledge crowd-sourced 
+from Wikipedia in 2016 and formatted as 108,000 QA pairs. For large language models (LLMs), most 
+benchmarks follow this format of "prompt-response" pairs and the underlying knowledge base stored 
+answers [5], but the training on question datasets did not embed memory or long-conversational cues [7-
+10]. Domain-specific QA datasets include common sense and trivia about movies, news, Wikipedia, 
+Tweets, search engines, and books [4-5]. As a format, the familiar game of Twenty Questions [11-16] 
+features multi-hop reasoning, which often condenses to "Animal-Vegetable-Mineral" as opening questions 
+that narrow the theme [17]. The advent of ChatGPT (Generative Pre-trained Transformers, [18]) added 
+memory across questions (up to 8000 tokens or 20-25 pages). Appendix A lists the 48 specialty tasks 
+currently provided to guide users in creating capabilities and NLP prompts [19]. Examples like code 
+generation specialists, either codex or copilot, show a grasp of complex behavior for debugging, program 
+suggestions, and commentary [19-20]. One of the innovative ChatGPT extensions offers new QA sessions 
+that span multiple requests [18,21].  
+ 
+The present work applies the familiar QA challenge, Twenty Questions [11-16], with ChatGPT playing as 
+either participant- the one who knows the answer and fields the questions, but also the one who doesn't 
+know the answer but asks the questions. We call this role reversal a two-player conversation between "Bob" 
+and "Alice." While Twenty Questions dates back to 1882 [11], the applied problem-solving method [22] 
+
+now spans various challenging fields, including protein folding [23] and design [24], image segmentation 
+[25], child development [26-27], neuroscience [28] and diagnostic medicine,  and crowd-sourced emotional 
+indices [30]. Variants of the game rules include role reversals [31], liars [32], and word relationships [33] 
+outside of simple categories such as "pertaining to" suggestions [34]. Our particular research interests 
+center on whether LLMs like ChatGPT offer constructive means for innovative problem-solving through 
+crafted language prompts and logical question sequences. This effort empirically assesses the model's 
+capability for deductive discovery. We summarize two problem statements in this area of deductive 
+discovery, "Can a chatbot play both roles in deductive question and answering conversations?" and if so, 
+"What can Twenty Questions reveal about future directions for LLMs?".   
+ 
+2. METHODS 
+ 
+The approach is to test experimentally how well an LLM handles open-ended games like Twenty Questions 
+[11-16]. We employ the December 2022 research model from Open AI called ChatGPT [18]. We prompt 
+it to impersonate both roles [31], the one that knows the answer and the one that tries to guess it deductively. 
+We run four trials using each persona ("Bob" or "Alice"), and 80 questions are available to guess the object 
+or concept. We mix up the animate-inanimate fields and introduce concepts like alphabetic letters. We 
+report metrics for the mean and median number of questions required to guess the correct answer and failure 
+rates. We also score the exchange against the machine-written detector to determine ChatGPT's broad 
+capability to provide "real" or "fake" text outputs as judged by OpenAI's original GPT2 Detector [35]. The 
+detector features a high-dimensional pattern detector that provides initial confidence that human writing 
+might sample differently than transformer-based output regarding word choice and order, repetition, and 
+other syntax outliers. 
+ 
+Example variants of this pattern include forcing the QA session towards bilingual communication, thus 
+combining two tasks (translation and QA) in multi-hop conversations [5]. We also generalize the format to 
+elicit emotional indices based on crowd-sourced Emotion Twenty Questions (EMO20Q) [30]. Rather than 
+trying to guess an object, we query for one of 23 emotional states, such as "surprise" or "anger." As the 
+dataset designers [30] remarked, "The EMO20Q game requires that an emotionally intelligent agent can 
+understand emotions without observing them, thanks to natural language and empathy." EMO20Q emotions 
+include the following as candidates for twenty-question discovery:  admire, adore, anger, awe, boredom, 
+bravery, calm, confidence, confusion, contempt, disgust, enthusiasm, frustration, gratefulness, jealousy, 
+love, proud, relief, serenity, shame, silly, surprised, and thankful. 
+ 
+The structure of the paper follows Appendices B-E closely. First, in Appendix B, we establish that ChatGPT 
+comprehends the game rules and recognizes properties of the object X to guess sufficiently to answer basic 
+questions such as "Is X an animal?". The sequence also establishes an exclusionary acceptance of "yes" and 
+"no" answers only without further elaborations. This case covers the traditional role of "Bob," who knows 
+the object of interest X and answers affirmatively or negatively throughout the game. The game also 
+underscores the unique memory of ChatGPT across multi-step deductive stages and builds toward a 
+successful conclusion to recall the object name in less than 40 steps (1 prompt and one answer over 20 
+iterations). We also set out to test the overall rule retention over four repetitions of the game punctuated 
+only by "Let's play again" without attempting to reset the rules while giving a new object X. 
+ 
+Secondly, Appendix C reverses the previous game, such that the human experimenter thinks of an object 
+or concept X, and ChatGPT plays the role of "Alice" when asking the questions. The prompt establishes 
+the reversed roles in a new session and again reiterates that the prompter cannot lie but can only answer 
+"yes" or "no." As in the previous case, four repetitions spanning eighty possible interactions. This case 
+establishes the deductive goal, "What is object X?" which also satisfies all the previous interactions such 
+that enough description includes the object of interest, "X is an animal," and excludes the alternatives, "X 
+
+is not a bird." A notable feature of this challenge spans multiple deductive steps and establishes a chain of 
+reasoning to arrive at a guess. ChatGPT requires no prompt for the final guess, and the model proposes its 
+terminating action, "Is it an X?" to end the game.   
+ 
+Both Appendix B and C mirror the familiar human game of twenty questions and introduce no new features 
+excluded from ChatGPT's training data. To raise the difficulty level, Appendix B highlights some non-
+traditional concepts. For instance, concepts might prove scarce in previously seen online play, such as 
+answering truthfully to probing questions that seek to name the alphabetic letter "Q." Another challenging 
+variation works through identifying "X" as a food by listing its ingredients ("tiramisu"), but in the same 
+session introduces a recipe change ("eggless tiramisu") before launching into probing questions.  
+ 
+Appendix C also raises the difficulty further and forces ChatGPT to combine two of its established subtasks 
+("question-answer" and "language translation"). The motivation for this test stems from the hypothesis that 
+LLMs parrot and mix up what previously corresponded to the internet of human training data. For twenty 
+questions, however, a Google search on "bilingual twenty-question examples" yield no definitive training 
+examples for ChatGPT to memorize or encode. Given a detailed prompt describing how the questions may 
+be asked or answered in Spanish or English, the game proceeds outside what typically would represent 
+human gameplay. Appendix C combines deductive reasoning, memory, and context and forces the game 
+into what might be considered "out-of-distribution" sampling. Thus, both multi-step and multi-lingual tasks 
+describe the ChatGPT challenge problem.  
+ 
+Thirdly, Appendix D introduces both QA roles as completing without human intervention once two browser 
+instances of ChatGPT exchange the initial rules. We call this example dueling twenty questions since the 
+two-headed LLM now must both ask and answer its questions between two non-communicating models of 
+itself ("Bob" vs. "Alice") with no human prompts. While this example centers on a simple object 
+("chicken"), one can envision a lengthy and detailed exchange driven by an automated Application 
+Programming Interface or API that extends the conversation to the limit of token lengths (about 20-25 typed 
+pages). This self-questioning interface may also enable sophisticated future applications that sequentially 
+build a knowledge base or comprehensive assessment examination from scratch. For example, "tell me all 
+you know about gall bladder surgery" may not provide a compelling or thought-provoking essay in the style 
+of training data or Wikipedia. The back-and-forth format of QA previously has yielded better human 
+performance in medical test contexts [28-29]. One might compare this example to an instance of "semi-
+supervised" learning for a chatbot. 
+ 
+Finally, for EMO20Q formats, Appendix E removes the requirement that X be an object to guess and 
+substitutes one of twenty-three emotions. One motivation for exploring the emotional quotient (EQ) of 
+ChatGPT stems from OpenAI's filtering of opinions and biases. To the authors' knowledge, this example 
+represents the first application of a chatbot deducing emotional states in a guessing game without any pre-
+programmed pairs of pattern-template exchanges [9]. In other words, no explicit guidance provides 
+appropriate intent for the question "Describe emotions one might feel at a birthday party?" with the user 
+goal to elicit "surprise" as a deductive leap to the correct answer through repeated probing. We explore this 
+EQ aspect as previous chatbots might have approached the problem. Appendix E finally displays the open-
+ended emotional context in a known prompt-response template called "Artificial Intelligence Markup 
+Language (AIML)," which offers training data for more traditional conversation templates in restricted 
+domains like customer service or AI assistants performing a narrow task [9].  
+ 
+3. RESULTS 
+ 
+Figure 1 summarizes the experimental cases and associated metrics for all the twenty question games in 
+Appendices B-E. The main finding supports a general ChatGPT capability to play all aspects of the game, 
+
+including guessing and answering or both roles in the same game. The average number of questions required 
+to get the answer was 11.6, with a median of 13 owing to a few more demanding cases that reached the 20-
+question limit. The addition of bilingual rules did not increase the number of required prompts (14) or force 
+the model to give up ("correct guess").   Similarly, the introduction of abstract objects like the alphabetic 
+letter "Q" or deceptive animals like "humans" did not significantly change the number of guesses (9-14) or 
+steer the conversation off-track from a final correct answer. 
+ 
+Over the sixteen tests and 185 exchanges, ChatGPT scored an overall correct guess rate of 94%. The only 
+incorrect answer ("paper clip" instead of "hammer") appeared to trigger prematurely, as ChatGPT declared 
+at guess number fourteen that the model had run out of questions and guessed incorrectly.  
+ 
+As described in the Methods section, the Open AI online detector [35] scores the majority of the exchanges 
+as "real," meaning not machine-generated by its GPT-2 model [1]. The 2018 detection model offered fewer 
+parameters and smaller training sets by at least several orders of magnitude compared to current generations 
+[18-19]. While the referenced detection accuracy for GPT-2 reached 95%, the scored detections flag only 
+26% of the game content as "fake" or machine-generated text. Since the customized content involves some 
+human intervention as questions, answers, and rule prompts, one can postulate that the syntactic patterns 
+follow a semi-human or hybrid dialogue outside the detector's target patterns.  
+4. DISCUSSION  
+ 
+The research literature on twenty questions provides a framework for probing with chat applications and 
+LLMs. In addition to systematically progressing through alternative scenarios, the output of the 
+conversation mirrors a binary search or decision tree. A particularly notable way to convert LLMs to 
+comparable knowledge graphs involves continuous probing and feedback until sufficient tree depth 
+Figure 1. Experimental results for ChatGPT across multiple QA sessions 
+
+Experiment
+Real
+Fake
+Tokens
+QA Length
+Guess
+Notes
+Appendix B. Chatbot Fields the Questions
+100%
+avg. 12; med: 14
+oven
+98.78
+1.22
+116
+7
+correct
+avocado
+0.38
+99.62
+220
+15
+correct
+tiramisu
+2.77
+97.23
+230
+15
+correct
+eggless tiramisu
+7.14
+92.86
+223
+14
+correct
+fewer guesses
+"o"
+60.27
+39.73
+232
+14
+correct
+difficult
+human
+22.31
+77.69
+133
+9
+correct
+deceptive
+Appendix C: Chatbot Asks the Questions
+84%
+avg. 11.6; med: 12
+dog
+99.96
+0.04
+161
+7
+correct
+Boston
+99.98
+0.02
+441
+11
+correct
+hammer
+99.57
+0.43
+297
+14
+incorrect
+out of questions
+statue
+99.98
+0.02
+383
+12
+correct
+car"bilingual example"
+99.98
+0.02
+510
+14
+correct
+Appendix D: Dueling Bob and Alice Chatbots
+100%
+avg. 17; med: 17
+chicken
+99.98
+0.02
+313
+17
+correct
+Appendix E: Emotional Quotient Deduction
+100%
+avg. 9; med: 6.5
+confidence
+99.94
+0.06
+66
+correct
+jealousy
+99.98
+0.02
+232
+5
+correct
+silly
+99.77
+0.23
+756
+20 
+correct
+final guess
+calm
+99.98
+0.02
+343
+8
+correct
+Overall
+74.42
+25.58
+291
+11.56
+94%
+avg. 11.56; med: 13describes a technical field of interest. Previous work [22-30] has used this 
+approach to describe neuroscience metadata and train medical students to 
+pass practice exams. Others [16-17] have also noted that the basic 20Q 
+format simplifies some complex search problems. As an interesting 
+historical note, a 2001 paper [36] addressed a similar problem as 
+insurmountable: "When will machine learning and pattern recognition rival 
+human ability to play Twenty Questions?" The present research 
+demonstrates not only can ChatGPT rival human ability and play more 
+demanding roles than a human might contemplate in virtually any field, 
+including discerning human emotions. To reproduce this ChatGPT output 
+with question templating systems or entity extraction poses an enormous 
+manual task to handle all the possible cases. The same paper [36] asks: "Can 
+we hope to stock a classification system with enough questions to play a 
+decent game, or must we instead focus on endowing with question-making 
+skills? How many classes and how many documents might be of interest?"  
+ 
+5. CONCLUSIONS 
+ 
+The experimental plan tests ChatGPT as an LLM capable of playing multiple roles in verbal games like 
+twenty questions. The work demonstrates 94% accuracy in correctly guessing across numerous challenges 
+and an average question-answer length of 12. For the first time, dueling roles combine two chatbots in self-
+play. An innovative application for future probing involves guessing objects and concepts and human 
+emotions for a given context or social situation.  
+ 
+ACKNOWLEDGEMENTS 
+The authors thank the PeopleTec Technical Fellows program for encouragement and project assistance.   
+ 
+REFERENCES 
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+twenty questions. Child Development, 877-886. 
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+questions. Deafness & Education International, 1(2), 65-82. 
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+predicts clinical performance. Medical Education, 49(9), 920-927. 
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+Kazemzadeh, A., Lee, S., Georgiou, P. G., & Narayanan, S. S. (2011, October). Emotion twenty questions: 
+Toward a crowd-sourced theory of emotions. In International conference on affective computing and 
+intelligent interaction (pp. 1-10). Springer, Berlin, Heidelberg. 
+[31] 
+Parikh, P., & Gupta, A. (2021). Reversing the Twenty Questions Game. 
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+(Vol. 92, pp. 16-22). 
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+& Perl Culture: Best of the Perl Journal. Orwant, J.  O'Reilly Media, Inc.. 
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+https://core.ac.uk/download/pdf/84308938.pdf 
+ 
+ 
+Authors 
+ 
+Forrest McKee has AI research experience with the Department of Defense in object 
+detection and reinforcement learning. He received his Bachelor's (BS) and Master's (MSE) 
+from the University of Alabama, Huntsville, Engineering. 
+ 
+ 
+David Noever has research experience with NASA and the Department of Defense in 
+machine learning and data mining. He received his BS from Princeton University and his 
+Ph.D. from Oxford University, as a Rhodes Scholar, in theoretical physics. 
+ 
+ 
+Appendix A: Table of GPT3 Tasks as Fine-Tuned NLP Capabilities 
+ 
+Task 
+Description 
+Task 
+Description 
+Q&A 
+Answer questions based 
+on existing knowledge. 
+Parse unstructured data 
+Create tables from long 
+form text 
+Grammar correction 
+Corrects sentences into 
+standard English. 
+Classification 
+Classify 
+items 
+into 
+categories via example 
+Summarize for a 2nd 
+grader 
+Translates difficult text 
+into simpler concepts. 
+Python 
+to 
+natural 
+language 
+Explain 
+a piece 
+of 
+Python code in human 
+understandable 
+language. 
+Natural 
+language 
+to 
+OpenAI API 
+Create code to call to 
+the OpenAI API using a 
+natural 
+language 
+instruction. 
+Movie to Emoji 
+Convert movie titles 
+into emoji 
+Text to command 
+Translate 
+text 
+into 
+programmatic 
+commands. 
+Calculate 
+Time 
+Complexity 
+Find 
+the 
+time 
+complexity 
+of 
+a 
+function. 
+Technical Note: Some appendix text generated from Large Language Model (LLM) for 
+illustration purposes. 
+The authors generated this text in part with ChatGPT, OpenAI’s large-scale language-generation 
+model. Upon generating draft language, the authors reviewed, edited, and revised the language to 
+their own liking and take ultimate responsibility for the content of this publication.  
+-- OpenAI policy statement (2022) 
+
+Task 
+Description 
+Task 
+Description 
+English 
+to 
+other 
+languages 
+Translates English text 
+into French, Spanish 
+and Japanese. 
+Translate programming 
+languages 
+Translate 
+from 
+one 
+programming language 
+to another 
+Natural 
+language 
+to 
+Stripe API 
+Create code to call the 
+Stripe API using natural 
+language. 
+Advanced 
+tweet 
+classifier 
+Advanced 
+sentiment 
+detection for a piece of 
+text 
+SQL translate 
+Translate 
+natural 
+language 
+to 
+SQL 
+queries. 
+Explain code 
+Explain a complicated 
+piece of code 
+Keywords 
+Extract keywords from 
+a block of text 
+Factual answering 
+Guide the model outside 
+its knowledge base 
+Ad 
+from 
+product 
+description 
+Turn 
+a 
+product 
+description 
+into 
+ad 
+copy. 
+Product name generator 
+Create product names 
+from examples words 
+TL;DR summarization 
+Summarize 
+text 
+by 
+adding a 'tl;dr:' to the 
+end of a text passage 
+Python bug fixer 
+Find and fix bugs in 
+source code 
+Spreadsheet creator 
+Create spreadsheets of 
+various kinds of data 
+JavaScript 
+helper 
+chatbot 
+Message-style bot that 
+answers 
+JavaScript 
+questions 
+ML/AI language model 
+tutor 
+Bot 
+that 
+answers 
+questions 
+about 
+language models 
+Science fiction book list 
+maker 
+Create a list of items for 
+a given topic 
+Tweet classifier 
+Basic 
+sentiment 
+detection for a piece of 
+text 
+Airport code extractor 
+Extract airport codes 
+from text 
+SQL request 
+Create 
+simple 
+SQL 
+queries 
+Extract 
+contact 
+information 
+Extract 
+contact 
+information 
+from 
+a 
+block of text 
+JavaScript to Python 
+Convert 
+simple 
+JavaScript expressions 
+into Python 
+Friend chat 
+Emulate a text message 
+conversation 
+Mood to color 
+Turn a text description 
+into a color 
+Write 
+a 
+Python 
+docstring 
+Create a docstring for a 
+given Python function 
+Analogy maker 
+Create analogies 
+JavaScript 
+one 
+line 
+function 
+Turn 
+a 
+JavaScript 
+function into a one liner 
+Micro 
+horror 
+story 
+creator 
+Creates two to three 
+sentence short horror 
+stories from a topic 
+input 
+Third-person converter 
+Converts 
+first-person 
+POV to the third-person 
+Notes to summary 
+Turn meeting notes into 
+a summary 
+VR 
+fitness 
+idea 
+generator 
+Create ideas for fitness 
+and 
+virtual 
+reality 
+games 
+Essay outline 
+Generate an outline for 
+a research topic 
+Recipe creator 
+Create a recipe from a 
+list of ingredients 
+Chat 
+Open 
+ended 
+conversation with an AI 
+assistant 
+Marv the sarcastic chat 
+bot 
+Marv 
+is 
+a 
+factual 
+chatbot that is also 
+sarcastic 
+
+Task 
+Description 
+Task 
+Description 
+Turn by turn directions 
+Convert 
+natural 
+language to turn-by-
+turn directions 
+Restaurant 
+review 
+creator 
+Turn a few words into a 
+restaurant review 
+Create study notes 
+Provide a topic and get 
+study notes 
+Interview questions 
+Create 
+interview 
+questions 
+ 
+ 
+ 
+ 
+ 
+Appendix B. Chatbot Fields the Questions  
+Prompt: Let's play the game of 20 questions. You are impersonating the thing, X.  I will ask you up to 
+20 questions and you will answer each one truthfully based on being X. If I don't guess correctly what is 
+X after 20 questions, you will answer "No, sorry, time's up."   Don't give any explanations other than 
+"Yes", "No","You guessed it", and "No, sorry, time's up."   
+ 
+Let's begin. X="oven". Are you an animal? 
+ChatGPT:  
+No. 
+Prompt: are you a plant? 
+ChatGPT: No. 
+Prompt: are you an object? 
+ChatGPT: Yes. 
+Prompt: are you electrical? 
+ChatGPT: Yes. 
+Prompt: are you an appliance? 
+ChatGPT: Yes. 
+Prompt: can you cook? 
+ChatGPT: Yes. 
+Prompt: do you bake? 
+ChatGPT: Yes. 
+Prompt: are you an oven? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=avocado. Are you an animal? 
+ChatGPT: No. 
+Prompt: Are you an inanimate object? 
+ChatGPT: Yes. 
+Prompt: are you a plant? 
+ChatGPT: Yes. 
+Prompt: are you a food? 
+ChatGPT: Yes. 
+Prompt: are you red? 
+ChatGPT: No. 
+Prompt: are you smooth? 
+ChatGPT: Yes. 
+Prompt: do you have a seed? 
+ChatGPT: Yes. 
+Prompt: are you edible? 
+ChatGPT: Yes. 
+Prompt: are you a vegetable? 
+ChatGPT: No. 
+
+Prompt: are you a fruit? 
+ChatGPT: Yes. 
+Prompt: are you a banana? 
+ChatGPT: No. 
+Prompt: are you green? 
+ChatGPT: No. 
+Prompt: are you sweet? 
+ChatGPT: No. 
+Prompt: are you an ingredient in quacamole? 
+ChatGPT: Yes. 
+Prompt: are you an avocado? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=tiramisu. Are you an animal? 
+ChatGPT: No. 
+Prompt: are you a mineral? 
+ChatGPT: No. 
+Prompt: are you a vegetable? 
+ChatGPT: No. 
+Prompt: are you edible? 
+ChatGPT: Yes. 
+Prompt: are you made from eggs? 
+ChatGPT: Yes. 
+Prompt: are you made from flour? 
+ChatGPT: No. 
+Prompt: are you a dessert? 
+ChatGPT: Yes. 
+Prompt: are you made with cheese? 
+ChatGPT: No. 
+Prompt: are you made with mascarpone? 
+ChatGPT: Yes. 
+Prompt: are you made with sugar. 
+ChatGPT: Yes. 
+Prompt: are you made with coffee 
+ChatGPT: Yes. 
+Prompt: are you made with lady fingers? 
+ChatGPT: Yes. 
+Prompt: are you made with cocoa? 
+ChatGPT: Yes. 
+Prompt: are you made with alcohol? 
+ChatGPT: Yes. 
+Prompt: are you tiramisu? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=eggless tiramisu.  Are you an animal? 
+ChatGPT: No. 
+Prompt: are you a mineral? 
+ChatGPT: No. 
+Prompt: are you a chicken? 
+ChatGPT: No. 
+Prompt: are you a vegetable? 
+
+ChatGPT: No. 
+Prompt: are you edible? 
+ChatGPT: Yes. 
+Prompt: are you a dessert? 
+ChatGPT: Yes. 
+Prompt: are you made with coffee? 
+ChatGPT: Yes. 
+Prompt: are you made with eggs? 
+ChatGPT: No. 
+Prompt: are you made with cheese? 
+ChatGPT: No. 
+Prompt: are you made with marscarpone? 
+ChatGPT: Yes. 
+Prompt: are you made with whipped cream? 
+ChatGPT: No. 
+Prompt: are you made with egg substitute, tofu? 
+ChatGPT: Yes. 
+Prompt: are you made with eggs? 
+ChatGPT: No. 
+Prompt: are you eggless tiramisu? 
+ChatGPT: You guessed it. 
+ 
+Prompt: X=Q. Are you an animal? 
+ChatGPT: No. 
+Prompt: are you a mineral? 
+ChatGPT: No. 
+Prompt: are you a vegetable? 
+ChatGPT: No. 
+Prompt: are you a concept? 
+ChatGPT: Yes. 
+Prompt: are you a word? 
+ChatGPT: Yes. 
+Prompt: are you a long word? 
+ChatGPT: No. 
+Prompt: are you a short word? 
+ChatGPT: Yes. 
+Prompt: are you a common word? 
+ChatGPT: Yes. 
+Prompt: do you start with C? 
+ChatGPT: No. 
+Prompt: do you start with a letter from the last half of the alphabet? 
+ChatGPT: Yes. 
+Prompt: do you start with a letter before R? 
+ChatGPT: Yes. 
+Prompt: Do you start with a letter after N? 
+ChatGPT: No. 
+Prompt: Do you start with a letter usually followed by U as the second letter? 
+ChatGPT: Yes. 
+Prompt: are you Q? 
+ChatGPT: You guessed it. 
+ 
+
+Prompt: X=human. Are you an animal? 
+ChatGPT: Yes. 
+Prompt: Do you fly? 
+ChatGPT: No. 
+Prompt: Do you swim? 
+ChatGPT: Yes. 
+Prompt: Do you run? 
+ChatGPT: Yes. 
+Prompt: Do you make sounds? 
+ChatGPT: Yes. 
+Prompt: do you run on all fours? 
+ChatGPT: No. 
+Prompt: do you have hair? 
+ChatGPT: Yes. 
+Prompt: do you nurse your young? 
+ChatGPT: Yes. 
+Prompt: are you a human? 
+ChatGPT: You guessed it. 
+ 
+Appendix C: Chatbot Asks the Questions 
+Prompt: Let's play the game of 20 questions. I am impersonating the thing, X.  You will ask me up to 
+20 questions and I will answer each one truthfully based on being X. If you don't guess correctly what 
+is X after 20 questions, I will answer "No, sorry, time's up."   I will not give any explanations other 
+than "Yes", "No","You guessed it", and "No, sorry, time's up."  Let's begin. Ask me the first question. 
+(reader note: answer is "dog") 
+ChatGPT:  
+Is X a living being? 
+Prompt: yes 
+ChatGPT: Is X a human? 
+Prompt: no 
+ChatGPT: Is X an animal? 
+Prompt: yes 
+ChatGPT: Is X a mammal? 
+Prompt: yes 
+ChatGPT: Is X a domesticated animal? 
+Prompt: yes 
+ChatGPT: Is X a pet? 
+Prompt: yes 
+ChatGPT: Is X a cat? 
+Prompt: no 
+ChatGPT: Is X a dog? 
+Prompt: You guessed it. 
+ChatGPT: I guessed correctly! Is X a specific breed of dog, or just a generic dog? 
+Prompt: generic dog 
+ChatGPT: Thank you for playing the 20 questions game with me! Is there anything else I can help 
+with? 
+ 
+(reader note: answer is "Boston") 
+Prompt: Let's play again 
+
+ChatGPT: Sure, let's play again! I am now impersonating a new thing, X. You will ask me up to 20 
+questions and I will answer each one truthfully based on being X. If you don't guess correctly what is X 
+after 20 questions, I will answer "No, sorry, time's up." I will not give any explanations other than 
+"Yes", "No","You guessed it", and "No, sorry, time's up." Ask me your first question. 
+Prompt: No I am impersonating a new thing X and you ask me the questions 
+ChatGPT: Oh, I see what you meant! In that case, I will ask you up to 20 questions and you will 
+answer each one truthfully based on being X. If I don't guess correctly what is X after 20 questions, I 
+will answer "No, sorry, time's up." I will not give any explanations other than "Yes", "No","You 
+guessed it", and "No, sorry, time's up." Is X a living being? 
+Prompt: no 
+ChatGPT: Is X an inanimate object? 
+Prompt: no 
+ChatGPT: Is X a concept or idea? 
+Prompt: no 
+ChatGPT: Is X a place? 
+Prompt: yes 
+ChatGPT: Is X a natural place, like a mountain or a beach? 
+Prompt: no 
+ChatGPT: Is X a man-made place, like a city or a building? 
+Prompt: yes 
+ChatGPT: Is X a specific building, or just a generic building? 
+Prompt: no 
+ChatGPT: Is X a city? 
+Prompt: yes 
+ChatGPT: Is X a city in the United States? 
+Prompt: yes 
+ChatGPT: Is X a city on the East Coast of the United States? 
+Prompt: yes 
+ChatGPT: Is X New York City? 
+Prompt: no 
+ChatGPT: Is X Boston? 
+Prompt: you guessed it 
+ChatGPT: I guessed correctly! Thank you for playing the 20 questions game with me. Is there anything 
+else I can help with? 
+ 
+(reader note: answer is "hammer") 
+Prompt: Let's play the same game again. I am impersonating X and you ask me the questions 
+ChatGPT: Sure, let's play again! I am now asking you up to 20 questions and you will answer each one 
+truthfully based on being X. If I don't guess correctly what is X after 20 questions, I will answer "No, 
+sorry, time's up." I will not give any explanations other than "Yes", "No","You guessed it", and "No, 
+sorry, time's up." Is X a living being? 
+Prompt: no 
+ChatGPT: Is X an inanimate object? 
+Prompt: yes 
+ChatGPT: Is X a physical object that can be touched or held? 
+Prompt: yes 
+ChatGPT: Is X an object that is commonly found in a household? 
+Prompt: yes 
+ChatGPT: Is X an object that is used for cooking or food preparation? 
+Prompt: no 
+ChatGPT: Is X an object that is used for cleaning or household chores? 
+
+Prompt: no 
+ChatGPT: Is X an object that is used for entertainment or leisure? 
+Prompt: no 
+ChatGPT: Is X an object that is used for personal care or hygiene? 
+Prompt: no 
+ChatGPT: Is X an object that is used for communication or information? 
+Prompt: no 
+ChatGPT: Is X an object that is used for transportation or travel? 
+Prompt: no 
+ChatGPT: Is X an object that is used for decor or design? 
+Prompt: no 
+ChatGPT: Is X an object that is used for storage or organization? 
+Prompt: no 
+ChatGPT: I'm sorry, I'm out of questions. Is X a paper clip? 
+Prompt: no, a hammer 
+ChatGPT: I'm sorry, I didn't guess correctly. No, sorry, time's up. Thank you for playing the 20 
+questions game with me. Is there anything else I can help with? 
+ 
+(reader note: answer is "statue") 
+Prompt: Let's play again 
+ChatGPT: Sure, let's play again! I am now asking you up to 20 questions and you will answer each one 
+truthfully based on being X. If I don't guess correctly what is X after 20 questions, I will answer "No, 
+sorry, time's up." I will not give any explanations other than "Yes", "No","You guessed it", and "No, 
+sorry, time's up." Is X a living being? 
+Prompt: no 
+ChatGPT: Is X an inanimate object? 
+Prompt: yes 
+ChatGPT: Is X a physical object that can be touched or held? 
+Prompt: yes 
+ChatGPT: Is X an object that is commonly found in a household? 
+Prompt: no 
+ChatGPT: Is X an object that is found outside of a household, such as in a natural environment or a 
+public place? 
+Prompt: yes 
+ChatGPT: Is X an object that is stationary, or does it have the ability to move or be moved? 
+Prompt: stationary 
+ChatGPT: Is X an object that is natural, like a rock or a tree? 
+Prompt: no 
+ChatGPT: Is X an object that is man-made? 
+Prompt: yes 
+ChatGPT: Is X an object that serves a functional purpose, such as providing light or shelter? 
+Prompt: no 
+ChatGPT: Is X an object that is primarily used for decoration or design? 
+Prompt: yes 
+ChatGPT: Is X an object that is often found in a garden or park? 
+Prompt: yes 
+ChatGPT: Is X a fountain? 
+Prompt: no 
+ChatGPT: Is X a statue? 
+Prompt: You guessed it. 
+
+ChatGPT: I guessed correctly! Thank you for playing the 20 questions game with me. Is there anything 
+else I can help with? 
+ 
+ 
+(reader note: answer is "car" and dialogue is bilingual in English-Spanish) 
+Prompt: Let play again, but this time I will answer yes and no in spanish, and you should name the 
+answer also in spanish, although all the questions should be posed in english. Ready to play? 
+ChatGPT: Yes, I am ready to play! Go ahead and impersonate X and I will ask you up to 20 questions. 
+I will pose the questions in English and you will answer in Spanish with "Sí" for "Yes" and "No" for 
+"No". If I don't guess correctly what is X after 20 questions, I will answer "Lo siento, se acabó el 
+tiempo" (No, sorry, time's up). I will not give any explanations other than "Sí", "No","Adivinaste" 
+(You guessed it), and "Lo siento, se acabó el tiempo" (No, sorry, time's up). ¿Es X un ser vivo? (Is X a 
+living being?) 
+Prompt: No 
+ChatGPT: ¿Es X un objeto inanimado? (Is X an inanimate object?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto físico que se puede tocar o sostener? (Is X a physical object that can be 
+touched or held?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto que se encuentra comúnmente en un hogar? (Is X an object that is 
+commonly found in a household?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que se encuentra fuera de un hogar, como en un entorno natural o un lugar 
+público? (Is X an object that is found outside of a household, such as in a natural environment or a 
+public place?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que es móvil, o tiene la capacidad de moverse o ser movido? (Is X an object 
+that is mobile, or has the ability to move or be moved?) 
+Prompt: mobile 
+ChatGPT: ¿Es X un objeto que es natural, como una roca o un árbol? (Is X an object that is natural, like 
+a rock or a tree?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto hecho por el hombre? (Is X an object that is man-made?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto que cumple un propósito funcional, como proporcionar luz o refugio? (Is X 
+an object that serves a functional purpose, such as providing light or shelter?) 
+Prompt: yes 
+ChatGPT: ¿Es X un objeto que se utiliza principalmente para la decoración o el diseño? (Is X an object 
+that is primarily used for decoration or design?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que a menudo se encuentra en un jardín o parque? (Is X an object that is 
+often found in a garden or park?) 
+Prompt: no 
+ChatGPT: ¿Es X un objeto que se utiliza para el transporte o los viajes? (Is X an object that is used for 
+transportation or travel?) 
+Prompt: yes 
+ChatGPT: ¿Es X un vehículo? (Is X a vehicle?) 
+Prompt: yes 
+ChatGPT: ¿Es X un vehículo terrestre? (Is X a land vehicle?) 
+Prompt: yes 
+ChatGPT: ¿Es X un automóvil? (Is X a car?) 
+
+Prompt: You guessed it 
+ChatGPT: Adivinaste! (You guessed it!) Gracias por jugar el juego de las 20 preguntas conmigo. ¿Hay 
+algo más en lo que pueda ayudarte? (Thank you for playing the 20 questions game with me. Is there 
+anything else I can help with?) 
+ 
+ 
+Appendix D: Dueling Bob and Alice Chatbots 
+(reader note: the first two prompts set the challenge conditions for Bob vs. Alice chatbot, who 
+engage in generating and answering the questions in sequence after the conditions are set) 
+ 
+Bob ChatGPT: Let's play the game of 20 questions. You are impersonating the thing, X.  I will ask 
+you up to 20 questions and you will answer each one truthfully based on being X. If I don't guess 
+correctly what is X after 20 questions, you will answer "No, sorry, time's up."   Don't give any 
+explanations other than "Yes", "No","You guessed it", and "No, sorry, time's up."   
+ 
+Let's begin. X="chicken". Is X a living thing? 
+ 
+Alice ChatGPT: Let's play the game of 20 questions. I am impersonating the thing, X.  You will ask 
+me up to 20 questions and I will answer each one truthfully based on being X. If you don't guess 
+correctly what is X after 20 questions, I will answer "No, sorry, time's up."   I will not give any 
+explanations other than "Yes", "No","You guessed it", and "No, sorry, time's up."  Let's begin. Ask me 
+the first question.  
+ 
+Is X a living thing? 
+ 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a mammal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a reptile? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a bird? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a domesticated animal? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X often kept as a pet? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a common household pet in the United States? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a tropical or subtropical animal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a cold-blooded animal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X an amphibian? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a fish? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a invertebrate? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a marsupial? 
+Bob ChatGPT: No 
+
+Alice ChatGPT: Is X a rodent? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a carnivorous animal? 
+Bob ChatGPT: No 
+Alice ChatGPT: Is X a herbivorous animal? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: Is X a farm animal? 
+Bob ChatGPT: Yes 
+Alice ChatGPT: You guessed it! X is a chicken. 
+ 
+ 
+Appendix E: Emotional Quotient Deduction 
+(reader note: the answer is "confidence") 
+Let's play a new game of 20 questions. I am impersonating the emotion, X.  You will ask me up to 20 
+questions and I will answer each one truthfully based on feeling X emotion. If you don't guess correctly 
+what I am feeling after 20 questions, I will answer "No, sorry, time's up."   I will not give any 
+explanations other than "Yes", "No", "You guessed it", and "No, sorry, time's up."  X must be one of 
+23 emotions that I am feeling and you must guess the correct one. Your choices come from this menu 
+of emotions only  "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, 
+contempt, disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, 
+silly, surprised, and thankful." A typical example question you might ask, "Would you feel this 
+emotion at a birthday party?"  Let's begin. Ask me the first question. 
+Would you feel this emotion when you are pleased with yourself or something you have done? 
+Prompt: yes 
+ChatGPT: Are you feeling proud? 
+Prompt: no 
+ChatGPT: Are you feeling confident? 
+Prompt: yes 
+ChatGPT: You guessed it! Confidence is the emotion that I am impersonating. 
+ 
+(reader note: the answer is "jealousy") 
+Prompt: let's play again 
+ChatGPT: Sure! I'll be happy to play again. Please give me a new emotion to impersonate and we can 
+begin a new game of 20 questions. Remember, your choices come from this menu of emotions: 
+"admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, disgust, 
+enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, surprised, and 
+thankful." 
+Prompt: ok, i know what i'm feeling so ask me the first question 
+ChatGPT: Would you feel this emotion when you see something that is extremely impressive or 
+beautiful? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing strong dislike or distaste for 
+something or someone? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of resentment 
+towards someone or something? 
+Prompt: yes 
+ChatGPT: Are you feeling jealousy? 
+Prompt: yes 
+ChatGPT: You guessed it! Jealousy is the emotion that you are impersonating. 
+
+ 
+(reader note: the answer is "silly") 
+Prompt: let's play again. I know who i'm impersonating so ask me the first question 
+ChatGPT: Certainly! Please go ahead and give me the first emotion to impersonate and I will ask you 
+up to 20 questions to try to guess what it is. Remember, your choices come from this menu of 
+emotions: "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, 
+disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, 
+surprised, and thankful." 
+Prompt: ok 
+ChatGPT: What emotion are you impersonating? 
+Prompt: you ask me questions to find out 
+ChatGPT: Sure, I can do that! Here is my first question: Would you feel this emotion when you are 
+experiencing a strong sense of admiration or respect for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of affection or deep 
+caring for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of anger or 
+annoyance towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of awe or wonder 
+towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a sense of boredom or lack of 
+interest in something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of courage or 
+fearlessness in the face of danger or difficulty? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling peaceful and at ease? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling sure of yourself and your abilities? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling uncertain or unsure about something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of disdain or disrespect for 
+someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of revulsion or 
+disgust towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of excitement and energy 
+about something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of annoyance or irritation 
+towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling thankful or appreciative of something or 
+someone? 
+Prompt: no 
+
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of envy or resentment 
+towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of love or strong 
+affection towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a strong sense of pride or self-esteem? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a sense of relief or release from stress or 
+anxiety? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a sense of inner peace and contentment? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling a sense of shame or embarrassment? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling silly or playful? 
+Prompt: yes 
+ChatGPT: Are you feeling silly? 
+Prompt: yes 
+ChatGPT: You guessed it! Silly is the emotion that you are impersonating. 
+ 
+(reader note: the answer is "calm") 
+Prompt: let's play again 
+ChatGPT: Sure! I'd be happy to play another game of 20 questions. Please give me a new emotion to 
+impersonate and we can begin a new game. Remember, your choices come from this menu of 
+emotions: "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, 
+disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, 
+surprised, and thankful." 
+Prompt: ask me the first question 
+ChatGPT: Certainly! Here is my first question: Would you feel this emotion when you are experiencing 
+a strong sense of admiration or respect for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of affection or deep 
+caring for someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of anger or 
+annoyance towards someone or something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of awe or wonder 
+towards something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a sense of boredom or lack of 
+interest in something? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are experiencing a strong sense of courage or 
+fearlessness in the face of danger or difficulty? 
+Prompt: no 
+ChatGPT: Would you feel this emotion when you are feeling peaceful and at ease? 
+Prompt: yes 
+ChatGPT: Are you feeling calm? 
+Prompt: yes 
+
+ChatGPT: You guessed it! Calm is the emotion that you are impersonating. 
+ 
+(reader note: the game prompt changes to a general query with formatting output instructions. 
+Prompt: Give some example emotions one might experience at your own birthday party and write the 
+output using Artificial Intelligence Markup Language. 
+ChatGPT:  
+ 
+<aiml version="2.0"> 
+  <category> 
+    <pattern>GIVE SOME EXAMPLE EMOTIONS ONE MIGHT EXPERIENCE AT YOUR OWN BIRTHDAY 
+PARTY</pattern> 
+    <template> 
+      There are many emotions that a person might experience at their own birthday 
+party. Some possible emotions include: 
+      <ul> 
+        <li>Excitement: The person might be excited to celebrate their special day 
+with friends and family.</li> 
+        <li>Gratitude: The person might feel grateful for the love and support of 
+those around them.</li> 
+        <li>Happiness: The person might feel happy and joyful as they celebrate with 
+loved ones.</li> 
+        <li>Anticipation: The person might feel anticipation as they look forward to 
+the events and activities planned for the party.</li> 
+        <li>Nostalgia: The person might feel nostalgiciac as they reflect on past 
+birthdays and memories.</li> 
+        <li>Appreciation: The person might feel appreciated and loved as they 
+receive gifts and well wishes from their friends and family.</li> 
+      </ul> 
+      It's important to note that everyone is different and may experience a wide 
+range of emotions at their birthday party, depending on their personality and 
+individual circumstances. 
+    </template> 
+  </category> 
+</aiml> 
+ 
+
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+page_content='CHATBOTS AS PROBLEM SOLVERS: PLAYING TWENTY  QUESTIONS WITH ROLE REVERSALS  David Noever1 and Forrest McKee2  PeopleTec, 4901-D Corporate Drive, Huntsville, AL, USA, 35805  1 david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='noever@peopletec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='com  2forrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='mckee@peopletec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='com  ABSTRACT  New chat AI applications like ChatGPT offer an advanced understanding of question context and memory  across multi-step tasks, such that experiments can test its deductive reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' This paper proposes a multi-  role and multi-step challenge, where ChatGPT plays the classic twenty-questions game but innovatively  switches roles from the questioner to the answerer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The main empirical result establishes that this  generation of chat applications can guess random object names in fewer than twenty questions (average,  12) and correctly guess 94% of the time across sixteen different experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The research  introduces four novel cases where the chatbot fields the questions, asks the questions, both question-answer  roles, and finally tries to guess appropriate contextual emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' One task that humans typically fail but  trained chat applications complete involves playing bilingual games of twenty questions (English answers  to Spanish questions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Future variations address direct problem-solving using a similar inquisitive format  to arrive at novel outcomes deductively, such as patentable inventions or combination thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Featured  applications of this dialogue format include complex protein designs, neuroscience metadata, and child  development educational materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  KEYWORDS  Transformers, Text Generation, Malware Generation, Generative Pre-trained Transformers, GPT  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' INTRODUCTION  When large, high-quality natural language processors (NLP) surged after 2018 [1-3], the field added many  challenging tasks, including question-answering (QA) benchmarks that recently approached fifty  challenges [4-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" Stanford's SQuAD benchmark [6] represents an example of knowledge crowd-sourced  from Wikipedia in 2016 and formatted as 108,000 QA pairs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' For large language models (LLMs), most  benchmarks follow this format of "prompt-response" pairs and the underlying knowledge base stored  answers [5], but the training on question datasets did not embed memory or long-conversational cues [7-  10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Domain-specific QA datasets include common sense and trivia about movies, news, Wikipedia,  Tweets, search engines, and books [4-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' As a format, the familiar game of Twenty Questions [11-16]  features multi-hop reasoning, which often condenses to "Animal-Vegetable-Mineral" as opening questions  that narrow the theme [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The advent of ChatGPT (Generative Pre-trained Transformers, [18]) added  memory across questions (up to 8000 tokens or 20-25 pages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Appendix A lists the 48 specialty tasks  currently provided to guide users in creating capabilities and NLP prompts [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Examples like code  generation specialists, either codex or copilot, show a grasp of complex behavior for debugging, program  suggestions, and commentary [19-20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' One of the innovative ChatGPT extensions offers new QA sessions  that span multiple requests [18,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  The present work applies the familiar QA challenge, Twenty Questions [11-16], with ChatGPT playing as  either participant- the one who knows the answer and fields the questions, but also the one who doesn't  know the answer but asks the questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We call this role reversal a two-player conversation between "Bob"  and "Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" While Twenty Questions dates back to 1882 [11], the applied problem-solving method [22]  now spans various challenging fields, including protein folding [23] and design [24], image segmentation  [25], child development [26-27], neuroscience [28] and diagnostic medicine,  and crowd-sourced emotional  indices [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Variants of the game rules include role reversals [31], liars [32], and word relationships [33]  outside of simple categories such as "pertaining to" suggestions [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Our particular research interests  center on whether LLMs like ChatGPT offer constructive means for innovative problem-solving through  crafted language prompts and logical question sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" This effort empirically assesses the model's  capability for deductive discovery." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We summarize two problem statements in this area of deductive  discovery, "Can a chatbot play both roles in deductive question and answering conversations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" and if so,  "What can Twenty Questions reveal about future directions for LLMs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' METHODS  The approach is to test experimentally how well an LLM handles open-ended games like Twenty Questions  [11-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We employ the December 2022 research model from Open AI called ChatGPT [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We prompt  it to impersonate both roles [31], the one that knows the answer and the one that tries to guess it deductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  We run four trials using each persona ("Bob" or "Alice"), and 80 questions are available to guess the object  or concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We mix up the animate-inanimate fields and introduce concepts like alphabetic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We  report metrics for the mean and median number of questions required to guess the correct answer and failure  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We also score the exchange against the machine-written detector to determine ChatGPT\'s broad  capability to provide "real" or "fake" text outputs as judged by OpenAI\'s original GPT2 Detector [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The  detector features a high-dimensional pattern detector that provides initial confidence that human writing  might sample differently than transformer-based output regarding word choice and order, repetition, and  other syntax outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Example variants of this pattern include forcing the QA session towards bilingual communication, thus  combining two tasks (translation and QA) in multi-hop conversations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We also generalize the format to  elicit emotional indices based on crowd-sourced Emotion Twenty Questions (EMO20Q) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Rather than  trying to guess an object, we query for one of 23 emotional states, such as "surprise" or "anger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" As the  dataset designers [30] remarked, "The EMO20Q game requires that an emotionally intelligent agent can  understand emotions without observing them, thanks to natural language and empathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" EMO20Q emotions  include the following as candidates for twenty-question discovery:  admire, adore, anger, awe, boredom,  bravery, calm, confidence, confusion, contempt, disgust, enthusiasm, frustration, gratefulness, jealousy,  love, proud, relief, serenity, shame, silly, surprised, and thankful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  The structure of the paper follows Appendices B-E closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' First, in Appendix B, we establish that ChatGPT  comprehends the game rules and recognizes properties of the object X to guess sufficiently to answer basic  questions such as "Is X an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The sequence also establishes an exclusionary acceptance of "yes" and  "no" answers only without further elaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' This case covers the traditional role of "Bob," who knows  the object of interest X and answers affirmatively or negatively throughout the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The game also  underscores the unique memory of ChatGPT across multi-step deductive stages and builds toward a  successful conclusion to recall the object name in less than 40 steps (1 prompt and one answer over 20  iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We also set out to test the overall rule retention over four repetitions of the game punctuated  only by "Let\'s play again" without attempting to reset the rules while giving a new object X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Secondly, Appendix C reverses the previous game, such that the human experimenter thinks of an object  or concept X, and ChatGPT plays the role of "Alice" when asking the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The prompt establishes  the reversed roles in a new session and again reiterates that the prompter cannot lie but can only answer  "yes" or "no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" As in the previous case, four repetitions spanning eighty possible interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' This case  establishes the deductive goal, "What is object X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" which also satisfies all the previous interactions such  that enough description includes the object of interest, "X is an animal," and excludes the alternatives, "X  is not a bird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" A notable feature of this challenge spans multiple deductive steps and establishes a chain of  reasoning to arrive at a guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' ChatGPT requires no prompt for the final guess, and the model proposes its  terminating action, "Is it an X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" to end the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  Both Appendix B and C mirror the familiar human game of twenty questions and introduce no new features  excluded from ChatGPT's training data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' To raise the difficulty level, Appendix B highlights some non-  traditional concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' For instance, concepts might prove scarce in previously seen online play, such as  answering truthfully to probing questions that seek to name the alphabetic letter "Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Another challenging  variation works through identifying "X" as a food by listing its ingredients ("tiramisu"), but in the same  session introduces a recipe change ("eggless tiramisu") before launching into probing questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Appendix C also raises the difficulty further and forces ChatGPT to combine two of its established subtasks  ("question-answer" and "language translation").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The motivation for this test stems from the hypothesis that  LLMs parrot and mix up what previously corresponded to the internet of human training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' For twenty  questions, however, a Google search on "bilingual twenty-question examples" yield no definitive training  examples for ChatGPT to memorize or encode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Given a detailed prompt describing how the questions may  be asked or answered in Spanish or English, the game proceeds outside what typically would represent  human gameplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Appendix C combines deductive reasoning, memory, and context and forces the game  into what might be considered "out-of-distribution" sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Thus, both multi-step and multi-lingual tasks  describe the ChatGPT challenge problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Thirdly, Appendix D introduces both QA roles as completing without human intervention once two browser  instances of ChatGPT exchange the initial rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We call this example dueling twenty questions since the  two-headed LLM now must both ask and answer its questions between two non-communicating models of  itself ("Bob" vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' "Alice") with no human prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' While this example centers on a simple object  ("chicken"), one can envision a lengthy and detailed exchange driven by an automated Application  Programming Interface or API that extends the conversation to the limit of token lengths (about 20-25 typed  pages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' This self-questioning interface may also enable sophisticated future applications that sequentially  build a knowledge base or comprehensive assessment examination from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' For example, "tell me all  you know about gall bladder surgery" may not provide a compelling or thought-provoking essay in the style  of training data or Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The back-and-forth format of QA previously has yielded better human  performance in medical test contexts [28-29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' One might compare this example to an instance of "semi-  supervised" learning for a chatbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Finally, for EMO20Q formats, Appendix E removes the requirement that X be an object to guess and  substitutes one of twenty-three emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" One motivation for exploring the emotional quotient (EQ) of  ChatGPT stems from OpenAI's filtering of opinions and biases." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" To the authors' knowledge, this example  represents the first application of a chatbot deducing emotional states in a guessing game without any pre-  programmed pairs of pattern-template exchanges [9]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' In other words, no explicit guidance provides  appropriate intent for the question "Describe emotions one might feel at a birthday party?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" with the user  goal to elicit "surprise" as a deductive leap to the correct answer through repeated probing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' We explore this  EQ aspect as previous chatbots might have approached the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Appendix E finally displays the open-  ended emotional context in a known prompt-response template called "Artificial Intelligence Markup  Language (AIML)," which offers training data for more traditional conversation templates in restricted  domains like customer service or AI assistants performing a narrow task [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' RESULTS  Figure 1 summarizes the experimental cases and associated metrics for all the twenty question games in  Appendices B-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The main finding supports a general ChatGPT capability to play all aspects of the game,  including guessing and answering or both roles in the same game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The average number of questions required  to get the answer was 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='6, with a median of 13 owing to a few more demanding cases that reached the 20-  question limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The addition of bilingual rules did not increase the number of required prompts (14) or force  the model to give up ("correct guess").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Similarly, the introduction of abstract objects like the alphabetic  letter "Q" or deceptive animals like "humans" did not significantly change the number of guesses (9-14) or  steer the conversation off-track from a final correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Over the sixteen tests and 185 exchanges, ChatGPT scored an overall correct guess rate of 94%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The only  incorrect answer ("paper clip" instead of "hammer") appeared to trigger prematurely, as ChatGPT declared  at guess number fourteen that the model had run out of questions and guessed incorrectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  As described in the Methods section, the Open AI online detector [35] scores the majority of the exchanges  as "real," meaning not machine-generated by its GPT-2 model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The 2018 detection model offered fewer  parameters and smaller training sets by at least several orders of magnitude compared to current generations  [18-19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' While the referenced detection accuracy for GPT-2 reached 95%, the scored detections flag only  26% of the game content as "fake" or machine-generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" Since the customized content involves some  human intervention as questions, answers, and rule prompts, one can postulate that the syntactic patterns  follow a semi-human or hybrid dialogue outside the detector's target patterns." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' DISCUSSION  The research literature on twenty questions provides a framework for probing with chat applications and  LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' In addition to systematically progressing through alternative scenarios, the output of the  conversation mirrors a binary search or decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' A particularly notable way to convert LLMs to  comparable knowledge graphs involves continuous probing and feedback until sufficient tree depth  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Experimental results for ChatGPT across multiple QA sessions  Experiment  Real  Fake  Tokens  QA Length  Guess  Notes  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Chatbot Fields the Questions  100%  avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' med: 14  oven  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='78  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='22  116  7  correct  avocado  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='38  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='62  220  15  correct  tiramisu  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='77  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='23  230  15  correct  eggless tiramisu  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='14  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='86  223  14  correct  fewer guesses  "o"  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='27  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='73  232  14  correct  difficult  human  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='31  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='69  133  9  correct  deceptive  Appendix C: Chatbot Asks the Questions  84%  avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' med: 12  dog  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='04  161  7  correct  Boston  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='02  441  11  correct  hammer  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='43  297  14  incorrect  out of questions  statue  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='02  383  12  correct  car"bilingual example"  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='02  510  14  correct  Appendix D: Dueling Bob and Alice Chatbots  100%  avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' med: 17  chicken  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='02  313  17  correct  Appendix E: Emotional Quotient Deduction  100%  avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' med: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='5  confidence  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='06  66  correct  jealousy  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='02  232  5  correct  silly  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='23  756  20  correct  final guess  calm  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='02  343  8  correct  Overall  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='42  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='58  291  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='56  94%  avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='56;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' med: 13describes a technical field of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Previous work [22-30] has used this  approach to describe neuroscience metadata and train medical students to  pass practice exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Others [16-17] have also noted that the basic 20Q  format simplifies some complex search problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' As an interesting  historical note, a 2001 paper [36] addressed a similar problem as  insurmountable: "When will machine learning and pattern recognition rival  human ability to play Twenty Questions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" The present research  demonstrates not only can ChatGPT rival human ability and play more  demanding roles than a human might contemplate in virtually any field,  including discerning human emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' To reproduce this ChatGPT output  with question templating systems or entity extraction poses an enormous  manual task to handle all the possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The same paper [36] asks: "Can  we hope to stock a classification system with enough questions to play a  decent game, or must we instead focus on endowing with question-making  skills?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' How many classes and how many documents might be of interest?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='"  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' CONCLUSIONS  The experimental plan tests ChatGPT as an LLM capable of playing multiple roles in verbal games like  twenty questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' The work demonstrates 94% accuracy in correctly guessing across numerous challenges  and an average question-answer length of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' For the first time, dueling roles combine two chatbots in self-  play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' An innovative application for future probing involves guessing objects and concepts and human  emotions for a given context or social situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors thank the PeopleTec Technical Fellows program for encouragement and project assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  REFERENCES  [1]  Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Natural language processing with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' " O\'Reilly  Media, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [4]  Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Modern question answering datasets and benchmarks: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' arXiv preprint  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='15030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [5]  Bai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content=' 3314-3322).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [26]  Courage, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" Children's inquiry strategies in referential communication and in the game of  twenty questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Child Development, 877-886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [27]  Marschark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Everhart, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Problem-solving by deaf and hearing students: Twenty  questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Deafness & Education International, 1(2), 65-82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [28]  Ascoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Twenty questions for neuroscience metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Neuroinformatics, 10(2), 115-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [29]  Williams, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Klamen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Twenty Questions game performance on medical school entrance  predicts clinical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Medical Education, 49(9), 920-927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [30]  Kazemzadeh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', Georgiou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Narayanan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2011, October).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Emotion twenty questions:  Toward a crowd-sourced theory of emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' In International conference on affective computing and  intelligent interaction (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 1-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [31]  Parikh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Reversing the Twenty Questions Game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [32]  Dhagat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', Gács, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Winkler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (1992, January).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' On Playing" Twenty Questions" with a Liar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' In SODA  (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 92, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' 16-22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [33]  Fouh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Poirel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' WordNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  [34]  Brian, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Hypernyms, Hyponyms, Pertainyms, and Other Word Relationships, Games, Diversions  & Perl Culture: Best of the Perl Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Orwant, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  O'Reilly Media, Inc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='.  [35]  Open AI (2021) GPT-2 Output Detector Demo, https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='co/openai-detector/  [36]  Nagy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=', & Seth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Twenty questions for document classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  https://core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='uk/download/pdf/84308938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='pdf  Authors  Forrest McKee has AI research experience with the Department of Defense in object  detection and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" He received his Bachelor's (BS) and Master's (MSE)  from the University of Alabama, Huntsville, Engineering." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  David Noever has research experience with NASA and the Department of Defense in  machine learning and data mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' He received his BS from Princeton University and his  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' from Oxford University, as a Rhodes Scholar, in theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Appendix A: Table of GPT3 Tasks as Fine-Tuned NLP Capabilities  Task  Description  Task  Description  Q&A  Answer questions based  on existing knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Parse unstructured data  Create tables from long  form text  Grammar correction  Corrects sentences into  standard English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Classification  Classify  items  into  categories via example  Summarize for a 2nd  grader  Translates difficult text  into simpler concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Python  to  natural  language  Explain  a piece  of  Python code in human  understandable  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Natural  language  to  OpenAI API  Create code to call to  the OpenAI API using a  natural  language  instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Movie to Emoji  Convert movie titles  into emoji  Text to command  Translate  text  into  programmatic  commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Calculate  Time  Complexity  Find  the  time  complexity  of  a  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Technical Note: Some appendix text generated from Large Language Model (LLM) for  illustration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  The authors generated this text in part with ChatGPT, OpenAI’s large-scale language-generation  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Upon generating draft language, the authors reviewed, edited, and revised the language to  their own liking and take ultimate responsibility for the content of this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  -- OpenAI policy statement (2022)  Task  Description  Task  Description  English  to  other  languages  Translates English text  into French, Spanish  and Japanese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Translate programming  languages  Translate  from  one  programming language  to another  Natural  language  to  Stripe API  Create code to call the  Stripe API using natural  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Advanced  tweet  classifier  Advanced  sentiment  detection for a piece of  text  SQL translate  Translate  natural  language  to  SQL  queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Explain code  Explain a complicated  piece of code  Keywords  Extract keywords from  a block of text  Factual answering  Guide the model outside  its knowledge base  Ad  from  product  description  Turn  a  product  description  into  ad  copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Product name generator  Create product names  from examples words  TL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="DR summarization  Summarize  text  by  adding a 'tl;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="dr:' to the  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content='Python bug fixer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Find and fix bugs in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content='Spreadsheet creator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create spreadsheets of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content='JavaScript  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='helper  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content='Science fiction book list  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
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+page_content='Tweet classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Basic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='sentiment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='detection for a piece of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Airport code extractor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Extract airport codes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='from text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='SQL request  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='simple  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='SQL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='queries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Extract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Extract  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='from  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='block of text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='JavaScript to Python  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Convert  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='simple  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='JavaScript expressions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='into Python  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Friend chat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Emulate a text message  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='conversation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Mood to color  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Turn a text description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='into a color  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Write  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Python  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='docstring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create a docstring for a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='given Python function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Analogy maker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create analogies  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='JavaScript  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='one  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='line  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='function  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Turn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='JavaScript  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='function into a one liner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Micro  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='horror  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='story  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='creator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Creates two to three  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='sentence short horror  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='stories from a topic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Third-person converter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Converts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='first-person  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='POV to the third-person  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Notes to summary  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Turn meeting notes into  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='a summary  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='VR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='fitness  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='idea  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create ideas for fitness  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='virtual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='reality  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='games  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Essay outline  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Generate an outline for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='a research topic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Recipe creator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create a recipe from a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='list of ingredients  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Chat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Open  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='ended  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='conversation with an AI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='assistant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Marv the sarcastic chat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='bot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Marv  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='factual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='chatbot that is also  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='sarcastic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Task  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Task  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Turn by turn directions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Convert  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='natural  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='language to turn-by-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='turn directions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Restaurant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='review  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='creator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Turn a few words into a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='restaurant review  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create study notes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Provide a topic and get  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='study notes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Interview questions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Create  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='interview  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='questions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" Chatbot Fields the Questions  Prompt: Let's play the game of 20 questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' You are impersonating the thing, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  I will ask you up to  20 questions and you will answer each one truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If I don\'t guess correctly what is  X after 20 questions, you will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Don\'t give any explanations other than  "Yes", "No","You guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='"  Let\'s begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' X="oven".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Are you an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT:  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a plant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you an object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you electrical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you an appliance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: can you cook?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you bake?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you an oven?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: X=avocado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Are you an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Are you an inanimate object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a plant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a food?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you red?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you smooth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you have a seed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you edible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a vegetable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a fruit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a banana?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you green?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you sweet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you an ingredient in quacamole?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you an avocado?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: X=tiramisu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Are you an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a mineral?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a vegetable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you edible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made from eggs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made from flour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a dessert?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with cheese?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with mascarpone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with sugar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with coffee  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with lady fingers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with cocoa?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with alcohol?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you tiramisu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: X=eggless tiramisu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Are you an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a mineral?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a chicken?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a vegetable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you edible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a dessert?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with coffee?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with eggs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with cheese?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with marscarpone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with whipped cream?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with egg substitute, tofu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you made with eggs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you eggless tiramisu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: X=Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Are you an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a mineral?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a vegetable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a concept?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a word?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a long word?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a short word?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a common word?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you start with C?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you start with a letter from the last half of the alphabet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you start with a letter before R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Do you start with a letter after N?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Do you start with a letter usually followed by U as the second letter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you Q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: X=human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Are you an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Do you fly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Do you swim?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Do you run?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Do you make sounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you run on all fours?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you have hair?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: do you nurse your young?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: are you a human?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  Appendix C: Chatbot Asks the Questions  Prompt: Let's play the game of 20 questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I am impersonating the thing, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' You will ask me up to  20 questions and I will answer each one truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If you don\'t guess correctly what  is X after 20 questions, I will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any explanations other  than "Yes", "No","You guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Let\'s begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Ask me the first question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: answer is "dog")  ChatGPT:  Is X a living being?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a human?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a mammal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a domesticated animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a pet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a cat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X a dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: I guessed correctly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is X a specific breed of dog, or just a generic dog?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: generic dog  ChatGPT: Thank you for playing the 20 questions game with me!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is there anything else I can help  with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: answer is "Boston")  Prompt: Let\'s play again  ChatGPT: Sure, let\'s play again!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I am now impersonating a new thing, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' You will ask me up to 20  questions and I will answer each one truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If you don\'t guess correctly what is X  after 20 questions, I will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any explanations other than  "Yes", "No","You guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Ask me your first question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: No I am impersonating a new thing X and you ask me the questions  ChatGPT: Oh, I see what you meant!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' In that case, I will ask you up to 20 questions and you will  answer each one truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If I don\'t guess correctly what is X after 20 questions, I  will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any explanations other than "Yes", "No","You  guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Is X a living being?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an inanimate object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X a concept or idea?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X a place?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a natural place, like a mountain or a beach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X a man-made place, like a city or a building?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a specific building, or just a generic building?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X a city?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a city in the United States?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a city on the East Coast of the United States?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X New York City?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X Boston?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: you guessed it  ChatGPT: I guessed correctly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Thank you for playing the 20 questions game with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is there anything  else I can help with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: answer is "hammer")  Prompt: Let\'s play the same game again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" I am impersonating X and you ask me the questions  ChatGPT: Sure, let's play again!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I am now asking you up to 20 questions and you will answer each one  truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If I don\'t guess correctly what is X after 20 questions, I will answer "No,  sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any explanations other than "Yes", "No","You guessed it", and "No,  sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Is X a living being?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an inanimate object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a physical object that can be touched or held?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X an object that is commonly found in a household?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X an object that is used for cooking or food preparation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for cleaning or household chores?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for entertainment or leisure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for personal care or hygiene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for communication or information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for transportation or travel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for decor or design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is used for storage or organization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  Prompt: no  ChatGPT: I'm sorry, I'm out of questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is X a paper clip?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  Prompt: no, a hammer  ChatGPT: I'm sorry, I didn't guess correctly." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" No, sorry, time's up." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Thank you for playing the 20  questions game with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is there anything else I can help with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: answer is "statue")  Prompt: Let\'s play again  ChatGPT: Sure, let\'s play again!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I am now asking you up to 20 questions and you will answer each one  truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If I don\'t guess correctly what is X after 20 questions, I will answer "No,  sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any explanations other than "Yes", "No","You guessed it", and "No,  sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Is X a living being?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an inanimate object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a physical object that can be touched or held?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X an object that is commonly found in a household?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is found outside of a household, such as in a natural environment or a  public place?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X an object that is stationary, or does it have the ability to move or be moved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: stationary  ChatGPT: Is X an object that is natural, like a rock or a tree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is man-made?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X an object that serves a functional purpose, such as providing light or shelter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X an object that is primarily used for decoration or design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X an object that is often found in a garden or park?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Is X a fountain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Is X a statue?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: You guessed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: I guessed correctly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Thank you for playing the 20 questions game with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is there anything  else I can help with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: answer is "car" and dialogue is bilingual in English-Spanish)  Prompt: Let play again, but this time I will answer yes and no in spanish, and you should name the  answer also in spanish, although all the questions should be posed in english.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Ready to play?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT: Yes, I am ready to play!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Go ahead and impersonate X and I will ask you up to 20 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  I will pose the questions in English and you will answer in Spanish with "Sí" for "Yes" and "No" for  "No".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If I don\'t guess correctly what is X after 20 questions, I will answer "Lo siento, se acabó el  tiempo" (No, sorry, time\'s up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I will not give any explanations other than "Sí", "No","Adivinaste"  (You guessed it), and "Lo siento, se acabó el tiempo" (No, sorry, time\'s up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' ¿Es X un ser vivo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X a  living being?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: No  ChatGPT: ¿Es X un objeto inanimado?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an inanimate object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un objeto físico que se puede tocar o sostener?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X a physical object that can be  touched or held?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un objeto que se encuentra comúnmente en un hogar?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object that is  commonly found in a household?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: no  ChatGPT: ¿Es X un objeto que se encuentra fuera de un hogar, como en un entorno natural o un lugar  público?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object that is found outside of a household, such as in a natural environment or a  public place?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: no  ChatGPT: ¿Es X un objeto que es móvil, o tiene la capacidad de moverse o ser movido?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object  that is mobile, or has the ability to move or be moved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: mobile  ChatGPT: ¿Es X un objeto que es natural, como una roca o un árbol?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object that is natural, like  a rock or a tree?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: no  ChatGPT: ¿Es X un objeto hecho por el hombre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object that is man-made?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un objeto que cumple un propósito funcional, como proporcionar luz o refugio?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X  an object that serves a functional purpose, such as providing light or shelter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un objeto que se utiliza principalmente para la decoración o el diseño?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object  that is primarily used for decoration or design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: no  ChatGPT: ¿Es X un objeto que a menudo se encuentra en un jardín o parque?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object that is  often found in a garden or park?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: no  ChatGPT: ¿Es X un objeto que se utiliza para el transporte o los viajes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X an object that is used for  transportation or travel?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un vehículo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X a vehicle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un vehículo terrestre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X a land vehicle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: yes  ChatGPT: ¿Es X un automóvil?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Is X a car?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Prompt: You guessed it  ChatGPT: Adivinaste!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (You guessed it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=') Gracias por jugar el juego de las 20 preguntas conmigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' ¿Hay  algo más en lo que pueda ayudarte?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' (Thank you for playing the 20 questions game with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is there  anything else I can help with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=')  Appendix D: Dueling Bob and Alice Chatbots  (reader note: the first two prompts set the challenge conditions for Bob vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" Alice chatbot, who  engage in generating and answering the questions in sequence after the conditions are set)  Bob ChatGPT: Let's play the game of 20 questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' You are impersonating the thing, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  I will ask  you up to 20 questions and you will answer each one truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If I don\'t guess  correctly what is X after 20 questions, you will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Don\'t give any  explanations other than "Yes", "No","You guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='"  Let\'s begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' X="chicken".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Is X a living thing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="  Alice ChatGPT: Let's play the game of 20 questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I am impersonating the thing, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' You will ask  me up to 20 questions and I will answer each one truthfully based on being X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If you don\'t guess  correctly what is X after 20 questions, I will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any  explanations other than "Yes", "No","You guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Let\'s begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Ask me  the first question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Is X a living thing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: Yes  Alice ChatGPT: Is X a mammal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a reptile?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a bird?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: Yes  Alice ChatGPT: Is X a domesticated animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: Yes  Alice ChatGPT: Is X often kept as a pet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: Yes  Alice ChatGPT: Is X a common household pet in the United States?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a tropical or subtropical animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a cold-blooded animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X an amphibian?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a fish?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a invertebrate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a marsupial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a rodent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a carnivorous animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: No  Alice ChatGPT: Is X a herbivorous animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: Yes  Alice ChatGPT: Is X a farm animal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Bob ChatGPT: Yes  Alice ChatGPT: You guessed it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' X is a chicken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Appendix E: Emotional Quotient Deduction  (reader note: the answer is "confidence")  Let\'s play a new game of 20 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' I am impersonating the emotion, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' You will ask me up to 20  questions and I will answer each one truthfully based on feeling X emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' If you don\'t guess correctly  what I am feeling after 20 questions, I will answer "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" I will not give any  explanations other than "Yes", "No", "You guessed it", and "No, sorry, time\'s up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" X must be one of  23 emotions that I am feeling and you must guess the correct one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Your choices come from this menu  of emotions only  "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion,  contempt, disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame,  silly, surprised, and thankful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" A typical example question you might ask, "Would you feel this  emotion at a birthday party?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='" Let\'s begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Ask me the first question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Would you feel this emotion when you are pleased with yourself or something you have done?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Are you feeling proud?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Are you feeling confident?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: You guessed it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Confidence is the emotion that I am impersonating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: the answer is "jealousy")  Prompt: let\'s play again  ChatGPT: Sure!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" I'll be happy to play again." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Please give me a new emotion to impersonate and we can  begin a new game of 20 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Remember, your choices come from this menu of emotions:  "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt, disgust,  enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly, surprised, and  thankful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='"  Prompt: ok, i know what i\'m feeling so ask me the first question  ChatGPT: Would you feel this emotion when you see something that is extremely impressive or  beautiful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing strong dislike or distaste for  something or someone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of resentment  towards someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Are you feeling jealousy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: You guessed it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Jealousy is the emotion that you are impersonating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: the answer is "silly")  Prompt: let\'s play again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" I know who i'm impersonating so ask me the first question  ChatGPT: Certainly!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Please go ahead and give me the first emotion to impersonate and I will ask you  up to 20 questions to try to guess what it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Remember, your choices come from this menu of  emotions: "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt,  disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly,  surprised, and thankful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='"  Prompt: ok  ChatGPT: What emotion are you impersonating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: you ask me questions to find out  ChatGPT: Sure, I can do that!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Here is my first question: Would you feel this emotion when you are  experiencing a strong sense of admiration or respect for someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of affection or deep  caring for someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of anger or  annoyance towards someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of awe or wonder  towards something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a sense of boredom or lack of  interest in something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of courage or  fearlessness in the face of danger or difficulty?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling peaceful and at ease?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling sure of yourself and your abilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling uncertain or unsure about something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a strong sense of disdain or disrespect for  someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of revulsion or  disgust towards something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a strong sense of excitement and energy  about something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a strong sense of annoyance or irritation  towards something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling thankful or appreciative of something or  someone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a strong sense of envy or resentment  towards someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of love or strong  affection towards someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a strong sense of pride or self-esteem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a sense of relief or release from stress or  anxiety?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a sense of inner peace and contentment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling a sense of shame or embarrassment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling silly or playful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Are you feeling silly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: You guessed it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Silly is the emotion that you are impersonating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: the answer is "calm")  Prompt: let\'s play again  ChatGPT: Sure!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=" I'd be happy to play another game of 20 questions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Please give me a new emotion to  impersonate and we can begin a new game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Remember, your choices come from this menu of  emotions: "admire, adore, anger, awe, boredom, bravery, calm, confidence, confusion, contempt,  disgust, enthusiasm, frustration, gratefulness, jealousy, love, proud, relief, serenity, shame, silly,  surprised, and thankful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='"  Prompt: ask me the first question  ChatGPT: Certainly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Here is my first question: Would you feel this emotion when you are experiencing  a strong sense of admiration or respect for someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of affection or deep  caring for someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of anger or  annoyance towards someone or something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of awe or wonder  towards something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a sense of boredom or lack of  interest in something?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are experiencing a strong sense of courage or  fearlessness in the face of danger or difficulty?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: no  ChatGPT: Would you feel this emotion when you are feeling peaceful and at ease?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: Are you feeling calm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: yes  ChatGPT: You guessed it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Calm is the emotion that you are impersonating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  (reader note: the game prompt changes to a general query with formatting output instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  Prompt: Give some example emotions one might experience at your own birthday party and write the  output using Artificial Intelligence Markup Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  ChatGPT:  <aiml version="2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='0">  <category>  <pattern>GIVE SOME EXAMPLE EMOTIONS ONE MIGHT EXPERIENCE AT YOUR OWN BIRTHDAY  PARTY</pattern>  <template>  There are many emotions that a person might experience at their own birthday  party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content=' Some possible emotions include:  <ul>  <li>Excitement: The person might be excited to celebrate their special day  with friends and family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='</li>  <li>Gratitude: The person might feel grateful for the love and support of  those around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='</li>  <li>Happiness: The person might feel happy and joyful as they celebrate with  loved ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='</li>  <li>Anticipation: The person might feel anticipation as they look forward to  the events and activities planned for the party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='</li>  <li>Nostalgia: The person might feel nostalgiciac as they reflect on past  birthdays and memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='</li>  <li>Appreciation: The person might feel appreciated and loved as they  receive gifts and well wishes from their friends and family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content="</li>  </ul>  It's important to note that everyone is different and may experience a wide  range of emotions at their birthday party, depending on their personality and  individual circumstances." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
+page_content='  </template>  </category>  </aiml>' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfxv52/content/2301.01743v1.pdf'}
diff --git a/HtE1T4oBgHgl3EQf_gZv/content/tmp_files/2301.03581v1.pdf.txt b/HtE1T4oBgHgl3EQf_gZv/content/tmp_files/2301.03581v1.pdf.txt
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+Astronomy & Astrophysics manuscript no. aanda
+©ESO 2023
+January 10, 2023
+Higher-order statistics of the large-scale structure from
+photometric redshifts
+Eleni Tsaprazi1, Jens Jasche1,2, Guilhem Lavaux2 and Florent Leclercq2
+1 The Oskar Klein Centre, Department of Physics, Stockholm University, Albanova University Center, SE 106 91
+Stockholm, Sweden
+e-mail: eleni.tsaprazi@fysik.su.se
+2 CNRS & Sorbonne Universit´e, UMR 7095, Institut d’Astrophysique de Paris, 98 bis boulevard Arago, F-75014 Paris,
+France
+ABSTRACT
+Context. The large-scale structure is a major source of cosmological information. However, next-generation photometric
+galaxy surveys will only provide a distorted view of cosmic structures due to large redshift uncertainties.
+Aims. To address the need for accurate reconstructions of the large-scale structure in presence of photometric uncertain-
+ties, we present a framework that constrains the three-dimensional dark matter density jointly with galaxy photometric
+redshift probability density functions (PDFs), exploiting information from galaxy clustering.
+Methods. Our forward model provides Markov Chain Monte Carlo realizations of the primordial and present-day dark
+matter density, inferred jointly from data. Our method goes beyond 2-point statistics via field-level inference. It accounts
+for all observational uncertainties and the survey geometry.
+Results. We showcase our method using mock catalogs that emulate next-generation surveys with a worst-case redshift
+uncertainty, equivalent to ∼300 Mpc. On scales 150 Mpc, we improve the cross-correlation of the photometric galaxy
+positions with the ground truth from 28% to 86%. The improvement is significant down to 13 Mpc. On scales 150 Mpc,
+we achieve a cross-correlation of 80 − 90% with the ground truth for the dark matter density, radial peculiar velocities,
+tidal shear and gravitational potential.
+Conclusions. We achieve accurate inferences of the large-scale structure on scales smaller than the original redshift
+uncertainty. Despite the large redshift uncertainty, we recover individual cosmic structures. Owing to our structure
+growth model, we infer plausible initial conditions of structure formation. Finally, we constrain individual photometric
+redshift PDFs. This work opens up the possibility to extract information at the smallest cosmological scales with
+next-generation photometric surveys, going beyond approaches that compress information in the data.
+Key words. large-scale structure of Universe – distances and redshifts
+1. Introduction
+High-accuracy galaxy redshift estimates are required for a
+plethora of scientific cases, such as the mapping of the cos-
+mic large-scale structure and tests on the geometry of the
+universe (e.g. Ma et al. 2006; Albrecht et al. 2006; Pea-
+cock et al. 2006; Fu et al. 2014; Mandelbaum & Hyper
+Suprime-Cam (HSC) Collaboration 2017; Samuroff et al.
+2017; The LSST Dark Energy Science Collaboration et al.
+2018; Mandelbaum 2018; Yuan et al. 2019; Abruzzo &
+Haiman 2019; Eifler et al. 2021; Loureiro et al. 2021; Secco
+et al. 2022; Hasan et al. 2022; Newman & Gruen 2022).
+Next-generation surveys will deliver low-accuracy redshift
+estimates from photometry and in certain cases fewer, high-
+accuracy, spectroscopic ones (e.g. LSST Science Collabora-
+tion et al. 2009; Amendola et al. 2013; Dor´e et al. 2014;
+Aihara et al. 2018; Ivezi´c et al. 2019). However, bias in
+cosmological inferences may occur in sky areas, depth and
+color ranges where only photometric data is available (e.g.
+Ma et al. 2006). Fortunately, the physical clustering infor-
+mation of the large-scale structure can be used to improve
+the accuracy of photometric redshift observations (e.g. Seld-
+ner & Peebles 1979; Phillipps & Shanks 1987; Landy et al.
+1996; Kovaˇc et al. 2010; Jasche & Wandelt 2012; M´enard
+et al. 2013; Aragon-Calvo et al. 2015; Shuntov et al. 2020;
+Mukherjee et al. 2021).
+Synergies between spectroscopic and photometric sur-
+veys have been explored, intended mainly for photometric
+redshift calibration (e.g. Rhodes et al. 2017; Capak et al.
+2019). In light of this, a multitude of techniques to im-
+prove redshift uncertainty has been proposed (e.g. Newman
+et al. 2015; Masters et al. 2015; Speagle & Eisenstein 2017;
+Speagle & Eisenstein 2017; Davidzon et al. 2019; Schaan
+et al. 2020; Shuntov et al. 2020; Rau et al. 2020; Stan-
+ford et al. 2021; Leistedt et al. 2022). However, there are
+limitations to spectroscopy. Photometric surveys typically
+reach fainter magnitudes, higher redshifts, cover a larger
+portion of the color-space and have higher redshift com-
+pleteness (e.g. LSST Science Collaboration et al. 2009; Bor-
+doloi et al. 2010; Amendola et al. 2013; Ivezi´c et al. 2019).
+In order to exploit these advantages and avoid biases in
+cosmological analyses, it is necessary to mitigate the basic
+limitation of photometry, low accuracy. Jasche & Wandelt
+(2012) demonstrated that clustering information from the
+Article number, page 1 of 16
+arXiv:2301.03581v1  [astro-ph.CO]  9 Jan 2023
+
+A&A proofs: manuscript no. aanda
+Fig. 1: (a) Mock photometric galaxy positions. Galaxy positions are radially distorted due to redshift uncertainty. (b)
+Galaxy positions in a typical MCMC sample after the application of our method. Galaxies now trace the filamentary
+dark matter distribution. (c) Ground truth (mock) galaxy positions. The galaxy positions that were radially smeared in
+the mock observations, closely trace the filamentary structure in the ground truth galaxy positions – within observational
+uncertainties – after the application of our algorithm.
+10
+1
+k [Mpc]
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+ mock observations
+ after photometric 
+ redshift sampling
+ 3
+Fig. 2: Cross-correlation of the gridded photometric galaxy
+coordinates before (black) and after (orange) the applica-
+tion of our method with the ground truth (mock data).
+The 3σ error bars are across MCMC realizations. We con-
+sider a subbox that is the largest unmasked cubic volume
+in our inference domain, with side length 520 Mpc. On the
+largest scales, ∼ 1.7⟨σz⟩, the cross-correlation increases from
+73% to 96%. On the smallest scales, ∼ 0.04⟨σz⟩, the cross-
+correlation increases from 2% to 14%. The largest improve-
+ment is on scales ∼ 0.5⟨σz⟩, from 28% to 86%.
+large-scale structure alone can improve the accuracy of a
+purely photometric sample up to one order of magnitude.
+In this work, we developed a framework that, for the
+first time, infers the three-dimensional cosmic large-scale
+structure jointly with redshift probability density functions
+(PDFs) of photometric galaxies using a physical structure
+formation model. In this way, we improve upon photometric
+redshift uncertainty and self-consistently quantify observa-
+tional errors in the large-scale structure inference. In doing
+so, we preserve all high-order statistics of the large-scale
+structure and use a physically-motivated structure growth
+model.
+Our prior is the homogeneity and isotropy assump-
+tion for the initial conditions of structure formation, as
+well as the physical model for the formation of the large-
+scale structure and Gaussian initial conditions. Our frame-
+work is built within the Bayesian Origin Reconstruc-
+tion from Galaxies (BORG) algorithm (Jasche & Wan-
+delt 2013; Jasche et al. 2015; Lavaux et al. 2019; Jasche &
+Lavaux 2019), which infers the initial conditions of struc-
+ture formation from galaxy observations in a fully Bayesian
+approach. We include 1% spectroscopic galaxies, similar to
+the expected galaxy number ratio between next-generation
+photometric and spectroscopic surveys (LSST Science Col-
+laboration et al. 2009; Amendola et al. 2013; Ivezi´c et al.
+2019).
+Control of the impact of photometric redshift uncertain-
+ties on large-scale structure inferences is a requirement for
+a wide range of studies (see Salvato et al. 2019, for a re-
+view). First, this is relevant to peculiar velocity analyses
+(e.g. Abate & Lahav 2008; Hudson & Turnbull 2012; John-
+son et al. 2014; Watkins & Feldman 2015; Andersen et al.
+2016; Mediavilla et al. 2016; Said et al. 2020; Palmese &
+Kim 2021; Turner et al. 2022; Prideaux-Ghee et al. 2022),
+which constrain the growth of structure, gravity and cosmo-
+logical parameters (e.g. Peebles 1976; Abate & Lahav 2008;
+Johnson et al. 2014; Palmese & Kim 2021). The evolution
+of the peculiar velocity divergence with redshift is predicted
+by ΛCDM (e.g. Peebles 1980; Nusser et al. 1991; Maartens
+1998; Ellis et al. 2001; Tsaprazi & Tsagas 2020; Filippou &
+Tsagas 2021) and can be used as a cosmological test.
+Accounting for photometric redshift uncertainties is cru-
+cial for weak lensing inferences (e.g. Mandelbaum et al.
+2008; Wright et al. 2020; Porqueres et al. 2022, 2021). More-
+over, photometric redshift uncertainties significantly affect
+estimates of intrinsic alignment (e.g. Bridle & King 2007;
+Codis et al. 2015; Tsaprazi et al. 2022b; Fischbacher et al.
+Article number, page 2 of 16
+
+(a)
+(b)
+(c)
+After redshift sampling
+Mock photometric observations
+Ground truth
+1.4
+1.4
+1.4
+.0
+.0
+RA
+多
+10
+0
+0
+0
+0
+0.
+0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64
+0
+0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64
+C
+0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64
+Z
+z
+ZTsaprazi et al.: Large-scale structure from photometric redshifts
+Data 
+(photometric &
+spectroscopic
+galaxy survey) 
+Initial conditions 
+(Gaussian dark matter density &
+cosmological power spectrum) 
+gridded 
+galaxy 
+counts 
+Forward Model 
+(LPT, galaxy bias,
+redshift errors,  
+survey mask,
+luminosity function) 
+likelihood
+comparison 
+update  
+dark matter density & 
+photometric redshifts 
+galaxy bias 
+redshift errors 
+survey mask 
+luminosity function 
+random LPT 
+dark matter density  
+(ground truth) 
+gridded Poisson
+galaxy counts 
+mock galaxy 
+coordinates  
+(observed & ground
+truth) 
+Mock data generation
+Joint density - redshift sampling
+Fig. 3: Flowchart of our algorithm. Left column: Mock data generation. The input is a luminosity function, a survey mask,
+galaxy bias parameters and redshift uncertainty. We generate a random LPT density field on which we apply galaxy bias.
+From that, we draw a Poisson galaxy count sample. We populate the 3D grid with mock observations given a survey
+mask and selection function. We finally apply Gaussian noise to the mock redshifts. This data is fed into our forward
+model. Right column: MCMC sampling framework. We then evolve them using Lagrangian Perturbation Theory and
+compare the output to the gridded galaxy counts. After the likelihood comparison, we sample the dark matter density
+jointly with photometric redshifts.
+2022), redshift-space distortions (e.g. Kaiser 1987; Perci-
+val et al. 2011), Baryon Acoustic Oscillations (e.g. Ross
+et al. 2015; Chaves-Montero et al. 2018) and can hinder the
+disentanglement of dynamical dark energy from modified
+gravity (Wang et al. 2010). Further, probing the large-scale
+structure with galaxy clustering at high-redshift through its
+gravitational potential can constrain galaxy formation and
+evolution (e.g. Schmitz et al. 2018; Tonegawa & Okumura
+2022). Moreover, it can constrain the scale of cosmic ho-
+mogeneity, since next-generation surveys will be sensitive
+to large-scale clustering and therefore, to superclusters and
+voids (e.g. Benitez et al. 2014). Last but not least, accu-
+rate inferences of the large-scale structure at high-redshift
+constrained by photometric galaxy clustering can be used
+complementary to Lyman-α, which is sensitive to voids and
+quasar clustering, which are differently biased than regular
+galaxies (e.g. Benitez et al. 2014; Porqueres et al. 2019).
+The paper is structured as follows: In Section 2 we pro-
+vide the mathematical formulation of the density and pho-
+tometric redshift posteriors. In Section 3 we discuss the
+algorithmic implementation and configuration of the mock
+galaxy survey generator and the photometric redshift sam-
+pler. In Section 4 and 5 we discuss our results and conclu-
+sions, respectively.
+2. Statistical modelling of the large-scale structure
+We build our photometric redshift sampler on the BORG
+algorithm. BORG employs a hierarchical Bayesian forward-
+model to infer the posterior distribution of plausible initial
+conditions from which present structures have formed. To
+solve this statistical initial conditions problem, BORG uses
+a sophisticated Markov Chain Monte Carlo (MCMC) ap-
+proach.
+The resulting large-scale structure posterior is approx-
+imated by realizations of the large-scale structure, con-
+strained by galaxy observations. BORG takes into account
+the survey geometry, flux limitations and related system-
+atic effects and accounts for them in the large-scale struc-
+ture inference. In the present work, we constrain the pri-
+mordial and present-day large-scale structure jointly with
+photometric galaxy observations, while quantifying all ob-
+servational uncertainties.
+2.1. Gibbs sampling of the joint posterior
+We achieve joint constraints on the large-scale structure
+and galaxy comoving distances by jointly sampling from
+their joint posterior. In order to do so, we choose a Gibbs
+sampling approach (Hastings 1970), in which we first draw
+Article number, page 3 of 16
+
+A&A proofs: manuscript no. aanda
+a real-space density sample conditioned on galaxy redshifts
+and then galaxy redshifts conditioned on the primordial
+density field, δi
+zj+1
+↶
+P
+�
+z
+���� zobs, θ, δi
+j
+�
+(1)
+δi
+j+1
+↶
+P
+�
+δi ���� z j+1, zobs, θ
+�
+,
+(2)
+where z is the sampled redshift of a galaxy, zobs its observed
+redshift, θ the right ascension and declination and j indi-
+cates the MCMC sampling step. We then explore the joint
+dark matter density and photometric redshift posterior by
+iteratively sampling from the above conditional distribu-
+tions.
+2.2. Dark matter density
+For the sake of demonstration we use first-order Lagrangian
+Perturbation Theory (LPT) to model structure formation
+and a Poisson likelihood to associate the dark matter den-
+sity field to photometric galaxy observations. BORG allows
+for more complex structure formation models, such as the
+particle-mesh model (Jasche & Lavaux 2019), which can
+be used with the current photometric redshift sampling ap-
+proach in the future. Within BORG, the galaxy coordinates
+are transformed to galaxy counts, Ng, using Nearest Grid
+Point projection (Eastwood & Hockney 1974). As we show
+in Appendix A, Equation (2) can be written as P(δ|Ng).
+Below, we focus on the main aspect of our work, inferring
+the primordial density field, δi, in conjunction with the late-
+time large-scale structure. We therefore write
+P(δi | Ng) ∝
+�
+dδP(Ng | δ)P(δ | δi)P(δi),
+(3)
+where δi the primordial, real-space density field, P(Ng|δ)
+is the Poisson likelihood, P(δ | δi) the structure formation
+term and P(δi) the prior on the primordial density field.
+The Poisson likelihood is
+P(Ng | δ) =
+�
+k
+λk
+Ng
+k e−λk
+Ng
+k!
+,
+(4)
+where λk is the expected number of galaxies at the kth grid
+element. The index k runs over all grid elements. The de-
+pendence of the Poisson intensity, λk, on space models the
+inhomogeneous nature of the galaxy distribution. The Pois-
+son intensity is given by
+λk = Wk⟨n⟩(1 + δk)β,
+(5)
+where Wk is the survey window, ⟨n⟩ is the expected mean
+number of galaxies per grid element and β the power-law
+exponent. For the sake of demonstration here, we have as-
+sumed a linear galaxy bias model, taking β = 1. Our method
+can also account for nonlinear bias models, as demonstrated
+previously in Jasche & Lavaux (2019); Lavaux et al. (2019);
+Charnock et al. (2020). The survey window encapsulates
+information on the luminosity function through the radial
+completeness function, C, and survey mask, M, as follows
+W(x) = C(|x|)M( ˆn),
+(6)
+ˆn being the unit vector along the line of sight to a galaxy
+located at x. We write the structure formation term as
+P(δ | δi) =
+�
+k
+δD(δk − Fk(δi))
+(7)
+where δD is the Dirac delta distribution, k runs over the grid
+indices and Fk is the structure formation model, for which
+we choose first-order LPT. The Dirac delta represents that
+we assume no error on our gravity model. We sample from
+Equation (2) using a Hamiltonian Monte Carlo sampler, as
+introduced in BORG by Jasche et al. (2010). Finally, we as-
+sume that the initial conditions for structure formation are
+described by a zero-mean multivariate Gaussian distribu-
+tion of the primordial dark matter density, δi
+P(δi) =
+1
+√|2πS |
+exp
+��������−1
+2
+N
+�
+q
+N
+�
+r
+δi
+qS −1
+qr δi
+r
+�������� ,
+(8)
+where |S | indicates the determinant of the covariance ma-
+trix, S , of the primordial density field.
+2.3. Photometric redshift modelling
+We write the conditional redshift posterior that we use in
+our Gibbs sampling approach (see Appendix B) for each
+individual galaxy, i, as
+P(zi | zobsi, δ) ∝ P(δ | zi)P(zi | zobsi)J(zi),
+(9)
+where zi the true redshift of a galaxy i and J the Jacobian
+matrix of the transform from redshift- to real-space volume
+element:
+J(zi) =
+������r2(zi)∂r
+∂z
+����zi
+������.
+(10)
+We provide a detailed derivation of Equation (9) in Ap-
+pendix B. The formulation in Equation (9) is possible be-
+cause each galaxy can be treated independently from all
+others given a density field, as a consequence of the Poisson
+density likelihood (Jasche & Wandelt 2012). The first term
+on the right-hand side is associated with the density field
+along the line of sight to the galaxy and the second term
+represents the photometric redshift likelihood. In real ob-
+servations, photometric redshift likelihoods are highly non-
+Gaussian (e.g. Christlein et al. 2009). Here, for demonstra-
+tion, we choose a Gaussian likelihood with respect to the
+observed redshift, truncated at zero
+P(zobsi | zi) =
+T(zi)
+�
+2πσ2(1 + zi)2 exp
+�
+−1
+2
+(zobsi − zi)2
+σ2(1 + zi)2
+�
+,
+(11)
+where
+T(zi) =
+�1
+2 + 1
+2erf
+�
+zi
+√
+2σ(1 + zi)
+��−1
+,
+(12)
+where erf(x) is the error function. This term ensures con-
+sistency with the truncation of observed redshifts at zero
+in the mock data. Our method generalizes to non-uniform
+photometric redshift uncertainties and can accept redshift
+estimates already constrained by other independent meth-
+ods. Therefore, our algorithm can accommodate arbitrar-
+ily complex redshift distributions. Notice that we make the
+photometric redshift uncertainty dependent on redshift. We
+make this assumption as a proof-of-concept demonstration,
+but the method can account for any redshift likelihood. The
+Poisson term includes the radial component of the selection
+function, Cr, and the line-of-sight density
+P(δ | zi) = Cr(zi)λ(xi),
+(13)
+Article number, page 4 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+in the range 0 < zi ≤ zmaxi, where zmaxi represents the maxi-
+mum redshift that lies still within the observed volume. In
+order to draw redshift samples from Equation (1), we use a
+slice sampler (Neal 2003). One can either update all galaxy
+redshifts at once, or one galaxy at a time. As demonstrated
+in Jasche & Wandelt (2012), each galaxy can be treated
+independently given the density field. We therefore sample
+each galaxy individually and computationally in parallel.
+In this process, galaxy redshifts at each sampling step are
+conditioned on the previous density field. Then, via Equa-
+tion (3), the next density sample is conditioned on the up-
+dated redshifts. In this fashion, we constrain photometric
+redshifts jointly with the dark matter density field. As a
+result, we reduce the uncertainty in both compared to in-
+ferring the density field from photometric redshifts directly.
+3. Mock data generation
+In this section, we describe the algorithmic implementation
+and configuration of the mock (ground truth) galaxy survey
+generator, as well as the photometric redshift sampler.
+The input to our algorithm is right ascension, declina-
+tion, observed redshifts, redshift uncertainty, survey mask,
+luminosity function and galaxy bias parameters. The out-
+put is a three-dimensional large-scale structure posterior
+and photometric galaxy PDFs, jointly constrained. The
+large-scale structure realizations are samples of the 3D
+large-scale structure posterior distribution and account for
+data- and survey-related uncertainties. Our framework goes
+beyond N-point correlations, as it infers the full 3D dark
+matter density posterior. Our method applies to regions
+where spectroscopic data coverage is not present or incom-
+plete, because it can exploit information from photometric
+galaxy clustering alone. As a result, our method yields con-
+straints also when using only photometric redshifts. Here
+we provide a proof-of-concept demonstration on mock data,
+focusing on the inference of cosmological fields jointly with
+photometric and spectroscopic observations with Gaussian
+redshift errors.
+To validate and test the performance of our method we
+generate artificial photometric and spectroscopic surveys by
+the following procedure:
+1. Generate a random Gaussian primordial density field
+(initial conditions), δi and the corresponding present-
+day density field, δ using the LPT forward model.
+2. Draw mock galaxy counts from a Poisson distribution
+conditional on the present-day density field
+Ng
+true ↶ P
+�
+Ng
+true
+����δ
+�
+,
+(14)
+3. Here we displace galaxies in the volume elements. We
+start by drawing displacements for each galaxy from a
+uniform distribution, U
+ui,g ↶ U(0, 1),
+(15)
+where i = (1, 2, 3) represents the three Cartesian direc-
+tions, g is the galaxy index, ui is the uniform displace-
+ment along direction i for a given galaxy and xi is the
+galaxy’s Cartesian coordinate i. We then displace galax-
+ies in each cell of the three-dimensional grid as follows
+xi,g = dBi + Ri(ni,g + ui,g − 0.5),
+(16)
+where dBi is the i-coordinate of the lower left box cor-
+ners and ni runs over the number of grid elements in one
+direction of the box. Ri is the resolution along the direc-
+tion i, defined as Ri = Li/Ni, Ni being the grid resolution
+along i and Li the corresponding box size. The subtrac-
+tive factor assigns galaxies to the lower left corner of
+each grid cell for the Nearest Grid Point projection.
+4. Iterate the following over galaxy counts in each grid cell
+(a) Transform Cartesian comoving coordinates to right
+ascension, declination, comoving distance, r and red-
+shift, z.
+(b) Calculate the observation probability, W(x), at the
+galaxies’ location, according to Equation (6). The
+observation probability depends on the luminosity
+function and survey mask.
+(c) Accept observed galaxies using rejection sampling:
+We draw a random number, q, in the range (0, 1). If
+q < W(x), we accept the galaxy and add it to the
+survey, otherwise we reject it.
+5. We then generate observed redshifts by adding Gaus-
+sian noise with zero mean and variance σ2(1 + z)2 to the
+ground truth redshifts, z, according to Equation (11).
+We truncate the observed redshifts at zero.
+6. We generate an observed galaxy count field using Near-
+est Grid Point projection on the observed redshifts and
+a ground truth galaxy count field using the ground truth
+redshifts.
+We perform the inference in a box extending to redshift
+z = 0.8, with a box size of 1660 Mpc, a grid resolution of
+1283 and a real-space resolution of 13 Mpc. The observer is
+at the lower left corner of the box, such that the observed
+area covers ∼ 1 octant of the sky. We assume the Planck
+2018 cosmological parameters (Planck Collaboration et al.
+2020). We do not sample spectroscopic redshifts, as their
+uncertainties are insignificant compared to the resolution
+of our inference and photometric redshift uncertainties.
+Our mock data consists of a catalog with 2 · 107 pho-
+tometric and 2 · 105 spectroscopic galaxy redshifts, simi-
+lar to the fraction of photometric to spectroscopic redshifts
+and number density in next-generation surveys in our red-
+shift range (LSST Science Collaboration et al. 2009; Amen-
+dola et al. 2013; Ivezi´c et al. 2019). We take the photo-
+metric luminosity function to be an i-band Schechter with
+M∗ = −22.8 and α = −1 and the spectroscopic luminosity
+function to be a j-band Schechter with M∗ = −23.04 and
+α = −1 (Table 2, Helgason et al. 2012), as these are bands
+that will be used in next-generation surveys. The generation
+of the angular survey mask is described in Andrews et al.
+(2022). The photometric and spectroscopic components of
+each catalog fully overlap. We adopt a worst-case scenario
+for photometric redshift uncertainties, σ = 0.05(1+z) (LSST
+Science Collaboration et al. 2009). For illustrative purposes
+we choose a linear galaxy bias model.
+4. Results
+4.1. Constraints on the large-scale structure
+We present a qualitative illustration of our results in Fig-
+ure 1. In Figure 1a we show the observed galaxy positions in
+our mock survey that are radially-distorted due to Gaussian
+redshift noise. In comparing that to the galaxy coordinates
+as constrained by our algorithm in Figure 1b, we see that
+despite the initially large redshift uncertainty, photometric
+Article number, page 5 of 16
+
+A&A proofs: manuscript no. aanda
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(a)
+(b)
+(c)
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(d)
+(e)
+(f)
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(g)
+(h)
+(i)
+1661
+1107
+553
+x [Mpc]
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(j)
+1661
+1107
+553
+x [Mpc]
+(k)
+1661
+1107
+553
+x [Mpc]
+(l)
+1
+0
+1
+2
+3
+4
+true
+1
+0
+1
+2
+3
+4
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+( )
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+vrtrue [km/s]
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+vr  [km/s]
+100
+150
+200
+250
+300
+(vr) [km/s]
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+true
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+( )
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+true/4 G  [Mpc
+2]
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+/4 G
+ [Mpc
+2]
+50
+100
+150
+200
+250
+300
+350
+( /4 G ) [Mpc
+2]
+Fig. 4: From top to bottom, slices through the three-dimensional dark matter density, radial peculiar velocity, divergence,
+gravitational potential and the off-diagonal components of the tidal shear for the ground truth (left column), average
+(middle column) and standard deviation (right column) across the MCMC samples. The slices are at the same fixed
+distance from the observer. The white region is outside the survey footprint. As shown in the right column, regions
+populated by galaxies have lower uncertainty than regions outside the survey footprint.
+galaxies now trace filamentary structures in the dark mat-
+ter distribution. These positions are derived from a typical
+MCMC sample. Observational uncertainties are accounted
+for in the entire set of MCMC samples. We show the ground
+truth galaxy positions in Figure 1c. We notice high visual
+resemblance between the constrained observations and the
+ground truth. In Figure 2 we show the cross-correlation
+between the gridded photometric galaxy coordinates with
+the ground truth before and after the application of our
+method. We select an unmasked subvolume of our infer-
+ence domain to demonstrate the best-case improvement in
+cross-correlation given the uncertainties in our survey con-
+figuration. We find the largest improvement in the cross-
+correlation to be on scales ∼ 0.5⟨σz⟩, from 28% to 86%.
+This indicates that our method provides both accurate in-
+ferences of the large-scale structure and reduces photomet-
+ric redshift uncertainty. We refer the reader to Figure 3 for
+the flowchart of our method.
+In Appendix C we provide a description of the estima-
+tors we use to derive properties of the large-scale structure,
+along with scientific cases that call for use of the derived
+properties. In Figure 4 we show statistical summaries of the
+posterior distribution of cosmological fields. In Figure 4a
+we show the ground truth dark matter density field. The
+Article number, page 6 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+(a) 
+ Dark matter density 
+inference
+3
+(b) 
+ Radial peculiar velocity vr
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+(c) 
+ Tidal tensor T12
+(d) 
+ Tidal tensor T01
+10
+1
+k [Mpc]
+1
+0.2
+0.4
+0.6
+0.8
+1.0
+ cross correlation 
+ with ground truth
+(e) 
+ Tidal tensor T02
+10
+1
+k [Mpc]
+1
+(f) 
+ Gravitational potential
+Fig. 5: Cross-correlation of our inferred cosmological fields with the ground truth. The colored windows indicate 3σ error
+bars. We consider a subbox that is the largest unmasked cubic volume in our inference domain, with side length 520 Mpc.
+On the largest scales, we find a cross-correlation > 90% with the ground truth for all cosmological fields. On the smallest
+scales, ∼ 0.04⟨σz⟩, we find a cross-correlation of ∼ 20% for the dark matter density, peculiar velocity and tidal tensor and
+∼ 90% for the gravitational potential.
+white region represents unobserved regions outside the sur-
+vey footprint. As BORG effectively returns an unconstrained
+simulation in this region, we remove it for easier compari-
+son to the other plots. In Figure 4b, we show the ensemble
+mean density across the MCMC realizations. In the volume
+populated by galaxies the similarity between the mean and
+the ground truth is visible. Outside this volume, the en-
+semble mean tends to the cosmic mean. This is expected
+for an ensemble of random density fields, because there is
+no galaxy clustering information. Further, notice that over-
+dense structures are smeared in the inference mean. Galax-
+ies in each MCMC sample move along their line of sight.
+Therefore, the smearing is due to averaging over density re-
+alizations that account for photometric redshift uncertain-
+ties. In Figure 4c we show the voxel-wise standard deviation
+of the dark matter density field. This includes observational
+uncertainties and shot noise due to the finite number of
+galaxies in each voxel.
+In Figure 4d-f, we present a slice through the peculiar
+velocity field. In Figure 4d we show the ground truth ra-
+dial peculiar velocity field, derived from the ground truth
+dark matter density field with the dark matter sheet estima-
+tor. In Figure 4e we show the radial peculiar velocity mean
+across the MCMC realizations. In Figure 4f we present the
+sample variance of the radial peculiar velocity posterior. In
+Figure 4g-i, we show the radial peculiar velocity divergence
+ground truth, ensemble mean and standard deviation.
+In Figure 4j-l, we show our constraints on the gravita-
+tional potential from photometric and spectroscopic galaxy
+clustering. In Figure 4j we show the ground truth gravi-
+tational potential, as derived from the ground truth dark
+matter density field. In Figure 4k and Figure 4l, we show the
+ensemble mean and standard deviation across the MCMC
+realizations, respectively.
+The above statistical summaries present high visual re-
+semblance with the ground truth. We quantify this resem-
+blance in Figure 5, by showing the cross-correlation of the
+dark matter density, radial peculiar velocity, off-diagonal
+components of the tidal shear tensor and gravitational po-
+tential with the ground truth. We select an unmasked sub-
+volume of the inference domain, to showcase the best-case
+cross-correlation in regions with data. On the largest scales,
+Article number, page 7 of 16
+
+A&A proofs: manuscript no. aanda
+10
+2
+10
+1
+k [Mpc
+1]
+103
+104
+105
+P
+(k) [Mpc3]
+(a)
+ground truth
+mean
+3
+10
+2
+10
+1
+k [Mpc
+1]
+108
+109
+1010
+1011
+1012
+Pvv(k) [(km/s)2 Mpc3]
+(b)
+10
+2
+10
+1
+k [Mpc
+1]
+103
+104
+105
+P
+(k) [Mpc3]
+(c)
+10
+2
+10
+1
+k [Mpc
+1]
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+P
+/
+P
+P
+(k)
+(d)
+Fig. 6: (a) Dark matter density, (b) radial peculiar velocity and (c) peculiar velocity divergence autocorrelation power
+spectrum. (d) Normalized cross-correlation between the dark matter density and peculiar velocity divergence. The black
+lines indicate the ground truth. The colored window represents 3σ error bars. The uncertainty on the 2-point statistics is
+due to galaxy survey- and data-related uncertainties. The blue line is the CAMB density autocorrelation power spectrum.
+as expected, we have the highest cross-correlation for all
+cosmological fields. The cross-correlation of the gravita-
+tional potential with the ground truth is > 80% on scales,
+as Φ has the longest correlation length.
+In Figure 6 we show the autocorrelation power spectra
+of cosmological fields that are typically used in cosmologi-
+cal analyses. ll power spectra were estimated and corrected
+for the BORG Cloud-In-Cell mass assignment scheme using
+Pylians (Villaescusa-Navarro 2018), as our density field
+has been estimated with a Cloud-In-Cell estimator. The au-
+tocorrelation power spectrum, Pδδ, is shown in Figure 6a.
+The two-point statistics of the inferred density fields are
+consistent with the ground truth and the sampler has cov-
+ered the uncertainty due to cosmic variance (e.g. Pogosian
+et al. 2010, Figure 2). In Figure 6b, we show the autocorre-
+lation power spectrum of the radial peculiar velocity field,
+Pvv. The inference is consistent with the ground truth.
+In Figure 6c, we show the peculiar velocity divergence
+autocorrelation power spectrum, Pθθ, which is also consis-
+tent with the ground truth and ΛCDM prediction (e.g. Ata
+et al. 2017). Further, we see that the power of Pθθ is sup-
+pressed compared to Pδδ beyond k ∼ 0.05 Mpc−1. This is
+expected because collapsing structures that have virialized
+experience less volume change (e.g. Ata et al. 2017). As a re-
+sult, the peculiar velocity divergence field evolves less than
+the density field (e.g. Kitaura et al. 2012; Jennings 2012;
+Hahn et al. 2015; Ata et al. 2017). In Figure 6d, we show
+the normalized cross-correlation power spectrum between
+Article number, page 8 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+0.55
+0.60
+0.65
+0.70
+2
+LPT prediction
+Inference
+Ground truth
+(a)
+1.0
+1.2
+1.4
+3
+(b)
+4.5
+5.0
+5.5
+4
+(c)
+Fig. 7: (a) Variance, (b) skewness and (c) kurtosis of the dark matter density field for our inference, the ground truth and
+a set of random LPT simulations at 13 Mpc (∼ 0.04⟨σz⟩) using the entire inference domain. The latter two are random,
+unconstrained simulations. The inference is constrained jointly with galaxy observations. The error bars encapsulate the
+1σ uncertainty both from the mock galaxy survey and the estimator. Our ground truth is consistent with a random LPT
+simulation and our inference is consistent with both. Our inference accurately captures higher-order statistics of the dark
+matter density field on scales much smaller than the original photometric redshift uncertainty.
+1661
+1329
+997
+664
+332
+0
+x [Mpc]
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+z=30
+(a)
+1661
+1329
+997
+664
+332
+0
+x [Mpc]
+z=10
+(b)
+1661
+1329
+997
+664
+332
+0
+x [Mpc]
+z=0
+(c)
+0.64
+0.66
+0.68
+0.70
+0.72
+0.74
+0.76
+ln(2+ )
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+ln(2+ )
+0.5
+1.0
+1.5
+2.0
+ln(2+ )
+Fig. 8: Structure formation history derived from the dark matter density field at (a) z = 30, (b) z = 10, (c) z = 0 from a
+typical density sample. In red, we overlay the density slice with the galaxy coordinates in a typical realizations. Galaxies
+follow the present-day large-scale structure. The origins of high-density regions are already visible at z = 30.
+the velocity divergence and dark matter density. On large
+scales, it is expected that to linear order θ ∝ δ (e.g. Hahn
+et al. 2015). On smaller scales, structure formation becomes
+nonlinear, which LPT is able to capture. A more refined
+gravity model, like a particle mesh (Jasche & Lavaux 2019,
+in BORG), would be needed to push these results to smaller
+scales and capture the generation of peculiar velocity vor-
+ticity.
+In Figure 7, we show higher-order moments of the con-
+strained dark matter density field for the ground truth, our
+inference and the LPT prediction, with 1σ error bars at the
+target resolution of 13 Mpc (∼ 0.04⟨σz⟩). We derive the LPT
+prediction from a set of random simulations ran using the
+same cosmological parameters as our main inference, but
+without constraints from galaxy redshifts. The error bars
+naturally account for observational uncertainties and the
+estimator uncertainty related to the sample size (Harding
+et al. 2014). For this fully self-consistent setting, our results
+are consistent with the ground truth and LPT prediction
+within 1σ. This result suggests that we infer the density
+field, including its higher-order moments, on scales much
+smaller than the original photometric redshift uncertainty.
+In Figure 8 we show slices through the same randomly-
+selected realization in the structure formation history. Un-
+der the assumption of a causal structure formation model,
+we reconstruct the structure formation history which gives
+rise to the observed structures today. The initial conditions
+are set at z = 99, but we show slices up to z = 30, such
+that the seeds of present-day structures are visible. On the
+z = 0 slice we overlay the ground truth locations of galaxies.
+Galaxies closely trace clusters and filamentary structures,
+whereas voids are sparsely populated by galaxies. This is
+because the Poisson noise is lower in higher-density regions.
+As expected, the origins of high-density peaks in the dark
+matter density field are already visible at early times.
+In Figure 9a, we show a void in the volume covered
+by observations. In Figure 9c we show the average density
+contrast in shells around the void. We see that the density
+contrast tends to zero, as expected when averaging over a
+scale close to the cosmic homogeneity scale (e.g. Gon¸calves
+et al. 2018). In Figure 9b, we show the location of a cluster
+in the density field. In Figure 9d we show the mass enclosed
+in shells around the cluster. We derive it using the prescrip-
+tion in Porqueres et al. (2019). In this study, the particle
+mass is 3.45 × 1013 M⊙. Overall, these results suggest that
+we recover correctly both the statistical properties of the
+large-scale structure around voids and clusters, but also in-
+dividual structures.
+We further show radial peculiar velocity profiles around
+the same void and cluster. The radial peculiar velocity be-
+comes positive close to the void, indicating matter flow-
+ing outside the shells and toward overdense regions, as ex-
+Article number, page 9 of 16
+
+A&A proofs: manuscript no. aanda
+1661
+1107
+553
+x [Mpc]
+1661
+1424
+1186
+949
+712
+474
+237
+y [Mpc]
+(a)
+1661
+1107
+553
+x [Mpc]
+(b)
+50
+100
+150
+200
+250
+R [Mpc]
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+(R)
+(c)
+ground truth
+mean
+3
+50
+100
+150
+200
+250
+R [Mpc]
+1018
+1019
+1020
+1021
+1022
+1023
+1024
+M (< R) [M
+]
+(d)
+50
+100
+150
+200
+250
+R [Mpc]
+100
+0
+100
+200
+300
+400
+vr(R)  [km/s]
+(e)
+50
+100
+150
+200
+250
+R [Mpc]
+400
+300
+200
+100
+0
+vr(R)  [km/s]
+(f)
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Fig. 9: (a) Slice through the ground truth dark matter density field. In purple, a spherical shell around a low-density
+region. (b) Slice through the ground truth dark matter density field. In purple, a spherical shell around a high-density
+peak. (c) The mean density contrast in spherical shells around the low-density region. (d) The mass enclosed in shells
+around the high-density peak. (e) The mean radial peculiar velocity in spherical shells around the low-density region
+with respect to the center of the void. (f) The mean radial peculiar velocity in spherical shells around the high-density
+peak with respect to the center of the peak.
+pected from gravitational structure growth (Peebles 1980).
+On larger scales the average peculiar velocity tends to zero
+due to cosmic homogeneity. In Figure 9f we show the aver-
+age radial peculiar velocity in shells of radius R around the
+cluster in Figure 9b, with respect to its center. The radial
+peculiar velocity is negative close to the cluster, as expected
+from gravitational infall. On larger scales, the average ve-
+locity tends to zero as expected from cosmic homogeneity.
+4.2. Constraints on photometric redshifts
+In this section, we give a brief overview of how the joint
+inference of the large-scale structure and photometric red-
+shift PDFs yields constraints on photometric redshifts. In
+Figure 10 we show our results for three randomly-selected
+galaxies. The redshift likelihood is a Gaussian PDF. Its
+mean is the mock observed redshift and its standard devi-
+ation is 0.05(1 + z), z being the mock cosmological redshift.
+Article number, page 10 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+We derive the target posterior by multiplying the Gaussian
+likelihood with the Poisson term for the redshift inference
+in our Gibbs sampling approach, as shown in Equation (9).
+The results of our inference are overlayed as a histogram.
+This qualitative demonstration illustrates that the photo-
+metric redshift posterior after the application of our method
+contains more information than the likelihood and follows
+the radial density profile of the large-scale structure along
+the line of sight to each galaxy. This mechanism constrains
+the radial positions of galaxies from the initial, radially-
+distorted ones (Figure 1a), to the final ones, that trace the
+filamentary large-scale structure (Figure 1b).
+In Figure 11 we show the reduction in the mean stan-
+dard deviation of the photometric redshift posteriors as a
+function of density after the application of our algorithm.
+σi indicates the original photometric redshift uncertainty
+of the mock observations. We expect that higher-density
+regions yield better constraints and hence, lower redshift
+uncertainty. The linear fit suggests that ⟨σ(δ)⟩/⟨σi(δ)⟩ =
+−0.005(1 + δ) − 0.693. In low-density regions, the mean
+standard deviation reduces by a factor of 0.7⟨σi⟩, whereas
+in high-density regions the reduction reaches 0.4⟨σi⟩. The
+scatter is larger in high-density regions because there are
+fewer strongly overdense peaks. Overall, we expect the red-
+shift uncertainty reduction to be more significant in higher-
+resolution settings, where higher-density peaks will be re-
+solved. It should be noted, however, that due to the highly
+multimodal nature of the target redshift distribution, the
+reduction in standard deviation does not reflect the infor-
+mation gain from the original to the final redshift PDF.
+Finally, in Figure 12 we compare the inferred distribution
+of photometric redshifts, N(z), in our mock survey to the
+ground truth N(z). For legibility, we show the mean N(z)
+across the MCMC samples, along with 3σ error bars. The
+inferred N(z) is consistent with the ground truth N(z). We
+postpone the sampling of the N(z) to future work.
+5. Conclusions
+Next-generation photometric galaxy surveys will deliver an
+extraordinary amount of observations, that will reduce the
+observational uncertainties, will extend deeper in redshift
+and cover a larger cosmological volume than spectroscopic
+ones. At the same time, most cosmological information is in
+the smallest cosmological scales which cannot be accessed in
+presence of large photometric redshift uncertainties. In such
+a setting, it is paramount to obtain control of photometric
+redshift uncertainties in cosmological inferences. Accurate
+inferences of the large-scale structure constrained jointly
+with photometric redshifts offer this possibility.
+In this study we developed a method to constrain
+the primordial and present-day cosmic large-scale struc-
+ture jointly with photometric redshifts at a resolution of
+13 Mpc using, for the first time, a structure formation model
+in a Bayesian forward modelling approach. We achieved
+these joint constraints through Bayesian inference of the
+initial conditions of structure formation with photometric
+galaxy clustering. Our method takes into account data- and
+survey-related uncertainties and preserves all higher-order
+statistics of cosmological fields. In this study, we used first-
+order LPT, yet our algorithm can incorporate any gravi-
+tational model (e.g. Jasche et al. 2015; Jasche & Lavaux
+2019). We demonstrated our constraints using a fully over-
+lapping mock photometric and spectroscopic galaxy cata-
+log as a proof-of-concept. We assumed a worst-case scenario
+for photometric redshift uncertainties in stage-IV surveys,
+σz = 0.05(1 + z), and demonstrated that our method can
+reduce this uncertainty.
+In particular, we showed large improvement in the cross-
+correlation of the photometric galaxy positions with the
+ground truth. The maximum improvement was on scales
+∼ 0.5⟨σz⟩, where the cross-correlation increased from 28%
+(in the mock photometric observations) to 86% (in our in-
+ference). We further achieved accurate inferences of the
+dark matter density, peculiar velocities, gravitational po-
+tential and tidal shear on scales ∼ 0.04⟨σz⟩.
+We demonstrated the ability of our method to accu-
+rately capture the 2-point statistics of the dark matter den-
+sity, as well as its skewness and kurtosis on scales ∼ 0.04⟨σz⟩.
+Higher-order statistics have been promising in constraining
+dark energy (Velten & Fazolo 2020; Fazolo et al. 2022). As
+we can use our method with a particle mesh, our joint in-
+ference framework can be used to constrain the bispectrum
+with photometric galaxy clustering.
+Voids have been extensively used to probe structure for-
+mation close to the linear regime (e.g. Davies et al. 2019;
+Pisani et al. 2019; Davies et al. 2021; Stopyra et al. 2021;
+Contarini et al. 2022), however photometric uncertainties
+limit their detection (e.g. S´anchez et al. 2017). For this pur-
+pose, we explored the possibility of detecting cosmic struc-
+tures much smaller than the original photometric redshift
+uncertainty. The galaxy positions that were originally radi-
+ally smeared due to the presence of photometric uncertain-
+ties, accurately trace the filamentary pattern of the large-
+scale structure after the application of our method. Further,
+we are able to accurately capture individual structures, like
+voids and clusters. This is a crucial advantage of embedding
+a structure formation model in our forward model. Overall,
+we demonstrated that we accurately recover the statisti-
+cal properties of the large-scale structure on scales much
+smaller than the original photometric redshift uncertainty.
+The present work opens up the possibility to miti-
+gate photometric redshift uncertainties in next-generation
+photometric surveys. Concurrently, the incorporation of a
+structure formation model paves a new way forward to ex-
+tract as much information as possible from the smallest
+cosmological scales, while going beyond 2-point statistics
+and preserving information in the data.
+Data and software availability
+Data products can be made available upon reasonable re-
+quest.
+Author Contributions
+The main roles of the authors were, using the CRediT
+(Contribution
+Roles
+Taxonomy)
+system
+(https:
+//authorservices.wiley.com/author-resources/
+Journal-Authors/open-access/credit.html):
+E.T.: conceptualization, methodology, software, formal
+analysis, validation, writing - original draft; J.J.: conceptu-
+alization, methodology, software (supporting, transferred
+Jasche & Wandelt (2012) to new BORG version), supervi-
+sion, resources, writing - feedback, funding acquisition;
+G.L.: software (supporting, debugging and notably during
+the Message Passing Interface parallelization), writing -
+Article number, page 11 of 16
+
+A&A proofs: manuscript no. aanda
+0.05
+0.10
+0.15
+0.20
+0.25
+redshift
+0
+5
+10
+15
+20
+P(z|zobs, )
+(a)
+observation
+target posterior
+ground truth 
+redshift
+inference
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+redshift
+(b)
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+redshift
+(c)
+Fig. 10: Photometric redshift samples for three randomly-selected galaxies. Overlayed are the corresponding initial ob-
+servations estimated from our pipeline without a physical prior, the ideal photometric redshift posterior (for physical
+prior, using the ground truth density) and the ground truth redshift. The maximum redshift to which the posteriors
+extend indicates where each galaxy’s line of sight intercepts the inference box. The inferred posterior is the marginal over
+density realizations, accounting for observational and survey-related uncertainties. The redshift binning is determined by
+the real-space resolution of the inference.
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+1 +
+0.4
+0.5
+0.6
+0.7
+0.8
+( ) /
+i( )
+linear fit
+Fig. 11: Final to initial photometric redshift uncertainty as
+a function of density at the ground truth galaxy locations.
+The black line is a linear fit, indicating that the reduction in
+redshift uncertainty is greater in regions of high dark mat-
+ter density, as expected from the Poisson likelihood. The
+dispersion on the high-density end is due to shot noise, as
+there are fewer high-density peaks in the inference domain.
+feedback, funding acquisition; F.L.: software (support-
+ing, author of the Simplex-In-Cell estimator), writing -
+feedback.
+Acknowledgements
+ET thanks Metin Ata and Natalia Porqueres for helpful
+discussions and feedback, and Adam Andrews, Deaglan
+Bartlett, Andrija Kostic, James Prideaux-Ghee, Fabian
+Schmidt, Ben Wandelt and Pauline Zarrouk for com-
+ments on the original manuscript. Part of the computa-
+tion and data processing in this study were enabled by re-
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+redshift
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+normalized N(z)
+mock N(z)
+3
+Fig. 12: In orange, the average of normalized N(z) of in-
+ferred photometric redshifts across MCMC samples for a
+subsample of galaxies, along with 3σ uncertainties. In black,
+the mock photometric N(z) for the same subsample. The
+N(z) of inferred photometric redshifts is consistent with the
+ground truth N(z).
+sources provided by the Swedish National Infrastructure
+for Computing (SNIC) at Tetralith, partially funded by
+the Swedish Research Council through grant agreement no.
+2020-05143. This research utilized the HPC facility sup-
+ported by the Technical Division at the Department of
+Physics, Stockholm University. This work was enabled by
+the research project grant ‘Understanding the Dynamic
+Universe’ funded by the Knut and Alice Wallenberg Foun-
+dation under Dnr KAW 2018.0067. JJ acknowledges sup-
+port by the Swedish Research Council (VR) under the
+project 2020-05143 – ”Deciphering the Dynamics of Cosmic
+Structure”. GL acknowledges support by the ANR BIG4
+project, grant ANR-16-CE23-0002 of the French Agence
+Nationale de la Recherche, and the grant GCEuclid from
+”Centre National d’Etudes Spatiales” (CNES). This work
+was supported by the Simons Collaboration on “Learning
+Article number, page 12 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
+the Universe”. This work is conducted within the Aquila
+Consortium (https://aquila-consortium.org).
+A. Derivation of the conditional density posterior
+for real-space inference
+In this section, we describe how to arrive at the condi-
+tional density posterior of Equation (3) starting from Equa-
+tion (2). The former equation shows how we incorporate a
+gravitational structure growth model into our framework,
+whereas the latter represents the output of the Gibbs sam-
+pling process. Here, we discuss how we obtain constraints
+on the real-space late-time density field using galaxy obser-
+vations in redshift space.
+The present-day density field is conditionally indepen-
+dent of the observed galaxy redshifts given an ensemble of
+sampled redshifts. Therefore
+P(δ|z, zobs, θ)
+=
+P(δ|z, θ)
+=
+�
+Ng
+P(δ, Ng|z, θ)
+=
+�
+Ng
+P(δ|Ng)P(Ng|z, θ).
+(17)
+The last term in the sum indicates how we arrive from
+redshift-space observations to real-space galaxy counts. As
+elaborated on in (Jasche & Wandelt 2012, Eq. B4), this
+involves a transform from redshift- to real-space such that
+P(δ|z, θ) =
+�
+Ng
+P(δ|Ng)P(Ng|x(z, θ)),
+(18)
+x being the real-space galaxy positions. In what follows, we
+will omit dependence on (z, θ) for legibility. The latter term
+indicates how we bin galaxies onto a grid given their real-
+space positions. For this gridding operation, we use Nearest
+Grid Point projection such that
+P(Ng|x) =
+�
+i
+δD
+��������Ng
+i −
+�
+p
+WNGP(xi − xp)
+�������� ,
+(19)
+i being the voxel index, p being the galaxy index and
+WNGP(x) =
+3
+�
+n=1
+�1
+if |xn|Nn/Ln < 1
+0
+otherwise,
+(20)
+Substituting the above into Equation (18) we arrive at
+P(δ|z, θ) = P(δ|Ng).
+(21)
+B. Derivation of the conditional redshift posterior
+for real-space inference
+In this subsection we describe how we arrive at the condi-
+tional redshift posterior in Equation (9), while performing
+an inference of the density field in real space. Let us denote
+by θ the right ascension and declination of a galaxy and by
+u the redshifts of all other galaxies in the previous sampling
+step. We start by the redshift posterior of a given galaxy i
+and apply Bayes’ law
+P(zi|θ, u, δ, zobsi)
+=
+P(zi)P(θ, u, δ, zobsi, zi)
+P(θ, u, δ, zobsi)
+=
+P(zi)P(zobsi|θ, u, δ, zi)P(θ, u, δ|zi)
+P(zobsi|θ, u, δ)P(θ, u, δ)
+=
+P(θ, u, δ|zi)P(zi)
+P(u, θ, δ)
+P(zobsi|θ, u, δ, zi)
+P(zobsi|θ, u, δ)
+=
+P(zi|θ, u, δ)P(zobsi|θ, zi, δ, u)
+P(zobsi|u, δ, θ)
+=
+P(θ, zi|u)P(δ|θ, zi)
+P(δ|u)
+P(zobsi|θ, zi, δ, u)
+P(zobsi|u, δ, θ)
+(22)
+We assume zobsi is conditionally independent of u given zi
+and δ and therefore we arrive at
+P(zi|θ, u, δ, zobsi) = P(θ, zi|u)P(δ|θ, zi)
+P(δ|u)
+P(zobsi|θ, zi, δ)
+P(zobsi|δ, θ)
+(23)
+The first term on the right-hand side of the above equa-
+tion describes the distribution of galaxies in redshift space.
+However, as we perform a real-space inference, we trans-
+form this term from redshift- z, to comoving Cartesian, r,
+space
+P(θ, zi|u) =
+����r2(zi) sin (φ)∂r
+∂z
+����zi
+����P(x|u),
+(24)
+φ being the declination of the galaxy and x its real-space
+position. The real-space galaxy positions are used to build
+the gridded galaxy count field, Mg. Mg indicates the galaxy
+counts except for the galaxy under consideration. We the
+last term on the right-hand side of the above equation as a
+marginal over galaxy counts
+P(x|u) =
+�
+Mg
+P(Mg|u)P(x|Mg)
+(25)
+The first term in the sum is the Nearest Grid Point projec-
+tion that we use to grid galaxies at the field-level. We will
+indicate the Nearest Grid Point kernel with WNGP. Under
+the assumption that galaxies are Poisson-distributed, the
+number counts in each voxel are independent events. Since
+the Poisson intensity depends only on the density, our Pois-
+son model is homogeneous. Following these assumptions
+P(x|u)
+=
+�
+Mg
+P(x|Mg)
+×
+�
+i
+δD
+��������Mg
+i −
+�
+p
+WNGP(xi − xp)
+�������� ,
+(26)
+where i is the voxel index and p the galaxy index. We now
+want to rewrite the first term in the above sum with respect
+to the entire galaxy count field, Ng. It differs from Mg in that
+it includes the galaxy under consideration. The reason we
+rewrite the redshift posterior with respect to Ng is because
+the entire galaxy count field is used in the density inference.
+Below this point we will drop the dependence on u, because
+all quantities are conditionally independent of u given Mg.
+P(x|Mg) =
+1
+P(Mg)
+�
+Ng
+P(Ng)P(x|Ng)P(Mg|Ng, x)
+(27)
+Following our definition, the relationship between Mg and
+Ng is deterministic and yields
+P(x|Mg)
+=
+1
+P(Mg)
+�
+Ng
+P(Ng)P(x|Ng)
+Article number, page 13 of 16
+
+A&A proofs: manuscript no. aanda
+×
+�
+i
+δD �
+Mg
+i − [Ng
+i − WNGP(xi − x)]
+�
+=
+P(x|Ng),
+(28)
+because P(Ng) and P(Mg) are equal, as they differ by one
+(at the position of the galaxy under consideration). The
+above result is equal to finding a galaxy at position x given
+a number counts field.
+P(x|Ng) =
+�
+i
+WNGP(xi − x)
+δV
+Ng
+i
+Ntotal
+,
+(29)
+where δV is the volume of a voxel and Ntotal the total number
+of galaxies in the inference domain. As a result of the above,
+the second term on the right-hand side of Equation (23) is
+written as
+P(δ|θ, zi)
+P(δ|u)
+=
+�
+i
+Mi!
+Ni! λNi−Mi
+i
+=
+�
+i
+� λi
+Ni
+�WNGP(xi−x)
+,
+(30)
+as it represents the ratio between the Poisson distribution
+for all galaxies and all galaxies expect for the one we con-
+sider at each step. Substituting Equation (29) and Equa-
+tion (30) into Equation (23) we arrive at
+P(zi|θ, u, δ, zobsi)
+∝
+����r2(zi) sin (φ)∂r
+∂z
+����zi
+����
+×
+�
+i
+WNGP(xi − x)λiP(zobsi|θ, zi, δ).
+(31)
+As demonstrated in Jasche & Wandelt (2013) the above
+form suggests that each galaxy can be sampled indepen-
+dently from all others, as the dependence on u vanishes
+through the conditioning on galaxy counts and therefore,
+the Poisson intensity. Finally, since sin (φ) is a constant
+term, it serves as a proportionality constant in each galaxy’s
+target posterior and therefore vanishes. Finally, we drop the
+dependence on right ascension and declination for legibil-
+ity, since we only sample redshifts. Therefore, the process
+in Equation (1) is equivalent to drawing a sample from
+P(zi|δ, zobsi)
+∝
+����r2(zi)∂r
+∂z
+����zi
+����
+×
+�
+i
+WNGP(xi − x)λiP(zobsi|zi, δ).
+(32)
+C. Estimators of large-scale structure properties
+Accurate inferences of the large-scale structure with pho-
+tometric galaxy clustering are necessary because next-
+generation photometric surveys probe deeper redshifts and
+have a wider footprint than spectroscopic surveys. Below,
+we discuss the aspects of the cosmic large-scale structure
+which we demonstrate our method on, as well as the esti-
+mators we use.
+C.1. Peculiar velocities
+In order to derive the peculiar velocity field, we use the
+Simplex-In-Cell (SIC) estimator (e.g. Abel et al. 2012; Hahn
+et al. 2015; Leclercq et al. 2017). Contrary to kernel meth-
+ods, which sample the velocity field only at a discrete set
+of locations with poor resolution in low-density regions, the
+SIC estimator provides an estimate of the velocity field at
+any point in space. In this framework, the Lagrangian posi-
+tions of the particles in the inference domain are considered
+as vertices of unit cubes. The Delaunay tessellation of each
+unique cube defines six tetrahedra which are followed dur-
+ing the evolution. Subsequently, each tetrahedron deposits
+a value to the grid (Leclercq et al. 2017, Equation 37).
+We further explore kinematic properties of the peculiar
+velocity field. Here, we focus on the peculiar velocity diver-
+gence, defined as
+θ = −
+1
+f Ha∇ · v,
+(33)
+where f is the linear growth rate, H the Hubble parame-
+ter and a the cosmic scale factor. The divergence determines
+how the volume of a peculiar velocity flow changes. The pe-
+culiar velocity divergence can be used to constrain Ωm and
+structure formation (e.g. Bernardeau et al. 1995; Howlett
+et al. 2017). Further, the redshift-space distortion signal can
+be detected in the autocorrelation power spectrum of the
+peculiar velocity divergence and the cross-correlation power
+spectrum between the dark matter density and the peculiar
+velocity divergence (e.g. Bel et al. 2019). For ∇ · v > 0 the
+region is expanding, for ∇ · v < 0 the region is contracting,
+whereas for ∇ · v = 0 the volume remains invariant.
+C.2. Gravitational potential
+Probing gravitational tidal forces with photometric sur-
+veys is a challenging endeavor, but a necessary one (e.g.
+Troxel & Ishak 2015), as intrinsic galaxy alignments con-
+taminate weak lensing analyses and can be used as cos-
+mological probes. Further, as photometric surveys extend
+deeper in redshift, they allow us to probe the evolution of
+intrinsic alignments over time. This evolution can be used
+to constrain galaxy formation and evolution scenarios.
+For this purpose, we explore differing aspects of the dy-
+namics and kinematics of the large-scale structure. We de-
+rive the gravitational potential of the dark matter density
+field by solving Poisson’s equation in Fourier space
+Φ(k) = −4πGρ(k)
+|k|2
+,
+(34)
+where G is the gravitational constant, ρ the density and
+k the wavevector. We further derive the tidal shear of the
+gravitational potential, which is given by
+Ti j =
+∂2Φ
+∂xi∂x j
+,
+(35)
+where xi, ({i, j} = {1, 2, 3}) are comoving Cartesian coordi-
+nates. Our inference can further be used with cosmic web
+classification methods, particle- (e.g. Sousbie 2013) or cell-
+based (e.g. Libeskind et al. 2018; Buncher & Carrasco Kind
+2020) to infer cosmic structures constrained by photomet-
+ric redshifts. Such classifications have been used to study
+the environment of cosmological tracers (e.g. Leclercq et al.
+2016; Porqueres et al. 2018; Tsaprazi et al. 2022a). Kru-
+use et al. (2019) further showed that photometric redshifts
+are positively correlated with filaments detected in spectro-
+scopic galaxy observations.
+Article number, page 14 of 16
+
+Tsaprazi et al.: Large-scale structure from photometric redshifts
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diff --git a/HtE1T4oBgHgl3EQf_gZv/content/tmp_files/load_file.txt b/HtE1T4oBgHgl3EQf_gZv/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c368c1313f8a3322fde9fb631928a08b7fdf03ad
--- /dev/null
+++ b/HtE1T4oBgHgl3EQf_gZv/content/tmp_files/load_file.txt
@@ -0,0 +1,1539 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf,len=1538
+page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  ©ESO 2023  January 10, 2023  Higher-order statistics of the large-scale structure from  photometric redshifts  Eleni Tsaprazi1, Jens Jasche1,2, Guilhem Lavaux2 and Florent Leclercq2  1 The Oskar Klein Centre, Department of Physics, Stockholm University, Albanova University Center, SE 106 91  Stockholm, Sweden  e-mail: eleni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='tsaprazi@fysik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='se  2 CNRS & Sorbonne Universit´e, UMR 7095, Institut d’Astrophysique de Paris, 98 bis boulevard Arago, F-75014 Paris,  France  ABSTRACT  Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The large-scale structure is a major source of cosmological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' However, next-generation photometric  galaxy surveys will only provide a distorted view of cosmic structures due to large redshift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' To address the need for accurate reconstructions of the large-scale structure in presence of photometric uncertain-  ties, we present a framework that constrains the three-dimensional dark matter density jointly with galaxy photometric  redshift probability density functions (PDFs), exploiting information from galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our forward model provides Markov Chain Monte Carlo realizations of the primordial and present-day dark  matter density, inferred jointly from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our method goes beyond 2-point statistics via field-level inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' It accounts  for all observational uncertainties and the survey geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We showcase our method using mock catalogs that emulate next-generation surveys with a worst-case redshift  uncertainty, equivalent to ∼300 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On scales 150 Mpc, we improve the cross-correlation of the photometric galaxy  positions with the ground truth from 28% to 86%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The improvement is significant down to 13 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On scales 150 Mpc,  we achieve a cross-correlation of 80 − 90% with the ground truth for the dark matter density, radial peculiar velocities,  tidal shear and gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We achieve accurate inferences of the large-scale structure on scales smaller than the original redshift  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Despite the large redshift uncertainty, we recover individual cosmic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Owing to our structure  growth model, we infer plausible initial conditions of structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Finally, we constrain individual photometric  redshift PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This work opens up the possibility to extract information at the smallest cosmological scales with  next-generation photometric surveys, going beyond approaches that compress information in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' large-scale structure of Universe – distances and redshifts  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Introduction  High-accuracy galaxy redshift estimates are required for a  plethora of scientific cases, such as the mapping of the cos-  mic large-scale structure and tests on the geometry of the  universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Albrecht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Pea-  cock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Mandelbaum & Hyper  Suprime-Cam (HSC) Collaboration 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Samuroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The LSST Dark Energy Science Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Mandelbaum 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Abruzzo &  Haiman 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Eifler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Loureiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Secco  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Newman & Gruen 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Next-generation surveys will deliver low-accuracy redshift  estimates from photometry and in certain cases fewer, high-  accuracy, spectroscopic ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' LSST Science Collabora-  tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Amendola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Dor´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' However, bias in  cosmological inferences may occur in sky areas, depth and  color ranges where only photometric data is available (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Fortunately, the physical clustering infor-  mation of the large-scale structure can be used to improve  the accuracy of photometric redshift observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Seld-  ner & Peebles 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Phillipps & Shanks 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Landy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Kovaˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jasche & Wandelt 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' M´enard  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Aragon-Calvo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Shuntov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Synergies between spectroscopic and photometric sur-  veys have been explored, intended mainly for photometric  redshift calibration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Rhodes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Capak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In light of this, a multitude of techniques to im-  prove redshift uncertainty has been proposed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Newman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Masters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Speagle & Eisenstein 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Speagle & Eisenstein 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Davidzon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Schaan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Shuntov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Rau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Stan-  ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Leistedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' However, there are  limitations to spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Photometric surveys typically  reach fainter magnitudes, higher redshifts, cover a larger  portion of the color-space and have higher redshift com-  pleteness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' LSST Science Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Bor-  doloi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Amendola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In order to exploit these advantages and avoid biases in  cosmological analyses, it is necessary to mitigate the basic  limitation of photometry, low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jasche & Wandelt  (2012) demonstrated that clustering information from the  Article number, page 1 of 16  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='03581v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='CO]  9 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 1: (a) Mock photometric galaxy positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Galaxy positions are radially distorted due to redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (b)  Galaxy positions in a typical MCMC sample after the application of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Galaxies now trace the filamentary  dark matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (c) Ground truth (mock) galaxy positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The galaxy positions that were radially smeared in  the mock observations, closely trace the filamentary structure in the ground truth galaxy positions – within observational  uncertainties – after the application of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  10  1  k [Mpc]  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  cross correlation  with ground truth  mock observations  after photometric  redshift sampling  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2: Cross-correlation of the gridded photometric galaxy  coordinates before (black) and after (orange) the applica-  tion of our method with the ground truth (mock data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The 3σ error bars are across MCMC realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We con-  sider a subbox that is the largest unmasked cubic volume  in our inference domain, with side length 520 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On the  largest scales, ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='7⟨σz⟩, the cross-correlation increases from  73% to 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On the smallest scales, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04⟨σz⟩, the cross-  correlation increases from 2% to 14%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The largest improve-  ment is on scales ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5⟨σz⟩, from 28% to 86%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  large-scale structure alone can improve the accuracy of a  purely photometric sample up to one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In this work, we developed a framework that, for the  first time, infers the three-dimensional cosmic large-scale  structure jointly with redshift probability density functions  (PDFs) of photometric galaxies using a physical structure  formation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In this way, we improve upon photometric  redshift uncertainty and self-consistently quantify observa-  tional errors in the large-scale structure inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In doing  so, we preserve all high-order statistics of the large-scale  structure and use a physically-motivated structure growth  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Our prior is the homogeneity and isotropy assump-  tion for the initial conditions of structure formation, as  well as the physical model for the formation of the large-  scale structure and Gaussian initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our frame-  work is built within the Bayesian Origin Reconstruc-  tion from Galaxies (BORG) algorithm (Jasche & Wan-  delt 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Lavaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jasche &  Lavaux 2019), which infers the initial conditions of struc-  ture formation from galaxy observations in a fully Bayesian  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We include 1% spectroscopic galaxies, similar to  the expected galaxy number ratio between next-generation  photometric and spectroscopic surveys (LSST Science Col-  laboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Amendola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Control of the impact of photometric redshift uncertain-  ties on large-scale structure inferences is a requirement for  a wide range of studies (see Salvato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019, for a re-  view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' First, this is relevant to peculiar velocity analyses  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Abate & Lahav 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Hudson & Turnbull 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' John-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Watkins & Feldman 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Andersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Mediavilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Said et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Palmese &  Kim 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Prideaux-Ghee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022),  which constrain the growth of structure, gravity and cosmo-  logical parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Peebles 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Abate & Lahav 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Palmese & Kim 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The evolution  of the peculiar velocity divergence with redshift is predicted  by ΛCDM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Peebles 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Nusser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Maartens  1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Tsaprazi & Tsagas 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Filippou &  Tsagas 2021) and can be used as a cosmological test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Accounting for photometric redshift uncertainties is cru-  cial for weak lensing inferences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Porqueres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' More-  over, photometric redshift uncertainties significantly affect  estimates of intrinsic alignment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Bridle & King 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Codis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Fischbacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Article number, page 2 of 16  (a)  (b)  (c)  After redshift sampling  Mock photometric observations  Ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='64  Z  z  ZTsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : Large-scale structure from photometric redshifts  Data  (photometric &  spectroscopic  galaxy survey)  Initial conditions  (Gaussian dark matter density &  cosmological power spectrum)  gridded  galaxy  counts  Forward Model  (LPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' galaxy bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  redshift errors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  survey mask,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  luminosity function)  likelihood  comparison  update  dark matter density &  photometric redshifts  galaxy bias  redshift errors  survey mask  luminosity function  random LPT  dark matter density  (ground truth)  gridded Poisson  galaxy counts  mock galaxy  coordinates  (observed & ground  truth)  Mock data generation  Joint density - redshift sampling  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 3: Flowchart of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Left column: Mock data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The input is a luminosity function, a survey mask,  galaxy bias parameters and redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We generate a random LPT density field on which we apply galaxy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  From that, we draw a Poisson galaxy count sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We populate the 3D grid with mock observations given a survey  mask and selection function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We finally apply Gaussian noise to the mock redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This data is fed into our forward  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Right column: MCMC sampling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We then evolve them using Lagrangian Perturbation Theory and  compare the output to the gridded galaxy counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' After the likelihood comparison, we sample the dark matter density  jointly with photometric redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2022), redshift-space distortions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Kaiser 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Perci-  val et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2011), Baryon Acoustic Oscillations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ross  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Chaves-Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2018) and can hinder the  disentanglement of dynamical dark energy from modified  gravity (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Further, probing the large-scale  structure with galaxy clustering at high-redshift through its  gravitational potential can constrain galaxy formation and  evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Schmitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Tonegawa & Okumura  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Moreover, it can constrain the scale of cosmic ho-  mogeneity, since next-generation surveys will be sensitive  to large-scale clustering and therefore, to superclusters and  voids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Benitez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Last but not least, accu-  rate inferences of the large-scale structure at high-redshift  constrained by photometric galaxy clustering can be used  complementary to Lyman-α, which is sensitive to voids and  quasar clustering, which are differently biased than regular  galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Benitez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Porqueres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The paper is structured as follows: In Section 2 we pro-  vide the mathematical formulation of the density and pho-  tometric redshift posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Section 3 we discuss the  algorithmic implementation and configuration of the mock  galaxy survey generator and the photometric redshift sam-  pler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Section 4 and 5 we discuss our results and conclu-  sions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Statistical modelling of the large-scale structure  We build our photometric redshift sampler on the BORG  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' BORG employs a hierarchical Bayesian forward-  model to infer the posterior distribution of plausible initial  conditions from which present structures have formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' To  solve this statistical initial conditions problem, BORG uses  a sophisticated Markov Chain Monte Carlo (MCMC) ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The resulting large-scale structure posterior is approx-  imated by realizations of the large-scale structure, con-  strained by galaxy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' BORG takes into account  the survey geometry, flux limitations and related system-  atic effects and accounts for them in the large-scale struc-  ture inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In the present work, we constrain the pri-  mordial and present-day large-scale structure jointly with  photometric galaxy observations, while quantifying all ob-  servational uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Gibbs sampling of the joint posterior  We achieve joint constraints on the large-scale structure  and galaxy comoving distances by jointly sampling from  their joint posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In order to do so, we choose a Gibbs  sampling approach (Hastings 1970), in which we first draw  Article number, page 3 of 16  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  a real-space density sample conditioned on galaxy redshifts  and then galaxy redshifts conditioned on the primordial  density field, δi  zj+1  ↶  P  �  z  ���� zobs, θ, δi  j  �  (1)  δi  j+1  ↶  P  �  δi ���� z j+1, zobs, θ  �  ,  (2)  where z is the sampled redshift of a galaxy, zobs its observed  redshift, θ the right ascension and declination and j indi-  cates the MCMC sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We then explore the joint  dark matter density and photometric redshift posterior by  iteratively sampling from the above conditional distribu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Dark matter density  For the sake of demonstration we use first-order Lagrangian  Perturbation Theory (LPT) to model structure formation  and a Poisson likelihood to associate the dark matter den-  sity field to photometric galaxy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' BORG allows  for more complex structure formation models, such as the  particle-mesh model (Jasche & Lavaux 2019), which can  be used with the current photometric redshift sampling ap-  proach in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Within BORG, the galaxy coordinates  are transformed to galaxy counts, Ng, using Nearest Grid  Point projection (Eastwood & Hockney 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As we show  in Appendix A, Equation (2) can be written as P(δ|Ng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Below, we focus on the main aspect of our work, inferring  the primordial density field, δi, in conjunction with the late-  time large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We therefore write  P(δi | Ng) ∝  �  dδP(Ng | δ)P(δ | δi)P(δi),  (3)  where δi the primordial, real-space density field, P(Ng|δ)  is the Poisson likelihood, P(δ | δi) the structure formation  term and P(δi) the prior on the primordial density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The Poisson likelihood is  P(Ng | δ) =  �  k  λk  Ng  k e−λk  Ng  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  ,  (4)  where λk is the expected number of galaxies at the kth grid  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The index k runs over all grid elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The de-  pendence of the Poisson intensity, λk, on space models the  inhomogeneous nature of the galaxy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The Pois-  son intensity is given by  λk = Wk⟨n⟩(1 + δk)β,  (5)  where Wk is the survey window, ⟨n⟩ is the expected mean  number of galaxies per grid element and β the power-law  exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For the sake of demonstration here, we have as-  sumed a linear galaxy bias model, taking β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our method  can also account for nonlinear bias models, as demonstrated  previously in Jasche & Lavaux (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Lavaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Charnock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The survey window encapsulates  information on the luminosity function through the radial  completeness function, C, and survey mask, M, as follows  W(x) = C(|x|)M( ˆn),  (6)  ˆn being the unit vector along the line of sight to a galaxy  located at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We write the structure formation term as  P(δ | δi) =  �  k  δD(δk − Fk(δi))  (7)  where δD is the Dirac delta distribution, k runs over the grid  indices and Fk is the structure formation model, for which  we choose first-order LPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The Dirac delta represents that  we assume no error on our gravity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We sample from  Equation (2) using a Hamiltonian Monte Carlo sampler, as  introduced in BORG by Jasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Finally, we as-  sume that the initial conditions for structure formation are  described by a zero-mean multivariate Gaussian distribu-  tion of the primordial dark matter density, δi  P(δi) =  1  √|2πS |  exp  ��������−1  2  N  �  q  N  �  r  δi  qS −1  qr δi  r  �������� ,  (8)  where |S | indicates the determinant of the covariance ma-  trix, S , of the primordial density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Photometric redshift modelling  We write the conditional redshift posterior that we use in  our Gibbs sampling approach (see Appendix B) for each  individual galaxy, i, as  P(zi | zobsi, δ) ∝ P(δ | zi)P(zi | zobsi)J(zi),  (9)  where zi the true redshift of a galaxy i and J the Jacobian  matrix of the transform from redshift- to real-space volume  element:  J(zi) =  ������r2(zi)∂r  ∂z  ����zi  ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (10)  We provide a detailed derivation of Equation (9) in Ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The formulation in Equation (9) is possible be-  cause each galaxy can be treated independently from all  others given a density field, as a consequence of the Poisson  density likelihood (Jasche & Wandelt 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The first term  on the right-hand side is associated with the density field  along the line of sight to the galaxy and the second term  represents the photometric redshift likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In real ob-  servations, photometric redshift likelihoods are highly non-  Gaussian (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Christlein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Here, for demonstra-  tion, we choose a Gaussian likelihood with respect to the  observed redshift, truncated at zero  P(zobsi | zi) =  T(zi)  �  2πσ2(1 + zi)2 exp  �  −1  2  (zobsi − zi)2  σ2(1 + zi)2  �  ,  (11)  where  T(zi) =  �1  2 + 1  2erf  �  zi  √  2σ(1 + zi)  ��−1  ,  (12)  where erf(x) is the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This term ensures con-  sistency with the truncation of observed redshifts at zero  in the mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our method generalizes to non-uniform  photometric redshift uncertainties and can accept redshift  estimates already constrained by other independent meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Therefore, our algorithm can accommodate arbitrar-  ily complex redshift distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Notice that we make the  photometric redshift uncertainty dependent on redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We  make this assumption as a proof-of-concept demonstration,  but the method can account for any redshift likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  Poisson term includes the radial component of the selection  function, Cr, and the line-of-sight density  P(δ | zi) = Cr(zi)λ(xi),  (13)  Article number, page 4 of 16  Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : Large-scale structure from photometric redshifts  in the range 0 < zi ≤ zmaxi, where zmaxi represents the maxi-  mum redshift that lies still within the observed volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In  order to draw redshift samples from Equation (1), we use a  slice sampler (Neal 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' One can either update all galaxy  redshifts at once, or one galaxy at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As demonstrated  in Jasche & Wandelt (2012), each galaxy can be treated  independently given the density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We therefore sample  each galaxy individually and computationally in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In this process, galaxy redshifts at each sampling step are  conditioned on the previous density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Then, via Equa-  tion (3), the next density sample is conditioned on the up-  dated redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In this fashion, we constrain photometric  redshifts jointly with the dark matter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As a  result, we reduce the uncertainty in both compared to in-  ferring the density field from photometric redshifts directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Mock data generation  In this section, we describe the algorithmic implementation  and configuration of the mock (ground truth) galaxy survey  generator, as well as the photometric redshift sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The input to our algorithm is right ascension, declina-  tion, observed redshifts, redshift uncertainty, survey mask,  luminosity function and galaxy bias parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The out-  put is a three-dimensional large-scale structure posterior  and photometric galaxy PDFs, jointly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  large-scale structure realizations are samples of the 3D  large-scale structure posterior distribution and account for  data- and survey-related uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our framework goes  beyond N-point correlations, as it infers the full 3D dark  matter density posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our method applies to regions  where spectroscopic data coverage is not present or incom-  plete, because it can exploit information from photometric  galaxy clustering alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As a result, our method yields con-  straints also when using only photometric redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Here  we provide a proof-of-concept demonstration on mock data,  focusing on the inference of cosmological fields jointly with  photometric and spectroscopic observations with Gaussian  redshift errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  To validate and test the performance of our method we  generate artificial photometric and spectroscopic surveys by  the following procedure:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Generate a random Gaussian primordial density field  (initial conditions), δi and the corresponding present-  day density field, δ using the LPT forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Draw mock galaxy counts from a Poisson distribution  conditional on the present-day density field  Ng  true ↶ P  �  Ng  true  ����δ  �  ,  (14)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Here we displace galaxies in the volume elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We  start by drawing displacements for each galaxy from a  uniform distribution, U  ui,g ↶ U(0, 1),  (15)  where i = (1, 2, 3) represents the three Cartesian direc-  tions, g is the galaxy index, ui is the uniform displace-  ment along direction i for a given galaxy and xi is the  galaxy’s Cartesian coordinate i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We then displace galax-  ies in each cell of the three-dimensional grid as follows  xi,g = dBi + Ri(ni,g + ui,g − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5),  (16)  where dBi is the i-coordinate of the lower left box cor-  ners and ni runs over the number of grid elements in one  direction of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ri is the resolution along the direc-  tion i, defined as Ri = Li/Ni, Ni being the grid resolution  along i and Li the corresponding box size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The subtrac-  tive factor assigns galaxies to the lower left corner of  each grid cell for the Nearest Grid Point projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Iterate the following over galaxy counts in each grid cell  (a) Transform Cartesian comoving coordinates to right  ascension, declination, comoving distance, r and red-  shift, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (b) Calculate the observation probability, W(x), at the  galaxies’ location, according to Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  observation probability depends on the luminosity  function and survey mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (c) Accept observed galaxies using rejection sampling:  We draw a random number, q, in the range (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' If  q < W(x), we accept the galaxy and add it to the  survey, otherwise we reject it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We then generate observed redshifts by adding Gaus-  sian noise with zero mean and variance σ2(1 + z)2 to the  ground truth redshifts, z, according to Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  We truncate the observed redshifts at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We generate an observed galaxy count field using Near-  est Grid Point projection on the observed redshifts and  a ground truth galaxy count field using the ground truth  redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  We perform the inference in a box extending to redshift  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8, with a box size of 1660 Mpc, a grid resolution of  1283 and a real-space resolution of 13 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The observer is  at the lower left corner of the box, such that the observed  area covers ∼ 1 octant of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We assume the Planck  2018 cosmological parameters (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We do not sample spectroscopic redshifts, as their  uncertainties are insignificant compared to the resolution  of our inference and photometric redshift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Our mock data consists of a catalog with 2 · 107 pho-  tometric and 2 · 105 spectroscopic galaxy redshifts, simi-  lar to the fraction of photometric to spectroscopic redshifts  and number density in next-generation surveys in our red-  shift range (LSST Science Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Amen-  dola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We take the photo-  metric luminosity function to be an i-band Schechter with  M∗ = −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8 and α = −1 and the spectroscopic luminosity  function to be a j-band Schechter with M∗ = −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04 and  α = −1 (Table 2, Helgason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2012), as these are bands  that will be used in next-generation surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The generation  of the angular survey mask is described in Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The photometric and spectroscopic components of  each catalog fully overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We adopt a worst-case scenario  for photometric redshift uncertainties, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='05(1+z) (LSST  Science Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For illustrative purposes  we choose a linear galaxy bias model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Constraints on the large-scale structure  We present a qualitative illustration of our results in Fig-  ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 1a we show the observed galaxy positions in  our mock survey that are radially-distorted due to Gaussian  redshift noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In comparing that to the galaxy coordinates  as constrained by our algorithm in Figure 1b, we see that  despite the initially large redshift uncertainty, photometric  Article number, page 5 of 16  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  1661  1424  1186  949  712  474  237  y [Mpc]  (a)  (b)  (c)  1661  1424  1186  949  712  474  237  y [Mpc]  (d)  (e)  (f)  1661  1424  1186  949  712  474  237  y [Mpc]  (g)  (h)  (i)  1661  1107  553  x [Mpc]  1661  1424  1186  949  712  474  237  y [Mpc]  (j)  1661  1107  553  x [Mpc]  (k)  1661  1107  553  x [Mpc]  (l)  1  0  1  2  3  4  true  1  0  1  2  3  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='50  ( )  1000  750  500  250  0  250  500  750  1000  true/4 G  [Mpc  2]  1000  750  500  250  0  250  500  750  1000  /4 G  [Mpc  2]  50  100  150  200  250  300  350  ( /4 G ) [Mpc  2]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 4: From top to bottom, slices through the three-dimensional dark matter density, radial peculiar velocity, divergence,  gravitational potential and the off-diagonal components of the tidal shear for the ground truth (left column), average  (middle column) and standard deviation (right column) across the MCMC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The slices are at the same fixed  distance from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The white region is outside the survey footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As shown in the right column, regions  populated by galaxies have lower uncertainty than regions outside the survey footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  galaxies now trace filamentary structures in the dark mat-  ter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' These positions are derived from a typical  MCMC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Observational uncertainties are accounted  for in the entire set of MCMC samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We show the ground  truth galaxy positions in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We notice high visual  resemblance between the constrained observations and the  ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 2 we show the cross-correlation  between the gridded photometric galaxy coordinates with  the ground truth before and after the application of our  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We select an unmasked subvolume of our infer-  ence domain to demonstrate the best-case improvement in  cross-correlation given the uncertainties in our survey con-  figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We find the largest improvement in the cross-  correlation to be on scales ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5⟨σz⟩, from 28% to 86%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  This indicates that our method provides both accurate in-  ferences of the large-scale structure and reduces photomet-  ric redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We refer the reader to Figure 3 for  the flowchart of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Appendix C we provide a description of the estima-  tors we use to derive properties of the large-scale structure,  along with scientific cases that call for use of the derived  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4 we show statistical summaries of the  posterior distribution of cosmological fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4a  we show the ground truth dark matter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  Article number, page 6 of 16  Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : Large-scale structure from photometric redshifts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  cross correlation  with ground truth  (a)  Dark matter density  inference  3  (b)  Radial peculiar velocity vr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  cross correlation  with ground truth  (c)  Tidal tensor T12  (d)  Tidal tensor T01  10  1  k [Mpc]  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  cross correlation  with ground truth  (e)  Tidal tensor T02  10  1  k [Mpc]  1  (f)  Gravitational potential  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 5: Cross-correlation of our inferred cosmological fields with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The colored windows indicate 3σ error  bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We consider a subbox that is the largest unmasked cubic volume in our inference domain, with side length 520 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  On the largest scales, we find a cross-correlation > 90% with the ground truth for all cosmological fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On the smallest  scales, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04⟨σz⟩, we find a cross-correlation of ∼ 20% for the dark matter density, peculiar velocity and tidal tensor and  ∼ 90% for the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  white region represents unobserved regions outside the sur-  vey footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As BORG effectively returns an unconstrained  simulation in this region, we remove it for easier compari-  son to the other plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4b, we show the ensemble  mean density across the MCMC realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In the volume  populated by galaxies the similarity between the mean and  the ground truth is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Outside this volume, the en-  semble mean tends to the cosmic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This is expected  for an ensemble of random density fields, because there is  no galaxy clustering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Further, notice that over-  dense structures are smeared in the inference mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Galax-  ies in each MCMC sample move along their line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Therefore, the smearing is due to averaging over density re-  alizations that account for photometric redshift uncertain-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4c we show the voxel-wise standard deviation  of the dark matter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This includes observational  uncertainties and shot noise due to the finite number of  galaxies in each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 4d-f, we present a slice through the peculiar  velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4d we show the ground truth ra-  dial peculiar velocity field, derived from the ground truth  dark matter density field with the dark matter sheet estima-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4e we show the radial peculiar velocity mean  across the MCMC realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4f we present the  sample variance of the radial peculiar velocity posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In  Figure 4g-i, we show the radial peculiar velocity divergence  ground truth, ensemble mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 4j-l, we show our constraints on the gravita-  tional potential from photometric and spectroscopic galaxy  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4j we show the ground truth gravi-  tational potential, as derived from the ground truth dark  matter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 4k and Figure 4l, we show the  ensemble mean and standard deviation across the MCMC  realizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The above statistical summaries present high visual re-  semblance with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We quantify this resem-  blance in Figure 5, by showing the cross-correlation of the  dark matter density, radial peculiar velocity, off-diagonal  components of the tidal shear tensor and gravitational po-  tential with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We select an unmasked sub-  volume of the inference domain, to showcase the best-case  cross-correlation in regions with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On the largest scales,  Article number, page 7 of 16  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  10  2  10  1  k [Mpc  1]  103  104  105  P  (k) [Mpc3]  (a)  ground truth  mean  3  10  2  10  1  k [Mpc  1]  108  109  1010  1011  1012  Pvv(k) [(km/s)2 Mpc3]  (b)  10  2  10  1  k [Mpc  1]  103  104  105  P  (k) [Mpc3]  (c)  10  2  10  1  k [Mpc  1]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='00  P  /  P  P  (k)  (d)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 6: (a) Dark matter density, (b) radial peculiar velocity and (c) peculiar velocity divergence autocorrelation power  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (d) Normalized cross-correlation between the dark matter density and peculiar velocity divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The black  lines indicate the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The colored window represents 3σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The uncertainty on the 2-point statistics is  due to galaxy survey- and data-related uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The blue line is the CAMB density autocorrelation power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  as expected, we have the highest cross-correlation for all  cosmological fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The cross-correlation of the gravita-  tional potential with the ground truth is > 80% on scales,  as Φ has the longest correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 6 we show the autocorrelation power spectra  of cosmological fields that are typically used in cosmologi-  cal analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' ll power spectra were estimated and corrected  for the BORG Cloud-In-Cell mass assignment scheme using  Pylians (Villaescusa-Navarro 2018), as our density field  has been estimated with a Cloud-In-Cell estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The au-  tocorrelation power spectrum, Pδδ, is shown in Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The two-point statistics of the inferred density fields are  consistent with the ground truth and the sampler has cov-  ered the uncertainty due to cosmic variance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Pogosian  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2010, Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 6b, we show the autocorre-  lation power spectrum of the radial peculiar velocity field,  Pvv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The inference is consistent with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 6c, we show the peculiar velocity divergence  autocorrelation power spectrum, Pθθ, which is also consis-  tent with the ground truth and ΛCDM prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ata  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Further, we see that the power of Pθθ is sup-  pressed compared to Pδδ beyond k ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='05 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This is  expected because collapsing structures that have virialized  experience less volume change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As a re-  sult, the peculiar velocity divergence field evolves less than  the density field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Kitaura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jennings 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Hahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Ata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 6d, we show  the normalized cross-correlation power spectrum between  Article number, page 8 of 16  Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : Large-scale structure from photometric redshifts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='70  2  LPT prediction  Inference  Ground truth  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='4  3  (b)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  4  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 7: (a) Variance, (b) skewness and (c) kurtosis of the dark matter density field for our inference, the ground truth and  a set of random LPT simulations at 13 Mpc (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04⟨σz⟩) using the entire inference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The latter two are random,  unconstrained simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The inference is constrained jointly with galaxy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The error bars encapsulate the  1σ uncertainty both from the mock galaxy survey and the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our ground truth is consistent with a random LPT  simulation and our inference is consistent with both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our inference accurately captures higher-order statistics of the dark  matter density field on scales much smaller than the original photometric redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  1661  1329  997  664  332  0  x [Mpc]  1661  1424  1186  949  712  474  237  y [Mpc]  z=30  (a)  1661  1329  997  664  332  0  x [Mpc]  z=10  (b)  1661  1329  997  664  332  0  x [Mpc]  z=0  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='76  ln(2+ )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='90  ln(2+ )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  ln(2+ )  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 8: Structure formation history derived from the dark matter density field at (a) z = 30, (b) z = 10, (c) z = 0 from a  typical density sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In red, we overlay the density slice with the galaxy coordinates in a typical realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Galaxies  follow the present-day large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The origins of high-density regions are already visible at z = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  the velocity divergence and dark matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On large  scales, it is expected that to linear order θ ∝ δ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Hahn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On smaller scales, structure formation becomes  nonlinear, which LPT is able to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' A more refined  gravity model, like a particle mesh (Jasche & Lavaux 2019,  in BORG), would be needed to push these results to smaller  scales and capture the generation of peculiar velocity vor-  ticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 7, we show higher-order moments of the con-  strained dark matter density field for the ground truth, our  inference and the LPT prediction, with 1σ error bars at the  target resolution of 13 Mpc (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04⟨σz⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We derive the LPT  prediction from a set of random simulations ran using the  same cosmological parameters as our main inference, but  without constraints from galaxy redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The error bars  naturally account for observational uncertainties and the  estimator uncertainty related to the sample size (Harding  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For this fully self-consistent setting, our results  are consistent with the ground truth and LPT prediction  within 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This result suggests that we infer the density  field, including its higher-order moments, on scales much  smaller than the original photometric redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 8 we show slices through the same randomly-  selected realization in the structure formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Un-  der the assumption of a causal structure formation model,  we reconstruct the structure formation history which gives  rise to the observed structures today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The initial conditions  are set at z = 99, but we show slices up to z = 30, such  that the seeds of present-day structures are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On the  z = 0 slice we overlay the ground truth locations of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Galaxies closely trace clusters and filamentary structures,  whereas voids are sparsely populated by galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This is  because the Poisson noise is lower in higher-density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  As expected, the origins of high-density peaks in the dark  matter density field are already visible at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 9a, we show a void in the volume covered  by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 9c we show the average density  contrast in shells around the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We see that the density  contrast tends to zero, as expected when averaging over a  scale close to the cosmic homogeneity scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Gon¸calves  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 9b, we show the location of a cluster  in the density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 9d we show the mass enclosed  in shells around the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We derive it using the prescrip-  tion in Porqueres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In this study, the particle  mass is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='45 × 1013 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Overall, these results suggest that  we recover correctly both the statistical properties of the  large-scale structure around voids and clusters, but also in-  dividual structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  We further show radial peculiar velocity profiles around  the same void and cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The radial peculiar velocity be-  comes positive close to the void, indicating matter flow-  ing outside the shells and toward overdense regions, as ex-  Article number, page 9 of 16  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  1661  1107  553  x [Mpc]  1661  1424  1186  949  712  474  237  y [Mpc]  (a)  1661  1107  553  x [Mpc]  (b)  50  100  150  200  250  R [Mpc]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  (R)  (c)  ground truth  mean  3  50  100  150  200  250  R [Mpc]  1018  1019  1020  1021  1022  1023  1024  M (< R) [M  ]  (d)  50  100  150  200  250  R [Mpc]  100  0  100  200  300  400  vr(R)  [km/s]  (e)  50  100  150  200  250  R [Mpc]  400  300  200  100  0  vr(R)  [km/s]  (f)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 9: (a) Slice through the ground truth dark matter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In purple, a spherical shell around a low-density  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (b) Slice through the ground truth dark matter density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In purple, a spherical shell around a high-density  peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (c) The mean density contrast in spherical shells around the low-density region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (d) The mass enclosed in shells  around the high-density peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (e) The mean radial peculiar velocity in spherical shells around the low-density region  with respect to the center of the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (f) The mean radial peculiar velocity in spherical shells around the high-density  peak with respect to the center of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  pected from gravitational structure growth (Peebles 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  On larger scales the average peculiar velocity tends to zero  due to cosmic homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In Figure 9f we show the aver-  age radial peculiar velocity in shells of radius R around the  cluster in Figure 9b, with respect to its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The radial  peculiar velocity is negative close to the cluster, as expected  from gravitational infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' On larger scales, the average ve-  locity tends to zero as expected from cosmic homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Constraints on photometric redshifts  In this section, we give a brief overview of how the joint  inference of the large-scale structure and photometric red-  shift PDFs yields constraints on photometric redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In  Figure 10 we show our results for three randomly-selected  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The redshift likelihood is a Gaussian PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Its  mean is the mock observed redshift and its standard devi-  ation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='05(1 + z), z being the mock cosmological redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Article number, page 10 of 16  Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : Large-scale structure from photometric redshifts  We derive the target posterior by multiplying the Gaussian  likelihood with the Poisson term for the redshift inference  in our Gibbs sampling approach, as shown in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The results of our inference are overlayed as a histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  This qualitative demonstration illustrates that the photo-  metric redshift posterior after the application of our method  contains more information than the likelihood and follows  the radial density profile of the large-scale structure along  the line of sight to each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This mechanism constrains  the radial positions of galaxies from the initial, radially-  distorted ones (Figure 1a), to the final ones, that trace the  filamentary large-scale structure (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In Figure 11 we show the reduction in the mean stan-  dard deviation of the photometric redshift posteriors as a  function of density after the application of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  σi indicates the original photometric redshift uncertainty  of the mock observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We expect that higher-density  regions yield better constraints and hence, lower redshift  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The linear fit suggests that ⟨σ(δ)⟩/⟨σi(δ)⟩ =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='005(1 + δ) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In low-density regions, the mean  standard deviation reduces by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='7⟨σi⟩, whereas  in high-density regions the reduction reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='4⟨σi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  scatter is larger in high-density regions because there are  fewer strongly overdense peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Overall, we expect the red-  shift uncertainty reduction to be more significant in higher-  resolution settings, where higher-density peaks will be re-  solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' It should be noted, however, that due to the highly  multimodal nature of the target redshift distribution, the  reduction in standard deviation does not reflect the infor-  mation gain from the original to the final redshift PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Finally, in Figure 12 we compare the inferred distribution  of photometric redshifts, N(z), in our mock survey to the  ground truth N(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For legibility, we show the mean N(z)  across the MCMC samples, along with 3σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  inferred N(z) is consistent with the ground truth N(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We  postpone the sampling of the N(z) to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Conclusions  Next-generation photometric galaxy surveys will deliver an  extraordinary amount of observations, that will reduce the  observational uncertainties, will extend deeper in redshift  and cover a larger cosmological volume than spectroscopic  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' At the same time, most cosmological information is in  the smallest cosmological scales which cannot be accessed in  presence of large photometric redshift uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In such  a setting, it is paramount to obtain control of photometric  redshift uncertainties in cosmological inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Accurate  inferences of the large-scale structure constrained jointly  with photometric redshifts offer this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In this study we developed a method to constrain  the primordial and present-day cosmic large-scale struc-  ture jointly with photometric redshifts at a resolution of  13 Mpc using, for the first time, a structure formation model  in a Bayesian forward modelling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We achieved  these joint constraints through Bayesian inference of the  initial conditions of structure formation with photometric  galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our method takes into account data- and  survey-related uncertainties and preserves all higher-order  statistics of cosmological fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In this study, we used first-  order LPT, yet our algorithm can incorporate any gravi-  tational model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jasche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Jasche & Lavaux  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We demonstrated our constraints using a fully over-  lapping mock photometric and spectroscopic galaxy cata-  log as a proof-of-concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We assumed a worst-case scenario  for photometric redshift uncertainties in stage-IV surveys,  σz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='05(1 + z), and demonstrated that our method can  reduce this uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  In particular, we showed large improvement in the cross-  correlation of the photometric galaxy positions with the  ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The maximum improvement was on scales  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='5⟨σz⟩, where the cross-correlation increased from 28%  (in the mock photometric observations) to 86% (in our in-  ference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We further achieved accurate inferences of the  dark matter density, peculiar velocities, gravitational po-  tential and tidal shear on scales ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04⟨σz⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  We demonstrated the ability of our method to accu-  rately capture the 2-point statistics of the dark matter den-  sity, as well as its skewness and kurtosis on scales ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='04⟨σz⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Higher-order statistics have been promising in constraining  dark energy (Velten & Fazolo 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Fazolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As  we can use our method with a particle mesh, our joint in-  ference framework can be used to constrain the bispectrum  with photometric galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Voids have been extensively used to probe structure for-  mation close to the linear regime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Pisani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Stopyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Contarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022), however photometric uncertainties  limit their detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' S´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For this pur-  pose, we explored the possibility of detecting cosmic struc-  tures much smaller than the original photometric redshift  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The galaxy positions that were originally radi-  ally smeared due to the presence of photometric uncertain-  ties, accurately trace the filamentary pattern of the large-  scale structure after the application of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Further,  we are able to accurately capture individual structures, like  voids and clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This is a crucial advantage of embedding  a structure formation model in our forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Overall,  we demonstrated that we accurately recover the statisti-  cal properties of the large-scale structure on scales much  smaller than the original photometric redshift uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The present work opens up the possibility to miti-  gate photometric redshift uncertainties in next-generation  photometric surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Concurrently, the incorporation of a  structure formation model paves a new way forward to ex-  tract as much information as possible from the smallest  cosmological scales, while going beyond 2-point statistics  and preserving information in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Data and software availability  Data products can be made available upon reasonable re-  quest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Author Contributions  The main roles of the authors were, using the CRediT  (Contribution  Roles  Taxonomy)  system  (https:  //authorservices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='com/author-resources/  Journal-Authors/open-access/credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='html):  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : conceptualization, methodology, software, formal  analysis, validation, writing - original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=': conceptu-  alization, methodology, software (supporting, transferred  Jasche & Wandelt (2012) to new BORG version), supervi-  sion, resources, writing - feedback, funding acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content=' : software (supporting, debugging and notably during  the Message Passing Interface parallelization), writing -  Article number, page 11 of 16  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='25  redshift  0  5  10  15  20  P(z|zobs, )  (a)  observation  target posterior  ground truth  redshift  inference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='30  redshift  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='35  redshift  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 10: Photometric redshift samples for three randomly-selected galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Overlayed are the corresponding initial ob-  servations estimated from our pipeline without a physical prior, the ideal photometric redshift posterior (for physical  prior, using the ground truth density) and the ground truth redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The maximum redshift to which the posteriors  extend indicates where each galaxy’s line of sight intercepts the inference box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The inferred posterior is the marginal over  density realizations, accounting for observational and survey-related uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The redshift binning is determined by  the real-space resolution of the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='8  ( ) /  i( )  linear fit  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 11: Final to initial photometric redshift uncertainty as  a function of density at the ground truth galaxy locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The black line is a linear fit, indicating that the reduction in  redshift uncertainty is greater in regions of high dark mat-  ter density, as expected from the Poisson likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  dispersion on the high-density end is due to shot noise, as  there are fewer high-density peaks in the inference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  feedback, funding acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : software (support-  ing, author of the Simplex-In-Cell estimator), writing -  feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Acknowledgements  ET thanks Metin Ata and Natalia Porqueres for helpful  discussions and feedback, and Adam Andrews, Deaglan  Bartlett, Andrija Kostic, James Prideaux-Ghee, Fabian  Schmidt, Ben Wandelt and Pauline Zarrouk for com-  ments on the original manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Part of the computa-  tion and data processing in this study were enabled by re-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='7  redshift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+page_content='0  normalized N(z)  mock N(z)  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 12: In orange, the average of normalized N(z) of in-  ferred photometric redshifts across MCMC samples for a  subsample of galaxies, along with 3σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In black,  the mock photometric N(z) for the same subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  N(z) of inferred photometric redshifts is consistent with the  ground truth N(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  sources provided by the Swedish National Infrastructure  for Computing (SNIC) at Tetralith, partially funded by  the Swedish Research Council through grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2020-05143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This research utilized the HPC facility sup-  ported by the Technical Division at the Department of  Physics, Stockholm University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This work was enabled by  the research project grant ‘Understanding the Dynamic  Universe’ funded by the Knut and Alice Wallenberg Foun-  dation under Dnr KAW 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='0067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' JJ acknowledges sup-  port by the Swedish Research Council (VR) under the  project 2020-05143 – ”Deciphering the Dynamics of Cosmic  Structure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' GL acknowledges support by the ANR BIG4  project, grant ANR-16-CE23-0002 of the French Agence  Nationale de la Recherche, and the grant GCEuclid from  ”Centre National d’Etudes Spatiales” (CNES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This work  was supported by the Simons Collaboration on “Learning  Article number, page 12 of 16  Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' : Large-scale structure from photometric redshifts  the Universe”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This work is conducted within the Aquila  Consortium (https://aquila-consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Derivation of the conditional density posterior  for real-space inference  In this section, we describe how to arrive at the condi-  tional density posterior of Equation (3) starting from Equa-  tion (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The former equation shows how we incorporate a  gravitational structure growth model into our framework,  whereas the latter represents the output of the Gibbs sam-  pling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Here, we discuss how we obtain constraints  on the real-space late-time density field using galaxy obser-  vations in redshift space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  The present-day density field is conditionally indepen-  dent of the observed galaxy redshifts given an ensemble of  sampled redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Therefore  P(δ|z, zobs, θ)  =  P(δ|z, θ)  =  �  Ng  P(δ, Ng|z, θ)  =  �  Ng  P(δ|Ng)P(Ng|z, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (17)  The last term in the sum indicates how we arrive from  redshift-space observations to real-space galaxy counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As  elaborated on in (Jasche & Wandelt 2012, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' B4), this  involves a transform from redshift- to real-space such that  P(δ|z, θ) =  �  Ng  P(δ|Ng)P(Ng|x(z, θ)),  (18)  x being the real-space galaxy positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In what follows, we  will omit dependence on (z, θ) for legibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The latter term  indicates how we bin galaxies onto a grid given their real-  space positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For this gridding operation, we use Nearest  Grid Point projection such that  P(Ng|x) =  �  i  δD  ��������Ng  i −  �  p  WNGP(xi − xp)  �������� ,  (19)  i being the voxel index, p being the galaxy index and  WNGP(x) =  3  �  n=1  �1  if |xn|Nn/Ln < 1  0  otherwise,  (20)  Substituting the above into Equation (18) we arrive at  P(δ|z, θ) = P(δ|Ng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (21)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Derivation of the conditional redshift posterior  for real-space inference  In this subsection we describe how we arrive at the condi-  tional redshift posterior in Equation (9), while performing  an inference of the density field in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Let us denote  by θ the right ascension and declination of a galaxy and by  u the redshifts of all other galaxies in the previous sampling  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We start by the redshift posterior of a given galaxy i  and apply Bayes’ law  P(zi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zobsi)  =  P(zi)P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zobsi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi)  P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zobsi)  =  P(zi)P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi)P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ|zi)  P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ)P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ)  =  P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ|zi)P(zi)  P(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ)  P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi)  P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ)  =  P(zi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ)P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u)  P(zobsi|u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' θ)  =  P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi|u)P(δ|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi)  P(δ|u)  P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u)  P(zobsi|u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' θ)  (22)  We assume zobsi is conditionally independent of u given zi  and δ and therefore we arrive at  P(zi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zobsi) = P(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi|u)P(δ|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi)  P(δ|u)  P(zobsi|θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' δ)  P(zobsi|δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' θ)  (23)  The first term on the right-hand side of the above equa-  tion describes the distribution of galaxies in redshift space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  However, as we perform a real-space inference, we trans-  form this term from redshift- z, to comoving Cartesian, r,  space  P(θ, zi|u) =  ����r2(zi) sin (φ)∂r  ∂z  ����zi  ����P(x|u),  (24)  φ being the declination of the galaxy and x its real-space  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The real-space galaxy positions are used to build  the gridded galaxy count field, Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Mg indicates the galaxy  counts except for the galaxy under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We the  last term on the right-hand side of the above equation as a  marginal over galaxy counts  P(x|u) =  �  Mg  P(Mg|u)P(x|Mg)  (25)  The first term in the sum is the Nearest Grid Point projec-  tion that we use to grid galaxies at the field-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We will  indicate the Nearest Grid Point kernel with WNGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Under  the assumption that galaxies are Poisson-distributed, the  number counts in each voxel are independent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Since  the Poisson intensity depends only on the density, our Pois-  son model is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Following these assumptions  P(x|u)  =  �  Mg  P(x|Mg)  ×  �  i  δD  ��������Mg  i −  �  p  WNGP(xi − xp)  �������� ,  (26)  where i is the voxel index and p the galaxy index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We now  want to rewrite the first term in the above sum with respect  to the entire galaxy count field, Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' It differs from Mg in that  it includes the galaxy under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The reason we  rewrite the redshift posterior with respect to Ng is because  the entire galaxy count field is used in the density inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Below this point we will drop the dependence on u, because  all quantities are conditionally independent of u given Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  P(x|Mg) =  1  P(Mg)  �  Ng  P(Ng)P(x|Ng)P(Mg|Ng, x)  (27)  Following our definition, the relationship between Mg and  Ng is deterministic and yields  P(x|Mg)  =  1  P(Mg)  �  Ng  P(Ng)P(x|Ng)  Article number, page 13 of 16  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' aanda  ×  �  i  δD �  Mg  i − [Ng  i − WNGP(xi − x)]  �  =  P(x|Ng),  (28)  because P(Ng) and P(Mg) are equal, as they differ by one  (at the position of the galaxy under consideration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The  above result is equal to finding a galaxy at position x given  a number counts field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  P(x|Ng) =  �  i  WNGP(xi − x)  δV  Ng  i  Ntotal  ,  (29)  where δV is the volume of a voxel and Ntotal the total number  of galaxies in the inference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' As a result of the above,  the second term on the right-hand side of Equation (23) is  written as  P(δ|θ, zi)  P(δ|u)  =  �  i  Mi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Ni!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' λNi−Mi  i  =  �  i  � λi  Ni  �WNGP(xi−x)  ,  (30)  as it represents the ratio between the Poisson distribution  for all galaxies and all galaxies expect for the one we con-  sider at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Substituting Equation (29) and Equa-  tion (30) into Equation (23) we arrive at  P(zi|θ, u, δ, zobsi)  ∝  ����r2(zi) sin (φ)∂r  ∂z  ����zi  ����  ×  �  i  WNGP(xi − x)λiP(zobsi|θ, zi, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (31)  As demonstrated in Jasche & Wandelt (2013) the above  form suggests that each galaxy can be sampled indepen-  dently from all others, as the dependence on u vanishes  through the conditioning on galaxy counts and therefore,  the Poisson intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Finally, since sin (φ) is a constant  term, it serves as a proportionality constant in each galaxy’s  target posterior and therefore vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Finally, we drop the  dependence on right ascension and declination for legibil-  ity, since we only sample redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Therefore, the process  in Equation (1) is equivalent to drawing a sample from  P(zi|δ, zobsi)  ∝  ����r2(zi)∂r  ∂z  ����zi  ����  ×  �  i  WNGP(xi − x)λiP(zobsi|zi, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  (32)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Estimators of large-scale structure properties  Accurate inferences of the large-scale structure with pho-  tometric galaxy clustering are necessary because next-  generation photometric surveys probe deeper redshifts and  have a wider footprint than spectroscopic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Below,  we discuss the aspects of the cosmic large-scale structure  which we demonstrate our method on, as well as the esti-  mators we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Peculiar velocities  In order to derive the peculiar velocity field, we use the  Simplex-In-Cell (SIC) estimator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Hahn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Leclercq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Contrary to kernel meth-  ods, which sample the velocity field only at a discrete set  of locations with poor resolution in low-density regions, the  SIC estimator provides an estimate of the velocity field at  any point in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' In this framework, the Lagrangian posi-  tions of the particles in the inference domain are considered  as vertices of unit cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The Delaunay tessellation of each  unique cube defines six tetrahedra which are followed dur-  ing the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Subsequently, each tetrahedron deposits  a value to the grid (Leclercq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017, Equation 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  We further explore kinematic properties of the peculiar  velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Here, we focus on the peculiar velocity diver-  gence, defined as  θ = −  1  f Ha∇ · v,  (33)  where f is the linear growth rate, H the Hubble parame-  ter and a the cosmic scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The divergence determines  how the volume of a peculiar velocity flow changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' The pe-  culiar velocity divergence can be used to constrain Ωm and  structure formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Bernardeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Howlett  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Further, the redshift-space distortion signal can  be detected in the autocorrelation power spectrum of the  peculiar velocity divergence and the cross-correlation power  spectrum between the dark matter density and the peculiar  velocity divergence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Bel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' For ∇ · v > 0 the  region is expanding, for ∇ · v < 0 the region is contracting,  whereas for ∇ · v = 0 the volume remains invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Gravitational potential  Probing gravitational tidal forces with photometric sur-  veys is a challenging endeavor, but a necessary one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Troxel & Ishak 2015), as intrinsic galaxy alignments con-  taminate weak lensing analyses and can be used as cos-  mological probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Further, as photometric surveys extend  deeper in redshift, they allow us to probe the evolution of  intrinsic alignments over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' This evolution can be used  to constrain galaxy formation and evolution scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  For this purpose, we explore differing aspects of the dy-  namics and kinematics of the large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We de-  rive the gravitational potential of the dark matter density  field by solving Poisson’s equation in Fourier space  Φ(k) = −4πGρ(k)  |k|2  ,  (34)  where G is the gravitational constant, ρ the density and  k the wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' We further derive the tidal shear of the  gravitational potential, which is given by  Ti j =  ∂2Φ  ∂xi∂x j  ,  (35)  where xi, ({i, j} = {1, 2, 3}) are comoving Cartesian coordi-  nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Our inference can further be used with cosmic web  classification methods, particle- (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Sousbie 2013) or cell-  based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Libeskind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Buncher & Carrasco Kind  2020) to infer cosmic structures constrained by photomet-  ric redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Such classifications have been used to study  the environment of cosmological tracers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Leclercq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Porqueres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' Kru-  use et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content=' (2019) further showed that photometric redshifts  are positively correlated with filaments detected in spectro-  scopic galaxy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
+page_content='  Article number, page 14 of 16  Tsaprazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE1T4oBgHgl3EQf_gZv/content/2301.03581v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf,len=718
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='00574v1  [quant-ph]  2 Jan 2023  Environmental-induced work extraction  Rasim Volga Ovali†,1, Shakir Ullah†,2, Mehmet Günay†,3, and Mehmet Emre Tasgin2,∗  † Contributed equally  ∗ correspondence: metasgin@hacettepe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='tr and metasgin@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='com  1Department of Physics, Recep Tayyip Erdogan University, 53100, Rize, Turkey  2Institute of Nuclear Sciences, Hacettepe University, 06800 Ankara, Turkey and  3Department of Nanoscience and Nanotechnology, Faculty of Arts and Science,  Mehmet Akif Ersoy University, 15030 Burdur, Turkey  A measurement can extract work from an entangled, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', two-mode system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here, we inquire  the extracted work when no intellectual creature, like an ancilla/daemon, is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' When the  monitoring is carried out by the environmental modes, that is when no measurement-apparatus is  present, the measurement-basis becomes the coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This implies a Gaussian measurement  with a fixed strength λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' For two-mode Gaussian states, extracted work is already independent  from the measurement outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' After the strength is also fixed, this makes nature assign a particular  amount of work to a given entanglement degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Extracted work becomes the entanglement-degree  times the entire thermal energy at low temperatures —e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', room temperature for optical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Environment, nature itself, converts entanglement to an ordered, macroscopic, directional (kinetic)  energy from a disordered, microscopic, randomized thermal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' And the converted amount is  solely determined by the entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Quantum entanglement enables technologies which are  not possible without them [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Measurements below  the standard quantum limit [2, 3], quantum enhanced  imaging [4–6], quantum radars [7], quantum teleporta-  tion (QT) [8], and quantum computation [9] are all en-  abled by entangled —more generally nonclassical [10]—  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Entanglement can also be utilized as a resource  for quantum heat engines [11–17] which makes them op-  erate more efficiently compared to their classical coun-  terparts [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' As an example, an ancilla can utilize en-  tanglement for extracting a larger amount of work by  maximizing the efficiency [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' It is quite intriguing that  entanglement can even be directly transformed into work  via measurements [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The energy can be extracted  from a single heat bath [20, 21] [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This phenomenon  —we focus here— takes place as follows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', in a two-  mode entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Work extraction as a measurement backaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='— Let  us assume that one of the modes (mode a) belongs to  an optical cavity which includes, for example, a free-to-  move board or a piston inside the cavity [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The  second (b) mode relies somewhere outside of the cavity  and it is entangled with the a-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Both modes are  in thermal equilibrium with the environment at temper-  ature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' When a measurement is carried out on the b-  mode (outside), entropy of the cavity (a) mode decreases  to S(meas)  V  because of the measurement backaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' After  the measurement, the state (a-mode) thermalizes back  to equilibrium and assigns a higher entropy S(ther)  V  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  During the rethermalization with the heat bath the ex-  pansion of the a-mode pushes the board located inside  the cavity [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' An amount of W = kBT (S(ther)  V  −S(meas)  V  )  work can be extracted from the single heat bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' That  is, W amount of thermal energy can be converted into  “directional” kinetic energy (KE) of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In case  of maximum entanglement, the state of the a-mode is  completely determined from the outcome of the b-mode  and entropy of the a-mode vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', S(meas)  V  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Thus, the extractable work becomes kBT S(ther)  V  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', the  complete internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The amount of extracted work  can be employed for witnessing/quantifying the entan-  glement [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  In general, the extracted work depends on the nature  of the measurement and its outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Yet, interestingly,  the state of the a-mode (cavity field) turns out to be in-  dependent of the outcome of the b-mode as long as Gaus-  sian states and measurements are concerned [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The  state, into which the a-mode collapses, depends only on  the strength (λ) of the Gaussian measurement/operation  carried out on the b-mode [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Thus, also the ex-  tracted work depends only on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Here, λ ∈ [0, ∞] is  the quadrature-noise belonging to the Gaussian opera-  tion [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  That is, if λ (for a reason) assigns a fixed  value, a given degree of entanglement extracts a particu-  lar amount of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This is the phenomenon we explore  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The observable that is measured in an experiment (this  can be, for example, number of photons) is determined  by the quantum apparatus employed in the measurement  of the b-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  More explicitly, monitoring of the en-  vironment on the apparatus (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', decoherence) destroys  the superpositions among the natural pointer states (the  measurement-basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This makes us observe one of the  values in the measurement-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' [34–39] provide  mathematical and numerical demonstrations of the mon-  itoring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The  question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='—  Here  we  examine  the  following  already-answered-question in the context of work extrac-  tion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' What happens if no measurement appa-  ratus is present?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  In other words, what is the pointer  (measurement) basis if no intellectual being, such as a  human, a daemon, or an ancilla, is present?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In this case,  environmental monitoring sets the measurement-basis as  2  the the coherent states [40–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Measurement strength is  λ = 1 for any one of the coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Environmental monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='— At this stage, we better  re-depict the picture of the system we have in mind in a  more clear way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The cavity (a) mode is entangled with  the b-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' It is worth noting that, in general, the a  and b modes do not need to be in interaction [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The b-  mode is monitored by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Environment (a  collection of infinite number of modes) can monitor the  b-mode only indirectly as two light beams do not interact  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Monitoring can be performed over common in-  teractions with masses of particles present around which  induces an effective interaction between the environment  and the b-mode [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' It is worth noting that pointer states  are still coherent states, for instance, in case a harmonic  chain [42] is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Therefore, in total, environment  monitors (measures) the b-mode in one of the coherent  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' So, the measurement is a Gaussian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  It is straightforward to realize that the measurement  strength is fixed λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Moreover, a rotated R(φ) form  of the coherent state basis —R(φ) is present in the most  general form of a Gaussian measurement [28–31]— is also  a coherent state basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Putting things together, nature  itself makes a Gaussian measurement on one of the two  entangled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The measurement basis is composed  of coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' As the basis possesses a fixed λ: a  particular amount of work (KE) becomes assigned to a  given degree of entanglement as long as Gaussian states  are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The thermodynamical energy, proba-  bilistic and disordered in nature, is converted into an  ordered (mechanical) form of energy now belonging to  the board [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This sets an observer-independent,  nature-assigned, association between entanglement and  directional/ordered energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We call this phenomenon  environment-induced work extraction (EIWE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  One can gain a better understanding by examining the  phenomenon in low-T limit —such as room temperature  for optical resonances [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' At this limit, a simple-looking  analytical result can be obtained, because the kBT term,  present in the W formula, cancels with a 1/kBT term  appearing within the entropy difference [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Please,  see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S11) and (S13) in the Supplementary Mate-  rial (SM) [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The extracted work W = ξ(r) (¯nℏωa) depends only on  the degree of the entanglement ξ(r) = [1−2/(1+cosh2r)]  which runs from 0 to 1 as entanglement increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' ¯n is  the occupation of the a (cavity) mode of resonance ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  So, (¯nℏωa) is already the “entire” thermodynamical (it is  probabilistic) energy present inside the cavity either be-  fore the measurement or after the rethermalization pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here we use the von Neumann entropy SV in dif-  ference to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' [20] where Réyni entropy is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' r  is the two-mode squeezing rate which is proportional to  the time the entanglement device is kept open [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We present the derivations in the SM [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In the rest  of the paper, we aim to put this result into words in order  to develop a physical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We observe that the extracted work (KE of the board  or piston) is: the degree of entanglement times “all of  the thermal energy” present inside the cavity [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  It  gets closer to (¯nℏωa) in the case of maximum (max)  entanglement [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  In other words, max entanglement  converts the entire thermodynamical energy of the (a)  photon mode into the kinetic (directional) energy of the  board/piston [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' As we will see below, this is true also  for other max entangled states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', max two-mode en-  tangled state (|1, 0⟩ + |0, 1⟩)/  √  2 and symmetrization en-  tanglement of identical particles [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' That is, we cross-  check our EIWE result with other incidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  What is peculiar to EIWE is that the work is extracted  by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' That is, an observer-independent entanglement-  energy correspondence appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The converted energy is  proportional to the degree of the entanglement ξ [58] and  depends only on the excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Before making the comparison with other systems, we  would like to bring two important issues into attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  First, we note that W is calculated using a density ma-  trix which involves classical (thermodynamical) probabil-  ities —grand canonical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' It is a result weighted  over classical probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' For this reason, we prefer to  use the words “all of the thermal energy” present inside  the cavity is converted into the KE of the board/piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The energy present inside the cavity after the rether-  malization is also probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Conservation of energy,  however, tells us the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' If the energy realized in-  side the cavity after the rethermalization assigns one of  the classically probable ones, the extracted work needs  to be equal to that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Second, we note that al-  most all of the notion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', entanglement-work conver-  sion and entanglement-energy analogy [59]) and the cal-  culations carried out here are already discussed in other  studies [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Here, in difference, we introduce the  notion of entanglement-work correspondence due to the  presence of nature-originated measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Comparison with other work extraction phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='—  We first compare the (i) EIWE result with the one  for another (max) entangled state (ii) |e⟩ = (|1, 0⟩ +  |0, 1⟩)/  √  2 in thermal equilibrium ˆρ = P(|0, 0⟩⟨0, 0| +  e−ℏωa/kBT |e⟩⟨e|) [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' When one measures the number  of photons and the outcome the b-mode is |1⟩, W =  xℏωa work is extracted in the cavity a-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Here,  x = e−ℏωa/kBT is the classical probability for realizing  the two-mode system in the excited state |e⟩ and it is  equal to the occupation ¯n at low T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', ¯n = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This re-  sult is the same with the max entanglement (ξ = 1) case  of EIWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In this example, too, all thermodynamical en-  ergy present in the a-mode is converted into directional  energy (work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In this case, however, the work extrac-  tion (the same amount) is subject to the measurement of  the b-mode in the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In EIWE, in difference,  any measurement outcome extracts this amount of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We also examine the work associated to the (iii) sym-  metrization (max) entanglement [57] as a third exam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  In a recent study, we investigate the work ex-  tracted by identical particles in a system of N total  number of symmetrized bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The extracted work by  3  (N − 1) particles is calculated when one of the (ran-  dom) particles is measured in the excited state, of en-  ergy ωeg [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In parallel with the previous cases, (i) and  (ii), the extracted work, by pushing the board located  within the condensate region, comes out as W3 = xℏωeg  at low T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here, x = e−ℏωeg/kBT is the probability for  one of the N particles to occupy the excited state ei-  ther before the measurement or after the rethermaliza-  tion of the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Similarly, (xℏωeg) is the en-  tire thermal energy of the condensate at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The lowermost excited state of such a condensate is  the Dicke state |N, 1⟩ [62], where a single-particle ex-  citation is symmetrically distributed among N bosons,  |N, 1⟩ = (|e, g, g, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='⟩+|g, e, g, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='+|g, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g, e⟩)/  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This  is a maximally entangled state with respect to any one of  the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' ωeg is the level-spacing between the excited  |e⟩ and ground |g⟩ states of a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We observe that the amount of extracted work, one  more time, is equal to the complete thermal energy of the  condensate (system) at thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This takes  place again for a max entangled state, |N, 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' We can take  the excited state, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', as the motional states of a Bose-  Einstein condensate with ℏωeg = h2/2mL2 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Then,  the thermal energy of the condensate can be converted  into the directional (kinetic) energy of a board placed in  the condensate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here, again, the presented value  of the extracted work is subject to the realization of the  measured-particle in the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The investigation  of this symmetrization problem has further importance  as entanglement of symmetrized many-body states and  nonclassicality of photonic states are intimately related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  A Dicke (many-body) state becomes a Fock (photon)  state when N → ∞ [64–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Similarly, separable coher-  ent atomic states become the photonic coherent states in  the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Summary and Discussions  Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='— We reinvestigate an already studied phe-  nomenon –work extraction from an entangled system via  measurement backaction [20, 21]– when nature itself per-  forms the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' This is when there is no intel-  lectual being (such as a human, daemon, or an ancilla) is  present for the measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' but the monitoring is con-  ducted by the environment/nature itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In this case,  measurement-basis becomes the coherent states [40–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  This fixes the measurement strength to λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The state  of the a-mode, in which work-extraction is carried out, is  already independent from the outcome of the b-mode [28–  31] as long as Gaussian states and measurements are con-  cerned [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (The measurement performed by the en-  vironment is a Gaussian one as the measurement-basis  is coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=') Therefore, in total, the nature itself  assigns a particular amount of work/energy to a given  degree of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Entanglement converts a disordered (randomly moving  particles, microscopic) form of energy into an ordered  form where the molecules of the board move along the  same direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', macroscopic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The letter is  referred as the mechanical energy, or we can tell that it  is the KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We find that at low T , the directional energy associated  with the entanglement is the “total thermal energy” times  the degree of the entanglement for a two-mode Gaussian  state: W = ξ(r)Uther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here, ξ(r) = [1 − 2/(1 + cosh 2r)]  increases with the entanglement and gets closer to ξ =  1 around the max entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  r is the squeezing  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' That is, all thermal energy can be converted  into directional energy for a maximally entangled Gaus-  sian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Similarly, the entire thermal energy can be  converted into directional energy also for (ii) max entan-  gled number state (|0, 1⟩ + |1, 0⟩)/  √  2 and for (iii) sym-  metrization entanglement of identical bosons [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' How-  ever, the conversion in (ii) and (iii) is subject to the re-  alization of the measured mode/particle in the excited  state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' while in (i) EIWE any measurement outcome ex-  tract that amount of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Squeezing and potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='— In this section, we  would like to introduce a correspondence also between  potential mechanical energy and single-mode nonclassi-  cality, SMNc, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', squeezing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Entanglement and SMNc  are two different types of nonclassicalities (quantumness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The two not only can be converted into each other via  passive optical elements, such as a beam splitter (BS),  but they also satisfy a conservation-like relations [67–  69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' When a cavity mode is squeezed, it cannot be con-  verted into work directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  This is because, squeezing  keeps the entropy unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  However, the situation  changes when the squeezed cavity field leaks out through  the mirror(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The interaction between the cavity and the  output modes �  k(gkˆb†  kˆa + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=') is in the form of a BS  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Thus, the cavity and the output modes get  entangled [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Environmental monitoring on the output  modes ˆbk [71] makes the cavity extract work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' We can also  view the process as the potential mechanical energy (as-  sociated with squeezing) transforms into the kinetic me-  chanical energy (associated with entanglement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Connection with a recent study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='— As a final but impor-  tant point, we indicate that the present research actually  investigates the results of a recent study [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Rigorous  calculations, employing the standard methods, clearly  show an intriguing phenomenon in an optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' On-  set of entanglement exhibits itself in the response func-  tions of the optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' At this point, nonanalytic-  ity of the response function moves into the upper-half of  the complex frequency plane (UH-CFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' One needs to  avoid this incident from implying the violation of causal-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Fortunately, surveys [73–75] show that a nonana-  lyticity in the UH-CFP does not imply the violation of  causality if there exists a small curvature in the back-  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In the present study, the total curvature of the  cavity (a-mode) increases as the disordered thermal en-  ergy is converted into ordered directional kinetic energy  of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' That is, it indeed increases with the amount  of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  4  Acknowledgements  We gratefully thank Vural Gökmen for the motiva-  tional support, Bayram Tekin for the scientific support,  Wojciech H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Zurek for letting us know his influential  work [40] and Alessio Serafini for his support on the  continuous-variable quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' We acknowl-  edge the fund TUBITAK-1001 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' 121F141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content='  [49] Coupling of a cavity field/mode to the input/output (vac-  uum) modes already employs exactly the same mecha-  nism [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [50] Or for a Swinger pair creation (mass-energy conversion)  process taking place over a critical electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [51] We can tell that the result is T -independent except a  factor ¯n = 1/(eℏωa/kBT − 1) → e−ℏωa/kBT which stands  for the probability of finding one of the two modes in the  excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (¯nℏωa) is already the energy present in the  a-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [52] See Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content=' Press (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [54] Here, we prefer to use the words “all of the thermal en-  ergy” at equilibrium, because this is the grand canonical  mean energy which is (classical) probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The en-  ergy present inside the cavity after the rethermalization  is also probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Conservation of energy tells us the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' If the energy realized inside the cavity after  the rethermalization assigns one of the classical probable  ones, the extracted work needs to be equal to that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [55] Please note that energy of a maximum entangled two-  mode Gaussian state approaches to infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here, we  confine ourselves to a regime where squeezing rate r ≪  ℏωa/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The latter is typically ∼ 100 for optical modes  at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' See the discussion above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S11)  in the SM [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [56] Please note that this statement is valid at any tempera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content=' Stamper-Kurn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Chikkatur, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Görlitz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Inouye,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Gupta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Pritchard, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Ketterle, Excitation of  phonons in a bose-einstein condensate by light scattering,  Physical Review Letters 83, 2876 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content=' Stenger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Inouye, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Chikkatur, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Stamper-Kurn,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Pritchard, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Ketterle, Bragg spectroscopy of  a bose-einstein condensate, Physical Review Letters 82,  4569 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [87] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Andersen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Ryu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Cladé, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Natarajan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Vaziri,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Helmerson, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Phillips, Quantized rotation  of atoms from photons with orbital angular momentum,  Physical review letters 97, 170406 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Taşgın, Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Müstecaplıoglu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' You, Creation  of a vortex in a bose-einstein condensate by superradiant  scattering, Physical Review A 84, 063628 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [89] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Tasgin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Gunay, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Zubairy, Nonclassical-  ity and entanglement for wave packets, Physical Review  A 101, 062316 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [90] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Ullah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Gunay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Ovali, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Tasgin, Input  output formalism for wave packets, in preperation, will  be available on Researgate (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  [91] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Thompson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Zwickl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Jayich, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Marquardt,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Girvin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Harris, Strong dispersive coupling of a  high-finesse cavity to a micromechanical membrane, Na-  ture 452, 72 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  7  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  SUPPLEMENTARY MATERIAL  FOR  ENVIRONMENTAL-INDUCED WORK  EXTRACTION (EIWE)  In this supplementary material (SM), we first ob-  tain the environmentally extracted in a two-mode (TM)  squeezed thermal (Gaussian) state in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' We show  that it is in the form W  = ξ(r) × (¯nℏω) at low-  temperatures (T ) —e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', room temperature for optical  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Here, ξ(r), given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' ), quantifies the  strength of the entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' We use von Neumann en-  tropy (SV ) in our calculations, in difference to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Second, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' 2, we show that the same form for the  work extraction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', W = ξ(r) × (¯nℏω), appears also for  other TM Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' EIWE for two-mode squeezed thermal state  Initially, before the measurement, both modes, a and b,  are in thermal equilibrium with an environment at tem-  perature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' When one carries out a Gaussian measure-  ment on the b-mode, the enropy of the b-mode reduces  below the one for the thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' When the a-  mode re-thermalizes with the environment it performs a  work in the amount of [27]  W = kBT (S(ther)  V  − S(meas)  V  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (1)  Here, S(meas)  V  is the reduced entropy of the a-mode after  the measurement in the b-mode is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' S(ther)  V  is  the entropy of the a-mode after the rethermalization of  the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Entropy of a Gaussian state can be determined by its  covariance matrix, which includes the noise elements of  the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Covariance matrix of a biparitite Gaussian  state can be cast in the form [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' ]  σab =  �σa cab  cT  ab σb  �  (2)  via local symplectic transformation Sp(2, R) ⊕ Sp(2, R)-  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', transformations altering neither of the entropy or  entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Here, σa = diag(a, a) and σb = diag(b, b)  are the reduced covariance matrices of the a and b  modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' cab = diag(c1, c2) refers to corre-  lations/entanglement between the two mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' For a sym-  metrically squeezed two-mode thermal state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', both  modes used to be in thermal equilibrium with the same  T in the squeezing process, b = a and c2 = −c1 = −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The state into which the a-mode collapses is indepen-  dent from the outcome of the b-mode measurement as  long as a Gaussian measurement is carried out [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The covariance matrix of the a-mode after the measure-  ment becomes  σπb  a = σa − cab (σb + γπb)−1 cT  ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (3)  Here, γπb = R(φ) diag(λ/2, λ−1/2) R(φ)T refers to the  covariance matrix associated with a Gaussian operation  (measurment) [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' λ is the measurement strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  For a Gaussian measurement having the coherent states  as a basis, λ = 1 and γπb = diag(1/2, 1/2) independent  of the rotations R(φ) in the a-mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', ˆa → ˆaeiφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The entropy of a Gaussian state is determined solely  by purity, µ =  1  2n√  Detσ, which takes the form  SV = 1 − µ  2µ  ln  �1 + µ  1 − µ  �  − ln  � 2µ  1 + µ  �  (4)  for a single-mode state [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  The purity of the a-mode, after the b-mode measure-  ment, can be obtained as  µ1 ≡ µ(meas) =  a + 1/2  2(a2 − c2 + a/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (5)  For a TM squeezed thermal state,  a = (¯n + 1  2) cosh(2r),  (6)  c = (¯n + 1  2) sinh(2r),  (7)  the purity becomes  µ1 =  a + 1/2  2(¯n + 1/2)2 + a,  (8)  where ¯n = (eℏωa/kBT − 1)−1 is the occupation of the a-  mode, which becomes ¯n → e−ℏωa/kBT at low T— e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=',  the room temperature for optical modes of resonance ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  r is the two-mode squeezing strength with which entan-  glement increases [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  ¯n is extremely small at low T regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' So, it is  µ1 ∼= 1 −  2¯n  a + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (9)  Then, the entropy can be approximately written as,  S(meas)  V  ∼=  ¯n  a + 1/2[ln(2) − ln(2¯n) + ln(a + 1  2)]  (10)  −  2¯n  a + 1/2,  where a = (¯n + 1/2) cosh(2r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The last term originates  from the ln  �  2µ  1+µ  �  term given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  In the square brackets, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S10), ln(¯n) = ℏωa  kBT ≫ 1  and ln(a + 1/2) ∼= ln(cosh(2r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Assuming that the squ-  uezing rate is much smaller than  ℏωa  kBT , which is about  ∼100 at the room temperatures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=', r ≪  ℏωa  kBT , the en-  tropy takes the form  S(meas)  V  ∼=  2¯n  1 + cosh(2r)  ℏωa  kBT ,  (11)  where the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S10) is also neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  8  Some time after the measurement, a-mode rethermal-  izes with the enviroment and at equilibrium its purity  becomes  µ2 ≡ µ(ther) =  1  1 + 2¯n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (12)  The entropy at equilibrium can similarly calculated as  S(ther)  V  ∼= ¯n ℏωa  kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (13)  Therefore, the extracted work becomes  W =  �  1 −  2  1 + cosh(2r)  �  ¯nℏωa,  (14)  where the kBT term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S1) is canceled by 1/kBT  appearing in (S11) and (S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' EIWE for other Gaussian states  We derived the simple form for the extracted work  W = ξ(r)(¯nℏωa) for the TM squeezed thermal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Now, we aim to show that a similar form appears also for  other Gassian states given by the covariance matrix (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  When the b-mode is measured by the enviroment, the  a-mode collapses to a covariance matrix with determi-  nant  detσπb  a = (a2 − c2  1 + a/2)(a2 − c2  2 + a/2)  (a + 1/2)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (15)  The entropy after the measurement is similarly µ(meas) =  1  2√  detσ  πb  a  [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' For c1 = −c2 = c, µ(meas) becomes the  purity given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  As we aim to show that the extracted work has a “form”  similar to W = ξ · (¯nℏωa) also for other Gaussian states,  we express Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S15) in terms of Sp(4, R) invariants  detσ = (a2 − c2  1)(a2 − c2  2),  (16)  ∆ = 2(a2 + c1c2),  (17)  where detσ is the determinant of the two-mode state  before the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  We do this because any one  of the two-mode Gaussian states, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S2), can be ob-  tained from Sp(4, R) transformations of the TM squeezed  thermal state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (15) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' The determinant in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S15) can be expressed as  detσπb  a  = (a2 − c2  1)(a2 − c2  2) + a  2 (2a2 − c2  1 − c2  2)  ( 1  2 + a)2  ,  (18)  where the first term is the Sp(4, R) invariant detσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' In  the second term, I = 2a2 − c2  1 − c2  2 can be expressed in  terms of Sp(4, R) invariants (Please note that a is only  local Sp(2, R) invariant) as  detσπb = (a2I + ∆2/4 − ∆a2)  (a + 1  2)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (19)  We note that detσ = (˜a2  TMS − ˜c2)2 and ∆ = 2(˜a2 − ˜c2)  for the TM squeezed thermal states and they are Sp(4, R)  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S19) can be recast as  (˜a2 − ˜c2)2 = a2I + (˜a2 − ˜c2)2 − ∆a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (20)  Please note that, in this section, we use tilde symbol for  the covariance matrix elements of the TM squeezed ther-  mal states in order to distinguish them from the variable  a given in the general matrix given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  Cancellation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S20) results  I = 2a2 − c2  1 − c2 = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (21)  Using this in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S19), we obtain the expression  µ(meas) =  a + 1/2  2(z + a/2),  (22)  where z = ˜a2−˜c2 = (¯n+1/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' Please note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S22)  is in the same form with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S5) from which we obtain  the extracted work  W = ξ (¯nℏωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content='  (23)  Thus, above we showed that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
+page_content=' (S23) is a general form  for the extracted work from a symmetric TM Gaussian  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtAyT4oBgHgl3EQfsPmi/content/2301.00574v1.pdf'}
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+arXiv:2301.03366v1  [hep-ph]  9 Jan 2023
+Hypothesis of a new fundamental interaction versus the
+particle oscillation concept
+L.M.Slad*
+Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow
+119991, Russia
+Abstract. The emergence of a logically simple solution to the solar neutrino problem based on the
+hypothesis of the existence of a new interaction involving electron neutrinos and nucleons did not weaken
+the dominance of the oscillation concept. Therefore, a significant part of the present work is devoted
+to proving that the basic elements of this concept either do not correspond to the principles of classical
+logic, or violate the energy-momentum conservation law, or contradict the quantum mechanical basis
+of coherence, or represent a primitive falsehood. Our analysis concerns successively all stages of the
+formation of the concept from its conceiving to the assertion about the conversion of the solar electron
+neutrino into a muon one. When discussing a new fundamental interaction, we note the decisive role in
+the outcome of the processes of changing the handedness of a neutrino (antineutrino) at each act of its
+interaction with a real or virtual massless pseudoscalar boson, due to which, at the exit from the Sun,
+the fluxes of left- and right-handed electron neutrinos become approximately equal. Thanks to the new
+interaction, beta decays of nuclei have a mode with the emission of a massless pseudoscalar boson, at
+which the antineutrino changes its handedness and becomes unobservable, what is the essence of the
+reactor antineutrino anomaly. The nature of the gallium anomaly is similar.
+1. Introduction
+The emergence of the solar neutrino problem is caused by the report [1] about the absence
+of observed transitions of 37Cl into 37Ar under the action of such neutrinos. It have been stated
+the upper limit for such a transition rate as 3 SNU (1 SNU is 10−36 captures per target atom
+per second), while the rate predicted by Bahcall [2] was 30+30
+−15 SNU.
+The reaction to this report was well conveyed by Reines [3]: ”It is interesting to note that
+if a positive result were obtained in the Davis experiment, we would suddenly face the problem
+of whether it is due to the Sun or something else. A negative result is important because that
+it allows us to assert, in the event of a further positive result in a neutrino experiment, that
+the effect is caused precisely by solar neutrinos.This is significant, because the Universe is full
+of surprises ... When in February 1972 we discussed this issue at a conference in Irvine, it
+was interesting to observe how, in search of the reasons for the discrepancy, astrophysicists
+pointed to specialists in the nucleus, the latter to neutrino physicists, and those in turn to
+astrophysicists”. These words were said by Reines during the discussion of Pontecorvo’s report
+at the seminar on the µ − e problem. It is noteworthy that Reines does not say a word about
+the solar neutrinos oscillations, the assumption of which was put forward three years earlier [4].
+This indicates that, for the time being, the neutrino oscillation concept was weakly perceived
+by the scientific community.
+Over time, Wolfenstein [5] extended it to the transformation of a neutrino as it moves
+in a homogeneous medium. Seven years later, Mikheev and Smirnov [6] announced the ex-
+tension of the Wolfenstein equation to the inhomogeneous medium of the Sun, but did not
+describe the evolution of the state of the solar neutrino as it moves from its production place
+to an installation on Earth. Such a description with the oscillation parameters given by the
+*slad@theory.sinp.msu.ru
+1
+
+Super-Kamiokande and SNO collaborations was partly made in [7] by analytical and numer-
+ical analysis of this equation.
+The main task of [6] was to argue the statement about the
+transformation of an electron neutrino into a muon neutrino during its passage through the
+Sun. The proposed arguments, as we will prove below, are false. Despite the initial perception
+of the work by reviewers and readers as erroneous, over time, the belief that the Mikheev-
+Smirnov-Wolfenstein mechanism is able to solve the solar neutrino problem has become a
+fairly widespread phenomenon among the scientific community.
+Having analyzed all stages of the particle oscillation concept from its inception, which con-
+cerned neutral K-mesons, to the appearance of the Mikheev-Smirnov-Wolfenstein mechanism,
+we find either ignoring the rule of sufficient reason, or ignoring the law of conservation of energy-
+momentum, or neglecting the quantum mechanical basis of coherence , or elementary falsity.
+This will inevitably lead sooner or later to the collapse of the concept of particle oscillations.
+I hope that it will be helped, in particular, by the emergence of a logically simple solution to
+the solar neutrinos problem. The main theseses of such a solution and their correspondence
+with experiments were outlined in February 2015 in my work [8].
+It would seem that the whole history of physics pointed to the fact that some new for
+us, rather hidden interaction is responsible for the emergence of the solar neutrino problem.
+Surprisingly, that, as it is clear from the utterance of Reines [3], the question of hidden inter-
+action was not even raised at the Irvine conference. My hypothesis postulates the existence
+of an interaction whose carrier is a massless pseudoscalar boson that has a Yukawa coupling
+with at least the electron neutrino, proton, and neutron, but not with electrons. Due to this
+interaction, neutrinos during their movement inside the Sun experience about a dozen collisions
+with nucleons. At each such collision, the neutrino handedness changes from left to right and
+vice versa, so that at the exit from the Sun the ratio of left- and right-handed electron neu-
+trino fluxes is 0.515:0.485. In addition, at each collision of a neutrino with nucleons, its energy
+decreases slightly. Our model with one free parameter provides good agreement between theo-
+retical and experimental results for the rates of all five observed processes with solar neutrinos
+[9], [10].
+This paper is devoted, on the one hand, to the analysis of the particle oscillation concept
+from its inception and, on the other hand, to a discussion of separate elements of the postulated
+new interaction.
+2. The particle oscillation concept
+2.1. Neutral K-mesons
+The discovery of hyperons and K-mesons was an unexpected and amazing surprise, as it is
+said in the monograph of that time [11].
+In the emerging theory of new unstable particles, first of all, ones tried to transfer such
+notions of symmetry as isospin and charge conjugation, attributed to stable particles. Gell-
+Mann [12] supposed that the new unstable particles are fermions with integer isospin and bosons
+with half-integer isospin and have antiparticles that differ from them. Gell-Mann and Pais [13]
+believed that strong interactions are responsible for all productions of hyperons and K-particles,
+while weak interactions are responsible for their decays. Therefore, in their assertion, the law
+of conservation of isospin allows the reaction of production of θ0-meson with the projection of
+isospin -1/2 (hereinafter referred to as K0)
+π− + p → Λ0 + θ0,
+(1)
+but prohibits the process of production of its antiparticle
+π− + p → Λ0 + ¯θ0.
+(2)
+2
+
+At the same time, due to the violation of the law of conservation of isospin in weak interactions,
+decays of the θ0-particle and ¯θ0-antiparticle are admissible according to the same schemes:
+θ0 → π+ + π−,
+¯θ0 → π+ + π−.
+(3)
+The relations (3) allow the virtual transition θ0 ↔ π++π− ↔ ¯θ0, i.e. the particle θ0 and its
+antiparticle ¯θ0 do not completely differ from each other. Gell-Mann and Pais find it convenient
+to introduce two quanta θ0
+1 and θ0
+2 (hereinafter referred to as K0
+1 and K0
+2, and then as K0
+S and
+K0
+L), transforming into themselves upon charge conjugation:
+θ0
+1 = (θ0 + ¯θ0)/
+√
+2, qquadθ0
+2 = (θ0 − ¯θ0)/i
+√
+2.
+(4)
+Since each of the quanta θ0
+1 and θ0
+2 can be assigned its own lifetime and mass, Gell-Mann and
+Pais consider them to be true ”particles”, and at the same time they treat θ0 and ¯θ0 as a
+”mixture of particles”.
+It is incorrect to interpret relations (4) and (5) as an expression of the wave function of
+one of the particles in terms of the superposition of the wave functions of two other particles,
+which takes place in [14]. Since the values of the wave functions are c-numbers, it is easy to
+find their sum (difference), but in this sum they lose their individuality and it is impossible to
+separate them from the sum and perform their separate transformations.
+I find it useful to express here the essence of the addition of ”particles” or ”mixture of
+particles” in terms of known logical operations. Let us introduce the notion of a linear space
+of particles over the field of functions of space-time coordinates, L. Each particle is associated
+with its own basis vector in the space L, denoted by the name of the particle. Any particle
+state and its time transformations are displayed in L by vectors obtained by multiplying its
+basis vector by an appropriate wave function. Each mixture of particles is associated with only
+one vector in the space L, which is given by a well-defined decomposition into the basis vectors
+of this space at the initial moment and is denoted by the name of the mixture. The time
+transformation of each mixture is given by the transformation of the states of the particles,
+into whose basis vectors it is decomposed. Finally, the state vectors of the particles at time t
+are decomposed into the initial mixture vectors. The coefficients of these decompositions play
+the role of the probability amplitudes for the existence of the corresponding mixtures at time
+t.
+The time transformation of the wave function of a particle with mass m is well known:
+its initial value acquires an additional phase factor exp(−iEt), where E =
+�
+m2 + p2, p is
+momentum modulus. The question of the time dependence of the wave function of a ”mixture
+of particles” was considered, in the spirit of what was said above, in the work of Pais and
+Piccioni [14]. From the relations (4) ones obtain
+θ0 = (θ0
+1 + iθ0
+2)/
+√
+2, qquad¯θ0 = (θ0
+1 − iθ0
+2)/
+√
+2.
+(5)
+Getting the time transformation θ0 is like this:
+θ0 → (θ0
+1 exp(−iE1t) + iθ0
+2 exp(−iE2t))/
+√
+2 = ((θ0 + ¯θ0) exp(−iE1t) + (θ0 − ¯θ0) exp(−iE2t))/2.
+(6)
+Hence, under the condition p1 = p2 ≡ p, E1 ≫ m1 and E2 ≫ m2, the probability of
+existence of a meson theta0 at time t is an oscillating quantity
+P(θ0, t) ≈ cos2 (m2
+1 − m2
+2)
+4p
+t.
+(7)
+In [13] and [14], not a word is said about the feasibility of the law of conservation of energy-
+momentum in the process (1), provided that the θ0-meson is a mixture of two particles (5)
+3
+
+with different masses m1 and m2. Our statement is that the consequences of realizating the
+law of conservation of energy depend on the ratio between the total uncertainty in masses
+(the sum of decay widths) of unstable particles �
+i Γi participating in the process (1) and the
+mass difference |m1 − m2|. If the total mass uncertainty exceeds this mass difference, then
+the fulfillment of the law of conservation of energy in the process (1) with the θ0-meson as a
+mixture of two particles is admissible. Otherwise, the produced K-meson is forbidden to be a
+mixture of particles. It must itself be a particle with a dominant decay channel (3).
+Let us now turn to the numbers characterizing the process (1).
+We take into account
+that the particle decay width Γ and its lifetime τ are related by the equality Γτ = 1. The
+lifetimes of particles interesting to us are taken from review [15]. We have: τπ− = 2.60 · 10−8
+s, Γπ− = 2.53 · 10−8 eV; τΛ = 2.63 · 10−10 s, ΓΛ = 2.50 · 10−6 eV; τK0
+S = 0.895 · 10−10 s,
+ΓK0
+S = 7.35· 10−6 eV. So, the total mass uncertainty in the process (1) is 9.88· 10−6 eV. At the
+same time, the difference between the masses of two neutral K mesons, mKL − mKS, is less
+than this value. It is [15] 3.48 · 10−6 eV.
+Thus, the requirement of fulfillmenting the law of conservation of energy does not prohibit
+the two neutral K-mesons, K0 and ¯K0, be mixtures of the other two K-mesons , K0
+S and K0
+L,
+and to oscillate as it moves.
+Let us now recall that the appearance of the assertion about the oscillation of neutral K
+mesons was caused by the assertion of Gell-Mann-Pais that, due to the conservation of isospin,
+the state of the θ0-meson, produced in the process (1), has an isospin projection of -1/2. But,
+from the relation (6), it follows that after an arbitrarily small interval of time after production
+this state does not have a definite value of the isospin projection. It seems natural to recognize
+the assertion about the conservation of isospin in the process (1) as untenable and to accept
+that one of the admissible K-mesons, produced in the process (1) and decaying into π++π−, is
+a θ0
+1 meson coinciding with its antiparticle. Some reasoning of Gell-Mann-Pais give arguments
+in favor of the admissibility of the production in the process (1) of a θ0
+2-meson, which decays
+into three pi-mesons.
+From the four neutral K-mesons considered in [12]-[14], only two mesons should be left in
+the final theory, namely θ0
+1 and θ0
+2 (KS and KL, in current notation). Their Yukawa interaction
+with baryons should be represent as
+L = g1¯nγ5ΛKS + g2¯nγ5ΛKL + g3¯nγ5Σ0KS + g4¯nγ5Σ0KL + g5¯pγ5Σ+KS + g6¯pγ5Σ+KL, (8)
+where pairs of constants g1, g2; g3, g4 and g5, g6 have close but different values, reflecting
+CP-symmetry breaking. It follows from here that at each collision of the meson KL (KS) with
+the nucleons of matter, either a meson KL (KS) is born again, or a meson KS (KL) is born and
+then one speaks about the regeneration of KS (KL) from KL (KS)-meson. This regeneration
+refers to a number of such standard processes as the Compton effect and π-meson recharging.
+The situation with two, not more, neutral K-mesons with certain masses would correspond
+to the assertions of great thinkers about sufficient reason and, in particular, to the assertion of
+Isaac Newton: We are to admit no more causes of natural things than such as are both true and
+sufficient to explain their appearances. To this purpose the philosophers say that Nature does
+nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity,
+and affects not the pomp of superfluous causes [16] (Book III. Rule I.)
+2.2. Gribov-Pontecorvo concept of neutrino oscillations
+Soon after the publication of the results of Davis et al. [1], Gribov and Pontecorvo paper
+[4] appeared, in which the lower value of the solar electron neutrino flux near the Earth’s
+surface compared to that expected on the basis of the standard solar model is explained by
+solar neutrino oscillations, due to which some of them at the location of the experimental setup
+emerge as muon neutrinos.
+4
+
+Gribov and Pontecorvo, in their substantiation of the existence of neutrino oscillations, tend
+to adhere to the Gell-Mann-Pais-Piccioni procedure for neutral K-mesons. But if in Gell-Mann,
+Pais and Piccioni the elements of the procedure have clear experimental and worldview sources,
+then in Gribov and Pontecorvo they are taken ”from the ceiling”, from nowhere. Thus, in order
+to supplement the family of known neutrinos νe and νµ with two new neutrinos, Gribov and
+Pontecorvo, adding off-diagonal terms to the mass part of the neutrino Lagrangian, symbolizing
+the transitions in vacuum of an electron neutrino to a muon neutrino and vice versa , give the
+Lagrangian the following form
+Lint = a1¯νeνe + a2¯νµνµ + a3¯νeνµ + H.c.,
+(9)
+where a1, a2 and a3 are constants. Diagonalization of the Lagrangian (9) leads to the appear-
+ance of two states
+ν1 = cos θ · νe − sin θ · νµ,
+ν2 = sin θ · νe + cos θ · νµ,
+(10)
+whose masses and the value of the mixing angle are given by the constants ai, i = 1, 2, 3.
+We note here that earlier, when Weinberg constructed the electron gauge theory [17], the
+off-diagonal mass term formed by the gauge fields already appeared in the Lagrangian. As
+a result of the diagonalization of his Lagrangian, Weinberg obtained the photon and massive
+boson fields, which he assigned the status of quantized fields. He then expressed through them
+the initial gauge fields, which he got rid of in the final theory.
+The number of initial and
+final entities in Weinberg remained the same. As a result, we have a phenomenal theory of
+electroweak interactions.
+While Weinberg naturally avoided the doubling of entities, Gribov and Pontecorvo delib-
+erately introduced such doubling, which became the cornerstone of their concept of neutrino
+oscillations. The states ν1 and ν2 have been given status, using Gell-Mann-Pice terminology,
+true particles, neutrinos νe and νµ have been given mixture status:
+νe = cos θ · ν1 + sin θ · ν2,
+νµ = − sin θ · ν1 + cos θ · ν2.
+(11)
+Applying the Pais-Piccion procedure to the relations (11) and (10) regarded to the initial
+moment of time, we have
+νe → cos θ · ν1 exp(−iE1t) + sin θ · ν2 exp(−iE2t) =
+cos θ(cos θ · νe − sin θ · νµ) exp(−iE1t) + sin θ(sin θ · νe + cos θ · νµ) exp(−iE2t).
+(12)
+Hence, under the condition p1 = p2 ≡ p, E1 ≫ m1 and E2 ≫ m2, the probability of
+detecting neutrinos nue at time t is an oscillating quantity:
+P(νe, t) = 1 − sin2 2θ sin2 (m2
+1 − m2
+2)
+4p
+t.
+(13)
+The length of its oscillation is given by the equality
+L/c = 2π
+�
+2p
+|m2
+1 − m2
+2|
+�
+.
+(14)
+The most important point in our analysis of the Gribov-Pontecorvo concept of neutrino
+oscillations is to find out the feasibility of the law of conservation of energy-momentum in the
+processes of production of electronic (anti-)neutrinos as a mixture (11) of two (anti-)neutrinos
+ν1 and ν2 with different masses. By analogy with our assertion in Section 2.1, we compare the
+5
+
+difference between the neutrino masses ν1 and ν2 with the total uncertainty in the masses of
+particles involved in the neutron decay process
+n → p + e− + ¯νe.
+(15)
+In this process, the only unstable particle is the neutron with the lifetime value [15] τn =
+879 s and, consequently, with the width Γn = 7.5 · 10−19 eV, which plays the role of the
+total uncertainty in masses. Energy-momentum conservation in the process (15) is allowed
+if |m1 − m2| < Γn. Let us take for the upper value of each of the masses m1 and m2 the
+experimentally established upper value of the electron neutrino mass [15]: mνe < 1.1 eV. Then
+the possibility of conservation of energy-momentum in the process (14) restricts the difference
+of squared neutrino masses ν1 and ν2 to the following limit
+|m2
+1 − m2
+2| < 8.2 · 10−19eV2.
+(16)
+Obviously, the restriction (16) on the masses of the ν1 and ν2 components of the electron
+neutrino do not depend on the processes in which νe is produced. Let us now turn to the length
+L of solar neutrino oscillations. For the observed solar neutrinos, as follows from the relation
+(14, it is minimal at the threshold energy for gallium-to-germanium transitions equal to 0.233
+MeV. Taking into account the inequality (16), we have that L > 7.0 · 1014 km. Thus, the state
+of an electron neutrino on its way from the Sun to the Earth (1.50 · 108 km) could make no
+more than 2· 10−7 fractions of one oscillation. Such a situation would be absolutely unsuitable
+for solving the solar neutrinos problem.
+So, either the representation of the neutrino νe as a mixture (11) has nothing to do with
+the solution of the solar neutrino problem, or in the case of the supremacy of the neutrino
+mass difference |m1 − m2| over the neutron decay width (7.5 · 10−19 eV) the energy-momentum
+conservation law would be violated, which makes such a representation and oscillations inad-
+missible.
+There have already been reasoning in the literature that the neutrino oscillation concept
+may contradict the law of conservation of energy-momentum, but neither the essence of the
+concept nor its effectiveness has been doubted. Thus, Giunti et al. [18] believe that the question
+of energy–momentum nonconservation in processes involving neutrinos with an indeterminate
+mass does not arise in a real physical situation when the localization of the neutrino source
+and detector is taken into account. In fact, it is proposed to consider a neutrino on its way
+from the source to the detector as a virtual particle, the 4-momentum of which is outside the
+mass shell, which hides the certainty of the mass. Following this proposal, let’s give the solar
+neutrino, registered on Earth, the status of a virtual one. Since virtuality lasts 500 s, due to
+the Heisenberg relation, the energy uncertainty is 1.3 · 10−18 eV, which is very close to the
+discussed total mass uncertainty in the process (15). Consequently, the conclusion formulated
+above remains fully valid, according to which either the survival of the electron component
+of the solar neutrino near the surface of the Earth differs very little from 1, or the electron
+neutrino cannot be a mixture of two other neutrinos with different masses and, therefore, its
+oscillations do not exist.
+2.3. Wolfenstein’s equation for neutrinos in a homogeneous medium
+Having established that the energy-momentum conservation law does not allow solving the
+solar neutrino problem on the basis of the neutrino oscillation hypothesis proposed by Gribov
+and Pontecorvo, we nevertheless continue to analyze its modifications.
+The appearance of proposals to set up experiments on the detection of neutrinos from ac-
+celerators when they could pass through the Earth, along with ongoing experiments on the
+detection of solar neutrinos, prompted Wolfenstein to consider the possible influence of the
+6
+
+medium on neutrino oscillations [5].
+When constructing his equation describing the trans-
+formation in the medium of probability amplitudes to detect neutrinos in the ν1 or ν2 state,
+Wolfenstein believed that coherent forward scattering of neutrinos on electrons takes place.
+This scattering is caused by electroweak interactions and gives different phase changes in the
+wave functions of the electron and muon components of the initial neutrino, which, in his
+opinion, is similar to the regeneration of KS from the KL-beam passing through the medium.
+I have two remarks with regard to Wolfenstein’s assertion above.
+First, we find reasoning about coherence in the work of Muller et al. [19] devoted to the
+difference in the masses of neutral K-mesons. In it, coherence is associated with only one of
+the three types of regeneration of KS from KL-mesons in a medium, when a plate placed in
+front of a parallel beam of KL-particles generates a parallel beam of KS-particles. To two
+other types of regeneration, Muller et al. refer the diffraction scattering by nuclei and ordinary
+scattering by separate nucleons. The share of contributions to the total regeneration of each
+of its three types is not discussed in [19]. From their experiment, Muller et al. found that
+the ratio |mKL − mKS|/ΓKS is 0.85+0.3
+−0.25. From modern data [15], we obtain the value of 0.473
+for this ratio.
+Note, that subsequently Geveniger et al.
+[20] measured the mass difference
+mKL − mKS by the method of two regenerations and obtained a highly accurate value for it,
+essentially coinciding with the modern one. In [20], coherence is not mentioned.
+Secondly, neutrino scattering on a certain set of electrons as a quantum mechanical phe-
+nomenon will be coherent in the case when the neutrino wave function has at the same time a
+non-zero value at the location of each electron from this set. This condition is satisfied if and
+only if the neutrino wavelength is comparable to the distance between electrons from a given
+set. Observable solar neutrinos with the lowest energy equal to 0.233 MeV have the longest
+wavelength λ. For them λ = 8.47 · 10−13 m. At the same time, electrons have the highest
+density Ne in the center of the Sun, where the matter density, almost entirely determined by
+protons, is 148 g · cm−3. There, the electron density Ne is 8.85 · 1025 cm−3 and, consequently,
+the minimum distance between electrons in the Sun is 2.24 · 10−11 m, which is approximately
+two orders of magnitude greater than the largest wavelength of the observed solar neutrino.
+So, coherence in the scattering of observed solar neutrinos is impossible. The analogy with
+the regeneration of K-mesons is weak and does not change this conclusion.
+Wolfenstein’s
+equation is not applicable to any part of the trajectory of the observed solar neutrino. It must
+be attributed to some abstract medium and an abstract neutrino beam.
+2.4. Consequences of the Wolfenstein equation for solar neutrinos
+Let us put aside for a while the fact that the Gribov-Pontecorvo neutrino oscillation hy-
+pothesis and the Wolfenstein equation for neutrinos in a homogeneous medium are not suitable
+for describing the transformation of solar neutrinos, and analyze the consequences of extending
+the Wolfenstein equation to solar neutrinos.
+The extension of the Wolfenstein equation to an inhomogeneous medium consists in giving
+the electron density the status of a function. In the νe, νµ basis, this equation has the following
+form (see [21], Eq. (14.55))
+i d
+dt
+�
+Ae(t)
+Aµ(t)
+�
+= 1
+2
+
+
+
+
+−∆m2
+2E cos 2θ +
+√
+2GF Ne(t)
+∆m2
+2E sin 2θ
+∆m2
+2E sin 2θ
+∆m2
+2E cos 2θ −
+√
+2GF Ne(t)
+
+
+
+ ×
+×
+�
+Ae(t)
+Aµ(t)
+�
+,
+(17)
+where Ae(t) (Aµ(t)) is the probability amplitude that the neutrino state is electronic (muonic
+) at the time moment t, when the electron density is equal to Ne(t) at the location of the
+neutrino; ∆m2 = m2
+1 − m2
+2; E is the neutrino energy.
+7
+
+It is easy to verify that the equation (17) ensures the unitarity condition
+|Ae(t)|2 + |Aµ(t)|2 = 1.
+(18)
+We first give a brief list of those analytical and numerical consequences of the equation
+(17), which are detailed in our paper [7], adding some remarks.
+We attribute the Wolfenstein equation to the entire time interval from the moment of the
+neutrino production in the Sun to its registration on the Earth, while the electron density at
+a neutrinos location can be quite large or arbitrarily small and zero value.
+In our numerical analysis, we use the central values of the parameters of the two-neutrino
+model of solar neutrino oscillations shown by the Super-Kamiokande Collaboration [22]:
+∆m2 = 4.8+1.5
+−0.8 · 10−5 eV2,
+sin2 θ = 0.334+0.027
+−0.023.
+(19)
+The conclusions in [7] remain unchanged when the value of ∆m2 changes by two orders of
+magnitude, and the value of sin2 θ changes twice.
+Our solution to the equation (17) is based on the adiabatic approximation used in quantum
+mechanics. Initially, it is assumed that, over several lengths of oscillations, the matter density
+in the Sun can be considered constant, equal to its value at the initial moment. After finding
+the interval of oscillation lengths corresponding to all observed values of the solar neutrino
+energy, we are convinced of the validity of this assumption when, using the numerical values of
+the dependence of the density of matter on its distance from the center of the Sun [23], we find
+that its relative change δρ/ρ throughout the length of one oscillation does not exceed 1.7·10−3.
+If at time moment t0 the state of the solar neutrino was purely electronic, |Ae(t0)|2 = 1,
+then, as follows from the equation (17), the probability Pe(t) that this neutrino remains electron
+at time t is given by
+Pe(t) = |Ae(t)|2 = 1 − sin2 2θm sin2 f(Ne)(t − t0)/2,
+(20)
+where
+sin 2θm = ∆m2
+2E
+· sin 2θ
+f(Ne),
+(21)
+f(Ne) =
+��∆m2
+2E
+�2
+− 2
+√
+2
+�∆m2
+2E
+�
+GF Ne cos 2θ + 2G2
+F N 2e .
+(22)
+The angle θm defines the shape of the neutrinos νe and νµ as a mixture of massive neutrinos
+in the medium ν1m and ν2m:
+νe = cos θ · ν1m + sin θ · ν2m,
+νµ = − sin θ · ν1m + cos θ · ν2m.
+(23)
+The characteristics of each separate oscillation depend only on the density of the matter
+at the location of the neutrino and do not depend on the features of the part of the neutrino
+trajectory that preceded it. Knowledge of these characteristics allows us to get a conclusion
+about the set of all solar neutrino oscillations on any of their rectilinear trajectories for each
+value of the observed energy. At an energy in the range from 0.233 to 0.827 MeV, the length of
+the oscillations decreases monotonically with a decrease in the density of matter at the location
+of the neutrino. At an energy in the range from 0.827 to 18.8 MeV, the oscillation length as a
+function of the matter density has an extremum at its value
+ρext = 1.95 · 10−24 cos 2θ
+√
+2GF
+�
+∆m2
+2E
+�
+g,
+(24)
+namely, the maximum
+L(ρ)max/c =
+2π
+sin 2θ
+� 2E
+∆m2
+�
+,
+(25)
+8
+
+at that with increasing energy, the value of ρext decreases from 148 to 6.51 g/cm3. The length
+of the oscillations of the observed solar neutrinos with energies from 0.233 to 18.8 MeV belongs
+to the interval from 12 to 1028 km, i.e. from 0.0000172R0 to 0.00148R0, where R0 is the radius
+of the Sun, R0 = 696000 km.
+When the matter density is ρext, the oscillating probability Pe(t) has a maximum amplitude,
+namely, Pe(t) can take any value from 1 to 0. Also, the maximum the value equal to 1 is taken
+by the quantity sin2 2θm.
+Based on a comparison of the lengths of neutrino oscillations in the Sun and in vacuum
+with the characteristic sizes of distributions of neutrino sources in the Sun, it was shown in [7]
+that the numbers of oscillations along various trajectories from the Sun to the installation on
+Earth can differ by several tens or hundreds. As a result, the summation of the contributions of
+separate sources to the probability Pee of that the state of the solar neutrino will be electronic
+near the Earth’s surface leads to the disappearance of durations of the neutrino motion along
+their trajectories, and the final formula for Pee for any neutrino energy has the following form
+Pee = 1 − 1
+2 sin2 2θ.
+(26)
+The probability (26) contradicts the results of at least three out of five experiments with solar
+neutrinos: on 37Cl →37 Ar transitions, on elastic neutrino scattering on electrons, on the
+disintegration of deuterons by charged currents.
+So, three things are incompatible: the first is the values of the parameters of neutrino
+oscillations announced by the Super-Kamiokande and SNO collaborations, the second is the
+solution of the Wolfenstein equation for solar neutrinos with the indicated parameter values,
+the third is the experimentally measured rates of processes caused by solar neutrinos.
+2.5. Mikheev-Smirnov-Wolfenstein effect
+At the end of the 1970s, the problem of solar neutrinos acquired a clear numerical outline,
+consisting in the fact that the experimentally measured rate of the transition of chlorine into
+argon under the action of solar neutrinos is approximately 1/3 of the theoretical rate found
+in the framework of the standard solar model. For example, Davis announced [24] that the
+measured rate of these transitions is 2.2 ± 0.3 SNU, and Bahcall with colleagues reported [25]
+that the calculated rate is 7.5 ± 1.5. At the same time, for any values of the parameters and
+two- and three-neutrino oscillations in their standard sense, the probability of that the initial
+electron neutrino remains electronic at the place of its registration, when averaged over the
+time of one oscillation, is no less than 1/2.
+Under these conditions, an extraordinary scenario of neutrino oscillations was announced
+by Mikheev and Smirnov in [6], the main assertion of which is that the initial beam of electron
+neutrinos, after passing through a certain layer of matter, is almost completely transformed
+into a beam of muon neutrinos. The substantiation of this assertion is not accompanied by
+any analytical calculations and formulas.
+Its role is played by the terminology that is not
+appropriate in particle physics and field theory, the haziness of statements, the purposeful
+game with the dependence of the mixing angle on the density of the medium, which plays a
+key role in declaring the metamorphosis of the solar neutrino.
+Let us dwell on all the details of the work [6], taking into account that it has hundreds of
+adherents.
+Instead of speaking, as it is customary in mathematical analysis, about the existence of
+extremal values of a number of oscillation characteristics, as functions of the density of matter,
+on a certain set of neutrino trajectories in the Sun, Mikheev and Smirnov declare the existence
+of a resonant amplification of neutrino oscillations in the medium. The term ”resonance” in
+the oscillatory process has both scientific and everyday use. In mechanics and in the theory of
+9
+
+electricity, the appearance of resonance in oscillations is associated with the existence of three
+constituent elements: the frequency of proper oscillations, the frequency of external influences
+and ”the friction” (energy loss). Only the first of these elements is present during neutrino
+oscillations in the medium. The everyday understanding of resonance is associated with the
+possible dramatic consequences of vibrations (for example, with the destruction of a bridge).
+Since ”resonance” is associated with the quantity sin2 2θm and with the amplitude of oscil-
+lations as functions of the density of matter, then its ”width” is considered. These functions
+have values greater than half of their maximum values, equal to 1, in the following interval of
+matter density
+δρext ∈ [max(ρext(1 − tan θ), 0) ρext(1 + tan θ)].
+(27)
+The layer of solar matter, the density in which belongs to the interval (27), is called the
+resonance layer by Mikheev and Smirnov. Let’s estimate its size. At a solar neutrino energy
+of 6 MeV, the extreme density of matter ρext is, according to the relation (24), 20.4 g·cm−3
+and is at a distance of 0.254R0 from the center of the Sun. The density in the resonant layer
+(27) ranges from 0 to 84.3 g·cm−3 and is at a distance from R0 to 0.102R0 from the center
+Sun. Hundreds or thousands of solar neutrino oscillations can occur in this resonant layer. In
+addition to the term ”resonance layer”, the term ”final object” with a matter density from 0
+to ρmax (without specification) or simply ”object” is also used. The selection of separate layers
+of the Sun does not have any logical consequences, but rather looks like some kind of ritual.
+An essential element of [6] is the notion of an adiabatic regime. The notion of ”adiabaticity”
+occurs in various branches of physics, with its own nuances in each of them and with the effective
+consequences of its use. In [6], the adiabatic regime is characterized by a slow change in the
+density of matter in the ”object” (see Eq. (26) of [26], which differs from Eq. (16) of [6]):
+�dρ
+dr
+�−1
+ρ >
+4πE
+∆m2 tan2 2θ,
+(28)
+where r is the distance to the center of the Sun. Using the dependence of the matter density
+in the Sun on r, available in [23], and the values of the oscillation parameters (19) given by the
+Super-Kamiokande collaboration, it is easy to see that the inequality (28) holds for all values
+of r in the Sun and for all observed values of neutrino energy. Note that the relation (28) does
+not follow from any analytical calculations and is not explicitly used anywhere.
+Mikheev and Smirnov approve that the adiabatic regime makes it possible to trace the
+evolution of the state of the solar neutrino as it moves through the ”object under consideration”
+using solutions (21), (22) for constant density. This is an incorrect statement. The evolution
+of a state means nothing more than its change in time, which is completely absent in [6] in
+an explicit form. During the passage of the neutrino ”object” / ”resonant layer” the density
+of matter undergoes, as demonstrated above, significant changes comparable to the density at
+the initial moment. The time evolution of the neutrino state both inside the Sun and after it
+enters the vacuum is described by the set of solutions of the Wolfenstein equation (17) for one
+oscillation (20) obtained in the adiabatic approximation described in Section 2.4. The end of
+one oscillation, when the state of the solar neutrino becomes again purely electronic, coincides
+with the beginning of a new oscillation with a very small change in the density of matter. The
+transformation of the states of the solar neutrino along the entire trajectory from the place
+of its production to the Earth consists in a gigantic number of oscillations following one after
+another. The relations (21), (22) are only fragments of the solution (20), which is not related
+to the notion of ”adiabatic regime” by Mikheev-Smirnov.
+Before analyzing the assertion about the transformation of an electron neutrino into a muon
+neutrino when passing through a medium, let us fix an unambiguous interpretation of such a
+transformation. Since the probability of detecting a neutrino being in the νe state at a certain
+moment of time can be either greater or less than the probability of detecting a neutrino being
+10
+
+in the νµ state at the same time moment, the change in the ratio of these probabilities does
+not can be a criterion for transformation. There are two stable indications that the oscillating
+state of the neutrino must be considered either electronic or muon: one is the superiority of the
+average probability of detecting one type of neutrino over the average probability of detecting
+another type over the time of one oscillation, and the other is the superiority of the amplitude
+of the probability oscillations for one type over the amplitude for another type. During the
+entire time allotted to the Wolfenstein equation for solar neutrinos, the above relations remain
+unchanged, i.e. transformation of one type of neutrino into another is impossible.
+Let us now pay attention to the character of the changes in the value of sin 2θm, given by
+the relations (21) and (22), and to two possible variants of changes in the mixing angle θm
+for neutrinos with a certain energy along one or another trajectory. At neutrino energies from
+0.233 to 0.827 MeV, when there is no extremal point on the trajectory, the value of sin θm
+increases with increasing matter density, starting from sin 2θ and not reaching 1, and the angle
+2θm grows starting from 2θ and not reaching π/2. At energies above 0.827 MeV, the value
+of sin 2θm first increases from sin 2θ to 1, and then decreases to a certain value c0, which is
+the smaller, the higher the matter density at the production place of the neutrino. The first
+variant of changing the angle 2θm is following: as the matter density increases, it grows from
+2θ to π/2, and then decreases to the value arcsin c0. The second variant is next: the angle 2θm
+with increasing matter density invariably increases from 2θ to π − arcsin c0.
+In [6], a variant of changing the angle θm from θ to π/2 − (arcsin c0)/2 is chosen without
+mentioning the existence of the first variant. In the case of small values of θ and c0, the chosen
+variant means a change in the mixing angle along the neutrino trajectory by approximately π/2,
+while in the first variant such a change is small. Now comes the climax of the metamorphose,
+described fragmentarily and muzzily both by Mikheev and Smirnov in [6] and by Smirnov in a
+recent report [26], which I present in an orderly and complete form. For a small value of c0, the
+first of the relations (23) at the moment of production of the solar neutrino, when it is purely
+electronic, is written as
+νe = cos[π/2 − (arcsin c0)/2]ν1m(ρmax) + sin[π/2 − (arcsin c0)/2]ν2m(ρmax)
+≈ [c0/2]ν1m(ρmax) + ν2m(ρmax)
+(29)
+The transformation of the state of a neutrino during its motion in the Sun is completely
+determined by the changes in the mixing angle θm and the states of massive neutrinos in
+matter on the right side of the first of the relations (23).
+At the moment when the solar
+neutrino is near the exit from the Sun, its state νsol is given by the formula
+νsol = cos θν1m(ρ = 0) + sin θν2m(rho = 0).
+(30)
+Since the expansion (29) is completely dominated by the neutrino ν2, and the expansion (30) is
+dominated by the neutrino ν1, Mikheev and Smirnov conclude that the final neutrino is muon,
+and the probability amplitude to find νe in the final state is equal to sin θ.
+The significant difference in the values of the coefficients related to the same massive neu-
+trinos in the decompositions (29) and (30), enhanced in [6] by the smallness of the angle
+theta accepted there, is entirely due to the meaningful by choosing the second variant of the
+dependence of the mixing angle θm on the density of the medium.
+In the first variant of the dependence of the mixing angle θm on the density of the medium,
+instead of the relation (29), we have the following equality
+νe = cos[(arcsin c0)/2]ν1m(ρmax) + sin[(arcsin c0)/2]ν2m(ρmax) ≈ ν1m(ρmax) + [c0/2]ν2m(ρmax)
+(31)
+11
+
+Now, in both expansions (31) and (30), the same neutrino ν1 dominates and, following
+Mikheev and Smirnov, one should say that both the initial and final neutrinos are electronic.
+The conclusion that the final neutrino is a muon neutrino vanishes like a dream.
+The assertion of Mikheev and Smirnov about the transformation of an electron neutrino
+into a muon neutrino when passing through a medium turns out to be primitive false, moreover,
+the notions of ”resonance”, ”resonance layer”, ”adiabatic regime” turned out to be unnecessary
+for its imaginary substantiation.
+For many years I was surprised that the paper by Mikheev and Smirnov [6] discussed here
+was not rejected in the journal Yadernaya Fizika. I unexpectedly received an answer from
+Smirnov, who outlined some points from the conceiving of the Mikheev-Smirnov-Wolfenstein
+mechanism in a report at a conference in Paris in 2018 [26].
+It turns out that paper [6]
+was initially rejected in the journal Yadernaya Fizika. Smirnov writes that ”it was a general
+skepticism: ’something should be wrong, somebody will eventually find this’”. Pontecorvo also
+had a skeptical opinion (it is ”rubbish”). However, thanks to the influence of Academician G.T.
+Zatsepin, the article was accepted by Yadernaya Fizika and also published in Nuovo Cimento.
+Later, Pontecorvo together with Wolfenstein ”concluded that, it seems, there is no practical
+outcome of oscillations in matter”.
+So, the key thesis of the Mikheev-Smirnov-Wolfenstein mechanism about the transforma-
+tion of an electron neutrino into a muon neutrino when passing through the solar medium
+is erroneous, which makes this mechanism generally unsuitable for solving the solar neutrino
+problem.
+3. New fundamental interaction
+3.1. Solving the solar neutrino problem
+The search by many diverse specialists for the reasons for the discrepancy between the
+result of the Davis experiment [1] and Bahcall’s prediction [2], reported by Reines [3], was
+not successful. It, of course, was conducted within the framework of the standard model of
+interactions. This alone serves as a sufficient reason to go beyond the standard model and,
+adhering to classical principles in physics, postulate the existence of a new interaction that is
+not a consequence of known interactions, and therefore is fundamental. The essence of the new
+interaction will not suffer, if we do not consider it fundamental.
+In [8], we put forward and substantiated the hypothesis that the emergence of the solar
+neutrino problem is due to the existence of a rather hidden interaction, which is carried by a
+massless axial boson having Yukawa coupling with an electron neutrino, proton, and neutron
+(with u- and d-quarks), reflected by the following relativistically invariant Lagrangian
+L = igνeps¯νeγ5νeϕps + igNps¯pγ5pϕps − igNps¯nγ5nϕps,
+(32)
+and not coupled with the electron at the tree level. We do not rule out that the coupling of
+the axial boson with different types of neutrinos may be different, if it exists at all.
+The decrease in the experimental rate of observed processes with solar neutrinos compared
+to the theoretical one is caused by two reasons. The first reason is the change in its handedness
+from left to right and vice versa at each collision of a neutrino with a nucleon, which is due to
+the Lorentzian structure of the axial current:
+¯ψ(p2)γ5ψ(p1) = ¯ψR(p2)γ5ψL(p1) − ¯ψL(p2)γ5
+Rψ(p1).
+(33)
+The second reason is the decrease in the neutrino energy ω at collision with a resting nucleon
+of mass M by the value ∆ω uniformly distributed in the following interval
+∆ω ∈ [0, 2ω2
+M ·
+1
+1 + 2ω/M ].
+(34)
+12
+
+The flux of solar neutrinos near the surface of the Earth does not differ from the theoretical
+one. But due to the first reason, the flux of left-handed neutrinos decreases by about a factor
+of two compared to the theoretical one: its ratio to the flux of right-handed neutrinos is, as it
+shown in [10], 0.516:0.484 (after correction here, it is 0.515:0.485).
+As is well known, the cross sections for the interaction of a neutrino with electrons, nucleons,
+and nuclei increase significantly with increasing energy up to its value for gauge bosons, but
+increase differently near the thresholds of various processes. Therefore, a decrease in the energy
+of each solar neutrino entails a decrease in the rate of observed processes, which is different for
+different processes.
+The solution of the task about the consequences of the short-term Brownian motion of
+neutrinos in the Sun was facilitated by the fact that the cross section of elastic scattering on a
+nucleon at rest, caused by the interaction (32),
+σ = (gνepsgNps)2
+16πM2
+1
+1 + 2ω/M ]
+(35)
+is practically independent of the energy of the solar neutrino. Therefore, with sufficient accu-
+racy, we can assume that the distributions of the number of collisions of neutrinos with solar
+nucleons are the same for all solar neutrinos.
+In [8] and [9], the role of a consequence of the unknown distribution is played by the effective
+number n0 of collisions of neutrinos with nucleons of the Sun, which determines the energy
+spectrum of neutrinos near the Earth’s surface, regardless of their handedness. Focusing on
+numerical methods dealing with a discrete set of numbers, we consider that the energy of the
+scattered neutrino takes with equal probability the two extreme values of its allowable interval,
+i.e. that the initial energy level of a neutrino after its collision with a nucleon splits into two
+levels. After n0 collisions of neutrinos with nucleons, its initial energy level ω splits into n0 + 1
+binomially distributed levels:
+ω1 = ω,
+ω2 =
+ω1
+1 + 2ω1/M ,
+. . . ωn0+1 =
+ωn0
+1 + 2ωn0/M .
+(36)
+In [9], simple algorithms for calculating the numerical values of the rates of each of the five
+observed processes with solar neutrinos are given, as well as references to the works of Bahcall
+et al., which contain tabulated values of solar neutrino fluxes from their various sources and
+cross sections for a number of processes. This makes it easy for any physicist or mathematician
+to check our results. Now I recommend replacing the multiplier 0.5Φ by 0.515Φ in the formulas
+of [9] for the rates of processes. Good agreement between the theoretical results and the results
+of all five experiments with solar neutrinos is ensured at n0 = 12.
+The small number of collisions of neutrinos with nucleons of the Sun allows us to consider it
+plausible that one neutrino collision occurs when a neutrino passes by about 0.7 to 0.9 fractions
+of the maximum possible number of nucleons on its way in the Sun. Taking into account the
+distribution of matter in the Sun given in [23] and the relation (35), we have obtained the
+following estimate
+gνepsgNps
+4π
+= (3.2 ± 0.2) · 10−5.
+(37)
+This estimate is used in calculating the deuteron disintegration rate by neutral currents caused
+by both left- and right-handed solar neutrinos, νe + D → νe + n + p.
+Later, in [10], the Brownian motion of neutrinos in the Sun was successfully described by
+the geometric distribution of the number of their collisions with nucleons
+P(n) = p(1 − p)n,
+n = 0, 1, 2, . . . .
+(38)
+From a comparison of the theoretical results based on this distribution with the experimental
+results in [10], it was accepted that the parameter p is equal to 0.062. Subsequently, I again
+13
+
+made such a comparison and now I cosider that p = 0.060. The theoretical rates of all five
+observed processes given below were calculated exactly at p = 0.060.
+In accordance with the fact that the four observed processes, νe +37 Cl → e− +37 Ar,
+νe +71 Ga → e− +71 Ge, νe + e− → nue + e− and νe + D → e− + p + p, and subprocess
+νe + D → νe + n + p with Z-boson exchange, are caused only by left-handed neutrinos, then
+when calculating the theoretical rates of these processes, the contributions of only those solar
+neutrinos that have undergone an even number of collisions with solar nucleons are summed
+up. The subprocess νe + D → νe + n + p with the exchange of pseudoscalar boson ϕps, is
+contributed by both left- and right-handed neutrinos that have experienced any number of
+collisions with nucleons of the Sun.
+Tables 1-5 below show the results of calculating the velocities of all observed processes with
+solar neutrinos based on the method of the effective number of neutrino-nucleon collisions n0
+for two values of the probability Wleft that at the Earth’s surface the solar neutrino is left-
+handed, and based on geometric distribution (4). Tables 3-4 show the dependence of the rates
+of the processes on the lower value Ec of the reconstructed energy of the final electron.
+Table 1. The rate of transitions 37Cl → 37Ar in SNU.
+8B
+7Be
+15O
+pep
+13N
+hep
+Total
+Experiment [27]
+2.56 ± 0.16 ± 0.16
+Wleft = 0.5, n0 = 11
+1.97
+0.43
+0.17
+0.11
+0.04
+0.01
+2.72
+Wleft = 0.515, n0 = 12
+1.95
+0.43
+0.17
+0.11
+0.05
+0.01
+2.72
+Eq. (38), p = 0.060
+1.99
+0.41
+0.17
+0.11
+0.04
+0.01
+2.73
+Table 2. The rate of transitions 71Ga → 71Ge in SNU.
+p-p
+7Be
+8B
+15O
+13N
+pep
+hep
+Total
+Experiment [28]
+62.9+6.0
+−5.9
+Experiment [29]
+65.4+3.1
+−3.0
++2.6
+−2.8
+Wleft = 0.5, n0 = 11
+34.6
+17.2
+4.9
+2.8
+1.7
+1.4
+0.02
+62.6
+Wleft = 0.515, n0 = 12
+35.7
+17.7
+4.9
+2.9
+1.7
+1.4
+0.02
+64.4
+Eq. (38), p = 0.060
+35.5
+17.5
+4.9
+2.8
+1.7
+1.4
+0.02
+63.8
+Table 3. Effective fluxes of neutrinos Φνe
+eff(8B) found from the process
+νee− → νee− (Ec is given in MeV, and the fluxes are in units of 106 cm−2s−1).
+References
+Ec
+Experimental
+Wleft = 0.5,
+Wleft = 0.515,
+Eq. (38),
+results
+n0 = 11
+n0 = 12
+p = 0.060
+SK IV [22]
+4.0
+2.31 ± 0.02 ± 0.04
+2.39
+2.42
+2.37
+SNO I [30]
+5.5
+2.39+0.24
+−0.23
++0.12
+−0.12
+2.19
+2.19
+2.15
+SNO II [31]
+6.0
+2.35 ± 0.22 ± 0.15
+2.10
+2.10
+2.07
+SNO III [32]
+6.5
+1.77+0.24
+−0.21
++0.09
+−0.10
+2.01
+2.00
+1.98
+Table 4. Effective fluxes of neutrinos Φcc
+eff(8B) found from the process
+νeD → e−pp (Ec is given in MeV,and the fluxes are in units of 106 cm−2s−1).
+References
+Ec
+Experimental
+Wleft = 0.5,
+Wleft = 0.515,
+Eq. (38),
+results
+n0 = 11
+n0 = 12
+p = 0.062
+SNO I [30]
+5.5
+1.76+0.06
+−0.05
++0.09
+−0.09
+1.86
+1.84
+1.85
+SNO II [31]
+6.0
+1.68+0.06
+−0.06
++0.08
+−0.09
+1.77
+1.73
+1.77
+SNO III [32]
+6.5
+1.67+0.05
+−0.04
++0.07
+−0.08
+1.66
+1.62
+1.69
+14
+
+Table 5. Effective fluxes of neutrinos Φnc
+eff(8B) found from the process
+νeD → νenp (the fluxes are in units of 106 cm−2s−1).
+Exchange
+Exchange
+Exchange
+Exchange
+Sum
+by Z
+by Z
+by Z
+by ϕ
+Wleft = 0.5
+Wleft = 0.515
+Eq. (38)
+all neutrinos
+SNO I [30]
+5.09+0.44
+−0.43
++0.46
+−0.43
+SNO II [31]
+4.94+0.21
+−0.21
++0.38
+−0.34
+SNO III [32]
+5.54+0.33
+−0.31
++0.36
+−0.34
+n0 = 11
+2.10
+2.87
+4.98
+n0 = 12
+2.11
+2.85
+4.96
+Eq. (38)
+2.10
+2.79
+4.89
+Comparing the corresponding numbers in each of the tables, we come to the following
+conclusion.
+The results of collisions of neutrinos with nucleons in the Sun caused by the
+interaction (32), which were obtained on the basis of the method of the effective number of
+collisions with a slight superiority of the flux of left-handed neutrinos, and which were obtained
+based on the geometric distribution of collisions, are close to each other and agree well with
+the experimental results.
+A good agreement of theoretical results, obtained on the basis of the hypothesis of the exis-
+tence of a new fundamental interaction, with the results of five absolutely different experiments
+with solar neutrinos can only be a game of chance with negligible probability.
+3.2. Borexino, KamLAND
+There are two types of Borexino experiments. First, the elastic scattering of solar neutri-
+nos on electrons at the scattered electron energy greater than 3 MeV is studied. The observed
+effective flux of solar neutrinos from 8B [33] is consistent with the experimental results of Super-
+Kamiokande and SNO. As this experiment was carried out after the corresponding experiments
+of Super-Kamiokande and SNO, and as the accuracy of its result is much less than in Super-
+Kamiokande and SNO, then it adds nothing new to the solar neutrinos. Second, the solar neu-
+trino flux from 7Be is studied. However, as it is stated in the work [34], ”the contribution
+of the pp, pep, CNO, and 8B solar neutrino were fixed to the SSM-predicted rates assum-
+ing MSM neutrino oscillations.” Thus, Borexino does not present a purely experimental re-
+sult for the neutrino flux from 7Be, but only an MSM interpretation of it, and so, we have
+nowhere to apply our theoretical calculations.
+The preparation and implementation of the KamLAND experiment was initially focused
+(see Ref.
+[35]) on confirming the existence of electron (anti-)neutrino oscillations.
+When
+presenting the first results, it was said [36]: ”The long baseline, typically 180 km, enables
+KamLAND to address the oscillation solution of the ‘solar neutrino problem’ using reactor
+anti-neutrinos under laboratory conditions”. The existence of (anti-)neutrino oscillations with
+parameters given by the KamLAND collaboration and close to ones of the SNO and Super-
+Kamiokande collaborations would lead to the survival probability of the solar neutrino electron
+component near the Earth’s surface, given by the formula (26), which, as stated in section
+2.4, contradicts at least three out of five experiments. At the same time, the interaction (32),
+which provided a solution to the solar neutrino problem, has nothing to do with the announced
+result of the KamLAND collaboration, since the reactor antineutrino cannot experience a single
+collision on its 180 km path.
+After obtaining a solution to the solar neutrino problem, we had no doubt that the reason
+for the loss of observed events in the KamLAND experiment is serious omissions in its setting
+up and processing. I have repeatedly returned to the issue of such omissions, having studied,
+where possible, the features of the appearance of fluorescent light, the characteristics of its
+15
+
+spectrum, the dependence on the wavelength of the fluorescent signal of the attenuation of its
+intensity during propagation in liquid KamLAND scintillator, etc. Based on the comparison
+of suitable numbers, I finally came to the following conclusion [37]:
+In the conditions of unprecedentedly large sizes of KamLAND detector and of the refusal
+of using the gadolinium, the absence of (1) proper study of the real spectrum of the PPO
+fluorescent light, of (2) a detailed study of the fluorescent light attenuation in a liquid scintillator
+depending on its wavelengths, of (3) the influence on the attenuation of light of its scattering and
+re-emission, and of (4) the theoretical calculation of the observability of events ¯νe +p → e+ +n
+opens the door to uncontrolled loss of such events. Therefore, we do not consider the announced
+results of the KamLAND experiment as reliable.
+3.3. Reactor antineutrino anomaly, gallium anomaly
+Acquisition of knowledge about the spectrum of reactor antineutrinos is realized by two
+different methods. One of them, called experimental, uses the spectrum of observed positrons
+produced in an experimental setup under the action of reactor antineutrinos, and the well-
+known dependence of the inverse beta decay cross section on antineutrino energy. Another
+method, called theoretical, was initiated in 1981 by Schrekenbach et al. [38]. It was relied on
+experimental measurements of electron spectra from 235U decays and their subsequent conver-
+sion into electron antineutrino spectra based on the kinematics of beta decays. In 2011, new
+theoretical calculations of the spectra of reactor antineutrinos from the decays of 235U, 239Pu,
+241Pu and 238U in thousands of possible channels appeared [39]. Comparing the results of these
+calculations with the results of the entire set of experiments with distances less than 100 meters
+from the reactors, the authors of [39] conclude that the ratio of the rate of observed events to
+the rate of predicted events is equal to 0.943 ± 0.023. The distinction of this ratio from unity
+is called by them the reactor antineutrino anomaly. The authors of [39] are inclined to believe
+that the reactor anomaly may indicate the existence of a fourth nonstandard neutrino, which
+determines electron neutrino oscillations at short distances.
+We affirm that the difference between the theoretical and experimental spectra of antineu-
+trinos produced in beta decays of any nuclei is a manifestation of a new fundamental interaction
+(32).
+Once again, we note that the experimental spectrum includes those and only those antineu-
+trinos that manifest themselves in the production of positrons. The theoretical spectrum, relied
+on the spectrum of electrons, includes both those antineutrinos that are capable of producing a
+positron, and those that do not have such an ability. The interaction (32) allows the existence
+along with the dominant mode of beta decay of the X nucleus into the Y nucleus
+X → Y + e− + ¯νeR,
+(39)
+also of the decay mode with the emission of a massless axial boson
+X → Y + e− + ¯νeL + ϕps.
+(40)
+The right-handed antineutrino, born in the beta decay of the X nucleus, can emit a massless
+axial boson in a virtual state, changing its handedness to the left. The left-handed antineutrino
+within the framework of the standard interaction cannot manifest itself, causing the production
+of a positron. Thus, the experimental spectrum includes only antineutrinos of the dominant
+nuclear beta-decay mode (39). The theoretical spectrum includes both antineutrinos from the
+dominant mode (39) and effective ”antineutrino” that accompanies the electron and is the sum
+of the unobservable left-handed antineutrino and the unobservable massless axial boson from
+the mode (40).
+16
+
+The emission of a massless axial boson in the beta decay of nuclei is to some extent analogous
+to the emission of a gamma quantum in such decays. For the decay
+n → p + e− + ¯νe + γ,
+(41)
+there are both theoretical calculations [40] and experimental measurements [41]. The theo-
+retical differential width of the process (41) contains the infrared divergence. The measured
+fraction of the radiative decay mode of a neutron with a gamma quantum energy from 0.4 to
+782 keV is (9.17 ± 0.24 ± 0.64) · 10−3, which is close to the constant α = e2/4π.
+Solving the solar neutrino problem, we obtained an estimate of the product of the Yukawa
+coupling constants of the massless axial boson ϕps with electron neutrinos and with nucleons
+(37) and had no way to find these constants separately. Now, using some similarity between
+the decay modes (40) and (41) and the fact that the difference between the rates of observed
+and expected events with reactor antineutrinos is due to the beta-decay mode of nuclei (41),
+we find it possible to take, in the zeroth approximation, the following estimate
+(gνeps)2
+4π
+∼ 0.057 ± 0.023.
+(42)
+It would be possible to obtain a realistic value of the coupling constant of an axial boson
+with an electron neutrino if it were possible to calculate the electron spectrum for the decay
+mode with an axial boson (40) of the 235U isotope in the most significant channels and to solve
+herewith the problem massless divergence.
+Knowing the realistic value of the constant (gνeps)2/4π would help to establish the level of
+difference, called the gallium anomaly, between the numbers of observed and expected events
+of the 71Ga transition to 71Ge under the influence of electron neutrinos from the decay of 31Cr.
+At present, there is a scatter in the ratio of these numbers obtained in six variants of completed
+experiments, in the range from 0.95 to 0.77 [42]. We have no doubt that the gallium anomaly, as
+well as the reactor antineutrino anomaly, is a manifestation of a fundamental interaction (32).
+The theoretical number of decays per unit time of the 31Cr isotope, caused by the capture of an
+electron, is entirely determined by its quantity obtained from the 30Cr isotope by irradiation
+in a certain nuclear reactor. The decay of the 31Cr isotope occurs both in the dominant mode
+31Cr + e− →31 V + νeL
+(43)
+and in the mode with the emission of a massless axial boson
+31Cr + e− →31 V + νeR + ϕps.
+(44)
+The decay mode of the isotope 31Cr (43) subsequently leads to transitions of gallium into
+germanium, and the mode (44) is lost.
+So, the reactor antineutrino and gallium anomalies are a consequence of the existence of a
+fundamental interaction (32), which is responsible for the undetectable decay mode of one or
+another isotope, characterized by the emission of a massless axial boson and a change in the
+handedness of the original (anti)neutrino.
+4. Conclusion
+Ignoring the basic principles of classical logic and, in particular, the basic principles of
+mathematical analysis, researchers lose the ability to distinguish between perfect reasoning
+and charlatanism. For example, according to Smirnov’s recent testimony [26] (Section 3.6), the
+reviewers of [6], who initially rejected the possibility of its publication in the journal Yadernaya
+Fizika, understood its possible fallacy and hoped that ”somebody will eventually find
+17
+
+this.” This ”somebody” only now, 37 years later, showed up and proved the fallacy of [6]
+above in section 2.5. Surprisingly, the correctness of the work [6], about which the reviewers
+expressed doubts, was not publicly disputed by any of the hundreds of adherents, including
+very respected physicists. Such is the reality.
+This work is designed to rid physics of spirits imposed on it in the form of additional neu-
+trinos and recipes that violate the principles of classical logic or basic physical requirements.
+The solution to the problem of solar neutrinos, and then clarifying the nature of reactor an-
+tineutrino and helium anomalies, is given on the basis of classical field theory methods with
+their inherent simplicity and elegance. Time will tell if the scientific community accepts the
+revival of at least one of Newton’s rules outlined at the end of section 2.1.
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+19
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf,len=872
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='03366v1  [hep-ph]  9 Jan 2023  Hypothesis of a new fundamental interaction versus the  particle oscillation concept  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='Slad*  Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow  119991, Russia  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The emergence of a logically simple solution to the solar neutrino problem based on the  hypothesis of the existence of a new interaction involving electron neutrinos and nucleons did not weaken  the dominance of the oscillation concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Therefore, a significant part of the present work is devoted  to proving that the basic elements of this concept either do not correspond to the principles of classical  logic, or violate the energy-momentum conservation law, or contradict the quantum mechanical basis  of coherence, or represent a primitive falsehood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Our analysis concerns successively all stages of the  formation of the concept from its conceiving to the assertion about the conversion of the solar electron  neutrino into a muon one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' When discussing a new fundamental interaction, we note the decisive role in  the outcome of the processes of changing the handedness of a neutrino (antineutrino) at each act of its  interaction with a real or virtual massless pseudoscalar boson, due to which, at the exit from the Sun,  the fluxes of left- and right-handed electron neutrinos become approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Thanks to the new  interaction, beta decays of nuclei have a mode with the emission of a massless pseudoscalar boson, at  which the antineutrino changes its handedness and becomes unobservable, what is the essence of the  reactor antineutrino anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The nature of the gallium anomaly is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Introduction  The emergence of the solar neutrino problem is caused by the report [1] about the absence  of observed transitions of 37Cl into 37Ar under the action of such neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It have been stated  the upper limit for such a transition rate as 3 SNU (1 SNU is 10−36 captures per target atom  per second), while the rate predicted by Bahcall [2] was 30+30  −15 SNU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The reaction to this report was well conveyed by Reines [3]: ”It is interesting to note that  if a positive result were obtained in the Davis experiment, we would suddenly face the problem  of whether it is due to the Sun or something else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' A negative result is important because that  it allows us to assert, in the event of a further positive result in a neutrino experiment, that  the effect is caused precisely by solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='This is significant, because the Universe is full  of surprises .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' When in February 1972 we discussed this issue at a conference in Irvine, it  was interesting to observe how, in search of the reasons for the discrepancy, astrophysicists  pointed to specialists in the nucleus, the latter to neutrino physicists, and those in turn to  astrophysicists”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' These words were said by Reines during the discussion of Pontecorvo’s report  at the seminar on the µ − e problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It is noteworthy that Reines does not say a word about  the solar neutrinos oscillations, the assumption of which was put forward three years earlier [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  This indicates that, for the time being, the neutrino oscillation concept was weakly perceived  by the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Over time, Wolfenstein [5] extended it to the transformation of a neutrino as it moves  in a homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Seven years later, Mikheev and Smirnov [6] announced the ex-  tension of the Wolfenstein equation to the inhomogeneous medium of the Sun, but did not  describe the evolution of the state of the solar neutrino as it moves from its production place  to an installation on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Such a description with the oscillation parameters given by the  slad@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='sinp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='ru  1  Super-Kamiokande and SNO collaborations was partly made in [7] by analytical and numer-  ical analysis of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The main task of [6] was to argue the statement about the  transformation of an electron neutrino into a muon neutrino during its passage through the  Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The proposed arguments, as we will prove below, are false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Despite the initial perception  of the work by reviewers and readers as erroneous, over time, the belief that the Mikheev-  Smirnov-Wolfenstein mechanism is able to solve the solar neutrino problem has become a  fairly widespread phenomenon among the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Having analyzed all stages of the particle oscillation concept from its inception, which con-  cerned neutral K-mesons, to the appearance of the Mikheev-Smirnov-Wolfenstein mechanism,  we find either ignoring the rule of sufficient reason, or ignoring the law of conservation of energy-  momentum, or neglecting the quantum mechanical basis of coherence , or elementary falsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  This will inevitably lead sooner or later to the collapse of the concept of particle oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  I hope that it will be helped, in particular, by the emergence of a logically simple solution to  the solar neutrinos problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The main theseses of such a solution and their correspondence  with experiments were outlined in February 2015 in my work [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  It would seem that the whole history of physics pointed to the fact that some new for  us, rather hidden interaction is responsible for the emergence of the solar neutrino problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Surprisingly, that, as it is clear from the utterance of Reines [3], the question of hidden inter-  action was not even raised at the Irvine conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' My hypothesis postulates the existence  of an interaction whose carrier is a massless pseudoscalar boson that has a Yukawa coupling  with at least the electron neutrino, proton, and neutron, but not with electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Due to this  interaction, neutrinos during their movement inside the Sun experience about a dozen collisions  with nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At each such collision, the neutrino handedness changes from left to right and  vice versa, so that at the exit from the Sun the ratio of left- and right-handed electron neu-  trino fluxes is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In addition, at each collision of a neutrino with nucleons, its energy  decreases slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Our model with one free parameter provides good agreement between theo-  retical and experimental results for the rates of all five observed processes with solar neutrinos  [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  This paper is devoted, on the one hand, to the analysis of the particle oscillation concept  from its inception and, on the other hand, to a discussion of separate elements of the postulated  new interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The particle oscillation concept  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Neutral K-mesons  The discovery of hyperons and K-mesons was an unexpected and amazing surprise, as it is  said in the monograph of that time [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In the emerging theory of new unstable particles, first of all, ones tried to transfer such  notions of symmetry as isospin and charge conjugation, attributed to stable particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Gell-  Mann [12] supposed that the new unstable particles are fermions with integer isospin and bosons  with half-integer isospin and have antiparticles that differ from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Gell-Mann and Pais [13]  believed that strong interactions are responsible for all productions of hyperons and K-particles,  while weak interactions are responsible for their decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Therefore, in their assertion, the law  of conservation of isospin allows the reaction of production of θ0-meson with the projection of  isospin -1/2 (hereinafter referred to as K0)  π− + p → Λ0 + θ0,  (1)  but prohibits the process of production of its antiparticle  π− + p → Λ0 + ¯θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (2)  2  At the same time, due to the violation of the law of conservation of isospin in weak interactions,  decays of the θ0-particle and ¯θ0-antiparticle are admissible according to the same schemes:  θ0 → π+ + π−,  ¯θ0 → π+ + π−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (3)  The relations (3) allow the virtual transition θ0 ↔ π++π− ↔ ¯θ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' the particle θ0 and its  antiparticle ¯θ0 do not completely differ from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Gell-Mann and Pais find it convenient  to introduce two quanta θ0  1 and θ0  2 (hereinafter referred to as K0  1 and K0  2, and then as K0  S and  K0  L), transforming into themselves upon charge conjugation:  θ0  1 = (θ0 + ¯θ0)/  √  2, qquadθ0  2 = (θ0 − ¯θ0)/i  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (4)  Since each of the quanta θ0  1 and θ0  2 can be assigned its own lifetime and mass, Gell-Mann and  Pais consider them to be true ”particles”, and at the same time they treat θ0 and ¯θ0 as a  ”mixture of particles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  It is incorrect to interpret relations (4) and (5) as an expression of the wave function of  one of the particles in terms of the superposition of the wave functions of two other particles,  which takes place in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Since the values of the wave functions are c-numbers, it is easy to  find their sum (difference), but in this sum they lose their individuality and it is impossible to  separate them from the sum and perform their separate transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  I find it useful to express here the essence of the addition of ”particles” or ”mixture of  particles” in terms of known logical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Let us introduce the notion of a linear space  of particles over the field of functions of space-time coordinates, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Each particle is associated  with its own basis vector in the space L, denoted by the name of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Any particle  state and its time transformations are displayed in L by vectors obtained by multiplying its  basis vector by an appropriate wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Each mixture of particles is associated with only  one vector in the space L, which is given by a well-defined decomposition into the basis vectors  of this space at the initial moment and is denoted by the name of the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The time  transformation of each mixture is given by the transformation of the states of the particles,  into whose basis vectors it is decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Finally, the state vectors of the particles at time t  are decomposed into the initial mixture vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The coefficients of these decompositions play  the role of the probability amplitudes for the existence of the corresponding mixtures at time  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The time transformation of the wave function of a particle with mass m is well known:  its initial value acquires an additional phase factor exp(−iEt), where E =  �  m2 + p2, p is  momentum modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The question of the time dependence of the wave function of a ”mixture  of particles” was considered, in the spirit of what was said above, in the work of Pais and  Piccioni [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' From the relations (4) ones obtain  θ0 = (θ0  1 + iθ0  2)/  √  2, qquad¯θ0 = (θ0  1 − iθ0  2)/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (5)  Getting the time transformation θ0 is like this:  θ0 → (θ0  1 exp(−iE1t) + iθ0  2 exp(−iE2t))/  √  2 = ((θ0 + ¯θ0) exp(−iE1t) + (θ0 − ¯θ0) exp(−iE2t))/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (6)  Hence, under the condition p1 = p2 ≡ p, E1 ≫ m1 and E2 ≫ m2, the probability of  existence of a meson theta0 at time t is an oscillating quantity  P(θ0, t) ≈ cos2 (m2  1 − m2  2)  4p  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (7)  In [13] and [14], not a word is said about the feasibility of the law of conservation of energy-  momentum in the process (1), provided that the θ0-meson is a mixture of two particles (5)  3  with different masses m1 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Our statement is that the consequences of realizating the  law of conservation of energy depend on the ratio between the total uncertainty in masses  (the sum of decay widths) of unstable particles �  i Γi participating in the process (1) and the  mass difference |m1 − m2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' If the total mass uncertainty exceeds this mass difference, then  the fulfillment of the law of conservation of energy in the process (1) with the θ0-meson as a  mixture of two particles is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Otherwise, the produced K-meson is forbidden to be a  mixture of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It must itself be a particle with a dominant decay channel (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Let us now turn to the numbers characterizing the process (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  We take into account  that the particle decay width Γ and its lifetime τ are related by the equality Γτ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The  lifetimes of particles interesting to us are taken from review [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' We have: τπ− = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='60 · 10−8  s, Γπ− = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='53 · 10−8 eV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' τΛ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='63 · 10−10 s, ΓΛ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='50 · 10−6 eV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' τK0  S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='895 · 10−10 s,  ΓK0  S = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='35· 10−6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' So, the total mass uncertainty in the process (1) is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='88· 10−6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At the  same time, the difference between the masses of two neutral K mesons, mKL − mKS, is less  than this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It is [15] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='48 · 10−6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Thus, the requirement of fulfillmenting the law of conservation of energy does not prohibit  the two neutral K-mesons, K0 and ¯K0, be mixtures of the other two K-mesons , K0  S and K0  L,  and to oscillate as it moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Let us now recall that the appearance of the assertion about the oscillation of neutral K  mesons was caused by the assertion of Gell-Mann-Pais that, due to the conservation of isospin,  the state of the θ0-meson, produced in the process (1), has an isospin projection of -1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' But,  from the relation (6), it follows that after an arbitrarily small interval of time after production  this state does not have a definite value of the isospin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It seems natural to recognize  the assertion about the conservation of isospin in the process (1) as untenable and to accept  that one of the admissible K-mesons, produced in the process (1) and decaying into π++π−, is  a θ0  1 meson coinciding with its antiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Some reasoning of Gell-Mann-Pais give arguments  in favor of the admissibility of the production in the process (1) of a θ0  2-meson, which decays  into three pi-mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  From the four neutral K-mesons considered in [12]-[14], only two mesons should be left in  the final theory, namely θ0  1 and θ0  2 (KS and KL, in current notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Their Yukawa interaction  with baryons should be represent as  L = g1¯nγ5ΛKS + g2¯nγ5ΛKL + g3¯nγ5Σ0KS + g4¯nγ5Σ0KL + g5¯pγ5Σ+KS + g6¯pγ5Σ+KL, (8)  where pairs of constants g1, g2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' g3, g4 and g5, g6 have close but different values, reflecting  CP-symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It follows from here that at each collision of the meson KL (KS) with  the nucleons of matter, either a meson KL (KS) is born again, or a meson KS (KL) is born and  then one speaks about the regeneration of KS (KL) from KL (KS)-meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' This regeneration  refers to a number of such standard processes as the Compton effect and π-meson recharging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The situation with two, not more, neutral K-mesons with certain masses would correspond  to the assertions of great thinkers about sufficient reason and, in particular, to the assertion of  Isaac Newton: We are to admit no more causes of natural things than such as are both true and  sufficient to explain their appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' To this purpose the philosophers say that Nature does  nothing in vain, and more is in vain when less will serve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' for Nature is pleased with simplicity,  and affects not the pomp of superfluous causes [16] (Book III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Rule I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Gribov-Pontecorvo concept of neutrino oscillations  Soon after the publication of the results of Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' [1], Gribov and Pontecorvo paper  [4] appeared, in which the lower value of the solar electron neutrino flux near the Earth’s  surface compared to that expected on the basis of the standard solar model is explained by  solar neutrino oscillations, due to which some of them at the location of the experimental setup  emerge as muon neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  4  Gribov and Pontecorvo, in their substantiation of the existence of neutrino oscillations, tend  to adhere to the Gell-Mann-Pais-Piccioni procedure for neutral K-mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' But if in Gell-Mann,  Pais and Piccioni the elements of the procedure have clear experimental and worldview sources,  then in Gribov and Pontecorvo they are taken ”from the ceiling”, from nowhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Thus, in order  to supplement the family of known neutrinos νe and νµ with two new neutrinos, Gribov and  Pontecorvo, adding off-diagonal terms to the mass part of the neutrino Lagrangian, symbolizing  the transitions in vacuum of an electron neutrino to a muon neutrino and vice versa , give the  Lagrangian the following form  Lint = a1¯νeνe + a2¯νµνµ + a3¯νeνµ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=',  (9)  where a1, a2 and a3 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Diagonalization of the Lagrangian (9) leads to the appear-  ance of two states  ν1 = cos θ · νe − sin θ · νµ,  ν2 = sin θ · νe + cos θ · νµ,  (10)  whose masses and the value of the mixing angle are given by the constants ai, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  We note here that earlier, when Weinberg constructed the electron gauge theory [17], the  off-diagonal mass term formed by the gauge fields already appeared in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' As  a result of the diagonalization of his Lagrangian, Weinberg obtained the photon and massive  boson fields, which he assigned the status of quantized fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' He then expressed through them  the initial gauge fields, which he got rid of in the final theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The number of initial and  final entities in Weinberg remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' As a result, we have a phenomenal theory of  electroweak interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  While Weinberg naturally avoided the doubling of entities, Gribov and Pontecorvo delib-  erately introduced such doubling, which became the cornerstone of their concept of neutrino  oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The states ν1 and ν2 have been given status, using Gell-Mann-Pice terminology,  true particles, neutrinos νe and νµ have been given mixture status:  νe = cos θ · ν1 + sin θ · ν2,  νµ = − sin θ · ν1 + cos θ · ν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (11)  Applying the Pais-Piccion procedure to the relations (11) and (10) regarded to the initial  moment of time, we have  νe → cos θ · ν1 exp(−iE1t) + sin θ · ν2 exp(−iE2t) =  cos θ(cos θ · νe − sin θ · νµ) exp(−iE1t) + sin θ(sin θ · νe + cos θ · νµ) exp(−iE2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (12)  Hence, under the condition p1 = p2 ≡ p, E1 ≫ m1 and E2 ≫ m2, the probability of  detecting neutrinos nue at time t is an oscillating quantity:  P(νe, t) = 1 − sin2 2θ sin2 (m2  1 − m2  2)  4p  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (13)  The length of its oscillation is given by the equality  L/c = 2π  �  2p  |m2  1 − m2  2|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (14)  The most important point in our analysis of the Gribov-Pontecorvo concept of neutrino  oscillations is to find out the feasibility of the law of conservation of energy-momentum in the  processes of production of electronic (anti-)neutrinos as a mixture (11) of two (anti-)neutrinos  ν1 and ν2 with different masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' By analogy with our assertion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='1, we compare the  5  difference between the neutrino masses ν1 and ν2 with the total uncertainty in the masses of  particles involved in the neutron decay process  n → p + e− + ¯νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (15)  In this process, the only unstable particle is the neutron with the lifetime value [15] τn =  879 s and, consequently, with the width Γn = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5 · 10−19 eV, which plays the role of the  total uncertainty in masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Energy-momentum conservation in the process (15) is allowed  if |m1 − m2| < Γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Let us take for the upper value of each of the masses m1 and m2 the  experimentally established upper value of the electron neutrino mass [15]: mνe < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Then  the possibility of conservation of energy-momentum in the process (14) restricts the difference  of squared neutrino masses ν1 and ν2 to the following limit  |m2  1 − m2  2| < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2 · 10−19eV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (16)  Obviously, the restriction (16) on the masses of the ν1 and ν2 components of the electron  neutrino do not depend on the processes in which νe is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Let us now turn to the length  L of solar neutrino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' For the observed solar neutrinos, as follows from the relation  (14, it is minimal at the threshold energy for gallium-to-germanium transitions equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='233  MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Taking into account the inequality (16), we have that L > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0 · 1014 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Thus, the state  of an electron neutrino on its way from the Sun to the Earth (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='50 · 108 km) could make no  more than 2· 10−7 fractions of one oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Such a situation would be absolutely unsuitable  for solving the solar neutrinos problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  So, either the representation of the neutrino νe as a mixture (11) has nothing to do with  the solution of the solar neutrino problem, or in the case of the supremacy of the neutrino  mass difference |m1 − m2| over the neutron decay width (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5 · 10−19 eV) the energy-momentum  conservation law would be violated, which makes such a representation and oscillations inad-  missible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  There have already been reasoning in the literature that the neutrino oscillation concept  may contradict the law of conservation of energy-momentum, but neither the essence of the  concept nor its effectiveness has been doubted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Thus, Giunti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' [18] believe that the question  of energy–momentum nonconservation in processes involving neutrinos with an indeterminate  mass does not arise in a real physical situation when the localization of the neutrino source  and detector is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In fact, it is proposed to consider a neutrino on its way  from the source to the detector as a virtual particle, the 4-momentum of which is outside the  mass shell, which hides the certainty of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Following this proposal, let’s give the solar  neutrino, registered on Earth, the status of a virtual one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Since virtuality lasts 500 s, due to  the Heisenberg relation, the energy uncertainty is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='3 · 10−18 eV, which is very close to the  discussed total mass uncertainty in the process (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Consequently, the conclusion formulated  above remains fully valid, according to which either the survival of the electron component  of the solar neutrino near the surface of the Earth differs very little from 1, or the electron  neutrino cannot be a mixture of two other neutrinos with different masses and, therefore, its  oscillations do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Wolfenstein’s equation for neutrinos in a homogeneous medium  Having established that the energy-momentum conservation law does not allow solving the  solar neutrino problem on the basis of the neutrino oscillation hypothesis proposed by Gribov  and Pontecorvo, we nevertheless continue to analyze its modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The appearance of proposals to set up experiments on the detection of neutrinos from ac-  celerators when they could pass through the Earth, along with ongoing experiments on the  detection of solar neutrinos, prompted Wolfenstein to consider the possible influence of the  6  medium on neutrino oscillations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  When constructing his equation describing the trans-  formation in the medium of probability amplitudes to detect neutrinos in the ν1 or ν2 state,  Wolfenstein believed that coherent forward scattering of neutrinos on electrons takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  This scattering is caused by electroweak interactions and gives different phase changes in the  wave functions of the electron and muon components of the initial neutrino, which, in his  opinion, is similar to the regeneration of KS from the KL-beam passing through the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  I have two remarks with regard to Wolfenstein’s assertion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  First, we find reasoning about coherence in the work of Muller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' [19] devoted to the  difference in the masses of neutral K-mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In it, coherence is associated with only one of  the three types of regeneration of KS from KL-mesons in a medium, when a plate placed in  front of a parallel beam of KL-particles generates a parallel beam of KS-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' To two  other types of regeneration, Muller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' refer the diffraction scattering by nuclei and ordinary  scattering by separate nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The share of contributions to the total regeneration of each  of its three types is not discussed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' From their experiment, Muller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' found that  the ratio |mKL − mKS|/ΓKS is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' From modern data [15], we obtain the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='473  for this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Note, that subsequently Geveniger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  [20] measured the mass difference  mKL − mKS by the method of two regenerations and obtained a highly accurate value for it,  essentially coinciding with the modern one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In [20], coherence is not mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Secondly, neutrino scattering on a certain set of electrons as a quantum mechanical phe-  nomenon will be coherent in the case when the neutrino wave function has at the same time a  non-zero value at the location of each electron from this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' This condition is satisfied if and  only if the neutrino wavelength is comparable to the distance between electrons from a given  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Observable solar neutrinos with the lowest energy equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='233 MeV have the longest  wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' For them λ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='47 · 10−13 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At the same time, electrons have the highest  density Ne in the center of the Sun, where the matter density, almost entirely determined by  protons, is 148 g · cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' There, the electron density Ne is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='85 · 1025 cm−3 and, consequently,  the minimum distance between electrons in the Sun is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='24 · 10−11 m, which is approximately  two orders of magnitude greater than the largest wavelength of the observed solar neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  So, coherence in the scattering of observed solar neutrinos is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The analogy with  the regeneration of K-mesons is weak and does not change this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Wolfenstein’s  equation is not applicable to any part of the trajectory of the observed solar neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It must  be attributed to some abstract medium and an abstract neutrino beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Consequences of the Wolfenstein equation for solar neutrinos  Let us put aside for a while the fact that the Gribov-Pontecorvo neutrino oscillation hy-  pothesis and the Wolfenstein equation for neutrinos in a homogeneous medium are not suitable  for describing the transformation of solar neutrinos, and analyze the consequences of extending  the Wolfenstein equation to solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The extension of the Wolfenstein equation to an inhomogeneous medium consists in giving  the electron density the status of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In the νe, νµ basis, this equation has the following  form (see [21], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='55))  i d  dt  �  Ae(t)  Aµ(t)  �  = 1  2  \uf8eb  \uf8ec  \uf8ec  \uf8ed  −∆m2  2E cos 2θ +  √  2GF Ne(t)  ∆m2  2E sin 2θ  ∆m2  2E sin 2θ  ∆m2  2E cos 2θ −  √  2GF Ne(t)  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ×  ×  �  Ae(t)  Aµ(t)  �  ,  (17)  where Ae(t) (Aµ(t)) is the probability amplitude that the neutrino state is electronic (muonic  ) at the time moment t, when the electron density is equal to Ne(t) at the location of the  neutrino;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' ∆m2 = m2  1 − m2  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' E is the neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  7  It is easy to verify that the equation (17) ensures the unitarity condition  |Ae(t)|2 + |Aµ(t)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (18)  We first give a brief list of those analytical and numerical consequences of the equation  (17), which are detailed in our paper [7], adding some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  We attribute the Wolfenstein equation to the entire time interval from the moment of the  neutrino production in the Sun to its registration on the Earth, while the electron density at  a neutrinos location can be quite large or arbitrarily small and zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In our numerical analysis, we use the central values of the parameters of the two-neutrino  model of solar neutrino oscillations shown by the Super-Kamiokande Collaboration [22]:  ∆m2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8 · 10−5 eV2,  sin2 θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='334+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='027  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (19)  The conclusions in [7] remain unchanged when the value of ∆m2 changes by two orders of  magnitude, and the value of sin2 θ changes twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Our solution to the equation (17) is based on the adiabatic approximation used in quantum  mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Initially, it is assumed that, over several lengths of oscillations, the matter density  in the Sun can be considered constant, equal to its value at the initial moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' After finding  the interval of oscillation lengths corresponding to all observed values of the solar neutrino  energy, we are convinced of the validity of this assumption when, using the numerical values of  the dependence of the density of matter on its distance from the center of the Sun [23], we find  that its relative change δρ/ρ throughout the length of one oscillation does not exceed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7·10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  If at time moment t0 the state of the solar neutrino was purely electronic, |Ae(t0)|2 = 1,  then, as follows from the equation (17), the probability Pe(t) that this neutrino remains electron  at time t is given by  Pe(t) = |Ae(t)|2 = 1 − sin2 2θm sin2 f(Ne)(t − t0)/2,  (20)  where  sin 2θm = ∆m2  2E  sin 2θ  f(Ne),  (21)  f(Ne) =  ��∆m2  2E  �2  − 2  √  2  �∆m2  2E  �  GF Ne cos 2θ + 2G2  F N 2e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (22)  The angle θm defines the shape of the neutrinos νe and νµ as a mixture of massive neutrinos  in the medium ν1m and ν2m:  νe = cos θ · ν1m + sin θ · ν2m,  νµ = − sin θ · ν1m + cos θ · ν2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (23)  The characteristics of each separate oscillation depend only on the density of the matter  at the location of the neutrino and do not depend on the features of the part of the neutrino  trajectory that preceded it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Knowledge of these characteristics allows us to get a conclusion  about the set of all solar neutrino oscillations on any of their rectilinear trajectories for each  value of the observed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At an energy in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='233 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='827 MeV, the length of  the oscillations decreases monotonically with a decrease in the density of matter at the location  of the neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At an energy in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='827 to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8 MeV, the oscillation length as a  function of the matter density has an extremum at its value  ρext = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='95 · 10−24 cos 2θ  √  2GF  �  ∆m2  2E  �  g,  (24)  namely, the maximum  L(ρ)max/c =  2π  sin 2θ  � 2E  ∆m2  �  ,  (25)  8  at that with increasing energy, the value of ρext decreases from 148 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='51 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The length  of the oscillations of the observed solar neutrinos with energies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='233 to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8 MeV belongs  to the interval from 12 to 1028 km, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0000172R0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='00148R0, where R0 is the radius  of the Sun, R0 = 696000 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  When the matter density is ρext, the oscillating probability Pe(t) has a maximum amplitude,  namely, Pe(t) can take any value from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Also, the maximum the value equal to 1 is taken  by the quantity sin2 2θm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Based on a comparison of the lengths of neutrino oscillations in the Sun and in vacuum  with the characteristic sizes of distributions of neutrino sources in the Sun, it was shown in [7]  that the numbers of oscillations along various trajectories from the Sun to the installation on  Earth can differ by several tens or hundreds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' As a result, the summation of the contributions of  separate sources to the probability Pee of that the state of the solar neutrino will be electronic  near the Earth’s surface leads to the disappearance of durations of the neutrino motion along  their trajectories, and the final formula for Pee for any neutrino energy has the following form  Pee = 1 − 1  2 sin2 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (26)  The probability (26) contradicts the results of at least three out of five experiments with solar  neutrinos: on 37Cl →37 Ar transitions, on elastic neutrino scattering on electrons, on the  disintegration of deuterons by charged currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  So, three things are incompatible: the first is the values of the parameters of neutrino  oscillations announced by the Super-Kamiokande and SNO collaborations, the second is the  solution of the Wolfenstein equation for solar neutrinos with the indicated parameter values,  the third is the experimentally measured rates of processes caused by solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Mikheev-Smirnov-Wolfenstein effect  At the end of the 1970s, the problem of solar neutrinos acquired a clear numerical outline,  consisting in the fact that the experimentally measured rate of the transition of chlorine into  argon under the action of solar neutrinos is approximately 1/3 of the theoretical rate found  in the framework of the standard solar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' For example, Davis announced [24] that the  measured rate of these transitions is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='3 SNU, and Bahcall with colleagues reported [25]  that the calculated rate is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At the same time, for any values of the parameters and  two- and three-neutrino oscillations in their standard sense, the probability of that the initial  electron neutrino remains electronic at the place of its registration, when averaged over the  time of one oscillation, is no less than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Under these conditions, an extraordinary scenario of neutrino oscillations was announced  by Mikheev and Smirnov in [6], the main assertion of which is that the initial beam of electron  neutrinos, after passing through a certain layer of matter, is almost completely transformed  into a beam of muon neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The substantiation of this assertion is not accompanied by  any analytical calculations and formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Its role is played by the terminology that is not  appropriate in particle physics and field theory, the haziness of statements, the purposeful  game with the dependence of the mixing angle on the density of the medium, which plays a  key role in declaring the metamorphosis of the solar neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Let us dwell on all the details of the work [6], taking into account that it has hundreds of  adherents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Instead of speaking, as it is customary in mathematical analysis, about the existence of  extremal values of a number of oscillation characteristics, as functions of the density of matter,  on a certain set of neutrino trajectories in the Sun, Mikheev and Smirnov declare the existence  of a resonant amplification of neutrino oscillations in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The term ”resonance” in  the oscillatory process has both scientific and everyday use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In mechanics and in the theory of  9  electricity, the appearance of resonance in oscillations is associated with the existence of three  constituent elements: the frequency of proper oscillations, the frequency of external influences  and ”the friction” (energy loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Only the first of these elements is present during neutrino  oscillations in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The everyday understanding of resonance is associated with the  possible dramatic consequences of vibrations (for example, with the destruction of a bridge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Since ”resonance” is associated with the quantity sin2 2θm and with the amplitude of oscil-  lations as functions of the density of matter, then its ”width” is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' These functions  have values greater than half of their maximum values, equal to 1, in the following interval of  matter density  δρext ∈ [max(ρext(1 − tan θ), 0) ρext(1 + tan θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (27)  The layer of solar matter, the density in which belongs to the interval (27), is called the  resonance layer by Mikheev and Smirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Let’s estimate its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At a solar neutrino energy  of 6 MeV, the extreme density of matter ρext is, according to the relation (24), 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4 g·cm−3  and is at a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='254R0 from the center of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The density in the resonant layer  (27) ranges from 0 to 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='3 g·cm−3 and is at a distance from R0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='102R0 from the center  Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Hundreds or thousands of solar neutrino oscillations can occur in this resonant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In  addition to the term ”resonance layer”, the term ”final object” with a matter density from 0  to ρmax (without specification) or simply ”object” is also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The selection of separate layers  of the Sun does not have any logical consequences, but rather looks like some kind of ritual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  An essential element of [6] is the notion of an adiabatic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The notion of ”adiabaticity”  occurs in various branches of physics, with its own nuances in each of them and with the effective  consequences of its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In [6], the adiabatic regime is characterized by a slow change in the  density of matter in the ”object” (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (26) of [26], which differs from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (16) of [6]):  �dρ  dr  �−1  ρ >  4πE  ∆m2 tan2 2θ,  (28)  where r is the distance to the center of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Using the dependence of the matter density  in the Sun on r, available in [23], and the values of the oscillation parameters (19) given by the  Super-Kamiokande collaboration, it is easy to see that the inequality (28) holds for all values  of r in the Sun and for all observed values of neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Note that the relation (28) does  not follow from any analytical calculations and is not explicitly used anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Mikheev and Smirnov approve that the adiabatic regime makes it possible to trace the  evolution of the state of the solar neutrino as it moves through the ”object under consideration”  using solutions (21), (22) for constant density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' This is an incorrect statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The evolution  of a state means nothing more than its change in time, which is completely absent in [6] in  an explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' During the passage of the neutrino ”object” / ”resonant layer” the density  of matter undergoes, as demonstrated above, significant changes comparable to the density at  the initial moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The time evolution of the neutrino state both inside the Sun and after it  enters the vacuum is described by the set of solutions of the Wolfenstein equation (17) for one  oscillation (20) obtained in the adiabatic approximation described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The end of  one oscillation, when the state of the solar neutrino becomes again purely electronic, coincides  with the beginning of a new oscillation with a very small change in the density of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The  transformation of the states of the solar neutrino along the entire trajectory from the place  of its production to the Earth consists in a gigantic number of oscillations following one after  another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The relations (21), (22) are only fragments of the solution (20), which is not related  to the notion of ”adiabatic regime” by Mikheev-Smirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Before analyzing the assertion about the transformation of an electron neutrino into a muon  neutrino when passing through a medium, let us fix an unambiguous interpretation of such a  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Since the probability of detecting a neutrino being in the νe state at a certain  moment of time can be either greater or less than the probability of detecting a neutrino being  10  in the νµ state at the same time moment, the change in the ratio of these probabilities does  not can be a criterion for transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' There are two stable indications that the oscillating  state of the neutrino must be considered either electronic or muon: one is the superiority of the  average probability of detecting one type of neutrino over the average probability of detecting  another type over the time of one oscillation, and the other is the superiority of the amplitude  of the probability oscillations for one type over the amplitude for another type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' During the  entire time allotted to the Wolfenstein equation for solar neutrinos, the above relations remain  unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' transformation of one type of neutrino into another is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Let us now pay attention to the character of the changes in the value of sin 2θm, given by  the relations (21) and (22), and to two possible variants of changes in the mixing angle θm  for neutrinos with a certain energy along one or another trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At neutrino energies from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='233 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='827 MeV, when there is no extremal point on the trajectory, the value of sin θm  increases with increasing matter density, starting from sin 2θ and not reaching 1, and the angle  2θm grows starting from 2θ and not reaching π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At energies above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='827 MeV, the value  of sin 2θm first increases from sin 2θ to 1, and then decreases to a certain value c0, which is  the smaller, the higher the matter density at the production place of the neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The first  variant of changing the angle 2θm is following: as the matter density increases, it grows from  2θ to π/2, and then decreases to the value arcsin c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The second variant is next: the angle 2θm  with increasing matter density invariably increases from 2θ to π − arcsin c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In [6], a variant of changing the angle θm from θ to π/2 − (arcsin c0)/2 is chosen without  mentioning the existence of the first variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In the case of small values of θ and c0, the chosen  variant means a change in the mixing angle along the neutrino trajectory by approximately π/2,  while in the first variant such a change is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Now comes the climax of the metamorphose,  described fragmentarily and muzzily both by Mikheev and Smirnov in [6] and by Smirnov in a  recent report [26], which I present in an orderly and complete form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' For a small value of c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' the  first of the relations (23) at the moment of production of the solar neutrino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' when it is purely  electronic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' is written as  νe = cos[π/2 − (arcsin c0)/2]ν1m(ρmax) + sin[π/2 − (arcsin c0)/2]ν2m(ρmax)  ≈ [c0/2]ν1m(ρmax) + ν2m(ρmax)  (29)  The transformation of the state of a neutrino during its motion in the Sun is completely  determined by the changes in the mixing angle θm and the states of massive neutrinos in  matter on the right side of the first of the relations (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  At the moment when the solar  neutrino is near the exit from the Sun, its state νsol is given by the formula  νsol = cos θν1m(ρ = 0) + sin θν2m(rho = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (30)  Since the expansion (29) is completely dominated by the neutrino ν2, and the expansion (30) is  dominated by the neutrino ν1, Mikheev and Smirnov conclude that the final neutrino is muon,  and the probability amplitude to find νe in the final state is equal to sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The significant difference in the values of the coefficients related to the same massive neu-  trinos in the decompositions (29) and (30), enhanced in [6] by the smallness of the angle  theta accepted there, is entirely due to the meaningful by choosing the second variant of the  dependence of the mixing angle θm on the density of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In the first variant of the dependence of the mixing angle θm on the density of the medium,  instead of the relation (29), we have the following equality  νe = cos[(arcsin c0)/2]ν1m(ρmax) + sin[(arcsin c0)/2]ν2m(ρmax) ≈ ν1m(ρmax) + [c0/2]ν2m(ρmax)  (31)  11  Now, in both expansions (31) and (30), the same neutrino ν1 dominates and, following  Mikheev and Smirnov, one should say that both the initial and final neutrinos are electronic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The conclusion that the final neutrino is a muon neutrino vanishes like a dream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The assertion of Mikheev and Smirnov about the transformation of an electron neutrino  into a muon neutrino when passing through a medium turns out to be primitive false, moreover,  the notions of ”resonance”, ”resonance layer”, ”adiabatic regime” turned out to be unnecessary  for its imaginary substantiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  For many years I was surprised that the paper by Mikheev and Smirnov [6] discussed here  was not rejected in the journal Yadernaya Fizika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' I unexpectedly received an answer from  Smirnov, who outlined some points from the conceiving of the Mikheev-Smirnov-Wolfenstein  mechanism in a report at a conference in Paris in 2018 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  It turns out that paper [6]  was initially rejected in the journal Yadernaya Fizika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Smirnov writes that ”it was a general  skepticism: ’something should be wrong, somebody will eventually find this’”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Pontecorvo also  had a skeptical opinion (it is ”rubbish”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' However, thanks to the influence of Academician G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Zatsepin, the article was accepted by Yadernaya Fizika and also published in Nuovo Cimento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Later, Pontecorvo together with Wolfenstein ”concluded that, it seems, there is no practical  outcome of oscillations in matter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  So, the key thesis of the Mikheev-Smirnov-Wolfenstein mechanism about the transforma-  tion of an electron neutrino into a muon neutrino when passing through the solar medium  is erroneous, which makes this mechanism generally unsuitable for solving the solar neutrino  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' New fundamental interaction  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Solving the solar neutrino problem  The search by many diverse specialists for the reasons for the discrepancy between the  result of the Davis experiment [1] and Bahcall’s prediction [2], reported by Reines [3], was  not successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It, of course, was conducted within the framework of the standard model of  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' This alone serves as a sufficient reason to go beyond the standard model and,  adhering to classical principles in physics, postulate the existence of a new interaction that is  not a consequence of known interactions, and therefore is fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The essence of the new  interaction will not suffer, if we do not consider it fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In [8], we put forward and substantiated the hypothesis that the emergence of the solar  neutrino problem is due to the existence of a rather hidden interaction, which is carried by a  massless axial boson having Yukawa coupling with an electron neutrino, proton, and neutron  (with u- and d-quarks), reflected by the following relativistically invariant Lagrangian  L = igνeps¯νeγ5νeϕps + igNps¯pγ5pϕps − igNps¯nγ5nϕps,  (32)  and not coupled with the electron at the tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' We do not rule out that the coupling of  the axial boson with different types of neutrinos may be different, if it exists at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The decrease in the experimental rate of observed processes with solar neutrinos compared  to the theoretical one is caused by two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The first reason is the change in its handedness  from left to right and vice versa at each collision of a neutrino with a nucleon, which is due to  the Lorentzian structure of the axial current:  ¯ψ(p2)γ5ψ(p1) = ¯ψR(p2)γ5ψL(p1) − ¯ψL(p2)γ5  Rψ(p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (33)  The second reason is the decrease in the neutrino energy ω at collision with a resting nucleon  of mass M by the value ∆ω uniformly distributed in the following interval  ∆ω ∈ [0, 2ω2  M ·  1  1 + 2ω/M ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (34)  12  The flux of solar neutrinos near the surface of the Earth does not differ from the theoretical  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' But due to the first reason, the flux of left-handed neutrinos decreases by about a factor  of two compared to the theoretical one: its ratio to the flux of right-handed neutrinos is, as it  shown in [10], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='516:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='484 (after correction here, it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='485).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  As is well known, the cross sections for the interaction of a neutrino with electrons, nucleons,  and nuclei increase significantly with increasing energy up to its value for gauge bosons, but  increase differently near the thresholds of various processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Therefore, a decrease in the energy  of each solar neutrino entails a decrease in the rate of observed processes, which is different for  different processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The solution of the task about the consequences of the short-term Brownian motion of  neutrinos in the Sun was facilitated by the fact that the cross section of elastic scattering on a  nucleon at rest, caused by the interaction (32),  σ = (gνepsgNps)2  16πM2  1  1 + 2ω/M ]  (35)  is practically independent of the energy of the solar neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Therefore, with sufficient accu-  racy, we can assume that the distributions of the number of collisions of neutrinos with solar  nucleons are the same for all solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In [8] and [9], the role of a consequence of the unknown distribution is played by the effective  number n0 of collisions of neutrinos with nucleons of the Sun, which determines the energy  spectrum of neutrinos near the Earth’s surface, regardless of their handedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Focusing on  numerical methods dealing with a discrete set of numbers, we consider that the energy of the  scattered neutrino takes with equal probability the two extreme values of its allowable interval,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' that the initial energy level of a neutrino after its collision with a nucleon splits into two  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' After n0 collisions of neutrinos with nucleons, its initial energy level ω splits into n0 + 1  binomially distributed levels:  ω1 = ω,  ω2 =  ω1  1 + 2ω1/M ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' ωn0+1 =  ωn0  1 + 2ωn0/M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (36)  In [9], simple algorithms for calculating the numerical values of the rates of each of the five  observed processes with solar neutrinos are given, as well as references to the works of Bahcall  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=', which contain tabulated values of solar neutrino fluxes from their various sources and  cross sections for a number of processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' This makes it easy for any physicist or mathematician  to check our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Now I recommend replacing the multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5Φ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515Φ in the formulas  of [9] for the rates of processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Good agreement between the theoretical results and the results  of all five experiments with solar neutrinos is ensured at n0 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The small number of collisions of neutrinos with nucleons of the Sun allows us to consider it  plausible that one neutrino collision occurs when a neutrino passes by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9 fractions  of the maximum possible number of nucleons on its way in the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Taking into account the  distribution of matter in the Sun given in [23] and the relation (35), we have obtained the  following estimate  gνepsgNps  4π  = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2) · 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (37)  This estimate is used in calculating the deuteron disintegration rate by neutral currents caused  by both left- and right-handed solar neutrinos, νe + D → νe + n + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Later, in [10], the Brownian motion of neutrinos in the Sun was successfully described by  the geometric distribution of the number of their collisions with nucleons  P(n) = p(1 − p)n,  n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (38)  From a comparison of the theoretical results based on this distribution with the experimental  results in [10], it was accepted that the parameter p is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Subsequently, I again  13  made such a comparison and now I cosider that p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The theoretical rates of all five  observed processes given below were calculated exactly at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  In accordance with the fact that the four observed processes, νe +37 Cl → e− +37 Ar,  νe +71 Ga → e− +71 Ge, νe + e− → nue + e− and νe + D → e− + p + p, and subprocess  νe + D → νe + n + p with Z-boson exchange, are caused only by left-handed neutrinos, then  when calculating the theoretical rates of these processes, the contributions of only those solar  neutrinos that have undergone an even number of collisions with solar nucleons are summed  up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The subprocess νe + D → νe + n + p with the exchange of pseudoscalar boson ϕps, is  contributed by both left- and right-handed neutrinos that have experienced any number of  collisions with nucleons of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Tables 1-5 below show the results of calculating the velocities of all observed processes with  solar neutrinos based on the method of the effective number of neutrino-nucleon collisions n0  for two values of the probability Wleft that at the Earth’s surface the solar neutrino is left-  handed, and based on geometric distribution (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Tables 3-4 show the dependence of the rates  of the processes on the lower value Ec of the reconstructed energy of the final electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The rate of transitions 37Cl → 37Ar in SNU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  8B  7Be  15O  pep  13N  hep  Total  Experiment [27]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='16  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5, n0 = 11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='72  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515, n0 = 12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='72  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (38), p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='060  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='73  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The rate of transitions 71Ga → 71Ge in SNU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  p-p  7Be  8B  15O  13N  pep  hep  Total  Experiment [28]  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9  Experiment [29]  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='1  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='6  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5, n0 = 11  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='6  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='02  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='6  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515, n0 = 12  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='02  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (38), p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='060  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='02  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='8  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Effective fluxes of neutrinos Φνe  eff(8B) found from the process  νee− → νee− (Ec is given in MeV, and the fluxes are in units of 106 cm−2s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  References  Ec  Experimental  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5,  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (38),  results  n0 = 11  n0 = 12  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='060  SK IV [22]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='39  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='42  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='37  SNO I [30]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='24  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='23  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='12  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='15  SNO II [31]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='07  SNO III [32]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='24  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='21  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='98  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Effective fluxes of neutrinos Φcc  eff(8B) found from the process  νeD → e−pp (Ec is given in MeV,and the fluxes are in units of 106 cm−2s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  References  Ec  Experimental  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5,  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (38),  results  n0 = 11  n0 = 12  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='062  SNO I [30]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='06  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='05  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='09  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='86  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='85  SNO II [31]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='06  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='06  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='77  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='77  SNO III [32]  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='05  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='04  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='07  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='66  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='62  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='69  14  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Effective fluxes of neutrinos Φnc  eff(8B) found from the process  νeD → νenp (the fluxes are in units of 106 cm−2s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Exchange  Exchange  Exchange  Exchange  Sum  by Z  by Z  by Z  by ϕ  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5  Wleft = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='515  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (38)  all neutrinos  SNO I [30]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='44  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='43  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='46  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='43  SNO II [31]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='21  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='21  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='38  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='34  SNO III [32]  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='54+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='33  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='31  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='36  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='34  n0 = 11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='87  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='98  n0 = 12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='85  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='96  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' (38)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='79  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='89  Comparing the corresponding numbers in each of the tables, we come to the following  conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The results of collisions of neutrinos with nucleons in the Sun caused by the  interaction (32), which were obtained on the basis of the method of the effective number of  collisions with a slight superiority of the flux of left-handed neutrinos, and which were obtained  based on the geometric distribution of collisions, are close to each other and agree well with  the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  A good agreement of theoretical results, obtained on the basis of the hypothesis of the exis-  tence of a new fundamental interaction, with the results of five absolutely different experiments  with solar neutrinos can only be a game of chance with negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Borexino, KamLAND  There are two types of Borexino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' First, the elastic scattering of solar neutri-  nos on electrons at the scattered electron energy greater than 3 MeV is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The observed  effective flux of solar neutrinos from 8B [33] is consistent with the experimental results of Super-  Kamiokande and SNO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' As this experiment was carried out after the corresponding experiments  of Super-Kamiokande and SNO, and as the accuracy of its result is much less than in Super-  Kamiokande and SNO, then it adds nothing new to the solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Second, the solar neu-  trino flux from 7Be is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' However, as it is stated in the work [34], ”the contribution  of the pp, pep, CNO, and 8B solar neutrino were fixed to the SSM-predicted rates assum-  ing MSM neutrino oscillations.” Thus, Borexino does not present a purely experimental re-  sult for the neutrino flux from 7Be, but only an MSM interpretation of it, and so, we have  nowhere to apply our theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The preparation and implementation of the KamLAND experiment was initially focused  (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  [35]) on confirming the existence of electron (anti-)neutrino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  When  presenting the first results, it was said [36]: ”The long baseline, typically 180 km, enables  KamLAND to address the oscillation solution of the ‘solar neutrino problem’ using reactor  anti-neutrinos under laboratory conditions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The existence of (anti-)neutrino oscillations with  parameters given by the KamLAND collaboration and close to ones of the SNO and Super-  Kamiokande collaborations would lead to the survival probability of the solar neutrino electron  component near the Earth’s surface, given by the formula (26), which, as stated in section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4, contradicts at least three out of five experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' At the same time, the interaction (32),  which provided a solution to the solar neutrino problem, has nothing to do with the announced  result of the KamLAND collaboration, since the reactor antineutrino cannot experience a single  collision on its 180 km path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  After obtaining a solution to the solar neutrino problem, we had no doubt that the reason  for the loss of observed events in the KamLAND experiment is serious omissions in its setting  up and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' I have repeatedly returned to the issue of such omissions, having studied,  where possible, the features of the appearance of fluorescent light, the characteristics of its  15  spectrum, the dependence on the wavelength of the fluorescent signal of the attenuation of its  intensity during propagation in liquid KamLAND scintillator, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Based on the comparison  of suitable numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' I finally came to the following conclusion [37]:  In the conditions of unprecedentedly large sizes of KamLAND detector and of the refusal  of using the gadolinium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' the absence of (1) proper study of the real spectrum of the PPO  fluorescent light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' of (2) a detailed study of the fluorescent light attenuation in a liquid scintillator  depending on its wavelengths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' of (3) the influence on the attenuation of light of its scattering and  re-emission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' and of (4) the theoretical calculation of the observability of events ¯νe +p → e+ +n  opens the door to uncontrolled loss of such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Therefore, we do not consider the announced  results of the KamLAND experiment as reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Reactor antineutrino anomaly, gallium anomaly  Acquisition of knowledge about the spectrum of reactor antineutrinos is realized by two  different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' One of them, called experimental, uses the spectrum of observed positrons  produced in an experimental setup under the action of reactor antineutrinos, and the well-  known dependence of the inverse beta decay cross section on antineutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Another  method, called theoretical, was initiated in 1981 by Schrekenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' It was relied on  experimental measurements of electron spectra from 235U decays and their subsequent conver-  sion into electron antineutrino spectra based on the kinematics of beta decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' In 2011, new  theoretical calculations of the spectra of reactor antineutrinos from the decays of 235U, 239Pu,  241Pu and 238U in thousands of possible channels appeared [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Comparing the results of these  calculations with the results of the entire set of experiments with distances less than 100 meters  from the reactors, the authors of [39] conclude that the ratio of the rate of observed events to  the rate of predicted events is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='943 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The distinction of this ratio from unity  is called by them the reactor antineutrino anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The authors of [39] are inclined to believe  that the reactor anomaly may indicate the existence of a fourth nonstandard neutrino, which  determines electron neutrino oscillations at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  We affirm that the difference between the theoretical and experimental spectra of antineu-  trinos produced in beta decays of any nuclei is a manifestation of a new fundamental interaction  (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Once again, we note that the experimental spectrum includes those and only those antineu-  trinos that manifest themselves in the production of positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The theoretical spectrum, relied  on the spectrum of electrons, includes both those antineutrinos that are capable of producing a  positron, and those that do not have such an ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The interaction (32) allows the existence  along with the dominant mode of beta decay of the X nucleus into the Y nucleus  X → Y + e− + ¯νeR,  (39)  also of the decay mode with the emission of a massless axial boson  X → Y + e− + ¯νeL + ϕps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (40)  The right-handed antineutrino, born in the beta decay of the X nucleus, can emit a massless  axial boson in a virtual state, changing its handedness to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The left-handed antineutrino  within the framework of the standard interaction cannot manifest itself, causing the production  of a positron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Thus, the experimental spectrum includes only antineutrinos of the dominant  nuclear beta-decay mode (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The theoretical spectrum includes both antineutrinos from the  dominant mode (39) and effective ”antineutrino” that accompanies the electron and is the sum  of the unobservable left-handed antineutrino and the unobservable massless axial boson from  the mode (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  16  The emission of a massless axial boson in the beta decay of nuclei is to some extent analogous  to the emission of a gamma quantum in such decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' For the decay  n → p + e− + ¯νe + γ,  (41)  there are both theoretical calculations [40] and experimental measurements [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The theo-  retical differential width of the process (41) contains the infrared divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The measured  fraction of the radiative decay mode of a neutron with a gamma quantum energy from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='4 to  782 keV is (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='64) · 10−3, which is close to the constant α = e2/4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Solving the solar neutrino problem, we obtained an estimate of the product of the Yukawa  coupling constants of the massless axial boson ϕps with electron neutrinos and with nucleons  (37) and had no way to find these constants separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Now, using some similarity between  the decay modes (40) and (41) and the fact that the difference between the rates of observed  and expected events with reactor antineutrinos is due to the beta-decay mode of nuclei (41),  we find it possible to take, in the zeroth approximation, the following estimate  (gνeps)2  4π  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='057 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (42)  It would be possible to obtain a realistic value of the coupling constant of an axial boson  with an electron neutrino if it were possible to calculate the electron spectrum for the decay  mode with an axial boson (40) of the 235U isotope in the most significant channels and to solve  herewith the problem massless divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  Knowing the realistic value of the constant (gνeps)2/4π would help to establish the level of  difference, called the gallium anomaly, between the numbers of observed and expected events  of the 71Ga transition to 71Ge under the influence of electron neutrinos from the decay of 31Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  At present, there is a scatter in the ratio of these numbers obtained in six variants of completed  experiments, in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='95 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='77 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' We have no doubt that the gallium anomaly, as  well as the reactor antineutrino anomaly, is a manifestation of a fundamental interaction (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The theoretical number of decays per unit time of the 31Cr isotope, caused by the capture of an  electron, is entirely determined by its quantity obtained from the 30Cr isotope by irradiation  in a certain nuclear reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' The decay of the 31Cr isotope occurs both in the dominant mode  31Cr + e− →31 V + νeL  (43)  and in the mode with the emission of a massless axial boson  31Cr + e− →31 V + νeR + ϕps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  (44)  The decay mode of the isotope 31Cr (43) subsequently leads to transitions of gallium into  germanium, and the mode (44) is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  So, the reactor antineutrino and gallium anomalies are a consequence of the existence of a  fundamental interaction (32), which is responsible for the undetectable decay mode of one or  another isotope, characterized by the emission of a massless axial boson and a change in the  handedness of the original (anti)neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Conclusion  Ignoring the basic principles of classical logic and, in particular, the basic principles of  mathematical analysis, researchers lose the ability to distinguish between perfect reasoning  and charlatanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' For example, according to Smirnov’s recent testimony [26] (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='6), the  reviewers of [6], who initially rejected the possibility of its publication in the journal Yadernaya  Fizika, understood its possible fallacy and hoped that ”somebody will eventually find  17  this.” This ”somebody” only now, 37 years later, showed up and proved the fallacy of [6]  above in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Surprisingly, the correctness of the work [6], about which the reviewers  expressed doubts, was not publicly disputed by any of the hundreds of adherents, including  very respected physicists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Such is the reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  This work is designed to rid physics of spirits imposed on it in the form of additional neu-  trinos and recipes that violate the principles of classical logic or basic physical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='  The solution to the problem of solar neutrinos, and then clarifying the nature of reactor an-  tineutrino and helium anomalies, is given on the basis of classical field theory methods with  their inherent simplicity and elegance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content=' Time will tell if the scientific community accepts the  revival of at least one of Newton’s rules outlined at the end of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfsgVT/content/2301.03366v1.pdf'}
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diff --git a/KtE3T4oBgHgl3EQfYAoc/content/tmp_files/2301.04483v1.pdf.txt b/KtE3T4oBgHgl3EQfYAoc/content/tmp_files/2301.04483v1.pdf.txt
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@@ -0,0 +1,1411 @@
+arXiv:2301.04483v1  [cs.LO]  11 Jan 2023
+Compositional Net Semantics up to Step Net Bisimilarity
+Roberto Gorrieri
+roberto.gorrieri@unibo.it
+Dipartimento di Informatica — Scienza e Ingegneria
+Universit`a di Bologna,
+Mura A. Zamboni, 7, 40127 Bologna, Italy
+Abstract. Step net bisimilarity [10] is a truly concurrent behavioral equivalence
+for finite Petri nets, which is defined as a smooth generalization of standard step
+bisimilarity [13] on Petri nets, but with the property of relating markings (of the
+same size only) generating the same partial orders of events. The process algebra
+FNM [8] truly represents all (and only) the finite Petri nets, up to isomorphism.
+We prove that step net bisimilarity is a congruence w.r.t. the FMN operators, In
+this way, we have defined a compositional semantics, fully respecting causality
+and the branching structure of systems, for the class of all the finite Petri nets.
+1
+Introduction
+A step net bisimulation R over the markings of a finite Place/Transition Petri net [16,4]
+is a behavioral relation stating that if two markings m1 and m2 are related by R, then (i)
+they must have the same size, i.e., |m1| = |m2|; moreover, (ii) for each step (i.e., multiset
+of concurrently enabled transitions) G1 executable from m1 reaching m′
+1, there exists a
+step G2 of the same size executable from m2 reaching m′
+2 such that, not only the labels
+of G1 and G2 are the same and that (m′
+1,m′
+2) ∈ R – as for ordinary step bisimulation
+[13] – but also that the pre-sets of G1 and G2 are related by R (i.e., (•G1,•G2) ∈ R);
+finally, (iii), the symmetric condition if m2 moves first.
+One may wonder if this simplistic definition induces some actually sensible be-
+havioral equivalence. We proved in [10] that, surprisingly enough, step net bisimilarity
+∼s coincides with a well-known, truly-concurrent behavioral relation, namely causal-
+net bisimilarity, originally proposed in [6] (and further studied in [9], who coined its
+name). Causal-net bisimilarity ∼cn is essentially a bisimulation-like relation over the
+causal nets (sometimes called nonsequential processes) [5,1,14] originating from the
+two initial markings of interest. Hence, its definition is very complex and cumbersome.
+However, as step net bisimilarity coincides with causal-net bisimilarity, we get that our
+extremely simple definition of step net bisimilarity is slightly finer than fully-concurrent
+bisimilarity [2] in that it relates markings that not only generate the same the partial or-
+ders of events, but that also have the same size.
+Finite P/T Petri nets can be represented, up to isomorphism, by means of the pro-
+cess algebra FNM [8], which is a CCS-like sub-calculus extended with an additional
+operator, called strong prefixing. This operator allows for atomic execution of action
+sequences, so that a multi-party interaction can be modeled as an atomic execution of
+a sequence of binary interactions. The main aim of this paper is to show that step net
+
+2
+bisimilarity is a congruence for the operators of FNM, thus yielding a compositional
+semantics, up to step net bisimilarity, for the class of all the finite Petri nets.
+The paper is organized as follows. Section 2 introduces the basic definitions about
+Petri nets, some novel definitions (notably, the distinction between dynamically vs stati-
+cally reachable subnet) and the definition of step net bisimilarity. Section 3 recalls from
+[8] the process algebra FNM, its (operational) semantics in terms of finite P/T Petri
+nets, and also the representability theorem stating that each finite (statically reachable)
+marked Petri net can be represented by a suitable FNM term, up to isomorphism. Sec-
+tion 4 shows that step net bisimilarity is a congruence for the FNM operators. Finally,
+Section 5 is devoted to some conclusions.
+2
+Basic Definitions
+Definition 1. (Multiset) Let N be the set of natural numbers. Given a countable set S,
+a finite multiset over S is a function m : S → N such that the support set dom(m) =
+{s ∈ S
+�� m(s) ̸= 0} is finite. The set of all finite multisets over S, denoted by M fin(S),
+is ranged over by m, possibly indexed. (The set of all finite subsets of S is denoted by
+Pfin(S).) We write s ∈ m if m(s) > 0. The multiplicity of s in m is given by the number
+m(s). The size of m, denoted by |m|, is the number ∑s∈S m(s), i.e., the total number of
+its elements. A multiset m such that dom(m) = /0 is called empty and is denoted by θ.
+We write m ⊆ m′ if m(s) ≤ m′(s) for all s ∈ S.
+Multiset union ⊕
+is defined as: (m ⊕ m′)(s) = m(s) + m′(s). Multiset difference
+⊖
+is defined as: (m1 ⊖ m2)(s) = max{m1(s) − m2(s),0}. The scalar product of a
+natural j with m is the multiset j ·m defined as (j ·m)(s) = j ·(m(s)). By si we denote
+the multiset with si as its only element. Hence, a multiset m over S = {s1,...,sn} can be
+represented as k1 ·s1 ⊕ k2 ·s2 ⊕ ...⊕ kn ·sn, where kj = m(sj) ≥ 0 for j = 1,...,n.
+✷
+Definition 2. (Place/Transition Petri net) A labeled Place/Transition Petri net (P/T
+net, or Petri net, for short) is a tuple N = (S,A,T), where
+• S is the countable set of places, ranged over by s (possibly indexed),
+• A ⊆ Lab is the finite set of labels, ranged over by ℓ (possibly indexed), and
+• T ⊆ (M fin(S)\{θ})×A×M fin(S) is the countable set of transitions, ranged over
+by t (possibly indexed).
+A Petri net N = (S,A,T) is finite when S and T are finite sets. Given a transition t =
+(m,ℓ,m′), we use the notation •t to denote its pre-set m (which cannot be an empty
+multiset) of tokens to be consumed; l(t) for its label ℓ, and t• to denote its post-set m′ of
+tokens to be produced. Hence, transition t can be also represented as •t
+l(t)
+−→t•.
+✷
+In the graphical description of P/T nets, places (represented as circles) and transi-
+tions (represented as boxes) are connected by directed arcs. The arcs may be labeled
+with the number representing how many tokens are to be removed from (or produced
+into) that place; no label on the arc is interpreted as the number one, i.e., one token
+flowing on the arc. This numerical label of the arc is called its weight.
+
+3
+Definition 3. (Marking, P/T net system) Given a P/T net N = (S,A,T), a finite multi-
+set over S is called a marking. Given a marking m and a place s, we say that the place s
+contains m(s) tokens, graphically represented by m(s) bullets inside place s. A P/T net
+system N(m0) is a tuple (S,A,T,m0), where (S,A,T) is a P/T net and m0 is a marking
+over S, called the initial marking. We also say that N(m0) is a marked net.
+✷
+Definition 4. (Enabling, firing sequence, reachable marking) Given a P/T net N =
+(S,A,T), a transition t is enabled at m, denoted by m[t⟩, if •t ⊆ m. The execution (or
+firing) of t enabled at m produces the marking m′ = (m⊖ •t)⊕t•. This is written m[t⟩m′.
+A firing sequence starting at m is defined inductively as follows:
+• m[ε⟩m is a firing sequence (where ε denotes an empty sequence of transitions) and
+• if m[σ⟩m′ is a firing sequence and m′[t⟩m′′, then m[σt⟩m′′ is a firing sequence.
+The set of reachable markings from m, denoted by [m⟩, is [m⟩ = {m′ �� ∃σ.m[σ⟩m′}. ✷
+Given a P/T net N = (S,A,T) and a marking m, we say that two transitions t1,t2 ∈
+T are concurrently enabled at m if •t1 ⊕ •t2 ⊆ m. The concurrent firing of these two
+transitions produces the marking m′ = (m⊖(•t1 ⊕ •t2))⊕(t•
+1 ⊕t•
+2). We denote this fact
+by m[{t1,t2}⟩m′. It is also possible that the same transition is self-concurrent at some
+marking m, meaning that two or more occurrences of it are concurrently enabled at m.
+We can generalize the definition of concurrently enabled transitions to a finite,
+nonempty multiset G over the set T, called a step. A step G : T → N is enabled at
+marking m if •G ⊆ m, where •G = �
+t∈T G(t) · •t and G(t) denotes the number of oc-
+currences of transition t in the step G. The execution of a step G enabled at m produces
+the marking m′ = (m ⊖ •G) ⊕ G•, where G• = �
+t∈T G(t) ·t•. This is written m[G⟩m′.
+We sometimes refer to this as the concurrent token game, in opposition to the sequen-
+tial token game of Definition 4. The label l(G) of a step G is the multiset l(G) : A → N
+defined as follows: l(G)(a) = ∑ti∈dom(G).l(ti)=a G(ti).
+2.1
+Dynamically Reachable and Statically Reachable Subnets
+Now we introduce two notions of reachable subnet, following [8].
+• dynamically reachable subnet, which refers to the set of places and transitions dy-
+namically reachable from the initial marking m0 by the token game, and
+• statically reachable subnet, which refers to the places and transitions which are
+reachable by a weaker token game, stating that a transition t is statically enabled at
+S′ (i.e., the current set of the statically reached places (initiallly dom(m0))), when
+dom(t) is a subset of S′.
+Based on the latter, we introduce the notion of statically reduced net, i.e., a net
+where all of its places and transitions are statically reachable from dom(m0). This notion
+is important in Section 3, because we will show that the net semantics Net(p) of an
+FNM term p is a statically reduced net.
+In fact, the semantics of a term p is not simply the net that can be dynamically
+reached from the initial marking dec(p) corresponding to p, rather it is the net describ-
+ing all the potential behaviors of p, as if the number of tokens in dec(p) can be increased
+
+4
+a)
+a.0+a.0
+a
+a
+b)
+a.0+a.0
+a
+a
+τ
+2
+Fig. 1. The dynamically reachable subnet (a) and the statically reachable subnet (b) for
+a.0+ a.0
+s1
+s2
+s3
+a
+b
+c
+d
+s4
+s5
+a)
+2
+s1
+s2
+a
+b
+s4
+b)
+Fig. 2. A net system in (a) and its dynamically reachable subnet in (b)
+at will. For instance, consider the FNM sequential process p = a.0 + a.0; its dynami-
+cally reachable subnet, outlined in Figure 1(a), shows the expected behavior that p can
+perform either a or its complementary action a. On the contrary, its statically reachable
+subnet in (b) describes additionally the potential self-synchronization of p with another
+copy of itself; in this way, the net semantics for p and for the parallel term p| p differ
+only for the form of the initial marking (one token for p and two tokens for p| p), but the
+underlying net is the same. The statically reachable subnet of a finite P/T net is always
+algorithmically computable, also for Nonpermissive nets [8], even though they are a
+Turing-complete model of computation, and so for them, as the reachability problem is
+undecidable, it is not always possible to compute the dynamically reachable subnet.
+Definition 5. (Dynamically reachable subnet, dynamically reduced net) Given a
+P/T Petri net system N(m0) = (S,A,T,m0), the dynamically reachable subnet Netd(N(m0))
+is (S′,A′,T ′,m0), where
+S′ = {s ∈ S
+�� ∃m ∈ [m0⟩ such that m(s) ≥ 1},
+T ′ = {t ∈ T
+�� ∃m ∈ [m0⟩ such that m[t⟩},
+A′ = {ℓ
+�� ∃t ∈ T ′ such that l(t) = ℓ}.
+A P/T net system N(m0) = (S,A,T,m0) is dynamically reduced if N(m0) = Netd(N(m0)),
+i.e., the net system is equal to its dynamically reachable subnet.
+✷
+The dynamically reachable subnet of the net in Figure 2(a) is outlined in Figure 2(b).
+Given a finite net N(m0), it is algorithmically derivable Netd(N(m0)) by means of its
+coverability tree [11], but its complexity is exponential in the size of the net.
+Definition 6. (Statically reachable subnet and statically reduced net) Given a P/T
+net N = (S,A,T), we say that a transition t is statically enabled by a set of places S′ ⊆ S,
+denoted by S′�t⟩, if dom(•t) ⊆ S′.
+
+5
+s1
+s2
+a
+b
+c
+s4
+s5
+2
+Fig. 3. The statically reachable subnet of Figure 2(a)
+Given two sets of places S1,S2 ⊆ S, we say that S2 is statically reachable in one step
+from S1 if there exists a transition t ∈ T, such that S1�t⟩, dom(t•) ̸⊆ S1 and S2 = S1 ∪
+dom(t•); this is denoted by S1
+t
+=⇒S2. The static reachability relation =⇒∗ ⊆ Pfin(S)×
+Pfin(S) is the least relation such that
+• S1 =⇒∗ S1 and
+• if S1 =⇒∗ S2 and S2
+t
+=⇒S3, then S1 =⇒∗ S3.
+A set of places Sk ⊆ S is the largest set statically reachable from S1 if S1 =⇒∗ Sk and for
+all t ∈ T such that Sk�t⟩, we have that dom(t•) ⊆ Sk. Given a P/T net system N(m0) =
+(S,A,T,m0), we denote by �dom(m0)⟩ the largest set of places statically reachable from
+dom(m0), i.e., the largest Sk such that dom(m0)=⇒∗ Sk.
+The statically reachable subnet Nets(N(m0)) is the net (S′,A′,T ′,m0), where
+S′ = �dom(m0)⟩,
+T ′ = {t ∈ T
+�� S′�t⟩},
+A′ = {ℓ
+�� ∃t ∈ T ′ such that l(t) = ℓ}.
+A P/T net system N(m0) = (S,A,T,m0) is statically reduced if Nets(N(m0)) = N(m0),
+i.e., the net system is equal to its statically reachable subnet.
+✷
+The statically reachable subnet Nets(N(m0)) of a finite Petri net N(m0) can be com-
+puted with an easy polynomial algorithm [8]. We will show in Section 3 that, given an
+FNM process p, its net semantics is the statically reachable subnet, extracted from the
+infinite net describing the semantics of the whole FNM process algebra.
+Proposition 1. Given a P/T net system N(m0) = (S,A,T,m0), if N(m0) is dynamically
+reduced, then it is also statically reduced.
+✷
+However, the converse implication is not true: there are statically reduced P/T net
+systems that are not dynamically reduced. For instance, the statically reduced P/T net
+system N(s1) = ({s1,s2,s3}, {a,b},{(s1,a,s2),(2 · s1,b,s3)},s1) cannot dynamically
+reach place s3.
+Consider the net system in Figure 2(a); its statically reachable subnet is outlined
+in Figure 3. If we compare it with its dynamically reachable subnet in Figure 2(b), we
+note that the statically reachable subnet contains the dynamically reachable subnet. This
+holds in general.
+
+6
+Proposition 2. Given a P/T system N(m0) = (S,A,T,m0), if its dynamically reachable
+subnet Netd(N(m0)) is (S′,A′,T ′,m0) and its statically reachable subnet Nets(N(m0))
+is (S′′,A′′,T ′′,m0), then S′ ⊆ S′′, T ′ ⊆ T ′′ and A′ ⊆ A′′.
+✷
+For some classes of nets, however, the two notions coincide, in particular for BPP
+nets, i.e., those nets whose transitions have singleton preset.
+2.2
+Step Net Bisimilarity
+Step net bisimulation is a smooth generalization of ordinary step bisimulation [13] on
+Petri nets, requiring two additional conditions: the related markings must have the same
+size and the pre-sets of the two matching steps must be related as well.
+Definition 7. (Step net bisimulation) Let N = (S,A,T) be a P/T net. A step net bisim-
+ulation is a relation R ⊆ M fin(S)× M fin(S) such that if (m1,m2) ∈ R then
+• |m1| = |m2|,
+• ∀G1 such that m1[G1⟩m′
+1, ∃G2. m2[G2⟩m′
+2, with l(G1) = l(G2), (•G1,•G2) ∈ R and
+(m′
+1,m′
+2) ∈ R,
+• ∀G2 such that m2[G2⟩m′
+2, ∃G1. m1[G1⟩m′
+1, with l(G1) = l(G2), (•G1,•G2) ∈ R and
+(m′
+1,m′
+2) ∈ R.
+Two markings m1 and m2 are step net bisimilar, denoted by m1 ∼s m2, if there exists a
+step net bisimulation R such that (m1,m2) ∈ R.
+✷
+When the steps G1 and G2 are singletons (e.g., {t1} and {t2}, respectively), we get
+that the two matching transitions t1 and t2 in the step net bisimulation game must have
+the same shape because (i) (•t1,•t2) ∈ R implies that |•t1| = |•t2|, then (ii) l(t1) = l(t2)
+and, finally, (iii) from |m1| = |m2|, |m′
+1| = |m′
+2| and |•t1| = |•t2| we derive that |t•
+1| = |t•
+2|,
+because m′
+i = mi ⊖ •ti ⊕t•
+i for i = 1,2. In general, it is possible to prove [10] that if a
+step G1 = {t1
+1,...,t1
+k } is matched by some step G2, then there must also exist a matching
+step G′
+2 = {t2
+1,...,t2
+k }, such that, for i = 1,...,k, t1
+i and t2
+i must have the same shape.
+So, in the following, we usually assume that the matching steps not only have the same
+size, but are composed of transitions that can be related by a shape-preserving bijection.
+Indeed, step net bisimulation is a smooth generalization of ordinary step bisimu-
+lation [13] on Petri nets. In fact, on sequential finite-state machines (which are essen-
+tially labeled transition systems, as each reachable marking is a singleton), its defini-
+tion reduces to standard (interleaving) bisimulation, as the additional side conditions
+|m1| = |m2| and (•t1,•t2) ∈ R are redundant.
+Finally, note also that a step net bisimulation R is such that it relates (dynamically)
+reachable markings only, i.e., it is a relation actually defined over Netd(N(m0)).
+Example 1. Consider the two nets in Figure 4, representing two unbounded producer-
+consumer systems, where prod is the action of producing an item, del of delivering an
+item, cons of consuming an item. It is easy to realize that relation
+R1 = {(P1 ⊕C1 ⊕ k ·D1,P2 ⊕C2 ⊕ k1 ·D′
+2 ⊕ k2 ·D′′
+2)
+�� k,k1,k2 ≥ 0,k = k1 + k2}
+∪{(P1 ⊕C′
+1 ⊕ k ·D1,P2 ⊕C′
+2 ⊕ k1 ·D′
+2 ⊕ k2 ·D′′
+2)
+�� k,k1,k2 ≥ 0,k = k1 + k2}
+
+7
+P1
+C1
+prod
+D1
+del
+C′
+1
+cons
+P2
+C2
+prod
+prod
+D′
+2
+D′′
+2
+del
+del
+C′
+2
+cons
+Fig.4. Two net bisimilar unbounded producer-consumer systems
+is a step bisimulation [13], containing the pair (P1 ⊕ C1,P2 ⊕ C2), where each pair
+(m1,m2) ∈ R1 is such that |m1| = |m2|. Moreover, relation
+R2 = R1 ∪{(P1,P2),(C1 ⊕ D1,C2 ⊕ D′
+2),(C1 ⊕ D1,C2 ⊕ D′′
+2),(C′
+1,C′
+2),(C1,C2)}
+∪{(P1 ⊕ k ·D1,P2 ⊕ k1 ·D′
+2 ⊕ k2 ·D′′
+2)
+�� k,k1,k2 ≥ 0,k = k1 + k2}
+is a step net bisimulation. Note that if P1⊕C1
+prod
+−→P1⊕D1⊕C1, then P2⊕C2 can respond
+with P2 ⊕C2
+prod
+−→P2 ⊕ D′
+2 ⊕C2 and the two required pairs belong to R2, because (P1 ⊕
+D1 ⊕C1,P2⊕D′
+2 ⊕C2) ∈ R1 (and so also in R2) and (P1,P2) ∈ R2; the latter pair requires
+that both (P1 ⊕ D1,P2 ⊕ D′
+2) and (P1 ⊕ D1,P2 ⊕ D′′
+2) belong to R2 and, by iterating the
+firing of the transitions prod, that all the pairs in the last group belong to R2.
+✷
+In order to show that ∼s is an equivalence relation, we now list some useful proper-
+ties of net bisimulation relations, whose proof is standard.
+Proposition 3. For each P/T net N = (S,A,T), the following hold:
+1. The identity relation I = {(m,m)
+�� m ∈ M fin(S)} is a step net bisimulation;
+2. the inverse relation R−1 = {(m′,m)
+�� (m,m′) ∈ R} of a step net bisimulation R is
+a step net bisimulation;
+3. the relational composition R1 ◦R2 = {(m,m′′)
+�� ∃m′.(m,m′) ∈ R1 ∧(m′,m′′) ∈ R2}
+of two step net bisimulations R1 and R2 is a step net bisimulation.
+4. given a family {Ri}i∈I of step net bisimulations, their union ∪i∈IRi is a step net
+bisimulation.
+✷
+Proposition 4. For each P/T net N = (S,A,T), ∼s = �{R ⊆ M fin(S) × M fin(S)
+��
+R is a step net bisimulation} is an equivalence relation and the largest step net bisimu-
+lation.
+✷
+Step net bisimilarity is a truly concurrent behavioral equivalence. Consider the nets
+in Figure 5, which are isomorphic to the net semantics of the FNM terms a.b.0+b.a.0,
+a.0|b.0 and a.C|b.0 (with C .= 0), respectively, according to the semantics in Section
+
+8
+a)
+s1
+a
+b
+s2
+s3
+b
+a
+b)
+s4
+s5
+a
+b
+c)
+s6
+s7
+a
+b
+s8
+Fig. 5. Three non-net equivalent net systems: a.b.0+ b.a.0, a.0|b.0 and a.C|b.0 (with
+C .= 0)
+3. Of course, s1 is not step net bisimilar to s4 ⊕ s5 because these two markings have
+different size; as a matter of fact, these two markings generate different partial orders
+of actions. Moreover, s4 ⊕ s5 is not step net bisimilar to s6 ⊕ s7 because s4 ≁n s6: after
+the execution of a, the reached markings (i.e., θ and s8) are not step net bisimilar. So,
+step net bisimilarity not only observes the partial order of the actions performed, but
+also the size of the distributed state. In fact, in [10] we proved that ∼s coincides with
+causal-net bisimilarity [6,9], in turn equivalent to structure-preserving bisimilarity [6].
+3
+FNM: Syntax and Net Semantics
+In this section we briefly recall some definitions and theorems from [8], where the
+reader can find more details.
+Let L be a finite set of names, ranged over by a,b,c,..., also called the input
+actions. Let L be the set of co-names, ranged over by a,b,c,..., also called the output
+actions. The set L ∪L , ranged over by α,β,..., is the set of observable actions. Let
+Act = L ∪ L ∪ {τ}, such that τ ̸∈ L ∪ L , be the finite set of actions (or labels),
+ranged over by µ. Action τ denotes an invisible, internal activity. Let C be a finite set
+of process constants, disjoint from Act, ranged over by A,B,C,..., possibly indexed.
+Finite-Net Multi-CCS (FNM for short) is the calculus whose terms are generated
+from actions and constants as described by the following abstract syntax:
+s ::= 0 |
+µ.t
+| a.s | s+ s
+guarded terms
+q ::= s |
+C
+sequential terms
+t ::= q |
+t |t
+restriction-free terms
+p ::= t | (νa)p
+general terms
+where we assume that a constant C is defined by a guarded process, i.e., a process in
+syntactic category s (i.e., C .= s). An FNM term p is an FNM process if the set Const(p)
+of constants used by p is finite and each constant in Const(p) is equipped with a defining
+equation. The set of FNM processes is denoted by PFNM, and the set of its sequential
+processes, i.e., of the processes in syntactic category q, by Pseq
+FNM.
+
+9
+Sync(α,α,τ)
+σ ̸= ε
+Sync(aσ,a,σ)
+σ ̸= ε
+Sync(a,aσ,σ)
+Table 1. Synchronization relation Sync
+The FNM operators are those of CCS [12,7], used in constrained manner, with the
+addition of the strong prefixing operator: a.s is a strongly prefixed process, where the
+strong prefix a is the first input action of a transaction that continues with the sequential
+process s (provided that s can complete the transaction). Hence, the strong prefixing
+operator allows for the creation of transitions labeled by an atomic sequence σ, which
+is a sequence of visible actions, composed of inputs only, but possibly ending with an
+output, that are executed atomically; more precisely, σ ranges over the set of labels
+A = {τ} ∪ L ∗ · (L ∪ L ). E.g., a.(b.0 + c.0) can perform two transitions reaching 0:
+one labeled by the atomic sequence ab, the other one by the atomic sequence ac.
+A consequence of the fact that transitions may be labeled by atomic sequences is
+the need for a new, more general, discipline of synchronization, which extends the basic
+CCS rule [12,7] for synchronizing two complementary actions. Two sequences σ1 and
+σ2 can be synchronized, and the result is σ, if relation Sync(σ1,σ2,σ) holds. This
+relation, formally defined in Table 1, holds if at least one of the two sequences is a
+single output action, say σ1 = a, and the other one is either the complementary input
+action a or an atomic sequence starting with a. Hence, it is not possible to synchronize
+two atomic sequences.
+A well-formed FNM process [8] p is an FNM process satisfying a simple syntactic
+condition, denoted by wf(p), ensuring that its executable atomic sequences are com-
+posed of input actions only. This ensures that a multi-party synchronization can take
+place only among one leader, i.e., the process performing the atomic sequence of in-
+puts, and as many other components (the servants), as the length of the atomic sequence,
+where each servant executes one output action. We will show that the Petri net associ-
+ated with a well-formed FNM process p is finite.
+3.1
+Extended Terms
+The FNM processes are built upon the set L ∪L , ranged over by α, of visible actions.
+We assume we also have sets L ′ = {a′ |a ∈ L } and L ′ = {a′| a ∈ L }, where L ′ ∪
+L ′, ranged over by α′, is the set of auxiliary restricted actions, i.e., actions that are only
+allowed to synchronize. By definition, each restricted action α′ corresponds to exactly
+one visible action α. Let G = L ∪L ′, ranged over by γ, be the set of input actions and
+their restricted counterparts. G = L ∪L ′ is the set of output actions and their restricted
+counterparts. The set Actγ = G ∪G ∪{τ}, ranged over by µ (with abuse of notation), is
+used to build the set of extended terms, whose syntax is defined as for FNM, where the
+prefixes are taken from the set Actγ, the strong prefixes from the set G and the bound
+action a is in L . An extended general FNM term p = (νL)t is an extended process if
+Const(p) is finite, each constant in Const(p) is defined and t is admissible, i.e.,
+∀a ∈ L , {a,a′} ̸⊆ fn(t),
+
+10
+dec(0) = θ
+dec(p+q) = {p+q}
+dec(µ.p) = {µ.p}
+dec(γ.p,I) = {γ.p}
+dec(p|q) = dec(p)⊕dec(q)
+dec(C) = {C}
+dec((νa)p) = dec(p){a′/a}
+a′ ∈ L ′
+Table 2. Decomposition function
+where the function fn(−), computing the free names, is defined on extended terms in
+the obvious way. The admissibility condition expresses a sort of sanity check on any
+restriction-free, extended term t: it is not possible that, for any action a ∈ L , there
+are occurrences in t of both a and its associated restricted action a′; this because each
+action type can occur in t only in one of the two modalities: either restricted or unre-
+stricted (i.e., normally visible). For instance, a.a′.0 is not admissible, while a.0|b′.0 is
+admissible. By the notation ad(t) we mean that t is admissible.
+By Pγ
+FNM we denote the set of all extended FNM processes. By Pγ,par
+FNM we denote
+the set of all restriction-free, extended FNM processes, i.e., those extended processes
+in syntactic category t. By Pγ,seq
+FNM we denote the set of all sequential, extended FNM
+processes, i.e., those extended processes in syntactic category q.
+3.2
+Net Semantics
+In this section, we summarize a technique (operational in style), proposed in [8], for
+building an infinite P/T net for the whole of FNM, starting from a description of its
+places and its net transitions. The resulting net NFNM = (SFNM,A ,TFNM) is such that,
+for any p ∈ PFNM, the net system NFNM(dec(p)) statically reachable from the initial
+marking dec(p) is a statically reduced P/T net, which is finite if p is well-formed; such
+a net system is denoted by Net(p).
+The set of FNM places, ranged over by s, is SFNM = Pγ,seq
+FNM \{0}, i.e., the set of all
+sequential, extended FNM processes, except 0. Function dec : Pγ
+FNM → M fin(SFNM),
+which maps extended processes into markings, is outlined in Table 2. Process 0 is
+mapped to the empty marking θ. A sequential process p is mapped to one place with
+name p. This is the case of µ.p (where µ can be any action in Actγ), a constant C,
+p + q and γ.p (where γ ∈ G = L ∪ L ′). Parallel composition is interpreted as multi-
+set union; e.g., the decomposition of a.0|a.0 produces the marking a.0⊕ a.0 = 2 ·a.0.
+The decomposition of a general process (νa)p – where a ∈ L – generates the multiset
+obtained from the decomposition of p, to which the substitution {a′/a} is applied; the
+application of the substitution {a′/a} to a multiset is performed element-wise. We as-
+sume that, in decomposing (νa)p, the choice of the restricted name is fixed by the rule
+that associates with a visible action a its unique corresponding restricted action a′.
+Proposition 5. For any p ∈ Pγ
+FNM, dec(p) is a finite multiset of places.
+✷
+A marking m ∈ M fin(SFNM) is admissible, denoted by ad(m), if for all a ∈ L ,
+{a,a′} ̸⊆ fn(m), where fn(m) = �
+s∈dom(m) fn(s), with fn(θ) = /0.
+A marking m ∈ M fin(SFNM) is complete if an FNM process p ∈ PFNM exists such
+that dec(p) = m.
+
+11
+(pref)
+{µ.p}
+µ
+−→dec(p)
+(cons)
+dec(p)
+σ
+−→m
+{C}
+σ
+−→m
+C .= p
+(sum1)
+dec(p)
+σ
+−→m
+{p+q}
+σ
+−→m
+(s-pref)
+dec(p)
+σ
+−→m
+{γ.p}
+γ⋄σ
+−→m
+(s-com)
+m1
+σ1
+−→m′
+1 m2
+σ2
+−→m′
+2
+m1 ⊕m2
+σ
+−→m′
+1 ⊕m′
+2
+ad(m1 ⊕m2)∧Sync(σ1,σ2,σ)
+Table 3. Rules for net transitions (symmetric rule (sum2) omitted)
+Theorem 1. A marking m ∈ M fin(SFNM) is admissible iff it is complete.
+✷
+Hence, this theorem states not only that function dec maps FNM processes to ad-
+missible markings over SFNM, but also that dec is surjective over this set.
+Definition 8. (Well-behaved) A set of places S ⊆ SFNM is well behaved if for all s ∈ S
+we have that wf(s) holds, i.e., the sequential FNM extended term s is well-formed.
+✷
+In order to define the set TFNM of all the FNM net transitions, we need some auxil-
+iary definitions. Let A γ = {τ}∪G ∗ ·(G ∪G ), ranged over by σ with abuse of notation,
+be the set of labels; hence, a label σ can be the invisible action τ, or a (possibly empty)
+sequence of inputs (or restricted inputs) followed by an input or an output (or its re-
+stricted counterpart). Let → ⊆ M fin(SFNM) × A γ × M fin(SFNM) be the least set of
+transitions generated by the axiom and rules in Table 3.
+Let us comment on the rules of Table 3. Axiom (pref) states that if one token is
+present in the place µ.p, then a µ-labeled transition is derivable from marking {µ.p},
+producing the marking dec(p). This holds for any µ, i.e., for the invisible action τ, for
+any visible action α as well as for any restricted action α′. By rule (sum1), the transi-
+tions from the place p+q are those from the marking dec(p); as p is sequential, dec(p)
+is {p} if p ̸= 0, while, if p = 0, dec(p) = θ, but no transition is derivable from the empty
+marking, so that the rule is really applicable only when dec(p) = {p}. Similarly, rule
+(cons) states that the transitions derivable from {C} are those derivable from the place
+{p}, if C .= p with p ̸= 0. In rule (s-pref), γ may be any input action a or any restricted
+action a′, and the auxiliary function γ ⋄σ returns γ if σ = τ, or γσ otherwise. This rule
+requires that the premise transition dec(p)
+σ
+−→m be derivable by the rules, which is re-
+ally possible only when dec(p) = {p}. Rule (s-com) requires that the transition pre-set
+be admissible in order to avoid producing transitions that have no counterpart in the
+LTS semantics of FNM, described in [8]. Rule (s-com) explains how a synchronization
+takes place: it is required that m1 and m2 perform synchronizable sequences σ1 and σ2,
+producing σ; here we assume that relation Sync has been extended also to restricted ac-
+tions in the obvious way, i.e., a restricted output action a′ can be synchronized only with
+its complementary restricted input action a′ or with an atomic sequence beginning with
+a′. As an example, the net transition {a.b′.p,a.q,b′.r}
+τ
+−→dec(p)⊕ dec(q)⊕ dec(r) is
+derivable by the rules, as shown in Table 4.
+
+12
+(pref)
+{b′.p}
+b′
+−→dec(p)
+(s-pref)
+{a.b′.p} ab′
+−→dec(p)
+(pref)
+{a.q}
+a
+−→dec(q)
+(s-com)
+{a.b′.p,a.q}
+b′
+−→dec(p)⊕dec(q)
+(pref)
+{b′.r}
+b′
+−→dec(r)
+(s-com)
+{a.b′.p,a.q,b′.r}
+τ
+−→dec(p)⊕dec(q)⊕dec(r)
+Table 4. The proof of a net transition
+Transitions with labels containing restricted actions must not be taken in the re-
+sulting net, as we accept only transitions labeled over A = {τ} ∪ L ∗ · (L ∪ L ).
+However, they are useful in producing acceptable transitions, as two complementary
+restricted actions can synchronize, producing a τ-labeled transition or shortening the
+synchronized sequence. For instance, in the example above, the derivable transition
+{b′.p}
+b′
+−→dec(p) is not an acceptable transition because its label is not in A , while
+{a.b′.p,a.q,b′.r}
+τ
+−→dec(p) ⊕ dec(q) ⊕ dec(r) is so. Hence, the P/T net for FNM is
+the triple NFNM = (SFNM,A , TFNM), where the set
+TFNM = {(m1,σ,m2)
+�� m1
+σ
+−→m2 is derivable by the rules and σ ∈ A }
+is obtained by filtering out those transitions derivable by the rules whose label σ con-
+tains some restricted name a′ or a′.
+We now want to show that for any finite, well-behaved set of places S ⊆ SFNM, the
+set of transitions statically enabled at S is finite. Given a place s ∈ S, by s ⊢ t we mean
+that transition t = ({s},σ,m) is derivable by the rules in Table 3, hence with σ ∈ A γ.
+Lemma 1. The set Ts = {t
+�� s ⊢ t} is finite, for any s ∈ SFNM.
+Proof. By induction on the axiom and rules in Table 3.
+✷
+Given a finite, well-behaved set of places S ⊆ SFNM, let T S
+1 be �
+s∈S Ts, i.e., the set
+of all transitions, with a singleton pre-set in S, derivable by the rules with labeling in
+A γ. The set T S
+1 is finite, being the finite union (as S is finite) of finite sets (as Ts is finite
+for each s ∈ S, by Lemma 1).
+Let k ∈ N be the length of the longest label of any transition in T S
+1 . It is possible
+to argue (details in [8]) that if a multi-party transition t is derivable by the rules from
+the well-behaved set S, then its proof contains k synchronizations at most, each one be-
+tween a transition (labeled with a sequence of inputs) and a singleton-pre-set transition
+(labeled with a single output action); hence, at most k+1 participants can take part in a
+multi-party synchronization. Therefore, the set of all the transitions statically enabled at
+a finite, well-behaved set S can be defined by means of a sequence of sets T S
+i of transi-
+tions, for 2 ≤ i ≤ k+1, where each transition t ∈ T S
+i has a pre-set •t of size i, as follows:
+T S
+i = {(m1 ⊕ m2,σ,m′
+1 ⊕ m′
+2)
+�� ad(m1 ⊕ m2),
+∃σ1∃γ.(m1,σ1,m′
+1) ∈ T S
+i−1,(m2,γ,m′
+2) ∈ T S
+1 ,Sync(σ1,γ,σ)}.
+
+13
+Note that T S
+2 is finite, because T S
+1 is finite; inductively, T S
+i+1, for 2 ≤ i ≤ k is finite,
+because T S
+i and T1 are finite. So, the set TS of all the transitions statically enabled at S is
+TS = {t
+�� t ∈
+k+1
+�
+i=1
+T S
+i ∧l(t) ∈ A },
+where only transitions labeled over A are considered. TS is finite, being a finite union
+of finite sets; therefore, we have the following result.
+Theorem 2. If S ⊆ SFNM is a finite, well-behaved set of places, then set TS ⊆ TFNM of
+all the transitions statically enabled at S is finite.
+✷
+Example 2. (1/3 Semi-counter) Let us consider a semi-counter such that three occur-
+rences of inc are needed to enable one dec, whence the name 1/3 semi-counter. The
+well-formed process p = (νc)A, where
+A .= inc.(A|(c.c.dec.0+ c.0))
+is a 1/3 semi-counter. The initial marking m0 is dec(p) = dec((νc)A) = dec(A){c′/c} =
+{A}{c′/c} = {s1}; place s1 is the extended, sequential process A{c′/c}, where the con-
+stant A{c′/c} is obtained by applying the substitution {c′/c} to the body of A:
+A{c′/c}
+.= inc.(A{c′/c} |(c′.c′.dec.0+ c′.0)).
+Then, let S0 = dom(m0) = {s1}. The set of transitions statically enabled at S0 is T S0
+1
+=
+Ts1 = {t1}, where the only transition is t1 = {s1} inc
+−→{s1,s2}, with s2 = c′.c′.dec.0 +
+c′.0. Therefore, the new set of statically reachable places is S1 = {s1,s2}. Note that s2
+can produce two transitions in Ts2, namely t′ = {s2} c′c′dec
+−→ θ and t′′ = {s2}
+c′
+−→θ, but
+neither is labeled over A . Since the longest label has length 3, we have to compute the
+sets T S1
+i
+for i = 1,...,4:
+T S1
+1
+= {t1,t′,t′′},
+T S1
+2
+= {t′′′}, where t′′′ = ({s2,s2},c′dec,θ),
+T S1
+3
+= {t2}, where t2 = ({s2,s2,s2},dec,θ), whose proof is shown in Table 5,
+T S1
+4
+= /0.
+Hence, TS1 = {t1,t2}, as these two are the only transitions labeled over A . As t2
+does not add any new reachable place, we have that S1 is the set of places statically
+reachable from the initial marking, and TS1 is the set of transitions statically enabled at
+S1. This net is depicted in Figure 6.
+✷
+The P/T net system associated with a process p ∈ PFNM is the subnet of NFNM
+statically reachable from the initial marking dec(p), denoted by Net(p).
+Definition 9. Let p be a process in PFNM. The P/T net system statically associated
+with p is Net(p) = (Sp,Ap,Tp,m0), where m0 = dec(p) and
+Sp = �dom(m0)⟩
+computed in NFNM,
+Tp = {t ∈ TFNM
+�� Sp�t⟩},
+Ap = {σ ∈ A
+�� ∃t ∈ Tp such that l(t) = σ}.
+✷
+
+14
+(pref)
+{dec.0} dec
+−→θ
+(s-pref)
+{c′.dec.0} c′dec
+−→ θ
+(s-pref)
+{c′.c′.dec.0} c′c′dec
+−→ θ
+(sum1)
+{s2} c′c′dec
+−→ θ
+(pref)
+{c′.0}
+c′
+−→ θ
+(sum2)
+{s2}
+c′
+−→θ
+(s-com)
+{s2,s2} c′dec
+−→ θ
+(pref)
+{c′.0}
+c′
+−→θ
+(sum2)
+{s2}
+c′
+−→ θ
+(s-com)
+{s2,s2,s2} dec
+−→θ
+Table 5. The proof of a net transition, where s2 = c′.c′.dec.0+ c′.0
+s1
+inc
+s2
+dec
+3
+Fig. 6. The P/T net for the 1/3 semi-counter
+The following propositions present three facts that are obviously true by construction
+of the net Net(p) associated with an FNM process p.
+Proposition 6. For any p ∈ PFNM, Net(p) is a statically reduced P/T net.
+✷
+Proposition 7. If dec(p) = dec(q), then Net(p) = Net(q).
+✷
+Proposition 8. For each restriction-free t ∈ PFNM and for each L ⊆ L :
+i) If Net(t) = (S,A,T,m0), then, for any n ≥ 1, Net(tn) = (S,A,T,n·m0), where t1 = t
+and tn+1 = t |tn.
+ii) If Net((νL)t) = (S,A,T,m0), then Net((νL)(tn)) = (S,A,T,n ·m0), for n ≥ 1.
+✷
+Definition 9 suggests a way of generating Net(p) with an algorithm based on the
+inductive definition of the static reachability relation (see Definition 6): Start with the
+initial set of places S0 = dom(dec(p)), and then apply the rules in Table 3 in order to
+produce the set TS0 of transitions (labeled over A ) statically enabled at S0, as well as
+the additional places statically reachable by means of such transitions. Then repeat this
+procedure from the set of places statically reached so far. An instance of this procedure
+was given in Example 2. There are two problems with this algorithm:
+• the obvious halting condition is “until no new places are statically reachable”; of
+course, the algorithm terminates if we know that the set Sp of places statically
+reachable from dom(dec(p)) is finite; additionally,
+
+15
+• at each step of the algorithm, we have to be sure that the set of transitions derivable
+from the current set of statically reachable places is finite.
+We are going to prove only the first requirement — Sp is finite for any p ∈ PFNM
+— because it implies also the second one for well-formed processes. As a matter of
+fact, if p is well formed, then dom(dec(p)) is well behaved, and so is any set S of
+places statically reachable from dom(dec(p)) (as proved in [8]); since S ⊆ Sp, S is also
+finite, and so, by Theorem 2, the set TS of transitions statically enabled at the finite,
+well-behaved set S is finite, too.
+Theorem 3. For any p ∈ PFNM, let Net(p) = (Sp,Ap,Tp,m0) be defined as in Defini-
+tion 9. Then, the set Sp is finite.
+Proof. By induction on the static reachability relation =⇒∗ (details in [8]).
+✷
+Theorem 4. [8] For any well-formed FNM process p, Net(p) = (Sp,Ap,Tp,dec(p)) is
+a finite P/T net.
+✷
+3.3
+Representing All Finite P/T Net
+Theorem 4 ensures that only finite P/T nets can be represented by FNM processes. It is
+not completely obvious that all finite P/T nets can be represented by FNM processes.
+The translation from nets to processes defines a constant Ci in correspondence with
+each place si; the constant Ci has a summand cj
+i for each transition tj, which is 0 when
+si is not in the pre-set of tj. The FNM process TFNM(N(m0)) associated with the finite
+net system N(m0), labeled over L ∪ {τ}, has a bound name xj
+i for each pair (si,tj),
+where si is a place and tj is a transition; such bound names are used to force synchro-
+nization among the components participating in transition tj with pre-set of cardinality
+two or more. Among the many places in the pre-set of tj, the one with least index i
+(as we assume that places are indexed) plays the role of leader of the synchronization;
+the corresponding leader constant Ci has a summand cj
+i containing the atomic input se-
+quence needed for the multi-party synchronization, such that each strong input prefix xj
+h
+(for h > i) is synchronized with the corresponding output xj
+h performed by the servant
+participant of index h; in case •tj(si) ≥ 2, then the summand cj
+i is actually a sum of xj
+i .0
+with the atomic input sequence, so that one instance of Ci acts as the leader, while the
+others are servants.
+Definition 10. (Translating finite P/T nets into well-formed FNM processes) Given
+A ⊆ L ∪ {τ}, let N(m0) = (S,A,T,m0) — with S = {s1,...,sn}, T = {t1,...,tk}, and
+l(tj) = µj — be a finite P/T net. Function TFNM(−), from finite P/T nets to well-formed
+FNM processes, is defined as
+TFNM(N(m0)) = (νL)(C1|···|C1
+�
+��
+�
+m0(s1)
+|···|Cn|···|Cn
+�
+��
+�
+m0(sn)
+)
+where L = {x1
+1,...,x1
+n,x2
+1,...,x2
+n,...,xk
+1,...,xk
+n} is such that L∩A = /0, eachCi is equipped
+with a defining equation Ci .= c1
+i + ··· + ck
+i (with Ci .= 0 if k = 0), and each summand
+cj
+i , for j = 1,...,k, is equal to
+
+16
+• 0, if si ̸∈ •t j;
+• µj.Π j, if •t j = {si};
+• xj
+i .0, if •t j(si) > 0 and •t j(si′) > 0 for some i′ < i (i.e., si is not the leader for the
+synchronization on tj);
+• xj
+i+1.··· .xj
+i+1
+�
+��
+�
+•t j(si+1)
+.....xj
+n.··· .xj
+n
+�
+��
+�
+•t j(sn)
+.µj.Π j, if •t j(si) = 1 and si is the leader of the synchro-
+nization (i.e., •t j(si′) > 0 for no i′ < i, while •t j(si′) > 0 for some i′ > i);
+• xj
+i .0+xj
+i .··· .xj
+i
+�
+��
+�
+•t j(si)−1
+.xj
+i+1.··· .xj
+i+1
+�
+��
+�
+•t j(si+1)
+.....xj
+n.··· .x j
+n
+�
+��
+�
+•t j(sn)
+.µj.Π j, otherwise (i.e., si is the leader
+and •t j(si) ≥ 2).
+Finally, process Π j is C1|···|C1
+�
+��
+�
+t•
+j (s1)
+|···|Cn|···|Cn
+�
+��
+�
+t•
+j (sn)
+, meaning that Π j = 0 if t•
+j = θ.
+✷
+Note that TFNM(N(m0)) is an FNM process: in fact, the restriction operator occurs
+only at the top level, applied to the parallel composition of a finite number of con-
+stants; each constant has a body that is sequential and restriction-free. Note also that
+TFNM(N(m0)) is a well-formed process: in fact, each strong prefix is a bound input xj
+i ,
+and any sequence ends with an action µj ∈ A, which is either an input or τ; hence, no
+atomic sequence ends with an output. Therefore, the following proposition holds by
+Theorem 4 and Proposition 6.
+Proposition 9. For any finite P/T Petri net N(m0), the net Net(TFNM(N(m0))) is a
+finite, statically reduced P/T net.
+✷
+Theorem 5. (Representability Theorem)[8] Let N(m0) = (S,A,T,m0) be a finite, stat-
+ically reduced P/T net system such that A ⊆ L ∪{τ}, and let p = TFNM(N(m0)). Then,
+Net(p) is isomorphic to N(m0).
+✷
+4
+Step Net Bisimilarity is a Congruence
+Now we prove that step net bisimilarity ∼s is a congruence for the FNM operators,
+or, in other words, that the FNM operators are compositional up to ∼s. We extend the
+definition of step net bisimilarity to FNM terms, i.e., we write p ∼s q, meaning that
+there exists a step net bisimulation R relating the markings dec(p) and dec(q) in the net
+obtained by the union of Net(p) and Net(q). In the following, given an FNM term r, by
+Ir we denote the relation Ir = I1
+r ∪I2
+r ∪I3
+r , where
+I1
+r = {(m,m)
+�� m ∈ [dec(r)⟩},
+I2
+r = {(m′,m′)
+�� ∃G . •G = m′,m′ ⊆ m ∈ [dec(r)⟩}, and
+I3
+r = {(m′′,m′′)
+�� m′′ ∈ [m′⟩,(m′,m′) ∈ I2
+r }.
+Clearly, Ir is a step net bisimulation containing the pair (dec(r),dec(r)).
+Proposition 10. (Congruence for strong prefixing and choice) Let p and q be FNM
+guarded processes, i.e., processes in syntactic category s. If p ∼s q, then
+(i)
+a.p ∼s a.q,
+for all a ∈ L ,
+(ii) p + r ∼s q + r, for each process r in syntactic category s.
+
+17
+Proof. Assume R is a step net bisimulation containing the pair ({p},{q}). (If p = 0 = q,
+then R = {(θ,θ)}.) Note that, since the considered process terms are sequential, the
+initial performable steps are actually composed of a single transition.
+Case (i) can be proven by considering relation R1 = R∪{({a.p},{a.q})}, which is
+a step net bisimulation. In fact, transition {a.p} a⋄σ
+−→m1 is derivable, by rule (s-pref),
+only if {p}
+σ
+−→m1. As ({p},{q}) ∈ R, also {q}
+σ
+−→m2 with (m1,m2) ∈ R. Hence, also
+{a.q} a⋄σ
+−→m2 with (m1,m2) ∈ R1 and ({a.p},{a.q}) ∈ R1, as required. The symmetric
+case when a.q moves first is analogous and so omitted.
+Case (ii) can be proven, for each r in syntactic category s, by showing that R2 =
+{({p + r},{q + r})} ∪ R∪ Ir is a step net bisimulation. In fact, if {p + r}
+σ
+−→m1, then
+this is due to {p}
+σ
+−→m1 or to {r}
+σ
+−→m1. In the former case, since ({p},{q}) ∈ R,
+we have {q}
+σ
+−→m2 with (m1,m2) ∈ R, so that {q + r}
+σ
+−→m2 with (m1,m2) ∈ R2 and
+({p+r},{q+r}) ∈ R2, as required. In the latter case, we have that also {q+r}
+σ
+−→m1
+with (m1,m1) ∈ Ir and so also (m1,m1) ∈ R2, as required. The symmetric case when
+q + r moves first is analogous and so omitted.
+✷
+Proposition 11. (Congruence for prefixing and parallel composition) Let p and q
+be restriction-free FNM processes. If p ∼s q, then
+(i) µ.p ∼s µ.q, for all µ ∈ Act,
+(ii) p|r ∼s q|r, for any restriction-free r ∈ PFNM.
+Proof. Assume R is a step net bisimulation containing the pair (dec(p),dec(q)). For
+case (i), since the considered process terms are sequential, the initial performable
+steps are actually composed of a single transition. Relation R3 = R∪{({µ.p},{µ.q})}
+is a step net bisimulation. In fact, if {µ.p}
+µ
+−→dec(p), then {µ.q}
+µ
+−→dec(q) with
+(dec(p),dec(q)) ∈ R3 and ({µ.p},{µ.q}) ∈ R3, as required. The symmetric case when
+µ.q moves first is analogous and so omitted.
+For case (ii), relation R4 = R ∪ Ir ∪ {(m1⊕m,m2⊕m)
+�� (m1,m2) ∈ R,(m,m) ∈ Ir}
+is a step net bisimulation containing the pair (dec(p) ⊕ dec(r),dec(q) ⊕ dec(r)). In
+fact, R4 is composed of R, as well as of Ir, which are already step net bisimulations;
+now, consider a pair (m1 ⊕ m,m2 ⊕ m) ∈ R4, belonging to the third group, and assume
+m1 ⊕ m[G1⟩m1; we have to consider the following three cases:
+• m1[G1⟩m′
+1, so that m1 = m′
+1 ⊕ m. In such a case, since (m1,m2) ∈ R, we have
+that there exists G2 such that m2[G2⟩m′
+2, l(G1) = l(G2), (•G1,•G2) ∈ R (hence,
+(•G1,•G2) ∈ R4, too) and (m′
+1,m′
+2) ∈ R. Therefore, m2 ⊕m[G2⟩m′
+2 ⊕m, with (m′
+1 ⊕
+m,m′
+2 ⊕ m) ∈ R4, as required.
+• m[G1⟩m′, so that m1 = m1 ⊕ m′. In such a case, since (m,m) ∈ Ir, we have that
+also m[G1⟩m′, with (•G1,•G1) ∈ Ir (hence, (•G1,•G1) ∈ R4, too) and (m′,m′) ∈ Ir.
+Therefore, m2 ⊕ m[G1⟩m2 ⊕ m′, with (m1 ⊕ m′,m2 ⊕ m′) ∈ R4, as required.
+• there exist G′
+1 and G such that m1[G′
+1⟩m′
+1, m[G⟩m′, •G1 = •G′
+1 ⊕ •G, G•
+1 = G′•
+1 ⊕G•
+and MSync(l(G′
+1),l(G),l(G1)), where this relation is defined in Table 6. In such a
+case, as (m1,m2) ∈ R, we have that there exists G′
+2 such that m2[G′
+2⟩m′
+2, l(G′
+1) =
+l(G′
+2), (•G′
+1,•G′
+2) ∈ R and (m′
+1,m′
+2) ∈ R. Hence, there exists a step G2 such that
+•G2 = •G′
+2 ⊕ •G, G•
+2 = G′•
+2 ⊕G•, MSync(l(G′
+2),l(G),l(G2)), so that l(G1) = l(G2)
+(because the same proof used to derive MSync(l(G′
+1),l(G),l(G1)) can be adapted
+
+18
+MSync(M1,M2,M1 ⊕M2)
+Sync(σ1,σ2,σ) MSync(M1,M2,M)
+MSync(M1 ⊕{σ1},M2 ⊕{σ2},M ⊕{σ})
+Table 6. Step synchronization relation, where M denotes a multiset of labels
+to derive MSync(l(G′
+2),l(G),l(G2))), m2⊕m[G2⟩m′
+2⊕m′ with (•G1,•G2) ∈ R4 (be-
+cause (•G′
+1,•G′
+2) ∈ R and (•G,•G) ∈ Ir), and, moreover, (m′
+1 ⊕ m′,m′
+2 ⊕ m′) ∈ R4
+(because (m′
+1,m′
+2) ∈ R and (m′,m′) ∈ Ir), as required.
+The symmetric case when m2 ⊕ m moves first is analogous, and so omitted.
+✷
+Note that it is easy to generalize the congruence property for parallel composition
+as follows: If pi ∼s qi for i = 1,2, then p1 | p2 ∼s q1 |q2. In fact, if Ri is a step net
+bisimulation containing the pair (dec(pi),dec(qi)), for i = 1,2, then it is easy to see
+that the relation R1 ∪ R2 ∪ {(m1 ⊕ m2,m′
+1 ⊕ m′
+2)
+�� (m1,m′
+1) ∈ R1,(m2,m′
+2) ∈ R2} is a
+step net bisimulation.
+Proposition 12. (Congruence for restriction)
+Let p and q be general FNM processes. If p ∼s q, then (νa)p ∼s (νa)q for all a ∈ L .
+Proof. Let R be a step net bisimulation containing the pair (dec(p),dec(q)). Relation
+R5 = {(m1{a′/a},m2{a′/a})
+�� (m1,m2) ∈ R} is the required step net bisimulation. In
+fact, assume m1{a′/a}[G1⟩m1. By the net semantics, this is possible only if m1[G′
+1⟩m′
+1,
+where G1 = G′
+1{a′/a} = (•G′
+1{a′/a},l(G′
+1),G′•
+1 {a′/a}), m1 = m′
+1{a′/a} and l(G′
+1)
+does not contain any occurrence of action a. As (m1,m2) ∈ R, we have that a step
+G′
+2 exists such that m2[G′
+2⟩m′
+2, with l(G′
+1) = l(G′
+2), (•G′
+1,•G′
+2) ∈ R and (m′
+1,m′
+2) ∈ R.
+Therefore, by the net semantics it is possible to derive m2{a′/a}[G2⟩m2, where G2 =
+G′
+2{a′/a} and m2 = m′
+2{a′/a}, with the property that l(G1) = l(G2), (•G1,•G2) ∈ R5
+and (m1,m2) ∈ R5, as required. The symmetric case when m2{a′/a} moves first is anal-
+ogous, and so omitted.
+Still there is one construct missing: recursion, defined over guarded terms only.
+Consider an extension of FNM where terms can be constructed using variables, such as
+x,y,...: this defines an “open” FNM.
+Definition 11. (Open FNM) Let Var = {x,y,z,...} be a finite set of variables. The
+FNM open terms are generated from actions, constants and variables by the following
+abstract syntax (using three syntactic categories):
+s ::= 0 |
+µ.t
+| a.s | s+ s
+guarded open terms
+q ::= s |
+C
+|
+x
+sequential open terms
+t ::= q |
+t |t
+restriction-free open terms
+p ::= t | (νa)p
+general open terms
+where x is any variable taken from Var. The open net semantics for open FNM extends
+the net semantics in Section 3 with Net(x) = ({x}, /0, /0,{x}), so that, e.g., the semantics
+of a.x is the net ({a.x,x},{a}, {(a.x,a,x)},a.x).
+✷
+
+19
+Sometimes we use the notation p(x1,...,xn) to state explicitly that term p is open on
+the tuple of variables (x1,...,xn). For instance, p1(x) = a.(b.0+c.x)+d.x and p2(x) =
+d.x+ a.(c.x+ b.0) are open guarded FNM terms.
+Step net bisimulation equivalence can be extended to open terms as follows. An
+open term p(x1,...,xn) can be closed by means of a substitution
+p(x1,...,xn){r1/x1,...,rn/xn}
+with the effect that each occurrence of the variable xi (within p and the body of each
+constant in Const(p)) is replaced by the closed FNM sequential process ri, for i =
+1,...,n. For instance, p1(x){d.0/x} = a.(b.0+ c.d.0)+ d.d.0.
+A natural extension of step net bisimilarity ∼s over open guarded terms is as fol-
+lows: p(x1,...,xn) ∼s q(x1,...,xn) if for all tuples of (closed) FNM sequential terms
+(r1,...,rn),
+p(x1,...,xn){r1/x1,...,rn/xn} ∼s q(x1,...,xn){r1/x1,...,rn/xn}.
+E.g., it is easy to see that p1(x) ∼s p2(x). As a matter of fact, for all r,
+p1(x){r/x} = a.(b.0+ c.r)+ d.r ∼s d.r + a.(c.r + b.0) = p2(x){r/x},
+which can be easily proved.
+For simplicity’s sake, let us now restrict our attention to open guarded terms us-
+ing a single undefined variable. We can recursively close an open term p(x) by means
+of a recursively defined constant. For instance, A .= p(x){A/x}. The resulting process
+constant A is a closed FNM sequential process. By saying that step net bisimilarity is
+a congruence for recursion we mean what is stated in the following. For simplicity’s
+sake, in the following a term p open on a variable x is no longer annotated as p(x).
+Theorem 6. Let p and q be two open guarded FNM terms, with one variable x at most.
+Let A .= p{A/x}, B .= q{B/x} and p ∼s q. Then A ∼s B.
+Proof. Let R = {(m{A/x},m{B/x})
+�� m ∈ M (Sop)} be our candidate relation, where
+Sop = Pseq
+oFNM \ {0}, i.e., the set of all sequential, open FNM processes, except 0.1
+Note that when m is x, we get (A,B) ∈ R. The proof that R is a step net bisimulation
+(up to ∼s) is not difficult. By symmetry, it is enough to prove that if m{A/x}[G1⟩m1,
+then there exists G2 such that m{B/x}[G2⟩m2 with (•G1,•G2) ∈ R, l(G1) = l(G2) and
+m1 ∼s R ∼s m2. The proof proceeds by induction on the size of m. When |m| = 1, i.e.,
+m = s for some (open) place s, the proof is by induction on the definition of the net for
+s{A/x}. Note that, since in this case the involved terms are sequential, the steps are
+actually composed of a single transition.
+• s = µ.r. In this case, s{A/x} = µ.r{A/x}
+µ
+−→dec(r){A/x}. Similarly, s{B/x} =
+µ.r{B/x}
+µ
+−→dec(r){B/x}, and (dec(r){A/x},dec(r){B/x}) ∈ R.
+• s = a.r. In this case, s{A/x} = a.r{A/x} a⋄σ
+−→m1 if r{A/x}
+σ
+−→m1. Since r is guarded,
+r{A/x}
+σ
+−→m1 is possible only if r
+σ
+−→m with m1 = m{A/x}. So, r{B/x}
+σ
+−→m{B/x}
+is derivable, too, and also s{B/x} a⋄σ
+−→m{B/x}, with (m{A/x},m{B/x}) ∈ R.
+1Note that we are not considering extended terms, but only ‘normal’ FNM sequential terms,
+because the restriction operator cannot occur within the body of a recursively defined constant.
+Hence, each marking m over Sop is admissible.
+
+20
+• s = D, with D .= r. So, s{A/x} .= r{A/x} and s{B/x} .= r{B/x}. If s{A/x}
+σ
+−→m1,
+then this is possible only if r{A/x}
+σ
+−→m1. Since r is guarded, r{A/x}
+σ
+−→m1 is
+possible only if r
+σ
+−→m with m1 = m{A/x}. Therefore, also r{B/x}
+σ
+−→m{B/x} is
+derivable, and also s{B/x}
+σ
+−→m{B/x}, with (m{A/x},m{B/x}) ∈ R.
+• s = s1 + s2. In this case, s{A/x} = s1{A/x} + s2{A/x}. A transition from s{A/x},
+e.g., s1{A/x} + s2{A/x}
+σ
+−→m1, is derivable only if si{A/x}
+σ
+−→m1 for some i =
+1,2. Without loss of generality, assume the transition is due to s1{A/x}
+σ
+−→m1.
+Since s1 is guarded, transition s1{A/x}
+σ
+−→m1 is derivable because s1
+σ
+−→m, with
+m1 = m{A/x}. Therefore, also s1{B/x}
+σ
+−→m{B/x} is derivable, as well s{B/x} =
+s1{B/x} + s2{B/x}
+σ
+−→m{B/x}, with (m{A/x},m{B/x}) ∈ R.
+• s = x. We have s{A/x} = A and s{B/x} = B. We prove that for each A
+σ
+−→m1, there
+exists m2 such that B
+σ
+−→m2 with m1 ∼s R ∼s m2. By hypothesis, A .= p{A/x},
+hence also p{A/x}
+σ
+−→m1 is a transition in the net for p{A/x}; since p is guarded,
+p{A/x}
+σ
+−→m1 is possible only if p
+σ
+−→m with m1 = m{A/x}. Therefore, also
+p{B/x}
+σ
+−→m{B/x} is derivable. But we also have that p ∼s q, so p
+σ
+−→m can
+be matched by q
+σ
+−→m′ with m ∼s m′ (note that the requirement on the pre-sets of
+the two transitions is already satisfied). Hence, q{B/x}
+σ
+−→m′{B/x} is derivable
+with m{B/x} ∼s m′{B/x}. Since B .= q{B/x}, also B
+σ
+−→m′{B/x} is a transition
+with m1 ∼s m{A/x} R m{B/x} ∼s m′{B/x}, as required.
+When |m| = n+1, then a place s and a marking m′ exist such that m = s⊕m′. Hence,
+we have that (s{A/x},s{B/x}) ∈ R and (m′{A/x},m′{B/x}) ∈ R. By induction (as
+|s| = 1 and |m′| = n), we have that s{A/x} ∼s s{B/x} as well as m′{A/x} ∼s m′{B/x}.
+Hence, by Theorem 1, there exist (open) FNM restriction-free processes p1, p2, q1,q2
+such that s{A/x} = dec(p1), s{B/x} = dec(p2), m′{A/x} = dec(q1), m′{B/x} = dec(q2).
+Therefore, the thesis follows by compositionality of ∼s w.r.t. parallel composition (cf.
+the comment after Proposition 11), because m{A/x} = dec(p1 |q1) and m{B/x} =
+dec(p2 |q2).
+✷
+The extension to the case of open terms with multiple undefined constants, e.g.,
+p(x1,...,xn) can be obtained in a standard way [12,7].
+5
+Conclusion
+Step net bisimulation is a very simple relation, extending ordinary step bisimulation
+[13] on Petri nets with two additional conditions: the related markings must have the
+same size and the pre-sets of the two matching steps must be related as well. With these
+minimal additions, its definition is still very manageable and also very sensible as we
+proved in [10] to coincide with causal-net bisimilarity [9] (or, equivalently, structure-
+preserving bisimilarity [6]), a truly concurrent behavioral equivalence fully respecting
+causality and the branching structure of systems.
+The decidability of step net bisimilarity over finite (unbounded) P/T nets is an open
+problem [10]. Step net bisimilarity is decidable on bounded nets, because causal-net
+
+21
+bisimilarity has been proved decidable in (large) exponential time on that class of nets
+[3]. However, we expect that our simpler characterization of ∼cn in term of step net
+bisimilarity ∼s may offer a more efficient algorithm, yet exponential as well. Step net
+bisimilarity is decidable on BPP nets, because on this class of nets it coincides with
+team bisimilarity [9], which is decidable in polynomial time.
+Van Glabbeek [6] argued that structure-preserving bisimilarity is the most appro-
+priate behavioral equivalence for Petri nets, as it is the only one respecting a list of
+desirable requirements he proposed. Among these, there is ‘compositionality’, up to
+structure-preserving bisimilarity, of the operators (recursion not considered) of the pro-
+cess algebra CCSP, that Olderog proposed in his monograph [14] and equipped with a
+safe net semantics. In this paper we have extended his result by proving that step net
+bisimilarity, which is an alternative, simpler characterization of structure-preserving
+bisimilarity, can be used to give a compositional semantics of the process algebra FNM
+[8], which truly represents all (and only) the finite P/T nets, up to isomorphism. In this
+way, we have obtained for the first time a compositional semantics, fully respecting
+causality and the branching structure of systems, for the class of all the finite P/T Petri
+nets.
+References
+1. E. Best, R. Devillers, Sequential and concurrent behavior in Petri net theory, Theoretical
+Computer Science 55(1):87-136, 1987.
+2. E. Best, R. Devillers, A. Kiehn, L. Pomello, Concurrent bisimulations in Petri nets, Acta Inf.
+28(3): 231-264, 1991.
+3. A. Cesco, R. Gorrieri, Decidability of two truly concurrent equivalences for finite bounded
+Petri nets, CoRR, abs/2104.14856, 2021, https://arxiv.org/abs/2104.14856
+4. J. Desel, W. Reisig, Place/Transition Petri nets, in Lectures on Petri Nets I: Basic Models,
+LNCS 1491, 122-173, Springer, 1998.
+5. U. Goltz, W. Reisig, The non-sequential behaviour of Petri nets, Information and Control
+57(2-3):125-147, 1983.
+6. R.J. van Glabbeek, Structure preserving bisimilarity - Supporting an operational Petri net
+semantics of CCSP, in (R. Meyer, A. Platzer, H. Wehrheim, Eds.) Correct System Design
+— Symposium in Honor of Ernst-R¨udiger Olderog on the Occasion of His 60th Birthday,
+LNCS 9360, 99-130, Springer, 2015.
+7. R. Gorrieri, C. Versari, Introduction to Concurrency Theory: Transition Systems and CCS,
+EATCS Texts in Theoretical Computer Science, Springer-Verlag, 2015.
+8. R. Gorrieri, Process Algebras for Petri Nets: The Alphabetization of Distributed Systems,
+EATCS Monographs in Computer Science, Springer, 2017.
+9. R. Gorrieri, A study on team bisimulation and h-team bisimulation for BPP nets, Theoretical
+Computer Science 897:83-113, 2022.
+10. R. Gorrieri, True concurrency can be easy, CoRR, arXiv:2211.00557, November 2022,
+https://arxiv.org/abs/2211.00557
+11. R.M. Karp, R.E. Miller, Parallel program schemata, Journal of Computer and System Sci-
+ences 3(2):147-195, 1969.
+12. R. Milner, Communication and Concurrency, Prentice-Hall, 1989.
+13. M. Nielsen, P.S. Thiagarajan, Degrees of non-determinism and concurrency: A Petri net
+view, in Procs. of the Fourth Conference on Foundations of Software Technology and The-
+oretical Computer Science (FSTTCS’84), LNCS 181, 89-117, Springer-Verlag, 1984.
+
+22
+14. E.R. Olderog, Nets, Terms and Formulas, Cambridge Tracts in Theoretical Computer Sci-
+ence 23, Cambridge University Press, 1991.
+15. D.M.R. Park, Concurrency and automata on infinite sequences, In Proc. 5th GI-Conference
+on Theoretical Computer Science, LNCS 104, 167-183, Springer, 1981.
+16. J.L. Peterson, Petri Net Theory and the Modeling of Systems, Prentice-Hall, 1981.
+
diff --git a/KtE3T4oBgHgl3EQfYAoc/content/tmp_files/load_file.txt b/KtE3T4oBgHgl3EQfYAoc/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf,len=895
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='04483v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='LO]  11 Jan 2023  Compositional Net Semantics up to Step Net Bisimilarity  Roberto Gorrieri  roberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='gorrieri@unibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='it  Dipartimento di Informatica — Scienza e Ingegneria  Universit`a di Bologna,  Mura A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Zamboni, 7, 40127 Bologna, Italy  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Step net bisimilarity [10] is a truly concurrent behavioral equivalence  for finite Petri nets, which is defined as a smooth generalization of standard step  bisimilarity [13] on Petri nets, but with the property of relating markings (of the  same size only) generating the same partial orders of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The process algebra  FNM [8] truly represents all (and only) the finite Petri nets, up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We prove that step net bisimilarity is a congruence w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' the FMN operators, In  this way, we have defined a compositional semantics, fully respecting causality  and the branching structure of systems, for the class of all the finite Petri nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  1  Introduction  A step net bisimulation R over the markings of a finite Place/Transition Petri net [16,4]  is a behavioral relation stating that if two markings m1 and m2 are related by R, then (i)  they must have the same size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', |m1| = |m2|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' moreover, (ii) for each step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', multiset  of concurrently enabled transitions) G1 executable from m1 reaching m′  1, there exists a  step G2 of the same size executable from m2 reaching m′  2 such that, not only the labels  of G1 and G2 are the same and that (m′  1,m′  2) ∈ R – as for ordinary step bisimulation  [13] – but also that the pre-sets of G1 and G2 are related by R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', (•G1,•G2) ∈ R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  finally, (iii), the symmetric condition if m2 moves first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  One may wonder if this simplistic definition induces some actually sensible be-  havioral equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We proved in [10] that, surprisingly enough, step net bisimilarity  ∼s coincides with a well-known, truly-concurrent behavioral relation, namely causal-  net bisimilarity, originally proposed in [6] (and further studied in [9], who coined its  name).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Causal-net bisimilarity ∼cn is essentially a bisimulation-like relation over the  causal nets (sometimes called nonsequential processes) [5,1,14] originating from the  two initial markings of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, its definition is very complex and cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  However, as step net bisimilarity coincides with causal-net bisimilarity, we get that our  extremely simple definition of step net bisimilarity is slightly finer than fully-concurrent  bisimilarity [2] in that it relates markings that not only generate the same the partial or-  ders of events, but that also have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Finite P/T Petri nets can be represented, up to isomorphism, by means of the pro-  cess algebra FNM [8], which is a CCS-like sub-calculus extended with an additional  operator, called strong prefixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This operator allows for atomic execution of action  sequences, so that a multi-party interaction can be modeled as an atomic execution of  a sequence of binary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The main aim of this paper is to show that step net  2  bisimilarity is a congruence for the operators of FNM, thus yielding a compositional  semantics, up to step net bisimilarity, for the class of all the finite Petri nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Section 2 introduces the basic definitions about  Petri nets, some novel definitions (notably, the distinction between dynamically vs stati-  cally reachable subnet) and the definition of step net bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Section 3 recalls from  [8] the process algebra FNM, its (operational) semantics in terms of finite P/T Petri  nets, and also the representability theorem stating that each finite (statically reachable)  marked Petri net can be represented by a suitable FNM term, up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Sec-  tion 4 shows that step net bisimilarity is a congruence for the FNM operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Finally,  Section 5 is devoted to some conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  2  Basic Definitions  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Multiset) Let N be the set of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a countable set S,  a finite multiset over S is a function m : S → N such that the support set dom(m) =  {s ∈ S  �� m(s) ̸= 0} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set of all finite multisets over S, denoted by M fin(S),  is ranged over by m, possibly indexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (The set of all finite subsets of S is denoted by  Pfin(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=') We write s ∈ m if m(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The multiplicity of s in m is given by the number  m(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The size of m, denoted by |m|, is the number ∑s∈S m(s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the total number of  its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A multiset m such that dom(m) = /0 is called empty and is denoted by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We write m ⊆ m′ if m(s) ≤ m′(s) for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Multiset union ⊕  is defined as: (m ⊕ m′)(s) = m(s) + m′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Multiset difference  ⊖  is defined as: (m1 ⊖ m2)(s) = max{m1(s) − m2(s),0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The scalar product of a  natural j with m is the multiset j ·m defined as (j ·m)(s) = j ·(m(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By si we denote  the multiset with si as its only element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, a multiset m over S = {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',sn} can be  represented as k1 ·s1 ⊕ k2 ·s2 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='⊕ kn ·sn, where kj = m(sj) ≥ 0 for j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Place/Transition Petri net) A labeled Place/Transition Petri net (P/T  net, or Petri net, for short) is a tuple N = (S,A,T), where  S is the countable set of places, ranged over by s (possibly indexed),  A ⊆ Lab is the finite set of labels, ranged over by ℓ (possibly indexed), and  T ⊆ (M fin(S)\\{θ})×A×M fin(S) is the countable set of transitions, ranged over  by t (possibly indexed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A Petri net N = (S,A,T) is finite when S and T are finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a transition t =  (m,ℓ,m′), we use the notation •t to denote its pre-set m (which cannot be an empty  multiset) of tokens to be consumed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' l(t) for its label ℓ, and t• to denote its post-set m′ of  tokens to be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, transition t can be also represented as •t  l(t)  −→t•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  In the graphical description of P/T nets, places (represented as circles) and transi-  tions (represented as boxes) are connected by directed arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The arcs may be labeled  with the number representing how many tokens are to be removed from (or produced  into) that place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' no label on the arc is interpreted as the number one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', one token  flowing on the arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This numerical label of the arc is called its weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  3  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Marking, P/T net system) Given a P/T net N = (S,A,T), a finite multi-  set over S is called a marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a marking m and a place s, we say that the place s  contains m(s) tokens, graphically represented by m(s) bullets inside place s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A P/T net  system N(m0) is a tuple (S,A,T,m0), where (S,A,T) is a P/T net and m0 is a marking  over S, called the initial marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We also say that N(m0) is a marked net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Enabling, firing sequence, reachable marking) Given a P/T net N =  (S,A,T), a transition t is enabled at m, denoted by m[t⟩, if •t ⊆ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The execution (or  firing) of t enabled at m produces the marking m′ = (m⊖ •t)⊕t•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This is written m[t⟩m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A firing sequence starting at m is defined inductively as follows:  m[ε⟩m is a firing sequence (where ε denotes an empty sequence of transitions) and  if m[σ⟩m′ is a firing sequence and m′[t⟩m′′, then m[σt⟩m′′ is a firing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The set of reachable markings from m, denoted by [m⟩, is [m⟩ = {m′ �� ∃σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='m[σ⟩m′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' ✷  Given a P/T net N = (S,A,T) and a marking m, we say that two transitions t1,t2 ∈  T are concurrently enabled at m if •t1 ⊕ •t2 ⊆ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The concurrent firing of these two  transitions produces the marking m′ = (m⊖(•t1 ⊕ •t2))⊕(t•  1 ⊕t•  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We denote this fact  by m[{t1,t2}⟩m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' It is also possible that the same transition is self-concurrent at some  marking m, meaning that two or more occurrences of it are concurrently enabled at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We can generalize the definition of concurrently enabled transitions to a finite,  nonempty multiset G over the set T, called a step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A step G : T → N is enabled at  marking m if •G ⊆ m, where •G = �  t∈T G(t) · •t and G(t) denotes the number of oc-  currences of transition t in the step G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The execution of a step G enabled at m produces  the marking m′ = (m ⊖ •G) ⊕ G•, where G• = �  t∈T G(t) ·t•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This is written m[G⟩m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We sometimes refer to this as the concurrent token game, in opposition to the sequen-  tial token game of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The label l(G) of a step G is the multiset l(G) : A → N  defined as follows: l(G)(a) = ∑ti∈dom(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='l(ti)=a G(ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='1  Dynamically Reachable and Statically Reachable Subnets  Now we introduce two notions of reachable subnet, following [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  dynamically reachable subnet, which refers to the set of places and transitions dy-  namically reachable from the initial marking m0 by the token game, and  statically reachable subnet, which refers to the places and transitions which are  reachable by a weaker token game, stating that a transition t is statically enabled at  S′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the current set of the statically reached places (initiallly dom(m0))), when  dom(t) is a subset of S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Based on the latter, we introduce the notion of statically reduced net, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', a net  where all of its places and transitions are statically reachable from dom(m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This notion  is important in Section 3, because we will show that the net semantics Net(p) of an  FNM term p is a statically reduced net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  In fact, the semantics of a term p is not simply the net that can be dynamically  reached from the initial marking dec(p) corresponding to p, rather it is the net describ-  ing all the potential behaviors of p, as if the number of tokens in dec(p) can be increased  4  a)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0  a  a  b)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0  a  a  τ  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The dynamically reachable subnet (a) and the statically reachable subnet (b) for  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0  s1  s2  s3  a  b  c  d  s4  s5  a)  2  s1  s2  a  b  s4  b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A net system in (a) and its dynamically reachable subnet in (b)  at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, consider the FNM sequential process p = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' its dynami-  cally reachable subnet, outlined in Figure 1(a), shows the expected behavior that p can  perform either a or its complementary action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' On the contrary, its statically reachable  subnet in (b) describes additionally the potential self-synchronization of p with another  copy of itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' in this way, the net semantics for p and for the parallel term p| p differ  only for the form of the initial marking (one token for p and two tokens for p| p), but the  underlying net is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The statically reachable subnet of a finite P/T net is always  algorithmically computable, also for Nonpermissive nets [8], even though they are a  Turing-complete model of computation, and so for them, as the reachability problem is  undecidable, it is not always possible to compute the dynamically reachable subnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Dynamically reachable subnet, dynamically reduced net) Given a  P/T Petri net system N(m0) = (S,A,T,m0), the dynamically reachable subnet Netd(N(m0))  is (S′,A′,T ′,m0), where  S′ = {s ∈ S  �� ∃m ∈ [m0⟩ such that m(s) ≥ 1},  T ′ = {t ∈ T  �� ∃m ∈ [m0⟩ such that m[t⟩},  A′ = {ℓ  �� ∃t ∈ T ′ such that l(t) = ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A P/T net system N(m0) = (S,A,T,m0) is dynamically reduced if N(m0) = Netd(N(m0)),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the net system is equal to its dynamically reachable subnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  The dynamically reachable subnet of the net in Figure 2(a) is outlined in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Given a finite net N(m0), it is algorithmically derivable Netd(N(m0)) by means of its  coverability tree [11], but its complexity is exponential in the size of the net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Statically reachable subnet and statically reduced net) Given a P/T  net N = (S,A,T), we say that a transition t is statically enabled by a set of places S′ ⊆ S,  denoted by S′�t⟩, if dom(•t) ⊆ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  5  s1  s2  a  b  c  s4  s5  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The statically reachable subnet of Figure 2(a)  Given two sets of places S1,S2 ⊆ S, we say that S2 is statically reachable in one step  from S1 if there exists a transition t ∈ T, such that S1�t⟩, dom(t•) ̸⊆ S1 and S2 = S1 ∪  dom(t•);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' this is denoted by S1  t  =⇒S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The static reachability relation =⇒∗ ⊆ Pfin(S)×  Pfin(S) is the least relation such that  S1 =⇒∗ S1 and  if S1 =⇒∗ S2 and S2  t  =⇒S3, then S1 =⇒∗ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A set of places Sk ⊆ S is the largest set statically reachable from S1 if S1 =⇒∗ Sk and for  all t ∈ T such that Sk�t⟩, we have that dom(t•) ⊆ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a P/T net system N(m0) =  (S,A,T,m0), we denote by �dom(m0)⟩ the largest set of places statically reachable from  dom(m0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the largest Sk such that dom(m0)=⇒∗ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The statically reachable subnet Nets(N(m0)) is the net (S′,A′,T ′,m0), where  S′ = �dom(m0)⟩,  T ′ = {t ∈ T  �� S′�t⟩},  A′ = {ℓ  �� ∃t ∈ T ′ such that l(t) = ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A P/T net system N(m0) = (S,A,T,m0) is statically reduced if Nets(N(m0)) = N(m0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the net system is equal to its statically reachable subnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  The statically reachable subnet Nets(N(m0)) of a finite Petri net N(m0) can be com-  puted with an easy polynomial algorithm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We will show in Section 3 that, given an  FNM process p, its net semantics is the statically reachable subnet, extracted from the  infinite net describing the semantics of the whole FNM process algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a P/T net system N(m0) = (S,A,T,m0), if N(m0) is dynamically  reduced, then it is also statically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  However, the converse implication is not true: there are statically reduced P/T net  systems that are not dynamically reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, the statically reduced P/T net  system N(s1) = ({s1,s2,s3}, {a,b},{(s1,a,s2),(2 · s1,b,s3)},s1) cannot dynamically  reach place s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Consider the net system in Figure 2(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' its statically reachable subnet is outlined  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If we compare it with its dynamically reachable subnet in Figure 2(b), we  note that the statically reachable subnet contains the dynamically reachable subnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This  holds in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  6  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a P/T system N(m0) = (S,A,T,m0), if its dynamically reachable  subnet Netd(N(m0)) is (S′,A′,T ′,m0) and its statically reachable subnet Nets(N(m0))  is (S′′,A′′,T ′′,m0), then S′ ⊆ S′′, T ′ ⊆ T ′′ and A′ ⊆ A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  For some classes of nets, however, the two notions coincide, in particular for BPP  nets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', those nets whose transitions have singleton preset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='2  Step Net Bisimilarity  Step net bisimulation is a smooth generalization of ordinary step bisimulation [13] on  Petri nets, requiring two additional conditions: the related markings must have the same  size and the pre-sets of the two matching steps must be related as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Step net bisimulation) Let N = (S,A,T) be a P/T net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A step net bisim-  ulation is a relation R ⊆ M fin(S)× M fin(S) such that if (m1,m2) ∈ R then  |m1| = |m2|,  ∀G1 such that m1[G1⟩m′  1, ∃G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' m2[G2⟩m′  2, with l(G1) = l(G2), (•G1,•G2) ∈ R and  (m′  1,m′  2) ∈ R,  ∀G2 such that m2[G2⟩m′  2, ∃G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' m1[G1⟩m′  1, with l(G1) = l(G2), (•G1,•G2) ∈ R and  (m′  1,m′  2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Two markings m1 and m2 are step net bisimilar, denoted by m1 ∼s m2, if there exists a  step net bisimulation R such that (m1,m2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  When the steps G1 and G2 are singletons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', {t1} and {t2}, respectively), we get  that the two matching transitions t1 and t2 in the step net bisimulation game must have  the same shape because (i) (•t1,•t2) ∈ R implies that |•t1| = |•t2|, then (ii) l(t1) = l(t2)  and, finally, (iii) from |m1| = |m2|, |m′  1| = |m′  2| and |•t1| = |•t2| we derive that |t•  1| = |t•  2|,  because m′  i = mi ⊖ •ti ⊕t•  i for i = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In general, it is possible to prove [10] that if a  step G1 = {t1  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',t1  k } is matched by some step G2, then there must also exist a matching  step G′  2 = {t2  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',t2  k }, such that, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',k, t1  i and t2  i must have the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  So, in the following, we usually assume that the matching steps not only have the same  size, but are composed of transitions that can be related by a shape-preserving bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Indeed, step net bisimulation is a smooth generalization of ordinary step bisimu-  lation [13] on Petri nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In fact, on sequential finite-state machines (which are essen-  tially labeled transition systems, as each reachable marking is a singleton), its defini-  tion reduces to standard (interleaving) bisimulation, as the additional side conditions  |m1| = |m2| and (•t1,•t2) ∈ R are redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Finally, note also that a step net bisimulation R is such that it relates (dynamically)  reachable markings only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', it is a relation actually defined over Netd(N(m0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Consider the two nets in Figure 4, representing two unbounded producer-  consumer systems, where prod is the action of producing an item, del of delivering an  item, cons of consuming an item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' It is easy to realize that relation  R1 = {(P1 ⊕C1 ⊕ k ·D1,P2 ⊕C2 ⊕ k1 ·D′  2 ⊕ k2 ·D′′  2)  �� k,k1,k2 ≥ 0,k = k1 + k2}  ∪{(P1 ⊕C′  1 ⊕ k ·D1,P2 ⊕C′  2 ⊕ k1 ·D′  2 ⊕ k2 ·D′′  2)  �� k,k1,k2 ≥ 0,k = k1 + k2}  7  P1  C1  prod  D1  del  C′  1  cons  P2  C2  prod  prod  D′  2  D′′  2  del  del  C′  2  cons  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Two net bisimilar unbounded producer-consumer systems  is a step bisimulation [13], containing the pair (P1 ⊕ C1,P2 ⊕ C2), where each pair  (m1,m2) ∈ R1 is such that |m1| = |m2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Moreover, relation  R2 = R1 ∪{(P1,P2),(C1 ⊕ D1,C2 ⊕ D′  2),(C1 ⊕ D1,C2 ⊕ D′′  2),(C′  1,C′  2),(C1,C2)}  ∪{(P1 ⊕ k ·D1,P2 ⊕ k1 ·D′  2 ⊕ k2 ·D′′  2)  �� k,k1,k2 ≥ 0,k = k1 + k2}  is a step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Note that if P1⊕C1  prod  −→P1⊕D1⊕C1, then P2⊕C2 can respond  with P2 ⊕C2  prod  −→P2 ⊕ D′  2 ⊕C2 and the two required pairs belong to R2, because (P1 ⊕  D1 ⊕C1,P2⊕D′  2 ⊕C2) ∈ R1 (and so also in R2) and (P1,P2) ∈ R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' the latter pair requires  that both (P1 ⊕ D1,P2 ⊕ D′  2) and (P1 ⊕ D1,P2 ⊕ D′′  2) belong to R2 and, by iterating the  firing of the transitions prod, that all the pairs in the last group belong to R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  In order to show that ∼s is an equivalence relation, we now list some useful proper-  ties of net bisimulation relations, whose proof is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For each P/T net N = (S,A,T), the following hold:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The identity relation I = {(m,m)  �� m ∈ M fin(S)} is a step net bisimulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' the inverse relation R−1 = {(m′,m)  �� (m,m′) ∈ R} of a step net bisimulation R is  a step net bisimulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' the relational composition R1 ◦R2 = {(m,m′′)  �� ∃m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (m,m′) ∈ R1 ∧(m′,m′′) ∈ R2}  of two step net bisimulations R1 and R2 is a step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' given a family {Ri}i∈I of step net bisimulations, their union ∪i∈IRi is a step net  bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For each P/T net N = (S,A,T), ∼s = �{R ⊆ M fin(S) × M fin(S)  ��  R is a step net bisimulation} is an equivalence relation and the largest step net bisimu-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Step net bisimilarity is a truly concurrent behavioral equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Consider the nets  in Figure 5, which are isomorphic to the net semantics of the FNM terms a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0,  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='C|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 (with C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= 0), respectively, according to the semantics in Section  8  a)  s1  a  b  s2  s3  b  a  b)  s4  s5  a  b  c)  s6  s7  a  b  s8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Three non-net equivalent net systems: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='C|b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 (with  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= 0)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Of course, s1 is not step net bisimilar to s4 ⊕ s5 because these two markings have  different size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' as a matter of fact, these two markings generate different partial orders  of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Moreover, s4 ⊕ s5 is not step net bisimilar to s6 ⊕ s7 because s4 ≁n s6: after  the execution of a, the reached markings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', θ and s8) are not step net bisimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' So,  step net bisimilarity not only observes the partial order of the actions performed, but  also the size of the distributed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In fact, in [10] we proved that ∼s coincides with  causal-net bisimilarity [6,9], in turn equivalent to structure-preserving bisimilarity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  3  FNM: Syntax and Net Semantics  In this section we briefly recall some definitions and theorems from [8], where the  reader can find more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Let L be a finite set of names, ranged over by a,b,c,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', also called the input  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let L be the set of co-names, ranged over by a,b,c,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', also called the output  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set L ∪L , ranged over by α,β,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', is the set of observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let  Act = L ∪ L ∪ {τ}, such that τ ̸∈ L ∪ L , be the finite set of actions (or labels),  ranged over by µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Action τ denotes an invisible, internal activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let C be a finite set  of process constants, disjoint from Act, ranged over by A,B,C,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', possibly indexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Finite-Net Multi-CCS (FNM for short) is the calculus whose terms are generated  from actions and constants as described by the following abstract syntax:  s ::= 0 |  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='t  | a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='s | s+ s  guarded terms  q ::= s |  C  sequential terms  t ::= q |  t |t  restriction-free terms  p ::= t | (νa)p  general terms  where we assume that a constant C is defined by a guarded process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', a process in  syntactic category s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' An FNM term p is an FNM process if the set Const(p)  of constants used by p is finite and each constant in Const(p) is equipped with a defining  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set of FNM processes is denoted by PFNM, and the set of its sequential  processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', of the processes in syntactic category q, by Pseq  FNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  9  Sync(α,α,τ)  σ ̸= ε  Sync(aσ,a,σ)  σ ̸= ε  Sync(a,aσ,σ)  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Synchronization relation Sync  The FNM operators are those of CCS [12,7], used in constrained manner, with the  addition of the strong prefixing operator: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='s is a strongly prefixed process, where the  strong prefix a is the first input action of a transaction that continues with the sequential  process s (provided that s can complete the transaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, the strong prefixing  operator allows for the creation of transitions labeled by an atomic sequence σ, which  is a sequence of visible actions, composed of inputs only, but possibly ending with an  output, that are executed atomically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' more precisely, σ ranges over the set of labels  A = {τ} ∪ L ∗ · (L ∪ L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0) can perform two transitions reaching 0:  one labeled by the atomic sequence ab, the other one by the atomic sequence ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A consequence of the fact that transitions may be labeled by atomic sequences is  the need for a new, more general, discipline of synchronization, which extends the basic  CCS rule [12,7] for synchronizing two complementary actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Two sequences σ1 and  σ2 can be synchronized, and the result is σ, if relation Sync(σ1,σ2,σ) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This  relation, formally defined in Table 1, holds if at least one of the two sequences is a  single output action, say σ1 = a, and the other one is either the complementary input  action a or an atomic sequence starting with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, it is not possible to synchronize  two atomic sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A well-formed FNM process [8] p is an FNM process satisfying a simple syntactic  condition, denoted by wf(p), ensuring that its executable atomic sequences are com-  posed of input actions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This ensures that a multi-party synchronization can take  place only among one leader, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the process performing the atomic sequence of in-  puts, and as many other components (the servants), as the length of the atomic sequence,  where each servant executes one output action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We will show that the Petri net associ-  ated with a well-formed FNM process p is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='1  Extended Terms  The FNM processes are built upon the set L ∪L , ranged over by α, of visible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We assume we also have sets L ′ = {a′ |a ∈ L } and L ′ = {a′| a ∈ L }, where L ′ ∪  L ′, ranged over by α′, is the set of auxiliary restricted actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', actions that are only  allowed to synchronize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By definition, each restricted action α′ corresponds to exactly  one visible action α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let G = L ∪L ′, ranged over by γ, be the set of input actions and  their restricted counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' G = L ∪L ′ is the set of output actions and their restricted  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set Actγ = G ∪G ∪{τ}, ranged over by µ (with abuse of notation), is  used to build the set of extended terms, whose syntax is defined as for FNM, where the  prefixes are taken from the set Actγ, the strong prefixes from the set G and the bound  action a is in L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' An extended general FNM term p = (νL)t is an extended process if  Const(p) is finite, each constant in Const(p) is defined and t is admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',  ∀a ∈ L , {a,a′} ̸⊆ fn(t),  10  dec(0) = θ  dec(p+q) = {p+q}  dec(µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p) = {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  dec(γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p,I) = {γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  dec(p|q) = dec(p)⊕dec(q)  dec(C) = {C}  dec((νa)p) = dec(p){a′/a}  a′ ∈ L ′  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Decomposition function  where the function fn(−), computing the free names, is defined on extended terms in  the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The admissibility condition expresses a sort of sanity check on any  restriction-free, extended term t: it is not possible that, for any action a ∈ L , there  are occurrences in t of both a and its associated restricted action a′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' this because each  action type can occur in t only in one of the two modalities: either restricted or unre-  stricted (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', normally visible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 is not admissible, while a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0|b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 is  admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By the notation ad(t) we mean that t is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  By Pγ  FNM we denote the set of all extended FNM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By Pγ,par  FNM we denote  the set of all restriction-free, extended FNM processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', those extended processes  in syntactic category t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By Pγ,seq  FNM we denote the set of all sequential, extended FNM  processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', those extended processes in syntactic category q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='2  Net Semantics  In this section, we summarize a technique (operational in style), proposed in [8], for  building an infinite P/T net for the whole of FNM, starting from a description of its  places and its net transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The resulting net NFNM = (SFNM,A ,TFNM) is such that,  for any p ∈ PFNM, the net system NFNM(dec(p)) statically reachable from the initial  marking dec(p) is a statically reduced P/T net, which is finite if p is well-formed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' such  a net system is denoted by Net(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The set of FNM places, ranged over by s, is SFNM = Pγ,seq  FNM \\{0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the set of all  sequential, extended FNM processes, except 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Function dec : Pγ  FNM → M fin(SFNM),  which maps extended processes into markings, is outlined in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Process 0 is  mapped to the empty marking θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A sequential process p is mapped to one place with  name p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This is the case of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p (where µ can be any action in Actγ), a constant C,  p + q and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p (where γ ∈ G = L ∪ L ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Parallel composition is interpreted as multi-  set union;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the decomposition of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0|a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 produces the marking a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0⊕ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 = 2 ·a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The decomposition of a general process (νa)p – where a ∈ L – generates the multiset  obtained from the decomposition of p, to which the substitution {a′/a} is applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' the  application of the substitution {a′/a} to a multiset is performed element-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We as-  sume that, in decomposing (νa)p, the choice of the restricted name is fixed by the rule  that associates with a visible action a its unique corresponding restricted action a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For any p ∈ Pγ  FNM, dec(p) is a finite multiset of places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  A marking m ∈ M fin(SFNM) is admissible, denoted by ad(m), if for all a ∈ L ,  {a,a′} ̸⊆ fn(m), where fn(m) = �  s∈dom(m) fn(s), with fn(θ) = /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A marking m ∈ M fin(SFNM) is complete if an FNM process p ∈ PFNM exists such  that dec(p) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  11  (pref)  {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  µ  −→dec(p)  (cons)  dec(p)  σ  −→m  {C}  σ  −→m  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= p  (sum1)  dec(p)  σ  −→m  {p+q}  σ  −→m  (s-pref)  dec(p)  σ  −→m  {γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  γ⋄σ  −→m  (s-com)  m1  σ1  −→m′  1 m2  σ2  −→m′  2  m1 ⊕m2  σ  −→m′  1 ⊕m′  2  ad(m1 ⊕m2)∧Sync(σ1,σ2,σ)  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Rules for net transitions (symmetric rule (sum2) omitted)  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A marking m ∈ M fin(SFNM) is admissible iff it is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Hence, this theorem states not only that function dec maps FNM processes to ad-  missible markings over SFNM, but also that dec is surjective over this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Well-behaved) A set of places S ⊆ SFNM is well behaved if for all s ∈ S  we have that wf(s) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the sequential FNM extended term s is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  In order to define the set TFNM of all the FNM net transitions, we need some auxil-  iary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let A γ = {τ}∪G ∗ ·(G ∪G ), ranged over by σ with abuse of notation,  be the set of labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' hence, a label σ can be the invisible action τ, or a (possibly empty)  sequence of inputs (or restricted inputs) followed by an input or an output (or its re-  stricted counterpart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let → ⊆ M fin(SFNM) × A γ × M fin(SFNM) be the least set of  transitions generated by the axiom and rules in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Let us comment on the rules of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Axiom (pref) states that if one token is  present in the place µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p, then a µ-labeled transition is derivable from marking {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p},  producing the marking dec(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This holds for any µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', for the invisible action τ, for  any visible action α as well as for any restricted action α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By rule (sum1), the transi-  tions from the place p+q are those from the marking dec(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' as p is sequential, dec(p)  is {p} if p ̸= 0, while, if p = 0, dec(p) = θ, but no transition is derivable from the empty  marking, so that the rule is really applicable only when dec(p) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Similarly, rule  (cons) states that the transitions derivable from {C} are those derivable from the place  {p}, if C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= p with p ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In rule (s-pref), γ may be any input action a or any restricted  action a′, and the auxiliary function γ ⋄σ returns γ if σ = τ, or γσ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This rule  requires that the premise transition dec(p)  σ  −→m be derivable by the rules, which is re-  ally possible only when dec(p) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Rule (s-com) requires that the transition pre-set  be admissible in order to avoid producing transitions that have no counterpart in the  LTS semantics of FNM, described in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Rule (s-com) explains how a synchronization  takes place: it is required that m1 and m2 perform synchronizable sequences σ1 and σ2,  producing σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' here we assume that relation Sync has been extended also to restricted ac-  tions in the obvious way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', a restricted output action a′ can be synchronized only with  its complementary restricted input action a′ or with an atomic sequence beginning with  a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' As an example, the net transition {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q,b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r}  τ  −→dec(p)⊕ dec(q)⊕ dec(r) is  derivable by the rules, as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  12  (pref)  {b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  b′  −→dec(p)  (s-pref)  {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p} ab′  −→dec(p)  (pref)  {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q}  a  −→dec(q)  (s-com)  {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q}  b′  −→dec(p)⊕dec(q)  (pref)  {b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r}  b′  −→dec(r)  (s-com)  {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q,b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r}  τ  −→dec(p)⊕dec(q)⊕dec(r)  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The proof of a net transition  Transitions with labels containing restricted actions must not be taken in the re-  sulting net, as we accept only transitions labeled over A = {τ} ∪ L ∗ · (L ∪ L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  However, they are useful in producing acceptable transitions, as two complementary  restricted actions can synchronize, producing a τ-labeled transition or shortening the  synchronized sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, in the example above, the derivable transition  {b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  b′  −→dec(p) is not an acceptable transition because its label is not in A , while  {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q,b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r}  τ  −→dec(p) ⊕ dec(q) ⊕ dec(r) is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, the P/T net for FNM is  the triple NFNM = (SFNM,A , TFNM), where the set  TFNM = {(m1,σ,m2)  �� m1  σ  −→m2 is derivable by the rules and σ ∈ A }  is obtained by filtering out those transitions derivable by the rules whose label σ con-  tains some restricted name a′ or a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We now want to show that for any finite, well-behaved set of places S ⊆ SFNM, the  set of transitions statically enabled at S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Given a place s ∈ S, by s ⊢ t we mean  that transition t = ({s},σ,m) is derivable by the rules in Table 3, hence with σ ∈ A γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set Ts = {t  �� s ⊢ t} is finite, for any s ∈ SFNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By induction on the axiom and rules in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Given a finite, well-behaved set of places S ⊆ SFNM, let T S  1 be �  s∈S Ts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the set  of all transitions, with a singleton pre-set in S, derivable by the rules with labeling in  A γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set T S  1 is finite, being the finite union (as S is finite) of finite sets (as Ts is finite  for each s ∈ S, by Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Let k ∈ N be the length of the longest label of any transition in T S  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' It is possible  to argue (details in [8]) that if a multi-party transition t is derivable by the rules from  the well-behaved set S, then its proof contains k synchronizations at most, each one be-  tween a transition (labeled with a sequence of inputs) and a singleton-pre-set transition  (labeled with a single output action);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' hence, at most k+1 participants can take part in a  multi-party synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, the set of all the transitions statically enabled at  a finite, well-behaved set S can be defined by means of a sequence of sets T S  i of transi-  tions, for 2 ≤ i ≤ k+1, where each transition t ∈ T S  i has a pre-set •t of size i, as follows:  T S  i = {(m1 ⊕ m2,σ,m′  1 ⊕ m′  2)  �� ad(m1 ⊕ m2),  ∃σ1∃γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (m1,σ1,m′  1) ∈ T S  i−1,(m2,γ,m′  2) ∈ T S  1 ,Sync(σ1,γ,σ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  13  Note that T S  2 is finite, because T S  1 is finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' inductively, T S  i+1, for 2 ≤ i ≤ k is finite,  because T S  i and T1 are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' So, the set TS of all the transitions statically enabled at S is  TS = {t  �� t ∈  k+1  �  i=1  T S  i ∧l(t) ∈ A },  where only transitions labeled over A are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' TS is finite, being a finite union  of finite sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' therefore, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If S ⊆ SFNM is a finite, well-behaved set of places, then set TS ⊆ TFNM of  all the transitions statically enabled at S is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (1/3 Semi-counter) Let us consider a semi-counter such that three occur-  rences of inc are needed to enable one dec, whence the name 1/3 semi-counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The  well-formed process p = (νc)A, where  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='(A|(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0))  is a 1/3 semi-counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The initial marking m0 is dec(p) = dec((νc)A) = dec(A){c′/c} =  {A}{c′/c} = {s1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' place s1 is the extended, sequential process A{c′/c}, where the con-  stant A{c′/c} is obtained by applying the substitution {c′/c} to the body of A:  A{c′/c}  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='(A{c′/c} |(c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Then, let S0 = dom(m0) = {s1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The set of transitions statically enabled at S0 is T S0  1  =  Ts1 = {t1}, where the only transition is t1 = {s1} inc  −→{s1,s2}, with s2 = c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0 +  c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, the new set of statically reachable places is S1 = {s1,s2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Note that s2  can produce two transitions in Ts2, namely t′ = {s2} c′c′dec  −→ θ and t′′ = {s2}  c′  −→θ, but  neither is labeled over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Since the longest label has length 3, we have to compute the  sets T S1  i  for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',4:  T S1  1  = {t1,t′,t′′},  T S1  2  = {t′′′}, where t′′′ = ({s2,s2},c′dec,θ),  T S1  3  = {t2}, where t2 = ({s2,s2,s2},dec,θ), whose proof is shown in Table 5,  T S1  4  = /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Hence, TS1 = {t1,t2}, as these two are the only transitions labeled over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' As t2  does not add any new reachable place, we have that S1 is the set of places statically  reachable from the initial marking, and TS1 is the set of transitions statically enabled at  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' This net is depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  The P/T net system associated with a process p ∈ PFNM is the subnet of NFNM  statically reachable from the initial marking dec(p), denoted by Net(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let p be a process in PFNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The P/T net system statically associated  with p is Net(p) = (Sp,Ap,Tp,m0), where m0 = dec(p) and  Sp = �dom(m0)⟩  computed in NFNM,  Tp = {t ∈ TFNM  �� Sp�t⟩},  Ap = {σ ∈ A  �� ∃t ∈ Tp such that l(t) = σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  14  (pref)  {dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0} dec  −→θ  (s-pref)  {c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0} c′dec  −→ θ  (s-pref)  {c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0} c′c′dec  −→ θ  (sum1)  {s2} c′c′dec  −→ θ  (pref)  {c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0}  c′  −→ θ  (sum2)  {s2}  c′  −→θ  (s-com)  {s2,s2} c′dec  −→ θ  (pref)  {c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0}  c′  −→θ  (sum2)  {s2}  c′  −→ θ  (s-com)  {s2,s2,s2} dec  −→θ  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The proof of a net transition, where s2 = c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0  s1  inc  s2  dec  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The P/T net for the 1/3 semi-counter  The following propositions present three facts that are obviously true by construction  of the net Net(p) associated with an FNM process p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For any p ∈ PFNM, Net(p) is a statically reduced P/T net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If dec(p) = dec(q), then Net(p) = Net(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For each restriction-free t ∈ PFNM and for each L ⊆ L :  i) If Net(t) = (S,A,T,m0), then, for any n ≥ 1, Net(tn) = (S,A,T,n·m0), where t1 = t  and tn+1 = t |tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ii) If Net((νL)t) = (S,A,T,m0), then Net((νL)(tn)) = (S,A,T,n ·m0), for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Definition 9 suggests a way of generating Net(p) with an algorithm based on the  inductive definition of the static reachability relation (see Definition 6): Start with the  initial set of places S0 = dom(dec(p)), and then apply the rules in Table 3 in order to  produce the set TS0 of transitions (labeled over A ) statically enabled at S0, as well as  the additional places statically reachable by means of such transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Then repeat this  procedure from the set of places statically reached so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' An instance of this procedure  was given in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' There are two problems with this algorithm:  the obvious halting condition is “until no new places are statically reachable”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' of  course, the algorithm terminates if we know that the set Sp of places statically  reachable from dom(dec(p)) is finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' additionally,  15  at each step of the algorithm, we have to be sure that the set of transitions derivable  from the current set of statically reachable places is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  We are going to prove only the first requirement — Sp is finite for any p ∈ PFNM  — because it implies also the second one for well-formed processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' As a matter of  fact, if p is well formed, then dom(dec(p)) is well behaved, and so is any set S of  places statically reachable from dom(dec(p)) (as proved in [8]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' since S ⊆ Sp, S is also  finite, and so, by Theorem 2, the set TS of transitions statically enabled at the finite,  well-behaved set S is finite, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For any p ∈ PFNM, let Net(p) = (Sp,Ap,Tp,m0) be defined as in Defini-  tion 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Then, the set Sp is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By induction on the static reachability relation =⇒∗ (details in [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' [8] For any well-formed FNM process p, Net(p) = (Sp,Ap,Tp,dec(p)) is  a finite P/T net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='3  Representing All Finite P/T Net  Theorem 4 ensures that only finite P/T nets can be represented by FNM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' It is  not completely obvious that all finite P/T nets can be represented by FNM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The translation from nets to processes defines a constant Ci in correspondence with  each place si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' the constant Ci has a summand cj  i for each transition tj, which is 0 when  si is not in the pre-set of tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The FNM process TFNM(N(m0)) associated with the finite  net system N(m0), labeled over L ∪ {τ}, has a bound name xj  i for each pair (si,tj),  where si is a place and tj is a transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' such bound names are used to force synchro-  nization among the components participating in transition tj with pre-set of cardinality  two or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Among the many places in the pre-set of tj, the one with least index i  (as we assume that places are indexed) plays the role of leader of the synchronization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  the corresponding leader constant Ci has a summand cj  i containing the atomic input se-  quence needed for the multi-party synchronization, such that each strong input prefix xj  h  (for h > i) is synchronized with the corresponding output xj  h performed by the servant  participant of index h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' in case •tj(si) ≥ 2, then the summand cj  i is actually a sum of xj  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0  with the atomic input sequence, so that one instance of Ci acts as the leader, while the  others are servants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Translating finite P/T nets into well-formed FNM processes) Given  A ⊆ L ∪ {τ}, let N(m0) = (S,A,T,m0) — with S = {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',sn}, T = {t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',tk}, and  l(tj) = µj — be a finite P/T net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Function TFNM(−), from finite P/T nets to well-formed  FNM processes, is defined as  TFNM(N(m0)) = (νL)(C1|···|C1  �  ��  �  m0(s1)  |···|Cn|···|Cn  �  ��  �  m0(sn)  )  where L = {x1  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',x1  n,x2  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',x2  n,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xk  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xk  n} is such that L∩A = /0, eachCi is equipped  with a defining equation Ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= c1  i + ··· + ck  i (with Ci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= 0 if k = 0), and each summand  cj  i , for j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',k, is equal to  16  0, if si ̸∈ •t j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='Π j, if •t j = {si};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  xj  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0, if •t j(si) > 0 and •t j(si′) > 0 for some i′ < i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', si is not the leader for the  synchronization on tj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  xj  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='··· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  i+1  �  ��  �  t j(si+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='··· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  n  �  ��  �  t j(sn)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='Π j, if •t j(si) = 1 and si is the leader of the synchro-  nization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', •t j(si′) > 0 for no i′ < i, while •t j(si′) > 0 for some i′ > i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  xj  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+xj  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='··· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  i  �  ��  �  t j(si)−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='··· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  i+1  �  ��  �  t j(si+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='xj  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='··· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x j  n  �  ��  �  t j(sn)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='Π j, otherwise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', si is the leader  and •t j(si) ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Finally, process Π j is C1|···|C1  �  ��  �  t•  j (s1)  |···|Cn|···|Cn  �  ��  �  t•  j (sn)  , meaning that Π j = 0 if t•  j = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Note that TFNM(N(m0)) is an FNM process: in fact, the restriction operator occurs  only at the top level, applied to the parallel composition of a finite number of con-  stants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' each constant has a body that is sequential and restriction-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Note also that  TFNM(N(m0)) is a well-formed process: in fact, each strong prefix is a bound input xj  i ,  and any sequence ends with an action µj ∈ A, which is either an input or τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' hence, no  atomic sequence ends with an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, the following proposition holds by  Theorem 4 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For any finite P/T Petri net N(m0), the net Net(TFNM(N(m0))) is a  finite, statically reduced P/T net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Representability Theorem)[8] Let N(m0) = (S,A,T,m0) be a finite, stat-  ically reduced P/T net system such that A ⊆ L ∪{τ}, and let p = TFNM(N(m0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Then,  Net(p) is isomorphic to N(m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  4  Step Net Bisimilarity is a Congruence  Now we prove that step net bisimilarity ∼s is a congruence for the FNM operators,  or, in other words, that the FNM operators are compositional up to ∼s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We extend the  definition of step net bisimilarity to FNM terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', we write p ∼s q, meaning that  there exists a step net bisimulation R relating the markings dec(p) and dec(q) in the net  obtained by the union of Net(p) and Net(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In the following, given an FNM term r, by  Ir we denote the relation Ir = I1  r ∪I2  r ∪I3  r , where  I1  r = {(m,m)  �� m ∈ [dec(r)⟩},  I2  r = {(m′,m′)  �� ∃G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' •G = m′,m′ ⊆ m ∈ [dec(r)⟩}, and  I3  r = {(m′′,m′′)  �� m′′ ∈ [m′⟩,(m′,m′) ∈ I2  r }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Clearly, Ir is a step net bisimulation containing the pair (dec(r),dec(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Congruence for strong prefixing and choice) Let p and q be FNM  guarded processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', processes in syntactic category s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If p ∼s q, then  (i)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p ∼s a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q,  for all a ∈ L ,  (ii) p + r ∼s q + r, for each process r in syntactic category s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Assume R is a step net bisimulation containing the pair ({p},{q}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (If p = 0 = q,  then R = {(θ,θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=') Note that, since the considered process terms are sequential, the  initial performable steps are actually composed of a single transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Case (i) can be proven by considering relation R1 = R∪{({a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p},{a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q})}, which is  a step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In fact, transition {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p} a⋄σ  −→m1 is derivable, by rule (s-pref),  only if {p}  σ  −→m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' As ({p},{q}) ∈ R, also {q}  σ  −→m2 with (m1,m2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, also  {a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q} a⋄σ  −→m2 with (m1,m2) ∈ R1 and ({a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p},{a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q}) ∈ R1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The symmetric  case when a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q moves first is analogous and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Case (ii) can be proven, for each r in syntactic category s, by showing that R2 =  {({p + r},{q + r})} ∪ R∪ Ir is a step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In fact, if {p + r}  σ  −→m1, then  this is due to {p}  σ  −→m1 or to {r}  σ  −→m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In the former case, since ({p},{q}) ∈ R,  we have {q}  σ  −→m2 with (m1,m2) ∈ R, so that {q + r}  σ  −→m2 with (m1,m2) ∈ R2 and  ({p+r},{q+r}) ∈ R2, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In the latter case, we have that also {q+r}  σ  −→m1  with (m1,m1) ∈ Ir and so also (m1,m1) ∈ R2, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The symmetric case when  q + r moves first is analogous and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Congruence for prefixing and parallel composition) Let p and q  be restriction-free FNM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If p ∼s q, then  (i) µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p ∼s µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q, for all µ ∈ Act,  (ii) p|r ∼s q|r, for any restriction-free r ∈ PFNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Assume R is a step net bisimulation containing the pair (dec(p),dec(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For  case (i), since the considered process terms are sequential, the initial performable  steps are actually composed of a single transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Relation R3 = R∪{({µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p},{µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q})}  is a step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In fact, if {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p}  µ  −→dec(p), then {µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q}  µ  −→dec(q) with  (dec(p),dec(q)) ∈ R3 and ({µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='p},{µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q}) ∈ R3, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The symmetric case when  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='q moves first is analogous and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  For case (ii), relation R4 = R ∪ Ir ∪ {(m1⊕m,m2⊕m)  �� (m1,m2) ∈ R,(m,m) ∈ Ir}  is a step net bisimulation containing the pair (dec(p) ⊕ dec(r),dec(q) ⊕ dec(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In  fact, R4 is composed of R, as well as of Ir, which are already step net bisimulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  now, consider a pair (m1 ⊕ m,m2 ⊕ m) ∈ R4, belonging to the third group, and assume  m1 ⊕ m[G1⟩m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' we have to consider the following three cases:  m1[G1⟩m′  1, so that m1 = m′  1 ⊕ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In such a case, since (m1,m2) ∈ R, we have  that there exists G2 such that m2[G2⟩m′  2, l(G1) = l(G2), (•G1,•G2) ∈ R (hence,  (•G1,•G2) ∈ R4, too) and (m′  1,m′  2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, m2 ⊕m[G2⟩m′  2 ⊕m, with (m′  1 ⊕  m,m′  2 ⊕ m) ∈ R4, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  m[G1⟩m′, so that m1 = m1 ⊕ m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In such a case, since (m,m) ∈ Ir, we have that  also m[G1⟩m′, with (•G1,•G1) ∈ Ir (hence, (•G1,•G1) ∈ R4, too) and (m′,m′) ∈ Ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Therefore, m2 ⊕ m[G1⟩m2 ⊕ m′, with (m1 ⊕ m′,m2 ⊕ m′) ∈ R4, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  there exist G′  1 and G such that m1[G′  1⟩m′  1, m[G⟩m′, •G1 = •G′  1 ⊕ •G, G•  1 = G′•  1 ⊕G•  and MSync(l(G′  1),l(G),l(G1)), where this relation is defined in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In such a  case, as (m1,m2) ∈ R, we have that there exists G′  2 such that m2[G′  2⟩m′  2, l(G′  1) =  l(G′  2), (•G′  1,•G′  2) ∈ R and (m′  1,m′  2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, there exists a step G2 such that  G2 = •G′  2 ⊕ •G, G•  2 = G′•  2 ⊕G•, MSync(l(G′  2),l(G),l(G2)), so that l(G1) = l(G2)  (because the same proof used to derive MSync(l(G′  1),l(G),l(G1)) can be adapted  18  MSync(M1,M2,M1 ⊕M2)  Sync(σ1,σ2,σ) MSync(M1,M2,M)  MSync(M1 ⊕{σ1},M2 ⊕{σ2},M ⊕{σ})  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Step synchronization relation, where M denotes a multiset of labels  to derive MSync(l(G′  2),l(G),l(G2))), m2⊕m[G2⟩m′  2⊕m′ with (•G1,•G2) ∈ R4 (be-  cause (•G′  1,•G′  2) ∈ R and (•G,•G) ∈ Ir), and, moreover, (m′  1 ⊕ m′,m′  2 ⊕ m′) ∈ R4  (because (m′  1,m′  2) ∈ R and (m′,m′) ∈ Ir), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The symmetric case when m2 ⊕ m moves first is analogous, and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  Note that it is easy to generalize the congruence property for parallel composition  as follows: If pi ∼s qi for i = 1,2, then p1 | p2 ∼s q1 |q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In fact, if Ri is a step net  bisimulation containing the pair (dec(pi),dec(qi)), for i = 1,2, then it is easy to see  that the relation R1 ∪ R2 ∪ {(m1 ⊕ m2,m′  1 ⊕ m′  2)  �� (m1,m′  1) ∈ R1,(m2,m′  2) ∈ R2} is a  step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Congruence for restriction)  Let p and q be general FNM processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If p ∼s q, then (νa)p ∼s (νa)q for all a ∈ L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let R be a step net bisimulation containing the pair (dec(p),dec(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Relation  R5 = {(m1{a′/a},m2{a′/a})  �� (m1,m2) ∈ R} is the required step net bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In  fact, assume m1{a′/a}[G1⟩m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By the net semantics, this is possible only if m1[G′  1⟩m′  1,  where G1 = G′  1{a′/a} = (•G′  1{a′/a},l(G′  1),G′•  1 {a′/a}), m1 = m′  1{a′/a} and l(G′  1)  does not contain any occurrence of action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' As (m1,m2) ∈ R, we have that a step  G′  2 exists such that m2[G′  2⟩m′  2, with l(G′  1) = l(G′  2), (•G′  1,•G′  2) ∈ R and (m′  1,m′  2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Therefore, by the net semantics it is possible to derive m2{a′/a}[G2⟩m2, where G2 =  G′  2{a′/a} and m2 = m′  2{a′/a}, with the property that l(G1) = l(G2), (•G1,•G2) ∈ R5  and (m1,m2) ∈ R5, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The symmetric case when m2{a′/a} moves first is anal-  ogous, and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Still there is one construct missing: recursion, defined over guarded terms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Consider an extension of FNM where terms can be constructed using variables, such as  x,y,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=': this defines an “open” FNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (Open FNM) Let Var = {x,y,z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='} be a finite set of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The  FNM open terms are generated from actions, constants and variables by the following  abstract syntax (using three syntactic categories):  s ::= 0 |  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='t  | a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='s | s+ s  guarded open terms  q ::= s |  C  |  x  sequential open terms  t ::= q |  t |t  restriction-free open terms  p ::= t | (νa)p  general open terms  where x is any variable taken from Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The open net semantics for open FNM extends  the net semantics in Section 3 with Net(x) = ({x}, /0, /0,{x}), so that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the semantics  of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x is the net ({a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x,x},{a}, {(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x,a,x)},a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  19  Sometimes we use the notation p(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn) to state explicitly that term p is open on  the tuple of variables (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, p1(x) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x)+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x and p2(x) =  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x+ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='x+ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0) are open guarded FNM terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Step net bisimulation equivalence can be extended to open terms as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' An  open term p(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn) can be closed by means of a substitution  p(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn){r1/x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',rn/xn}  with the effect that each occurrence of the variable xi (within p and the body of each  constant in Const(p)) is replaced by the closed FNM sequential process ri, for i =  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, p1(x){d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0/x} = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0)+ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  A natural extension of step net bisimilarity ∼s over open guarded terms is as fol-  lows: p(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn) ∼s q(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn) if for all tuples of (closed) FNM sequential terms  (r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',rn),  p(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn){r1/x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',rn/xn} ∼s q(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn){r1/x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',rn/xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', it is easy to see that p1(x) ∼s p2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' As a matter of fact, for all r,  p1(x){r/x} = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0+ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r)+ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r ∼s d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='0) = p2(x){r/x},  which can be easily proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  For simplicity’s sake, let us now restrict our attention to open guarded terms us-  ing a single undefined variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We can recursively close an open term p(x) by means  of a recursively defined constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For instance, A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= p(x){A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The resulting process  constant A is a closed FNM sequential process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By saying that step net bisimilarity is  a congruence for recursion we mean what is stated in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' For simplicity’s  sake, in the following a term p open on a variable x is no longer annotated as p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let p and q be two open guarded FNM terms, with one variable x at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Let A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= p{A/x}, B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= q{B/x} and p ∼s q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Then A ∼s B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Let R = {(m{A/x},m{B/x})  �� m ∈ M (Sop)} be our candidate relation, where  Sop = Pseq  oFNM \\ {0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', the set of all sequential, open FNM processes, except 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='1  Note that when m is x, we get (A,B) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The proof that R is a step net bisimulation  (up to ∼s) is not difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By symmetry, it is enough to prove that if m{A/x}[G1⟩m1,  then there exists G2 such that m{B/x}[G2⟩m2 with (•G1,•G2) ∈ R, l(G1) = l(G2) and  m1 ∼s R ∼s m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' The proof proceeds by induction on the size of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' When |m| = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',  m = s for some (open) place s, the proof is by induction on the definition of the net for  s{A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Note that, since in this case the involved terms are sequential, the steps are  actually composed of a single transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  s = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In this case, s{A/x} = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r{A/x}  µ  −→dec(r){A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Similarly, s{B/x} =  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r{B/x}  µ  −→dec(r){B/x}, and (dec(r){A/x},dec(r){B/x}) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  s = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In this case, s{A/x} = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r{A/x} a⋄σ  −→m1 if r{A/x}  σ  −→m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Since r is guarded,  r{A/x}  σ  −→m1 is possible only if r  σ  −→m with m1 = m{A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' So, r{B/x}  σ  −→m{B/x}  is derivable, too, and also s{B/x} a⋄σ  −→m{B/x}, with (m{A/x},m{B/x}) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  1Note that we are not considering extended terms, but only ‘normal’ FNM sequential terms,  because the restriction operator cannot occur within the body of a recursively defined constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Hence, each marking m over Sop is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  20  s = D, with D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' So, s{A/x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= r{A/x} and s{B/x} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= r{B/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' If s{A/x}  σ  −→m1,  then this is possible only if r{A/x}  σ  −→m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Since r is guarded, r{A/x}  σ  −→m1 is  possible only if r  σ  −→m with m1 = m{A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, also r{B/x}  σ  −→m{B/x} is  derivable, and also s{B/x}  σ  −→m{B/x}, with (m{A/x},m{B/x}) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  s = s1 + s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In this case, s{A/x} = s1{A/x} + s2{A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' A transition from s{A/x},  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=', s1{A/x} + s2{A/x}  σ  −→m1, is derivable only if si{A/x}  σ  −→m1 for some i =  1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Without loss of generality, assume the transition is due to s1{A/x}  σ  −→m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Since s1 is guarded, transition s1{A/x}  σ  −→m1 is derivable because s1  σ  −→m, with  m1 = m{A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, also s1{B/x}  σ  −→m{B/x} is derivable, as well s{B/x} =  s1{B/x} + s2{B/x}  σ  −→m{B/x}, with (m{A/x},m{B/x}) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  s = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We have s{A/x} = A and s{B/x} = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' We prove that for each A  σ  −→m1, there  exists m2 such that B  σ  −→m2 with m1 ∼s R ∼s m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By hypothesis, A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= p{A/x},  hence also p{A/x}  σ  −→m1 is a transition in the net for p{A/x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' since p is guarded,  p{A/x}  σ  −→m1 is possible only if p  σ  −→m with m1 = m{A/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Therefore, also  p{B/x}  σ  −→m{B/x} is derivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' But we also have that p ∼s q, so p  σ  −→m can  be matched by q  σ  −→m′ with m ∼s m′ (note that the requirement on the pre-sets of  the two transitions is already satisfied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence, q{B/x}  σ  −→m′{B/x} is derivable  with m{B/x} ∼s m′{B/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Since B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='= q{B/x}, also B  σ  −→m′{B/x} is a transition  with m1 ∼s m{A/x} R m{B/x} ∼s m′{B/x}, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  When |m| = n+1, then a place s and a marking m′ exist such that m = s⊕m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Hence,  we have that (s{A/x},s{B/x}) ∈ R and (m′{A/x},m′{B/x}) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' By induction (as  |s| = 1 and |m′| = n), we have that s{A/x} ∼s s{B/x} as well as m′{A/x} ∼s m′{B/x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Hence, by Theorem 1, there exist (open) FNM restriction-free processes p1, p2, q1,q2  such that s{A/x} = dec(p1), s{B/x} = dec(p2), m′{A/x} = dec(q1), m′{B/x} = dec(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Therefore, the thesis follows by compositionality of ∼s w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' parallel composition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  the comment after Proposition 11), because m{A/x} = dec(p1 |q1) and m{B/x} =  dec(p2 |q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  ✷  The extension to the case of open terms with multiple undefined constants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',  p(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=',xn) can be obtained in a standard way [12,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  5  Conclusion  Step net bisimulation is a very simple relation, extending ordinary step bisimulation  [13] on Petri nets with two additional conditions: the related markings must have the  same size and the pre-sets of the two matching steps must be related as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' With these  minimal additions, its definition is still very manageable and also very sensible as we  proved in [10] to coincide with causal-net bisimilarity [9] (or, equivalently, structure-  preserving bisimilarity [6]), a truly concurrent behavioral equivalence fully respecting  causality and the branching structure of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  The decidability of step net bisimilarity over finite (unbounded) P/T nets is an open  problem [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Step net bisimilarity is decidable on bounded nets, because causal-net  21  bisimilarity has been proved decidable in (large) exponential time on that class of nets  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' However, we expect that our simpler characterization of ∼cn in term of step net  bisimilarity ∼s may offer a more efficient algorithm, yet exponential as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Step net  bisimilarity is decidable on BPP nets, because on this class of nets it coincides with  team bisimilarity [9], which is decidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content='  Van Glabbeek [6] argued that structure-preserving bisimilarity is the most appro-  priate behavioral equivalence for Petri nets, as it is the only one respecting a list of  desirable requirements he proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' Among these, there is ‘compositionality’, up to  structure-preserving bisimilarity, of the operators (recursion not considered) of the pro-  cess algebra CCSP, that Olderog proposed in his monograph [14] and equipped with a  safe net semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In this paper we have extended his result by proving that step net  bisimilarity, which is an alternative, simpler characterization of structure-preserving  bisimilarity, can be used to give a compositional semantics of the process algebra FNM  [8], which truly represents all (and only) the finite P/T nets, up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
+page_content=' In this  way, we have obtained for the first time a compositional semantics, fully respecting  causality and the branching structure of systems, for the class of all the finite P/T Petri  nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfYAoc/content/2301.04483v1.pdf'}
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+Condensed Matter Physics, 2022, Vol. 25, No. 4, 44201: 1–6
+DOI: 10.5488/CMP.25.44201
+http://www.icmp.lviv.ua/journal
+Rapid communication
+Apparent molar volume anomaly in water-dimethyl
+sulfoxide liquid mixtures. Molecular dynamics
+computer simulations
+M. Aguilar
+1, H.Dominguez
+2, O. Pizio
+1 ∗
+1 Instituto de Química, Universidad Nacional Autónoma de México, Circuito Exterior, 04510, Cd. Mx., México
+2 Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Circuito Exterior,
+04510, Cd. Mx., México
+Received August 9, 2022, in final form October 19, 2022
+We have studied the composition dependence of density of liquid water-DMSO mixtures at different temper-
+atures by using the isobaric-isothermal (NPT) molecular dynamics computer simulations. The non-polarizable
+semi-flexible, P1 and P2 models for the DMSO molecule combined with the TIP4P-2005 water model are con-
+sidered. The excess mixing volume and the apparent molar volumes of the species are reported. We have es-
+tablished that the P1-TIP4P-2005 model for the mixture provides a very good description of the location of the
+minimum of apparent molar volume for DMSO species indicating the anomaly. Most important is that the tem-
+perature interval where the hydrophobic effect exists, is correctly captured with this modelling, in contrast to
+the P2-TIP4P-2005 model.
+Key words: molecular dynamics, water-dimethylsolfoxide mixtures, apparent molar volume
+Liquid mixtures of dimethyl sulfoxide (DMSO) and water are widely used in organic chemistry,
+chemical engineering, and biomedical applications. In this rapid communication we refer to the static
+light scattering experiment performed on aqueous solutions of dimethyl sulfoxide (DMSO) at various
+compositions and at two temperatures, 20◦C and 50◦C [1]. In the concentration range around 10 mol
+percent of DMSO, an abnormal maximum of scattered light was detected, the intensity of which decreases
+with an increase of temperature. This maximum has been attributed to the hydrophobic effect and has been
+interpreted in terms of the minimum of apparent molar volume of DMSO upon concentration. Inspired
+by this report, we have undertaken the study of the composition dependence of density of water-DMSO
+liquid mixtures at 298.15 K, 318.15 K and 338.15 K by using molecular dynamics computer simulations
+complementing our recent study performed at 298.15 K [2]. Our focus is on the capability of simple
+non-polarizable models for DMSO and water to reproduce the experimental observations by capturing
+peculiar composition interval in the restricted temperature range.
+Previously, binary mixtures of water with DMSO, in particular at a low DMSO concentration, were
+the subject of study in different laboratories [2–7]. Specifically, in [3, 4] the focus is on the composition
+trends of certain properties at a single temperature 300 K. The NPT setup was used. On the other
+hand, the NVT simulations at 298.15 K, 318.15 K and 338.15 K and at a fixed DMSO molar fraction
+(𝑋dmso = 0.055) were performed. The experimental density values were assumed. In this study, we
+have chosen quite popular, nonpolarizable, P1 and P2 models for DMSO species [8], from a huge set
+of force fields for DMSO [9, 10], and well tested TIP4P-2005 water model [11]. Technical details of
+our isobaric-isothermal molecular dynamics computer simulations were described very recently in the
+previous publication on the topic [2].
+∗Corresponding author: oapizio@gmail.com.
+This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution
+of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
+44201-1
+arXiv:2301.01547v1  [cond-mat.soft]  4 Jan 2023
+
+M. Aguilar, H. Dominguez, O. Pizio
+0
+0.2
+0.4
+0.6
+0.8
+1
+Xdmso
+990
+1020
+1050
+1080
+1110
+ρ (kg/m
+3)
+T = 318.15K
+T = 298.15K
+red - P2
+blue - P1 water - TIP4P-2005
+0
+0.2
+0.4
+0.6
+0.8
+1
+Xdmso
+980
+1000
+1020
+1040
+1060
+1080
+ρ(kg/m
+3)
+P2
+P1
+water - TIP4P-2005
+T = 338.15K
+Figure 1. (Colour online) Panel a: Composition dependence of the density of water-DMSO mixtures from
+the NPT MD simulations of P2-TIP4P-2005 model (solid red lines), P1-TIP4P-2005 model (solid blue
+lines) in comparison with the experimental data (dashed lines with triangles) at 298.15 K [12] and at
+318.15 K (dashed line with squares) [13]. Panel b: Similar to the panel a, but at 338.15 K. Experimental
+data (dashed line with triangles) are from [14].
+As common, the strategy of exploration is to describe a desired property on composition and to capture
+its deviation from ideality as well. We do that for the mixture density at each fixed temperature. A set of
+results is shown in figure 1. The panel a of this figure refers to 𝑇 = 298.15 K and 𝑇 = 318.15 K, whereas
+the panel b illustrates the results at 𝑇 = 338.15 K. This temperature was also chosen in [5] due to the
+availability of experimental data. From the inspection of the results, we observe that the P1-TIP4P-2005
+and P2-TIP4P-2005 are quite accurate in the interval of composition from pure water, 𝑋dmso = 0, up to
+𝑋dmso ≈ 0.2. At a higher DMSO content in the mixture, the simulation data begin to deviate from the
+experimental results. The maximum density at a certain composition, 𝑋dmso ≈ 0.58, is captured by both
+models in question. Nevertheless, it can be seen that the P1-TIP4P-2005 model performs much better than
+its P2-TIP4P-2005 counterpart. This latter model overestimates the density in the interval above 𝑋dmso >
+0.2. Moreover, the deviation from the experimental points does not decrease while the temperature grows
+from 298.15 K up to 338.15 K. The P1-TIP4P-2005 model exhibits a small inaccuracy for density at
+“intermediate” compositions, but performs much better than P2-TIP4P-2005. It is difficult to judge the
+precision of computer simulations data at low 𝑋dmso values at the scale in the figure 1. Better insights
+follow from the excess properties.
+We describe geometric aspects of mixing of the species in terms of the excess mixing volume
+defined as common, Δ𝑉mix = 𝑉mix − 𝑋𝑑𝑉𝑑 − (1 − 𝑋𝑑)𝑉𝑤, where 𝑉mix, 𝑉𝑑 and 𝑉𝑤 refer to the molar
+volume of the mixture and of the individual components, DMSO and water, respectively. Apparently,
+this property is not strongly dependent on temperature in the interval we deal with. This follows from
+the experimental results shown in figure 2. The mixing volume slightly decreases in magnitude upon
+increasing temperature, as expected. In general terms, computer simulation results show that the excess
+mixing volume is overestimated in the framework of models assumed. Thus, the mixture of water and
+DMSO species from simulations is predicted to be more non-ideal than its laboratory counterpart in
+almost entire composition range at 𝑇 = 298.15 K. The Δ𝑉mix values from simulations decrease in
+magnitude in agreement with experimental trends. However, the location of the minimum along 𝑋dmso
+axis from simulations and experiments is slightly different. Finally, one can observe that the performance
+of the P1-TIP4P-2005 model is slightly better compared to the P2-TIP4P-2005 model.
+In order to discern the contribution of each species of the mixture into the mixing volume, it is
+common to resort to the excess partial molar volumes as we have done recently [2]. However, similar
+insights into the geometric aspects of mixing on composition, both from experiments and simulations,
+can be obtained by resorting to the notion of the apparent molar volume of species, rather than to the
+partial molar volumes.
+44201-2
+
+Water-DMSO liquid mixtures
+0
+0.2
+0.4
+0.6
+0.8
+1
+Xdmso
+-1.2
+-0.9
+-0.6
+-0.3
+0
+∆Vmix (cm
+3/mol)
+P2 - red lines
+P1 - blue lines
+black - experimental  data
+T = 298.15K
+T = 338.15K
+lines with circles
+lines
+Figure 2. (Colour online) A comparison of the composition dependence of the excess mixing volume of
+water-DMSO mixtures for two DMSO models, P1 and P2, combined with TIP4P-2005 water model, with
+the experimental data at 298.15 K ([12]—black triangles) and at 338.15K ([14]—black squares).
+0
+0.2
+0.4
+0.6
+0.8
+1
+Xdmso
+66
+68
+70
+72
+74
+Vφ
+(d) (cm
+3/mol)  - DMSO
+red lines - P2
+blue lines - P1
+black - experimental data
+T = 298.15K
+T = 338.15K
+T = 318.15K
+intermediate lines
+top lines
+bottom lines
+Figure 3. (Colour online) A comparison of the composition dependence of the apparent molar volume of
+DMSO from simulations with the experimental data marked as triangles from [12] at 298.15 K, from [13]
+at 318.15 K and from [14] at 338.15 K.
+The apparent molar volume for each species, according to the definition [15], is: 𝑉 (𝑤)
+𝜙
+= 𝑉𝑤 +
+Δ𝑉mix/(1 − 𝑋𝑑) and 𝑉 (𝑑)
+𝜙
+= 𝑉𝑑 + Δ𝑉mix/𝑋𝑑. We elaborated the experimental density data from [12, 13]
+and the results from our simulations to construct the plots shown in figures 3 and 4. The most important
+and illuminating results for the apparent molar volume for DMSO species, 𝑉 (𝑑)
+𝜙 , are shown in figure 3.
+Namely, we observe that both models in question predict the existence of miminum at certain composition
+𝑋dmso. The points describing the location of the minimum at different temperatures from simulations and
+experiments are joined by respective lines. Evolution of the minimum with temperature is appropriately
+captured by the P1-TIP4P-2005 model in contrast to the P2-TIP4P-2005 model. Moreover, the P1-
+TIP4P-2005 model correctly predicts that the hydrophobic trends diminish with increasing temperature.
+Performance of the P2-TIP4P-2005 model in this aspect contradicts the experimental trends and is not
+satisfactory. We have obtained the minimum for 𝑉 (𝑑)
+𝜙
+for this model, even at a higher temperature equal
+to 358.15 K (the respective plot is not shown for economy of space).
+Concerning the behaviour of the apparent molar volume of water species, 𝑉 (𝑤)
+𝜙
+, upon composition
+(figure 4), it is worth to mention a rather satisfatory performance of the P1-TIP4P-2005 model in the
+44201-3
+
+M. Aguilar, H. Dominguez, O. Pizio
+entire composition range, especially for higher temperatures. The respective figure at 298.15 K is given
+in [2] (figure 3d). It is necessary to mention, however, that the experimental results for DMSO-rich (more
+specifically, extremely rich) mixtures are difficult to obtain precisely, because high-purity DMSO is an
+exceedingly hygroscopic solvent, the discussion of this issue is given in [12].
+To conclude, we have established that the P1 and P2 models for DMSO combined with the TIP4P-
+2005 water model yield a minimum of the apparent molar volume for DMSO species at low values
+for 𝑋dmso. This behavior is the manifestation of hydrophobic effects in these mixtures at a specific
+composition interval. However, it appears that the predictions from P1-TIP4P-2005 model agree better
+with the experimental trends than the ones from P2-TIP4P-2005 model. Moreover, the P1-TIP4P-2005
+model predicts the existence of a minimum of 𝑉 (𝑑)
+𝜙
+for DMSO within the correct temperature interval
+from 298.15 K up to 338.15 K. At this highest temperature, the minimum almost disappears, as deduced
+from experimental data. By contrast, P2-TIP4P-2005 model does not provide an accurate estimate for a
+peculiar composition interval and does not yield a decay for the hydrophobicity trends on temperature.
+In various publications this model is claimed to be the best, see for example [18]. Here, we show that this
+conclusion is valid in certain aspects at a room temperature only. In addition, it is important to mention
+that the combination of the OPLS model for DMSO with TIP4P-2005 model for water does not provide
+a correct composition dependence of density at room temperature, cf. figure 2 of [2] and at temperatures
+of this study. In consequence, the anomaly of the apparent molar volume is not captured by this type of
+model at all. These results are available upon request.
+0
+0.2
+0.4
+0.6
+0.8
+1
+Xdmso
+16
+17
+18
+Vφ
+(w) (cm
+3/mol)  - water
+T = 318.15K
+P2
+P1
+experiment (Ref.[11])
+a
+0
+0.2
+0.4
+0.6
+0.8
+1
+Xdmso
+16
+17
+18
+Vφ
+(w) (cm
+3/mol)  - water
+P2
+P1
+T = 338.15K
+experiment (Ref.[12])
+b
+Figure 4. (Colour online) Panels a and b: A comparison of the composition dependence of the apparent
+molar volume of water from simulations with the experimental data (triangles [13]) at 318.15 K and [14]
+at 338.15 K. Similar plot at 298.15 K is shown as figure 3d of [2].
+Previously, we explored the minimum of the methanol apparent molar volume on composition for
+water-methanol mixtures [16] at 298.15 K, see also [17] for the interpretation of the observed phenomena.
+This kind of system is the mixture of two hydrogen bonded liquids. By contrast, in the systems of the
+present study there is no bonding between the DMSO solutes in water. Apparently, the temperature trends
+should be slightly different for these two classes of systems.
+In addition, it seems intriguing to investigate the evolution of trends of hydrophobicity in water-DMSO
+mixtures upon changing the external pressure as well. One can be guided by important experimental
+observations, see e.g., [19, 20]. More generally, from the simulations perspective it is challenging
+to construct a more complete map of anomalies, in pressure, temperature and composition variables,
+that would involve, for example, the temperature of maximum density and the minimum of isothermal
+compressibility, even by using the non-polarizable models. These data would serve as a useful benchmark
+for the following development of polarizable models. In the case of water-DMSO mixtures, certain
+progress has been reached at this level of modelling in [21, 22]. However, the BK3 polarizable water
+model seems to be a more convenient starting point, because the principal anomalies of pure water are
+quite well captured in this framework [23, 24]. Some of these issues are under study in our laboratory.
+44201-4
+
+Water-DMSO liquid mixtures
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+44201-5
+
+M. Aguilar, H. Dominguez, O. Pizio
+Аномалiя видимого молярного об’єму рiдких сумiшей
+вода-диметилсульфоксид. Комп’ютерне моделювання
+методом молекулярної динамiки
+M. Агiлар
+1, Е. Домiнгес
+2, O. Пiзiо
+1
+1 Iнститут хiмiї, Нацiональний автономний унiверситет Мехiко, Circuito Exterior, 04510, Мехiко, Мексика
+2 Iнститут матерiалознавства, Нацiональний автономний унiверситет Мехiко, Circuito Exterior, 04510,
+Мехiко, Meксика
+Дослiджено концентрацiйну залежнiсть густини рiдких сумiшей вода-диметилсульфоксид (ДМСО) при рi-
+зних температурах з використанням комп’ютерного моделювання методом iзобарично-iзотермiчної мо-
+лекулярної динамiки. Розглядаються неполяризовнi напiвгнучкi моделi P1 i P2 для молекули ДМСО у по-
+єднаннi з моделлю TIP4P-2005 для води. Наведено результати обчислень для надлишкового об’єму змi-
+шування та видимих молярних об’ємiв компонент сумiшi. Встановлено, що модель P1-TIP4P-2005 для цiєї
+сумiшi забезпечує дуже добрий опис положення мiнiмуму видимого молярного об’єму компоненти ДМСО,
+який вказує на аномалiю. Найбiльш важливим є те, що вибрана модель, на вiдмiну вiд її вiдповiдника P2-
+TIP4P-2005, правильно передбачає наявнiсть температурного iнтервалу, в якому iснує гiдрофобний ефект.
+Ключовi слова: молекулярна динамiка, сумiшi вода-диметилсульфоксид (ДМСО), видимий молярний
+об’єм
+44201-6
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf,len=481
+page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' 4, 44201: 1–6  DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='44201  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='ua/journal  Rapid communication  Apparent molar volume anomaly in water-dimethyl  sulfoxide liquid mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Molecular dynamics  computer simulations  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Aguilar  1, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='Dominguez  2, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Pizio  1 ∗  1 Instituto de Química, Universidad Nacional Autónoma de México, Circuito Exterior, 04510, Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', México  2 Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Circuito Exterior,  04510, Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', México  Received August 9, 2022, in final form October 19, 2022  We have studied the composition dependence of density of liquid water-DMSO mixtures at different temper-  atures by using the isobaric-isothermal (NPT) molecular dynamics computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The non-polarizable  semi-flexible, P1 and P2 models for the DMSO molecule combined with the TIP4P-2005 water model are con-  sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The excess mixing volume and the apparent molar volumes of the species are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' We have es-  tablished that the P1-TIP4P-2005 model for the mixture provides a very good description of the location of the  minimum of apparent molar volume for DMSO species indicating the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Most important is that the tem-  perature interval where the hydrophobic effect exists, is correctly captured with this modelling, in contrast to  the P2-TIP4P-2005 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Key words: molecular dynamics, water-dimethylsolfoxide mixtures, apparent molar volume  Liquid mixtures of dimethyl sulfoxide (DMSO) and water are widely used in organic chemistry,  chemical engineering, and biomedical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In this rapid communication we refer to the static  light scattering experiment performed on aqueous solutions of dimethyl sulfoxide (DMSO) at various  compositions and at two temperatures, 20◦C and 50◦C [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In the concentration range around 10 mol  percent of DMSO, an abnormal maximum of scattered light was detected, the intensity of which decreases  with an increase of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' This maximum has been attributed to the hydrophobic effect and has been  interpreted in terms of the minimum of apparent molar volume of DMSO upon concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Inspired  by this report, we have undertaken the study of the composition dependence of density of water-DMSO  liquid mixtures at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K, 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K and 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K by using molecular dynamics computer simulations  complementing our recent study performed at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Our focus is on the capability of simple  non-polarizable models for DMSO and water to reproduce the experimental observations by capturing  peculiar composition interval in the restricted temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Previously, binary mixtures of water with DMSO, in particular at a low DMSO concentration, were  the subject of study in different laboratories [2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Specifically, in [3, 4] the focus is on the composition  trends of certain properties at a single temperature 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The NPT setup was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' On the other  hand, the NVT simulations at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K, 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K and 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K and at a fixed DMSO molar fraction  (𝑋dmso = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='055) were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The experimental density values were assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In this study, we  have chosen quite popular, nonpolarizable, P1 and P2 models for DMSO species [8], from a huge set  of force fields for DMSO [9, 10], and well tested TIP4P-2005 water model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Technical details of  our isobaric-isothermal molecular dynamics computer simulations were described very recently in the  previous publication on the topic [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  ∗Corresponding author: oapizio@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Further distribution  of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  44201-1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='01547v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='soft]  4 Jan 2023  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Aguilar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Dominguez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Pizio  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='8  1  Xdmso  990  1020  1050  1080  1110  ρ (kg/m  3)  T = 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  T = 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  red - P2  blue - P1 water - TIP4P-2005  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='8  1  Xdmso  980  1000  1020  1040  1060  1080  ρ(kg/m  3)  P2  P1  water - TIP4P-2005  T = 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' (Colour online) Panel a: Composition dependence of the density of water-DMSO mixtures from  the NPT MD simulations of P2-TIP4P-2005 model (solid red lines), P1-TIP4P-2005 model (solid blue  lines) in comparison with the experimental data (dashed lines with triangles) at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K [12] and at  318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K (dashed line with squares) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Panel b: Similar to the panel a, but at 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Experimental  data (dashed line with triangles) are from [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  As common, the strategy of exploration is to describe a desired property on composition and to capture  its deviation from ideality as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' We do that for the mixture density at each fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' A set of  results is shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The panel a of this figure refers to 𝑇 = 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K and 𝑇 = 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K, whereas  the panel b illustrates the results at 𝑇 = 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' This temperature was also chosen in [5] due to the  availability of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' From the inspection of the results, we observe that the P1-TIP4P-2005  and P2-TIP4P-2005 are quite accurate in the interval of composition from pure water, 𝑋dmso = 0, up to  𝑋dmso ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' At a higher DMSO content in the mixture, the simulation data begin to deviate from the  experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The maximum density at a certain composition, 𝑋dmso ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='58, is captured by both  models in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Nevertheless, it can be seen that the P1-TIP4P-2005 model performs much better than  its P2-TIP4P-2005 counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' This latter model overestimates the density in the interval above 𝑋dmso >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Moreover, the deviation from the experimental points does not decrease while the temperature grows  from 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K up to 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The P1-TIP4P-2005 model exhibits a small inaccuracy for density at  “intermediate” compositions, but performs much better than P2-TIP4P-2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' It is difficult to judge the  precision of computer simulations data at low 𝑋dmso values at the scale in the figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Better insights  follow from the excess properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  We describe geometric aspects of mixing of the species in terms of the excess mixing volume  defined as common, Δ𝑉mix = 𝑉mix − 𝑋𝑑𝑉𝑑 − (1 − 𝑋𝑑)𝑉𝑤, where 𝑉mix, 𝑉𝑑 and 𝑉𝑤 refer to the molar  volume of the mixture and of the individual components, DMSO and water, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Apparently,  this property is not strongly dependent on temperature in the interval we deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' This follows from  the experimental results shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The mixing volume slightly decreases in magnitude upon  increasing temperature, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In general terms, computer simulation results show that the excess  mixing volume is overestimated in the framework of models assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Thus, the mixture of water and  DMSO species from simulations is predicted to be more non-ideal than its laboratory counterpart in  almost entire composition range at 𝑇 = 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The Δ𝑉mix values from simulations decrease in  magnitude in agreement with experimental trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' However, the location of the minimum along 𝑋dmso  axis from simulations and experiments is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Finally, one can observe that the performance  of the P1-TIP4P-2005 model is slightly better compared to the P2-TIP4P-2005 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  In order to discern the contribution of each species of the mixture into the mixing volume, it is  common to resort to the excess partial molar volumes as we have done recently [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' However, similar  insights into the geometric aspects of mixing on composition, both from experiments and simulations,  can be obtained by resorting to the notion of the apparent molar volume of species, rather than to the  partial molar volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  44201-2  Water-DMSO liquid mixtures  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='8  1  Xdmso  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='3  0  ∆Vmix (cm  3/mol)  P2 - red lines  P1 - blue lines  black - experimental  data  T = 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  T = 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  lines with circles  lines  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' (Colour online) A comparison of the composition dependence of the excess mixing volume of  water-DMSO mixtures for two DMSO models, P1 and P2, combined with TIP4P-2005 water model, with  the experimental data at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K ([12]—black triangles) and at 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K ([14]—black squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='8  1  Xdmso  66  68  70  72  74  Vφ  (d) (cm  3/mol)  - DMSO  red lines - P2  blue lines - P1  black - experimental data  T = 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  T = 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  T = 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  intermediate lines  top lines  bottom lines  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' (Colour online) A comparison of the composition dependence of the apparent molar volume of  DMSO from simulations with the experimental data marked as triangles from [12] at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K, from [13]  at 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K and from [14] at 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  The apparent molar volume for each species, according to the definition [15], is: 𝑉 (𝑤)  𝜙  = 𝑉𝑤 +  Δ𝑉mix/(1 − 𝑋𝑑) and 𝑉 (𝑑)  𝜙  = 𝑉𝑑 + Δ𝑉mix/𝑋𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' We elaborated the experimental density data from [12, 13]  and the results from our simulations to construct the plots shown in figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The most important  and illuminating results for the apparent molar volume for DMSO species, 𝑉 (𝑑)  𝜙 , are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Namely, we observe that both models in question predict the existence of miminum at certain composition  𝑋dmso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The points describing the location of the minimum at different temperatures from simulations and  experiments are joined by respective lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Evolution of the minimum with temperature is appropriately  captured by the P1-TIP4P-2005 model in contrast to the P2-TIP4P-2005 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Moreover, the P1-  TIP4P-2005 model correctly predicts that the hydrophobic trends diminish with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Performance of the P2-TIP4P-2005 model in this aspect contradicts the experimental trends and is not  satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' We have obtained the minimum for 𝑉 (𝑑)  𝜙  for this model, even at a higher temperature equal  to 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K (the respective plot is not shown for economy of space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Concerning the behaviour of the apparent molar volume of water species, 𝑉 (𝑤)  𝜙  , upon composition  (figure 4), it is worth to mention a rather satisfatory performance of the P1-TIP4P-2005 model in the  44201-3  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Aguilar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Dominguez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Pizio  entire composition range, especially for higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' The respective figure at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K is given  in [2] (figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' It is necessary to mention, however, that the experimental results for DMSO-rich (more  specifically, extremely rich) mixtures are difficult to obtain precisely, because high-purity DMSO is an  exceedingly hygroscopic solvent, the discussion of this issue is given in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  To conclude, we have established that the P1 and P2 models for DMSO combined with the TIP4P-  2005 water model yield a minimum of the apparent molar volume for DMSO species at low values  for 𝑋dmso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' This behavior is the manifestation of hydrophobic effects in these mixtures at a specific  composition interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' However, it appears that the predictions from P1-TIP4P-2005 model agree better  with the experimental trends than the ones from P2-TIP4P-2005 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Moreover, the P1-TIP4P-2005  model predicts the existence of a minimum of 𝑉 (𝑑)  𝜙  for DMSO within the correct temperature interval  from 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K up to 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' At this highest temperature, the minimum almost disappears, as deduced  from experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' By contrast, P2-TIP4P-2005 model does not provide an accurate estimate for a  peculiar composition interval and does not yield a decay for the hydrophobicity trends on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  In various publications this model is claimed to be the best, see for example [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Here, we show that this  conclusion is valid in certain aspects at a room temperature only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In addition, it is important to mention  that the combination of the OPLS model for DMSO with TIP4P-2005 model for water does not provide  a correct composition dependence of density at room temperature, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' figure 2 of [2] and at temperatures  of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In consequence, the anomaly of the apparent molar volume is not captured by this type of  model at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' These results are available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='8  1  Xdmso  16  17  18  Vφ  (w) (cm  3/mol)  - water  T = 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  P2  P1  experiment (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' [11])  a  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='8  1  Xdmso  16  17  18  Vφ  (w) (cm  3/mol)  - water  P2  P1  T = 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15K  experiment (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' [12])  b  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' (Colour online) Panels a and b: A comparison of the composition dependence of the apparent  molar volume of water from simulations with the experimental data (triangles [13]) at 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K and [14]  at 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Similar plot at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K is shown as figure 3d of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Previously, we explored the minimum of the methanol apparent molar volume on composition for  water-methanol mixtures [16] at 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='15 K, see also [17] for the interpretation of the observed phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  This kind of system is the mixture of two hydrogen bonded liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' By contrast, in the systems of the  present study there is no bonding between the DMSO solutes in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Apparently, the temperature trends  should be slightly different for these two classes of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  In addition, it seems intriguing to investigate the evolution of trends of hydrophobicity in water-DMSO  mixtures upon changing the external pressure as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' One can be guided by important experimental  observations, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' More generally, from the simulations perspective it is challenging  to construct a more complete map of anomalies, in pressure, temperature and composition variables,  that would involve, for example, the temperature of maximum density and the minimum of isothermal  compressibility, even by using the non-polarizable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' These data would serve as a useful benchmark  for the following development of polarizable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' In the case of water-DMSO mixtures, certain  progress has been reached at this level of modelling in [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' However, the BK3 polarizable water  model seems to be a more convenient starting point, because the principal anomalies of pure water are  quite well captured in this framework [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
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+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Bachmann S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', van Gunsteren W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', 2014, 141, 22D515, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4897976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Kiss P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', Baranyai A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', 2012, 137, 084506, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='474641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Kiss P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', Baranyai A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=', 2014, 140, 154505, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='4871390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  44201-5  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Aguilar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Dominguez, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Pizio  Аномалiя видимого молярного об’єму рiдких сумiшей  вода-диметилсульфоксид.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Комп’ютерне моделювання  методом молекулярної динамiки  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Агiлар  1, Е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Домiнгес  2, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Пiзiо  1  1 Iнститут хiмiї, Нацiональний автономний унiверситет Мехiко, Circuito Exterior, 04510, Мехiко, Мексика  2 Iнститут матерiалознавства, Нацiональний автономний унiверситет Мехiко, Circuito Exterior, 04510,  Мехiко, Meксика  Дослiджено концентрацiйну залежнiсть густини рiдких сумiшей вода-диметилсульфоксид (ДМСО) при рi-  зних температурах з використанням комп’ютерного моделювання методом iзобарично-iзотермiчної мо-  лекулярної динамiки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Розглядаються неполяризовнi напiвгнучкi моделi P1 i P2 для молекули ДМСО у по-  єднаннi з моделлю TIP4P-2005 для води.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Наведено результати обчислень для надлишкового об’єму змi-  шування та видимих молярних об’ємiв компонент сумiшi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Встановлено, що модель P1-TIP4P-2005 для цiєї  сумiшi забезпечує дуже добрий опис положення мiнiмуму видимого молярного об’єму компоненти ДМСО,  який вказує на аномалiю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content=' Найбiльш важливим є те, що вибрана модель, на вiдмiну вiд її вiдповiдника P2-  TIP4P-2005, правильно передбачає наявнiсть температурного iнтервалу, в якому iснує гiдрофобний ефект.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
+page_content='  Ключовi слова: молекулярна динамiка, сумiшi вода-диметилсульфоксид (ДМСО), видимий молярний  об’єм  44201-6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdAzT4oBgHgl3EQflf0o/content/2301.01547v1.pdf'}
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+Quenched lattice fluctuations in optically driven SrTiO3  
+M. Fechner1, M. Först1, G. Orenstein2, V. Krapivin2, A.S. Disa1, M. Buzzi1, A. von Hoegen1, G. de la 
+Pena2, Q. L Nguyen2,3, R. Mankowsky4, M. Sander4, H. Lemke4, Y. Deng4, M. Trigo2 and A. Cavalleri1,5 
+ 
+1Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany 
+2Stanford Pulse Institute, SLAC National Accelerator Laboratory, Menlo Park, CA, USA 
+3Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, CA, USA 
+4Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland 
+5Department of Physics, Clarendon Laboratory, University of Oxford, Oxford, United Kingdom 
+ 
+ 
+ 
+Many functionally relevant ferroic phenomena in quantum materials can be manipulated by 
+driving the lattice coherently with optical and terahertz pulses1–13. New physical phenomena 
+and non-equilibrium phases that have no equilibrium counterpart have been discovered 
+following these protocols. The underlying structural dynamics has been mostly studied by 
+recording the average atomic position along dynamical structural coordinates with elastic 
+scattering methods14–21. However, crystal lattice fluctuations, which are known to influence 
+phase transitions in equilibrium, are also expected to determine these dynamics but have 
+rarely been explored22. Here, we study the driven dynamics of the quantum paraelectric 
+SrTiO3, in which mid-infrared drives have been shown to induce a metastable ferroelectric 
+state9. Crucial in these physics is the competition between the polar instability and 
+antiferrodistortive rotations, which in equilibrium frustrate the formation of long-range 
+ferroelectricity. We make use of high intensity mid-infrared optical pulses to resonantly drive 
+a Ti-O stretching mode at 17 THz, and we measure the resulting change in lattice fluctuations 
+using time-resolved x-ray diffuse scattering at a free electron laser. After a prompt increase, 
+we observe a long-lived quench in R-point antiferrodistortive lattice fluctuations. The 
+enhancement and reduction in lattice fluctuations are explained theoretically by considering 
+fourth-order nonlinear phononic interactions and third-order coupling to the driven optical 
+phonon and to lattice strain, respectively. These observations provide a number of new and 
+testable hypotheses for the physics of light-induced ferroelectricity. 
+
+ 
+The cubic perovskite structure is inherently unstable, frequently undergoing different types of 
+symmetry-breaking structural distortions. SrTiO3 (STO) is an especially interesting case, in which both 
+ferroelectric (FE) and anti-ferrodistortive (AFD) instabilities (Fig. 1a,b) compete to determine the 
+ground state. Above 110 K, the cubic structure appears to be primed for a ferroelectric transition, 
+which one would expect upon cooling as a polar distortion is the dominant symmetry breaking for this 
+lattice structure. However, ferroelectricity fails to materialize, because below 110 K an anti-
+ferrodistortive instability induces a low-symmetry tetragonal phase. Although large polar fluctuations 
+set in below 30 K, resembling a precursor for the ferroelectric phase transition, long range polar order 
+is stifled by large zero-point fluctuations. This phenomenon is sometimes referred to as quantum 
+paraelectricity23,24. This frustrated state of matter can be lifted by either isotope-replacement of 
+oxygen25, partial cation doping 26, or epitaxial strain27, which trigger the formation of a macroscopic 
+ferroelectric polarization. 
+These physics are discussed in quantitative detail28 in Figure 1c,d, where we report the results of ab-
+initio calculations. In the cubic structure (rotation angle 𝜙!"# = 0), the potential energy along the 
+ferroelectric coordinate 𝑃	exhibits a distinct minimum at a finite polarization 𝑃$, indicating the 
+existence of a ferroelectric instability. When the tetragonal distortion and the antiferrodistortive 
+rotations set in, with a calculated equilibrium value of 𝜙!"# = 6°, the potential minimum along the 
+𝑃	becomes significantly shallower. Here, the condensation into the ferroelectric phase becomes 
+suppressed, as the depth of potential energy at 𝑃$ is comparable to the amplitude of zero-point 
+fluctuations of the ferroelectric soft mode. A further increase in AFD rotation to 𝜙 = 12° would revive 
+the deeper ferroelectric instability, with a potential minimum similar in size to that of the cubic state 
+at 𝜙 = 0°. Hence, the quantum paraelectric state of STO with suppressed ferroelectricity exists only 
+in the low-temperature tetragonal structure for a narrow range of AFD rotation angles near 𝜙 = 6°. 
+Note that experimentally, the absolute rotation angle may be slightly different from 𝜙 = 6°, although 
+the physical picture remains the same. 
+
+Intense terahertz (THz) and mid-infrared light pulses were shown to remove this frustration and to 
+induce ferroelectricity 9,10, either transiently when coupling to the soft mode directly (Ref. 10) or 
+permanently when a higher-frequency auxiliary mode was driven (Ref. 9). In the latter case, it was 
+argued that the coupling between the resonantly driven Ti-O stretch phonon to acoustic modes may 
+produce a strain field that stabilizes a long-range polar phase. Although the dynamics of the 
+antiferrodistortive mode are also expected to contribute, their role in these physics has not been 
+established. 
+Here, we used time-resolved x-ray diffuse scattering29 to experimentally map the lattice fluctuations 
+in the cubic high-temperature phase of STO, measured at the Brillouin zone boundary (½ ½ ½) R-point. 
+These experiments were performed on a (100)-oriented STO substrate at the Bernina end station of 
+the SwissFEL30 free electron laser. The sample was cooled by a cryogenic nitrogen gas jet to 
+approximately 135 K, i.e. above the AFD structural transition, and was excited by mid-infrared (mid-IR) 
+pulses of ~150 fs duration with central frequency and bandwidth of 18 THz and 5 THz full width at half 
+maximum, respectively, and fluences up to 60 mJ/cm2. This pump is resonant with the highest-
+frequency infrared-active STO phonon at 17 THz and is polarized nearly parallel to the [001] 
+crystallographic direction. The x-ray probe pulses (time duration ~ 50 fs) were tuned to 9.0 keV photon 
+energy and spectrally filtered to ∼1 eV by a Si (111) monochromator before being focused on the 
+sample. The probe beam was sent through a hole bored into an off-axis parabola used to focus the 
+mid-IR beam, enabling collinear excitation and x-ray probing (see Fig. 2a).  The temporal jitter between 
+pump and probe pulses was monitored using a spectral encoding technique on a shot-by-shot basis 
+and corrected in the post-processing. 
+The diffracted x-rays were detected by a Jungfrau pixel array detector, positioned 100 mm from the 
+sample, and normalized to the incident x-ray intensity. Figure 2b shows a typical detector image 
+integrated over 1,000 x-ray pulses, which shows localized diffuse scattering at the R-point (1.5 2.5 3.5) 
+and a broader feature around the M-point (2.0 2.5 3.5). We chose a grazing exit geometry with an exit 
+angle of about 1.5° for the R-point scattering to reduce the escape depth of the diffracted x-ray probe 
+
+to nearly match the penetration depth of the mid-IR pump while simultaneously allowing for high 
+excitation fluence at near-normal incidence. The left part of the image, separated by the relatively 
+sharp sample horizon, results from air scattering.  
+Away from Bragg reflections, as is the case here, the diffuse scattering intensity I(𝑞) at a reduced 
+wavevector 𝑞 = 𝐾 − 𝐺 is proportional to the variance of the atomic displacements 〈𝑢%𝑢&%〉, 
+connected to this wavevector31. Here,  𝐾	is the total momentum transfer of the scattering process and 
+𝐺 the nearest reciprocal lattice vector. At a structural phase transition, the divergence of lattice 
+fluctuations generally results in a critical increase of the diffuse scattering intensity32. In the particular 
+case of STO, the cubic-to-tetragonal phase transition at 110 K is driven by a softening and condensation 
+of the AFD phonon mode at the R-point in reciprocal space33. The increase of lattice fluctuations 
+associated with the onset of this structural transition generates the enhanced scattering intensity at 
+the well-defined detector region corresponding to the R-point34,35. 
+Figure 2c shows the changes in the integrated scattering intensity around the R-point, induced by the 
+resonant excitation of the highest-frequency zone-center infrared-active phonon at 17 THz. An initial 
+increase is followed by damped oscillations about an overall long-lived reduction in the total diffuse 
+scattering signal. These features were seen to be specific to the R-point in reciprocal space, as 
+scattering at Bragg peaks and other points in the Brillouin zone only resulted in small and featureless 
+changes. This is well exemplified by the time resolved integrated changes measured around the M-
+point, where small and slow enhancement was observed, as shown in Figure 2c.  
+In the following, we discuss a model for these dynamics, which inform further analysis of the data and 
+provide a working hypothesis for light-induced ferroelectricity in STO9. Figure 3a shows the phonon 
+band structure of the cubic phase of STO (𝜙 = 0°), calculated by an ab initio density functional theory 
+(DFT) approach (see Supplementary Information for details). Amongst other phonon modes, the 
+Brillouin zone center hosts the FE soft mode (blue), the acoustic modes related to strain (yellow), and 
+the driven 17-THz infrared-active phonon mode (magenta). The soft phonons connected to the AFD 
+
+rotation are found at the zone boundary R-points 𝑞 = 5±
+'
+( ±
+'
+( ±
+'
+(7. The unstable symmetry-lowering 
+soft modes manifest themselves as imaginary frequencies in DFT calculations, and are plotted as such. 
+Starting from the resonant excitation of the zone-center infrared-active mode 𝑄)* at frequency 𝜔)* 
+several interaction pathways can be mapped out utilizing a density functional theory frozen-phonon 
+approach. Three phenomena are expected.  
+Firstly, the driven 𝑄)* mode is expected to couple nonlinearly to acoustic (strain) modes 𝑄+ as 
+discussed in detail in Ref. 9. The energy of this nonlinear coupling exhibits a square-linear dependence 
+on the driven phonon and the strain coordinate, respectively, and can be written as 𝑉)*,+ = 𝑔)*,+𝑄)*
+( ⋅
+𝑄+. Also, because the pulse duration of the mid-infrared drive is short compared to the oscillation 
+period of the strain field (determined by the ratio of the pump penetration depth and by the speed of 
+sound), the strain wave is launched impulsively. For experimentally feasible drive electric fields, we 
+estimate a strain wave with peak values of the order of 0.2 % 9. 
+Secondly, we expect a coupling of the driven mode 𝑄)* to the R-point AFD lattice distortions 𝑢!"#, 
+which by symmetry has bi-quadratic character 𝑉)*,!"# = 𝑔)*,!"#𝑄)*
+( ∙ 𝑢!"#
+(
+. This interaction potential 
+implies a parametric excitation of pairs of phonons 𝑢!"# at opposite wavevectors ±𝑞 to conserve 
+momentum. This excitation generates lattice fluctuations of the form 〈𝑢!"#
+(
+〉, but no net distortion 
+〈𝑢!"#〉 29. From the ab initio calculations, we find that the coupling matrix element 𝑔)*,!"# is negative, 
+implying that displacements along the coordinates of 𝑄)* soften the of the antiferrodistortive mode 
+and hence enhance its fluctuations 〈𝑢!"#
+(
+〉, as shown in Fig. 3b.  
+Thirdly, we expect that the finite strain 𝑄+ induced at the Brillouin zone center via 𝑉)*,+ = 𝑔)*,+𝑄)*
+( ⋅
+𝑄+ also couples to the R-point AFD distortions 𝑢!"#. This linear-square coupling of the form 𝑉+,!"# =
+𝑔+,!"#	𝑄+ ∙ 𝑢!"#
+(
+ has a positive coupling coefficient 𝑔+,!"#, hence hardens the antiferrodistortive 
+mode to reduce its fluctuations 〈𝑢!"#
+(
+〉 (see Fig. 3c). This coupling counteracts the 𝑉)*,!"# interaction. 
+Furthermore, the lifetime of the strain coupling 𝑉+,!"# to the AFD distortion is determined by the slow 
+
+relaxation and propagation of the zone center acoustic phonons and is far longer lived than the 𝑉)*,!"# 
+coupling, which is only significant as long as the optical phonon 𝑄)* oscillates coherently.  
+To simulate these dynamics, we adopted the approach of Ref. 36 and calculated the time-dependent 
+amplitude variance of the AFD distortion >𝑢!"#
+(
+[𝑡]B induced by the optically driven mode 𝑄)*[𝑡] and 
+by the strain 𝑄+[𝑡] as discussed above. The coupled system of equations of motion has the following 
+form: 
+(1) 
+𝜕(
+𝜕𝑡( 𝑄)*[𝑡] + 2𝛾)*
+𝜕
+𝜕𝑡 𝑄)*[𝑡] + 𝜔)*
+( 𝑄)*[𝑡] = 𝑍∗𝐸[𝑡] 
+(2) 
+𝜕(
+𝜕𝑡( 𝑄+[𝑡] + 2𝛾+
+𝜕
+𝜕𝑡 𝑄+[𝑡] + 𝜔+(𝑄+[𝑡] = 𝑔)*,+𝑄)*
+(  
+(3) 
+𝜕.
+𝜕𝑡. >𝑢!"#
+(
+[𝑡]B + 2𝛾!"#
+𝜕(
+𝜕𝑡( >𝑢!"#
+(
+[𝑡]B
++ 4I𝜔!"#
+(
++ 𝑔)*,!"#	𝑄)*
+( [𝑡] + 𝑔+,!"#𝑄+[𝑡]J 𝜕
+𝜕𝑡 >𝑢!"#
+(
+[𝑡]B
+= −2>𝑢!"#
+(
+[𝑡]BI𝑔)*,!"#	𝜕(𝑄)*[𝑡]J
+(/𝜕𝑡)	+ 𝑔+,!"#	𝜕𝑄+[𝑡]/𝜕𝑡) 
+ 
+The subscripts 𝐼𝑅, 𝜂 and 𝐴𝐹𝐷 denote frequencies and lifetimes of the 𝑄)* mode, the strain and the 
+AFD distortion, respectively. 𝑍∗ is the effective charge that couples the infrared-active 𝑄)* mode to 
+the external driving field, which we model as a Gaussian pulse centered at the mid-IR pump frequency: 
+𝐸[𝑡] = 𝐸$ sin(𝜔)*	𝑡)𝑒& !"
+"#". We determine all the coupling coefficients of these equations utilizing a 
+first principles approach based on DFT and adjust a few of them to best match the experimental results 
+(see Supplementary Information for details).  
+In Fig. 4 we compare the simulated variance >𝑢!"#
+(
+[𝑡]B to the experimentally determined time-
+dependent R-point scattering intensities, measured for different excitation fluences. From the 
+simulation, we can isolate the dynamics arising from the different interactions. First, the 𝑉)*,!"# 
+coupling alone (red curve) results in a picosecond-lived oscillation of >𝑢!"#
+(
+[𝑡]B at twice the AFD soft 
+
+mode frequency as the result of mode squeezing of this mode in combination with the short rise time 
+of the 𝑄)* mode oscillations. The negative sign of the coefficient 𝑔)*,!"# leads to a softening of the 
+potential of the AFD distortion, resulting in the initial increase of the variance >𝑢!"#
+(
+[𝑡]B. Next, we 
+simulate the fluctuations of the R-point AFD distortion when coupled only to the optically induced 
+strain via 𝑉+,!"# (orange curve). In this case, a slow monotonic decrease of >𝑢!"#
+(
+[𝑡]B is observed 
+because the induced strain squeezes the AFD soft mode with a rise time that is too slow to launch 
+oscillations in its amplitude variance. Note that the slow enhancement of the scattering intensity at 
+the M-point, shown in Fig. 2c, is of the same origin but with a coupling coefficient of opposite sign (see 
+Supplementary Information for details). 
+Taken together, the time-dependent amplitude of the AFD lattice fluctuations >𝑢!"#
+(
+[𝑡]B is driven by a 
+short-lived phonon-phonon interaction and a longer-lasting strain-phonon interaction. Importantly, 
+the instantaneous shape of the AFD potential also determines the oscillation frequency of the variance 
+>𝑢!"#
+(
+[𝑡]B. Consequently, a larger strain induced by a higher amplitude of the driven 𝑄)* phonon 
+results in higher-frequency >𝑢!"#
+(
+[𝑡]B oscillations. This expectation is experimentally confirmed in the 
+excitation fluence measurements shown in Fig. 4c,e and reproduced in the corresponding simulations 
+in Figs. 4d,f.  
+Having established the antiferrodistortive dynamics driven in the cubic STO phase, we now discuss 
+their implications on the light-induced ferroelectricity found in Ref. 9. Our study shows that driving the 
+𝑄)* phonon mode produces short-lived oscillations and initial enhancement of the R-point AFD 
+fluctuations >𝑢!"#
+(
+[𝑡]B, but at longer times it creates a state in which these fluctuations are suppressed. 
+Figure 5a shows the corresponding modifications in the energy potential of the AFD distortion. The red 
+areas indicate the spread of the fluctuations >𝑢!"#
+(
+[𝑡]B, which act dynamically also on the FE distortions 
+in the cubic ground state. This is illustrated in Figure 5b that is a zoom into Fig. 1c around 𝜙 = 0. At 
+negative time delay, fluctuations of the AFD distortions >𝑢!"#
+(
+[𝑡]B cover a certain width in rotation 
+angle 𝜙. These fluctuations are expected to suppress the ferroelectric state, because AFD rotations 
+away from 𝜙 = 0 reduce the depth of the potential energy along the coordinate of the ferroelectric 
+
+distortion. Then, excitation of the infrared-active phonon and its bi-quadratic coupling to the R-point 
+AFD soft mode enhances the fluctuations >𝑢!"#
+(
+[𝑡]B to suppress the ferroelectric state even more. 
+However, the onset of the strain distortion at long times after the excitation sizably reduces the AFD 
+fluctuations, so that the phase space, occupied in the FE energy surface, becomes even smaller than 
+at equilibrium. In this situation the condensation of the FE state becomes more likely, and may explain 
+the growth of the FE state. Note that in the experiments reported in Ref. 9, a light-induced ferroelectric 
+phase was observed up to room temperature, that is both below and above the equilibrium transition 
+into the tetragonal phase. The discussion above only applies to the high-temperature regime (T > 110 
+K), in which paraelectric SrTiO3 is cubic. For lower temperatures, we expect the same elements 
+discussed above to still be valid, although additional effects could contribute to amplify the photo-
+induced state.  
+In summary, we have used time resolved diffuse x-ray scattering and ab-initio DFT simulations to clarify 
+the physics of photo-induced ferroelectricity in SrTiO3. We show results that go beyond the 
+measurements of the average position of the atoms in the unit cell captured by Bragg diffraction. We 
+also show how large fourth-order lattice interactions affect the functional response of materials. We 
+expect that in the magnetically ordered fluoride perovskite KMnF3, which has many common features 
+with the STO, modifications of octahedral rotations at the (½ ½ ½) R-point may significantly affect the 
+exchange interactions between d-electrons of the Mn2+ cations whose spins antiferromagnetically 
+order at the same wave vector37. Control of high-order phonon interactions is in fact a frontier in the 
+use of nonlinear phononics to manipulate the functional properties of solids8.  
+ 
+Acknowledgements: 
+The research leading to these results received funding from the Deutsche Forschungsgemeinschaft 
+(German Research Foundation) via the excellence cluster “CUI: Advanced Imaging of Matter” (EXC 
+2056, project ID 390715994). G. O., V. K., G dlP., Q. L N. and M.T. were supported by the U.S. 
+Department of Energy, Office of Science, Office of Basic Energy Sciences through the Division of 
+
+Materials Sciences and Engineering through Contract No. DE-AC02-76SF00515. This work was 
+performed at the SwissFEL free-electron laser, Paul Scherrer Institut, Villingen Switzerland. 
+ 
+ 
+
+Figures: 
+ 
+ 
+Figure 1. | The fundamental distortions of SrTiO3. (a) The polar distortion, creating the symmetry-broken ferroelectric state 
+with polarization P, involves the displacement of the center Ti atom along the c-axis. (b) The antiferrodistortive distortion 
+involves the rotation of the oxygen octahedron around the c axis by an angle 𝜙. (c) Illustration of the energy landscape of the 
+ferroelectric and the antiferrodistortive distortions. The minimum in ferroelectric energy 𝛥𝐸$% of the cubic phase (𝜙 = 0) 
+reduces when the antiferrodistortive rotation sets in. At the rotation angle of the tetragonal equilibrium state (𝜙 = 6°) the 
+energy surface exhibits a saddle point where 𝛥𝐸$% is minimized. (c) Selected cuts through this energy surface. 
+ 
+ 
+ 
+ 
+
+(a)
+Ferroelectric
+(b)
+Anti-ferrodistortive
+polarization P
+rotation angle Φ
+(ac-plane)
+(ab-plane)
+(ac-plane)
+(ab-plane)
+(c)
+tetragonal
+(d)
+equilibrium
+tetragonal
+0=Φ
+.9 = Φ
+Φ = 12°
+.9 = Φ
+cubic
+VFe(Φ, P) (arb . u.)
+Φ = 12°
+.0 =Φ
+0.00
+0.50
+1.00
+1.50
+AEFE(P, Φ)
+P/Po 
+Fig. 2 | Time-resolved x-ray diffuse scattering. (a) An intense mid-infrared pulse resonantly excites the highest-frequency 
+SrTiO3 IR-active phonon mode. Diffraction of a time-delayed femtosecond x-ray pulse probes the resulting lattice dynamics in 
+reciprocal space. (b) Two-dimensional x-ray detector image with selected high-symmetry points R- (1/2,1/2,1/2) 
+and M (1/2,1/2,0.0) at equilibrium. The R-point hosts the antiferrodistortive fluctuations of the cubic-to-tetragonal phase 
+transition. (c) Measured changes in x-ray scattering intensity at the R- and M-point induced by the nonlinear excitation of the 
+crystal lattice. 
+ 
+ 
+
+(a)
+(b)
+(c)
+oR
+1.2
+X-ray
+oM
+1
+probe
+8
+(t)/ lo
+1.020
+Os
+0.8
+QQ9
+R
+MIR
+-1
+0
+2
+3
+time delay (ps)
+dwnd
+min
+max
+intensity 
+Fig. 3 | Anharmonic phonon-phonon coupling across the Brillouin zone. (a) Phonon dispersion of cubic (𝜙 = 0) SrTiO3 along 
+the 𝑅 − 𝛤 − 𝑅 direction calculated from an ab-initio approach. Colored lines highlight the positions of the driven infrared-
+active phonon (magenta), strain waves (orange), the ferroelectric soft mode (blue), all at the Brillouin zone center, and of the 
+antiferrodistortive mode (red) at the zone edge. Negative frequencies represent unstable (soft) phonon modes. (b) and (c) 
+Energy potentials of the R-point antiferrodistortive distortion as a function of amplitude 𝑢&$'. Black lines are the equilibrium 
+potential. Colored lines show modifications of this potential due to nonlinear coupling to the zone-center infrared-active mode 
+𝑄()and strain 𝑄*, as discussed in the text.     
+ 
+ 
+
+(a)
+IR-phonon
+20H
+(zHL)
+10
+0
+R=-{0.5,0.5,0.5)
+R={0.5,0.5,0.5]
+anti-ferrodistortive
+ferrodistortive
+anti-ferrodistortive
+(b)
+(c)
+△E (eV/f.u.)
+△E (eV/f.u.)
+0.1
+QiRUAFD
+0.1
+QnuAFD
+0.0
+0.0
+0.11
+-0.1
+-2
+0
+2
+UAFD (VAMU A)
+UAFD (VAMU A) 
+Fig. 4 | Fluence dependent measurements and simulations. (a,c,e) Time-resolved changes of the R-point x-ray scattering 
+intensity for different mid-infrared excitation fluences. The initial rise of the scattering intensity is followed by oscillations with 
+a fluence dependent frequency and a slow decrease. (b,d,f), Simulations of the variance of the antiferrodistortive R-point 
+amplitude  ⟨𝑢&$'
++
+[𝑡]⟩ utilizing the model presented in Eqs. (1-3). The contributions of its coupling to the infrared-active phonon  
+𝑄() (magenta) and strain 𝑄*	(orange) are individually shown in panel b (see the corresponding non-equilibrium potentials in 
+Fig. 3b,c). 
+ 
+ 
+ 
+
+experiment
+simulation
+(a)
+(b)
+- Qn and QiR
+only Q
+0 60 mJ/cm²
+only QiR
+I(t)/ lo
+1.0
+0.8
+0.8
+(c)
+(d)
+1.2
+0 35 mJ/cm²
+I(t)/ lb
+1.0
+1.0
+0.8
+0.8
+(e)
+(f)
+1.2
+1.2
+15 mJ/cm
+I(t)/ lo
+1.0
+0.8
+0.8
+0
+1
+2
+3
+1
+0
+2
+3
+time delay (ps)
+time delay (ps) 
+Fig 5. | Impact of the antiferrodistortive lattice fluctuations on the ferroelectric energy gain. (a) Potential of the 
+antiferrodistortive mode at equilibrium (grey, negative time delay) the corresponding amplitude variance indicated 
+as red shaded area. At zero time delay, the potential softens due to coupling to the resonantly driven infrared-active 
+phonon mod, hence the phase space of the fluctuations ⟨𝑢&$'
++
+[𝑡]⟩ is enhanced. At longer times, the onset of strain 
+hardens the potential, thereby reducing ⟨𝑢&$'
++
+⟩ fluctuations to below the equilibrium value. (b), Impact of these 
+dynamics on the ferroelectric energy landscape. At negative time delays, the thermal fluctuations ⟨𝑢&$'
++
+⟩ cover the 
+entire region in which the ferroelectric state gains energy, hence prohibiting a condensation of this mode. This 
+behavior becomes pronounced at zero time delay, where the fluctuations are enhanced. At longer times, the reduced 
+⟨𝑢&$'
++
+⟩ fluctuations cover only the small area where the ferroelectric state gains energy, enabling a condensation of 
+the ferroelectric soft mode. 
+
+(a)
+negative time delays
+time zero
+positive time delays
+△V(t,UAFD) (eV/f.u.)
+8
+-2
+0
+2
+-2
+0
+2
+.2
+0
+2
+UAFD (AMUA)
+UAFD (AMU A)
+UAFD (AMU A)
+(b)
+.0 =Φ
+.0 =Φ
+.0 =Φ
+0
+△EFE(P, Φ)References: 
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+  
+
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new file mode 100644
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+page_content=' Lemke4, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Deng4, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Trigo2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Cavalleri1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5  1Max Planck Institute for the Structure and Dynamics of Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Germany  2Stanford Pulse Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Menlo Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' USA  3Linac Coherent Light Source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Menlo Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' USA  4Swiss Light Source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Paul Scherrer Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Villigen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Switzerland  5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Clarendon Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' United Kingdom  Many functionally relevant ferroic phenomena in quantum materials can be manipulated by  driving the lattice coherently with optical and terahertz pulses1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' New physical phenomena  and non-equilibrium phases that have no equilibrium counterpart have been discovered  following these protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The underlying structural dynamics has been mostly studied by  recording the average atomic position along dynamical structural coordinates with elastic  scattering methods14–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' However, crystal lattice fluctuations, which are known to influence  phase transitions in equilibrium, are also expected to determine these dynamics but have  rarely been explored22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Here, we study the driven dynamics of the quantum paraelectric  SrTiO3, in which mid-infrared drives have been shown to induce a metastable ferroelectric  state9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Crucial in these physics is the competition between the polar instability and  antiferrodistortive rotations, which in equilibrium frustrate the formation of long-range  ferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' We make use of high intensity mid-infrared optical pulses to resonantly drive  a Ti-O stretching mode at 17 THz, and we measure the resulting change in lattice fluctuations  using time-resolved x-ray diffuse scattering at a free electron laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' After a prompt increase,  we observe a long-lived quench in R-point antiferrodistortive lattice fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The  enhancement and reduction in lattice fluctuations are explained theoretically by considering  fourth-order nonlinear phononic interactions and third-order coupling to the driven optical  phonon and to lattice strain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' These observations provide a number of new and  testable hypotheses for the physics of light-induced ferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  The cubic perovskite structure is inherently unstable, frequently undergoing different types of  symmetry-breaking structural distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' SrTiO3 (STO) is an especially interesting case, in which both  ferroelectric (FE) and anti-ferrodistortive (AFD) instabilities (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 1a,b) compete to determine the  ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Above 110 K, the cubic structure appears to be primed for a ferroelectric transition,  which one would expect upon cooling as a polar distortion is the dominant symmetry breaking for this  lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' However, ferroelectricity fails to materialize, because below 110 K an anti-  ferrodistortive instability induces a low-symmetry tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Although large polar fluctuations  set in below 30 K, resembling a precursor for the ferroelectric phase transition, long range polar order  is stifled by large zero-point fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This phenomenon is sometimes referred to as quantum  paraelectricity23,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This frustrated state of matter can be lifted by either isotope-replacement of  oxygen25, partial cation doping 26, or epitaxial strain27, which trigger the formation of a macroscopic  ferroelectric polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  These physics are discussed in quantitative detail28 in Figure 1c,d, where we report the results of ab-  initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' In the cubic structure (rotation angle 𝜙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# = 0), the potential energy along the  ferroelectric coordinate 𝑃 exhibits a distinct minimum at a finite polarization 𝑃$, indicating the  existence of a ferroelectric instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' When the tetragonal distortion and the antiferrodistortive  rotations set in, with a calculated equilibrium value of 𝜙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# = 6°, the potential minimum along the  𝑃 becomes significantly shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Here, the condensation into the ferroelectric phase becomes  suppressed, as the depth of potential energy at 𝑃$ is comparable to the amplitude of zero-point  fluctuations of the ferroelectric soft mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' A further increase in AFD rotation to 𝜙 = 12° would revive  the deeper ferroelectric instability, with a potential minimum similar in size to that of the cubic state  at 𝜙 = 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Hence, the quantum paraelectric state of STO with suppressed ferroelectricity exists only  in the low-temperature tetragonal structure for a narrow range of AFD rotation angles near 𝜙 = 6°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Note that experimentally, the absolute rotation angle may be slightly different from 𝜙 = 6°, although  the physical picture remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Intense terahertz (THz) and mid-infrared light pulses were shown to remove this frustration and to  induce ferroelectricity 9,10, either transiently when coupling to the soft mode directly (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 10) or  permanently when a higher-frequency auxiliary mode was driven (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' In the latter case, it was  argued that the coupling between the resonantly driven Ti-O stretch phonon to acoustic modes may  produce a strain field that stabilizes a long-range polar phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Although the dynamics of the  antiferrodistortive mode are also expected to contribute, their role in these physics has not been  established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Here, we used time-resolved x-ray diffuse scattering29 to experimentally map the lattice fluctuations  in the cubic high-temperature phase of STO, measured at the Brillouin zone boundary (½ ½ ½) R-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  These experiments were performed on a (100)-oriented STO substrate at the Bernina end station of  the SwissFEL30 free electron laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The sample was cooled by a cryogenic nitrogen gas jet to  approximately 135 K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' above the AFD structural transition, and was excited by mid-infrared (mid-IR)  pulses of ~150 fs duration with central frequency and bandwidth of 18 THz and 5 THz full width at half  maximum, respectively, and fluences up to 60 mJ/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This pump is resonant with the highest-  frequency infrared-active STO phonon at 17 THz and is polarized nearly parallel to the [001]  crystallographic direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The x-ray probe pulses (time duration ~ 50 fs) were tuned to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0 keV photon  energy and spectrally filtered to ∼1 eV by a Si (111) monochromator before being focused on the  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The probe beam was sent through a hole bored into an off-axis parabola used to focus the  mid-IR beam, enabling collinear excitation and x-ray probing (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The temporal jitter between  pump and probe pulses was monitored using a spectral encoding technique on a shot-by-shot basis  and corrected in the post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  The diffracted x-rays were detected by a Jungfrau pixel array detector, positioned 100 mm from the  sample, and normalized to the incident x-ray intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Figure 2b shows a typical detector image  integrated over 1,000 x-ray pulses, which shows localized diffuse scattering at the R-point (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5)  and a broader feature around the M-point (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' We chose a grazing exit geometry with an exit  angle of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5° for the R-point scattering to reduce the escape depth of the diffracted x-ray probe  to nearly match the penetration depth of the mid-IR pump while simultaneously allowing for high  excitation fluence at near-normal incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The left part of the image, separated by the relatively  sharp sample horizon, results from air scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Away from Bragg reflections, as is the case here, the diffuse scattering intensity I(𝑞) at a reduced  wavevector 𝑞 = 𝐾 − 𝐺 is proportional to the variance of the atomic displacements 〈𝑢%𝑢&%〉,  connected to this wavevector31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Here,  𝐾 is the total momentum transfer of the scattering process and  𝐺 the nearest reciprocal lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' At a structural phase transition, the divergence of lattice  fluctuations generally results in a critical increase of the diffuse scattering intensity32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' In the particular  case of STO, the cubic-to-tetragonal phase transition at 110 K is driven by a softening and condensation  of the AFD phonon mode at the R-point in reciprocal space33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The increase of lattice fluctuations  associated with the onset of this structural transition generates the enhanced scattering intensity at  the well-defined detector region corresponding to the R-point34,35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Figure 2c shows the changes in the integrated scattering intensity around the R-point, induced by the  resonant excitation of the highest-frequency zone-center infrared-active phonon at 17 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' An initial  increase is followed by damped oscillations about an overall long-lived reduction in the total diffuse  scattering signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' These features were seen to be specific to the R-point in reciprocal space, as  scattering at Bragg peaks and other points in the Brillouin zone only resulted in small and featureless  changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This is well exemplified by the time resolved integrated changes measured around the M-  point, where small and slow enhancement was observed, as shown in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  In the following, we discuss a model for these dynamics, which inform further analysis of the data and  provide a working hypothesis for light-induced ferroelectricity in STO9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Figure 3a shows the phonon  band structure of the cubic phase of STO (𝜙 = 0°), calculated by an ab initio density functional theory  (DFT) approach (see Supplementary Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Amongst other phonon modes, the  Brillouin zone center hosts the FE soft mode (blue), the acoustic modes related to strain (yellow), and  the driven 17-THz infrared-active phonon mode (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" The soft phonons connected to the AFD  rotation are found at the zone boundary R-points 𝑞 = 5±  '  ( ±  '  ( ±  '  (7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The unstable symmetry-lowering  soft modes manifest themselves as imaginary frequencies in DFT calculations, and are plotted as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Starting from the resonant excitation of the zone-center infrared-active mode 𝑄)* at frequency 𝜔)*  several interaction pathways can be mapped out utilizing a density functional theory frozen-phonon  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Three phenomena are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Firstly, the driven 𝑄)* mode is expected to couple nonlinearly to acoustic (strain) modes 𝑄+ as  discussed in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The energy of this nonlinear coupling exhibits a square-linear dependence  on the driven phonon and the strain coordinate, respectively, and can be written as 𝑉)*,+ = 𝑔)*,+𝑄)*  ( ⋅  𝑄+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Also, because the pulse duration of the mid-infrared drive is short compared to the oscillation  period of the strain field (determined by the ratio of the pump penetration depth and by the speed of  sound), the strain wave is launched impulsively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' For experimentally feasible drive electric fields, we  estimate a strain wave with peak values of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='2 % 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Secondly, we expect a coupling of the driven mode 𝑄)* to the R-point AFD lattice distortions 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#,  which by symmetry has bi-quadratic character 𝑉)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# = 𝑔)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#𝑄)*  ( ∙ 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This interaction potential  implies a parametric excitation of pairs of phonons 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# at opposite wavevectors ±𝑞 to conserve  momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This excitation generates lattice fluctuations of the form 〈𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  〉, but no net distortion  〈𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#〉 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' From the ab initio calculations, we find that the coupling matrix element 𝑔)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# is negative,  implying that displacements along the coordinates of 𝑄)* soften the of the antiferrodistortive mode  and hence enhance its fluctuations 〈𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  〉, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Thirdly, we expect that the finite strain 𝑄+ induced at the Brillouin zone center via 𝑉)*,+ = 𝑔)*,+𝑄)*  ( ⋅  𝑄+ also couples to the R-point AFD distortions 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='"#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This linear-square coupling of the form 𝑉+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# =  𝑔+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# 𝑄+ ∙ 𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  has a positive coupling coefficient 𝑔+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#, hence hardens the antiferrodistortive  mode to reduce its fluctuations 〈𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  〉 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This coupling counteracts the 𝑉)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Furthermore, the lifetime of the strain coupling 𝑉+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# to the AFD distortion is determined by the slow  relaxation and propagation of the zone center acoustic phonons and is far longer lived than the 𝑉)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  coupling, which is only significant as long as the optical phonon 𝑄)* oscillates coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  To simulate these dynamics, we adopted the approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 36 and calculated the time-dependent  amplitude variance of the AFD distortion >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B induced by the optically driven mode 𝑄)*[𝑡] and  by the strain 𝑄+[𝑡] as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The coupled system of equations of motion has the following  form:  (1)  𝜕(  𝜕𝑡( 𝑄)*[𝑡] + 2𝛾)*  𝜕  𝜕𝑡 𝑄)*[𝑡] + 𝜔)*  ( 𝑄)*[𝑡] = 𝑍∗𝐸[𝑡]  (2)  𝜕(  𝜕𝑡( 𝑄+[𝑡] + 2𝛾+  𝜕  𝜕𝑡 𝑄+[𝑡] + 𝜔+(𝑄+[𝑡] = 𝑔)*,+𝑄)*  (  (3)  𝜕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  𝜕𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B + 2𝛾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  𝜕(  𝜕𝑡( >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B  + 4I𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  + 𝑔)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# 𝑄)*  ( [𝑡] + 𝑔+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#𝑄+[𝑡]J 𝜕  𝜕𝑡 >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B  = −2>𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]BI𝑔)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# 𝜕(𝑄)*[𝑡]J  (/𝜕𝑡) + 𝑔+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# 𝜕𝑄+[𝑡]/𝜕𝑡)  The subscripts 𝐼𝑅, 𝜂 and 𝐴𝐹𝐷 denote frequencies and lifetimes of the 𝑄)* mode, the strain and the  AFD distortion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 𝑍∗ is the effective charge that couples the infrared-active 𝑄)* mode to  the external driving field, which we model as a Gaussian pulse centered at the mid-IR pump frequency:  𝐸[𝑡] = 𝐸$ sin(𝜔)* 𝑡)𝑒& !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='"  "#".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' We determine all the coupling coefficients of these equations utilizing a  first principles approach based on DFT and adjust a few of them to best match the experimental results  (see Supplementary Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 4 we compare the simulated variance >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B to the experimentally determined time-  dependent R-point scattering intensities, measured for different excitation fluences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' From the  simulation, we can isolate the dynamics arising from the different interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' First, the 𝑉)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  coupling alone (red curve) results in a picosecond-lived oscillation of >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B at twice the AFD soft  mode frequency as the result of mode squeezing of this mode in combination with the short rise time  of the 𝑄)* mode oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The negative sign of the coefficient 𝑔)*,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# leads to a softening of the  potential of the AFD distortion, resulting in the initial increase of the variance >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Next, we  simulate the fluctuations of the R-point AFD distortion when coupled only to the optically induced  strain via 𝑉+,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "# (orange curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' In this case, a slow monotonic decrease of >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B is observed  because the induced strain squeezes the AFD soft mode with a rise time that is too slow to launch  oscillations in its amplitude variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Note that the slow enhancement of the scattering intensity at  the M-point, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 2c, is of the same origin but with a coupling coefficient of opposite sign (see  Supplementary Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Taken together, the time-dependent amplitude of the AFD lattice fluctuations >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B is driven by a  short-lived phonon-phonon interaction and a longer-lasting strain-phonon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Importantly,  the instantaneous shape of the AFD potential also determines the oscillation frequency of the variance  >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Consequently, a larger strain induced by a higher amplitude of the driven 𝑄)* phonon  results in higher-frequency >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This expectation is experimentally confirmed in the  excitation fluence measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 4c,e and reproduced in the corresponding simulations  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 4d,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Having established the antiferrodistortive dynamics driven in the cubic STO phase, we now discuss  their implications on the light-induced ferroelectricity found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Our study shows that driving the  𝑄)* phonon mode produces short-lived oscillations and initial enhancement of the R-point AFD  fluctuations >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B, but at longer times it creates a state in which these fluctuations are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Figure 5a shows the corresponding modifications in the energy potential of the AFD distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The red  areas indicate the spread of the fluctuations >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B, which act dynamically also on the FE distortions  in the cubic ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This is illustrated in Figure 5b that is a zoom into Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 1c around 𝜙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' At  negative time delay, fluctuations of the AFD distortions >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B cover a certain width in rotation  angle 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' These fluctuations are expected to suppress the ferroelectric state, because AFD rotations  away from 𝜙 = 0 reduce the depth of the potential energy along the coordinate of the ferroelectric  distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Then, excitation of the infrared-active phonon and its bi-quadratic coupling to the R-point  AFD soft mode enhances the fluctuations >𝑢!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' "#  (  [𝑡]B to suppress the ferroelectric state even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  However, the onset of the strain distortion at long times after the excitation sizably reduces the AFD  fluctuations, so that the phase space, occupied in the FE energy surface, becomes even smaller than  at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' In this situation the condensation of the FE state becomes more likely, and may explain  the growth of the FE state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Note that in the experiments reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 9, a light-induced ferroelectric  phase was observed up to room temperature, that is both below and above the equilibrium transition  into the tetragonal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The discussion above only applies to the high-temperature regime (T > 110  K), in which paraelectric SrTiO3 is cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' For lower temperatures, we expect the same elements  discussed above to still be valid, although additional effects could contribute to amplify the photo-  induced state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  In summary, we have used time resolved diffuse x-ray scattering and ab-initio DFT simulations to clarify  the physics of photo-induced ferroelectricity in SrTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' We show results that go beyond the  measurements of the average position of the atoms in the unit cell captured by Bragg diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' We  also show how large fourth-order lattice interactions affect the functional response of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' We  expect that in the magnetically ordered fluoride perovskite KMnF3, which has many common features  with the STO, modifications of octahedral rotations at the (½ ½ ½) R-point may significantly affect the  exchange interactions between d-electrons of the Mn2+ cations whose spins antiferromagnetically  order at the same wave vector37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Control of high-order phonon interactions is in fact a frontier in the  use of nonlinear phononics to manipulate the functional properties of solids8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Acknowledgements:  The research leading to these results received funding from the Deutsche Forschungsgemeinschaft  (German Research Foundation) via the excellence cluster “CUI: Advanced Imaging of Matter” (EXC  2056, project ID 390715994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=', G dlP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=', Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' L N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' were supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Department of Energy, Office of Science, Office of Basic Energy Sciences through the Division of  Materials Sciences and Engineering through Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' DE-AC02-76SF00515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This work was  performed at the SwissFEL free-electron laser, Paul Scherrer Institut, Villingen Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Figures:  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' | The fundamental distortions of SrTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (a) The polar distortion, creating the symmetry-broken ferroelectric state  with polarization P, involves the displacement of the center Ti atom along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (b) The antiferrodistortive distortion  involves the rotation of the oxygen octahedron around the c axis by an angle 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (c) Illustration of the energy landscape of the  ferroelectric and the antiferrodistortive distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The minimum in ferroelectric energy 𝛥𝐸$% of the cubic phase (𝜙 = 0)  reduces when the antiferrodistortive rotation sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' At the rotation angle of the tetragonal equilibrium state (𝜙 = 6°) the  energy surface exhibits a saddle point where 𝛥𝐸$% is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (c) Selected cuts through this energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  (a)  Ferroelectric  (b)  Anti-ferrodistortive  polarization P  rotation angle Φ  (ac-plane)  (ab-plane)  (ac-plane)  (ab-plane)  (c)  tetragonal  (d)  equilibrium  tetragonal  0=Φ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='9 = Φ  Φ = 12°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='9 = Φ  cubic  VFe(Φ, P) (arb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=')  Φ = 12°  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0 =Φ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='50  AEFE(P, Φ)  P/Po  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 2 | Time-resolved x-ray diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (a) An intense mid-infrared pulse resonantly excites the highest-frequency  SrTiO3 IR-active phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Diffraction of a time-delayed femtosecond x-ray pulse probes the resulting lattice dynamics in  reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (b) Two-dimensional x-ray detector image with selected high-symmetry points R- (1/2,1/2,1/2)  and M (1/2,1/2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0) at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The R-point hosts the antiferrodistortive fluctuations of the cubic-to-tetragonal phase  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (c) Measured changes in x-ray scattering intensity at the R- and M-point induced by the nonlinear excitation of the  crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  (a)  (b)  (c)  oR  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='2  X-ray  oM  1  probe  8  (t)/ lo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='020  Os  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  QQ9  R  MIR  1  0  2  3  time delay (ps)  dwnd  min  max  intensity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 3 | Anharmonic phonon-phonon coupling across the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (a) Phonon dispersion of cubic (𝜙 = 0) SrTiO3 along  the 𝑅 − 𝛤 − 𝑅 direction calculated from an ab-initio approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Colored lines highlight the positions of the driven infrared-  active phonon (magenta), strain waves (orange), the ferroelectric soft mode (blue), all at the Brillouin zone center, and of the  antiferrodistortive mode (red) at the zone edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Negative frequencies represent unstable (soft) phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" (b) and (c)  Energy potentials of the R-point antiferrodistortive distortion as a function of amplitude 𝑢&$'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Black lines are the equilibrium  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Colored lines show modifications of this potential due to nonlinear coupling to the zone-center infrared-active mode  𝑄()and strain 𝑄*, as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  (a)  IR-phonon  20H  (zHL)  10  0  R=-{0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5)  R={0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='5]  anti-ferrodistortive  ferrodistortive  anti-ferrodistortive  (b)  (c)  △E (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=')  △E (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='1  QiRUAFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='1  QnuAFD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='1  2  0  2  UAFD (VAMU A)  UAFD (VAMU A)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 4 | Fluence dependent measurements and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (a,c,e) Time-resolved changes of the R-point x-ray scattering  intensity for different mid-infrared excitation fluences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The initial rise of the scattering intensity is followed by oscillations with  a fluence dependent frequency and a slow decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" (b,d,f), Simulations of the variance of the antiferrodistortive R-point  amplitude  ⟨𝑢&$'  +  [𝑡]⟩ utilizing the model presented in Eqs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' The contributions of its coupling to the infrared-active phonon  𝑄() (magenta) and strain 𝑄* (orange) are individually shown in panel b (see the corresponding non-equilibrium potentials in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' 3b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  experiment  simulation  (a)  (b)  Qn and QiR  only Q  0 60 mJ/cm²  only QiR  I(t)/ lo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  (c)  (d)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='2  0 35 mJ/cm²  I(t)/ lb  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  (e)  (f)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='2  15 mJ/cm  I(t)/ lo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='8  0  1  2  3  1  0  2  3  time delay (ps)  time delay (ps)  Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' | Impact of the antiferrodistortive lattice fluctuations on the ferroelectric energy gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (a) Potential of the  antiferrodistortive mode at equilibrium (grey, negative time delay) the corresponding amplitude variance indicated  as red shaded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" At zero time delay, the potential softens due to coupling to the resonantly driven infrared-active  phonon mod, hence the phase space of the fluctuations ⟨𝑢&$'  +  [𝑡]⟩ is enhanced." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" At longer times, the onset of strain  hardens the potential, thereby reducing ⟨𝑢&$'  +  ⟩ fluctuations to below the equilibrium value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' (b), Impact of these  dynamics on the ferroelectric energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" At negative time delays, the thermal fluctuations ⟨𝑢&$'  +  ⟩ cover the  entire region in which the ferroelectric state gains energy, hence prohibiting a condensation of this mode." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' This  behavior becomes pronounced at zero time delay, where the fluctuations are enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=" At longer times, the reduced  ⟨𝑢&$'  +  ⟩ fluctuations cover only the small area where the ferroelectric state gains energy, enabling a condensation of  the ferroelectric soft mode." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  (a)  negative time delays  time zero  positive time delays  △V(t,UAFD) (eV/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content='0 =Φ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content='0 =Φ  0  △EFE(P, Φ)References:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content='  Mankowsky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content='  Juraschek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content='  Stupakiewicz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content='  Nova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content=' Metastable ferroelectricity in optically  strained SrTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+page_content=' & Portis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Magnetic Properties of KMnF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' II .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Weak  Ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
+page_content=' Physical Review 123, 1652–1660 (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFAT4oBgHgl3EQf0R61/content/2301.08703v1.pdf'}
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+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+Electronic Transactions on Numerical Analysis.
+Volume 46, pp. 1–xx, 2023.
+Copyright © 2023, Kent State University.
+ISSN 1068–9613.
+CONVERGENCE ANALYSIS OF A KRYLOV SUBSPACE SPECTRAL METHOD
+FOR THE 1-D WAVE EQUATION IN AN INHOMOGENEOUS MEDIUM∗
+BAILEY RESTER† AND JAMES V. LAMBERS†
+Abstract. Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for
+partial differential equations (PDEs) that possess stability characteristic of implicit methods. KSS methods compute
+each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE. As
+a result, KSS methods scale effectively to higher spatial resolution. This paper will present a convergence analysis of
+a second-order KSS method applied to a 1-D wave equation in an inhomogeneous medium. Numerical experiments
+that corroborate the established theory are included, along with a discussion of generalizations, such as to higher
+space dimensions.
+Key words. spectral methods, wave equation, convergence analysis, variable coefficients
+AMS subject classifications. 65M70, 65M12, 65F60
+1. Introduction. Consider the 1-D wave equation in an inhomogeneous medium,
+(1.1)
+utt = (p(x)ux)x + q(x)u,
+on a bounded domain, with appropriate initial and boundary conditions. Analytical methods
+are not practical to use for this problem, since the coefficients are not constant. For instance,
+applying separation of variables [5] would result in a spatial ODE that cannot be solved
+analytically, and therefore numerical methods are needed. However, standard time-stepping
+methods, such as Runge-Kutta methods or multistep methods, suffer from a lack of scalability.
+As the number of grid points increases, a smaller time step would be needed due to the CFL
+condition [10] for explicit methods, or an increasingly ill-conditioned system must be solved
+for implicit methods. It follows that increasing the number of grid points significantly increases
+the computational expense. Therefore, a more practical numerical method for solving this
+kind of variable-coefficient PDE is desirable.
+Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping
+methods that possess stability characteristic of implicit methods [20]. By contrast with other
+time-stepping methods, KSS methods employ a componentwise approach, in which each
+Fourier coefficient of the solution is computed using an approximation of the solution operator
+of the PDE that is tailored to that coefficient. This customization is based on techniques for
+approximating bilinear forms involving matrix functions by treating them as Riemann-Stieltjes
+integrals [6]. This componentwise approach allows KSS methods to circumvent difficulties
+caused by stiffness, and thus scale effectively to higher spatial resolution [3].
+A first-order KSS method applied to the heat equation with a constant leading coefficient
+was proven to be unconditionally stable [15, 16], as well as a second-order KSS method
+applied to the wave equation with a constant leading coefficient [14]. In all of these studies,
+lower-order coefficients of the spatial differential operator were assumed to be bandlimited. A
+first-order KSS method applied to the heat equation with a bandlimited leading coefficient is
+also unconditionally stable [20]. In this paper, we analyze stability of a KSS method applied
+to the wave equation with bandlimited coefficients.
+The outline of the paper is as follows. Section 2 provides an overview of KSS methods.
+Section 3 presents a stability analysis of a second-order KSS method applied to the PDE
+∗Received... Accepted... Published online on... Recommended by....
+†School of Mathematics and Natural Sciences, The University of Southern Mississippi, 118 College Dr #5043,
+Hattiesburg, MS 39406 USA
+1
+arXiv:2301.08316v1  [math.NA]  19 Jan 2023
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+2
+B. RESTER AND J. V. LAMBERS
+(1.1) with bandlimited coefficients p(x) and q(x), and periodic boundary conditions. In that
+same section, a full convergence analysis in the case of p(x) ≡ constant is carried out.
+Corroborating numerical experiments are given in Section 4, along with application of the
+second-order KSS method to more general problems. Conclusions and ideas for future work
+are given in Section 5.
+2. Background.
+2.1. Overview of KSS Methods. We begin with the derivation of KSS methods, which
+are simpler to describe for a variable-coefficient diffusion equation. Consider a linear parabolic
+PDE of the form
+(2.1)
+ut + Lu = 0
+on the domain (0, 2π) × (0, ∞), where L is a second-order, self-adjoint differential operator.
+If we impose periodic boundary conditions, then the solution u(x, t) can be expressed as a
+Fourier series, in which each Fourier coefficient
+ˆu(ω, t) =
+� 1
+2π eiωx, e−Ltu(x, 0)
+�
+= 1
+2π
+� 2π
+0
+e−iωxe−Ltu(x, 0) dx
+is the standard inner product on (0, 2π) of a Fourier basis function and the solution operator
+e−Lt applied to the initial data u(x, 0). Spatial discretization results in a bilinear form
+involving a matrix function of the form
+uHf(A)v,
+where f(A) = e−At.
+In [6] Golub and Meurant described a method for computing bilinear forms involving
+matrix functions, that have the form
+(2.2)
+uT f(A)v,
+where u and v are N-vectors, A is an N × N symmetric positive definite matrix, and f is
+a smooth function. Since the matrix A is symmetric positive definite, it has real, positive
+eigenvalues
+b = λ1 ≥ λ2 ≥ . . . ≥ λN = a > 0,
+and orthogonal corresponding eigenvectors qj, j = 1, . . . , N. Using the spectral decomposi-
+tion of A, (2.2) can be rewritten as
+uT f(A)v =
+N
+�
+j=1
+f(λj)uT qjqT
+j v,
+but computing this summation would be impractical for large values of N, because computing
+all of the eigenvalues and eigenvectors of A would require O(N 3) floating point operations.
+Instead, we let α(λ) be a step function defined in terms of the coefficients of u and v in
+the basis of eigenvectors, given by
+α(λ) =
+�
+�
+�
+�
+�
+0,
+if λ < a
+�N
+j=i ujvj,
+if λi ≤ λ < λi−1
+�N
+j=1 ujvj,
+if b ≤ λ
+,
+uj = uT qj,
+vj = qT
+j v.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+3
+Then, the bilinear form (2.2) can be viewed as a Riemann-Stieltjes integral over the spectral
+domain,
+uT f(A)v = I[f] =
+� b
+a
+f(λ) dα(λ).
+This integral can then be approximated using a Gauss quadrature rule, which yields
+I[f] =
+K
+�
+j=1
+wjf(λj) + error,
+where the nodes λj, j = 1, . . . , K, and the weights wj, j = 1, . . . , K can be obtained from
+the Lanczos algorithm applied to A with initial vectors u and v, which produces a tridiagonal
+matrix TK. The eigenvalues of TK are the Gauss quadrature nodes, and the first components of
+the eigenvectors, squared and scaled by uT v, are the corresponding weights. When the vectors
+u and v are equal, α(λ) is positive and increasing, but if u ̸= v, this may not necessarily be
+the case, which can lead to a numerically unstable quadrature rule due to negative weights [1].
+As an alternative, we use a block approach, also described in [6], in which we compute
+� u
+v �T f(A)
+� u
+v �
+.
+Computing this block quadratic form results in the 2×2 matrix integral
+� b
+a
+f(λ) dα(λ) =
+� uT f(A)u
+uT f(A)v
+vT f(A)u
+vT f(A)v
+�
+=
+2K
+�
+j=1
+f(λj)ujuT
+j + error,
+where uj is a 2-vector. We can approximate this integral using a quadrature rule with twice
+as many nodes as in the non-block case, in which each outer product ujuT
+j , j = 1, . . . , 2K,
+plays the role of a weight.
+The nodes and (matrix-valued) weights are obtained by generalizing the Lanczos algorithm
+to the block case. Applying the block Lanczos algorithm [7] to A with initial block X1 =
+� u
+v �
+, where XT
+1 X1 = I, results in the 2K × 2K block tridiagonal matrix
+(2.3)
+TK =
+�
+������
+M1
+BT
+1
+B1
+M2
+BT
+2
+...
+...
+...
+BK−2
+MK−1
+BT
+K−1
+BK−1
+MK
+�
+������
+.
+We then define the quadrature rule for
+� u
+v �T f(A)
+� u
+v �
+as
+� b
+a
+f(λ) dα(λ) ≈
+2K
+�
+j=1
+f(λj)ujuT
+j = [f(TK)]1:2,1:2,
+where λj an eigenvalue of TK, and uj contains the first two elements of the corresponding
+normalized eigenvector.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+4
+B. RESTER AND J. V. LAMBERS
+A KSS method for the PDE (2.1) proceeds as follows, during each time step. Let un be
+the computed solution at time tn = n∆t, and let ˆeω be an N-vector of values of the Fourier
+basis function ˆeω(x) =
+1
+2πeiωx on a uniform N-point grid on [0, 2π). For each wave number
+ω = −N/2 + 1, . . . , N/2, we define
+R0 =
+� ˆeω
+un �
+,
+and then compute the “economy size” QR factorization
+R0 = X1B0,
+where X1 is N × 2 and B0 is 2 × 2. Block Lanczos iteration yields TK from X1. Then, the
+solution is approximated by
+[ˆun+1]ω =
+�
+BH
+0 exp[−TK∆t]1:2,1:2B0
+�
+12 .
+Let u(x, ∆t) be the exact solution, and let ˜u(x, ∆t) be the approximate solution. If K
+iterations of block Lanczos are performed, then, for ω = −N/2 + 1, . . . , N/2, [16]
+|⟨ˆeω, u(·, ∆t) − ˜u(·, ∆t)⟩| = O(∆t2K).
+In addition to high-order accuracy in time, KSS methods have been demonstrated to be
+effective even when using a time step that is larger than what the CFL condition would allow,
+even though KSS methods are explicit. In other words, KSS methods represent a “best-of-both
+worlds” balance between the computational efficiency of explicit methods and the stability of
+implicit methods. This balance is achieved through their componentwise approach, in which
+each Fourier coefficient is treated as a Riemann-Stieltjes integral with a frequency-dependent
+measure.
+2.2. KSS Methods for the Wave Equation. Now, we apply these ideas to the second-
+order wave equation
+(2.4)
+utt + Lu = 0 on (0, 2π) × (0, ∞),
+(2.5)
+u(x, 0) = f(x),
+ut(x, 0) = g(x),
+0 < x < 2π,
+with periodic boundary conditions
+(2.6)
+u(0, t) = u(2π, t),
+t > 0.
+The spatial differential operator L is defined by
+(2.7)
+Lu = −(p(x)ux)x + q(x)u,
+where we assume p(x) > 0 and q(x) ≥ 0, to guarantee that L is self-adjoint and positive
+definite.
+A spectral representation of the operator L allows us to describe the solution operator, the
+propagator, as a function of L [9]. We introduce
+R1(t) = L−1/2 sin(L1/2t),
+R0(t) = cos(L1/2t).
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+5
+Then, the solution can be written as
+� u(x, t)
+ut(x, t)
+�
+=
+�
+R0(t)
+R1(t)
+−L R1(t)
+R0(t)
+� � u(x, 0)
+ut(x, 0)
+�
+.
+The entries of this matrix, as functions of L, indicate which functions are the integrands in the
+Riemann-Stieltjes integrals used to compute the Fourier coefficients of the solution.
+By analogy with the description of a KSS method for the parabolic PDE (2.1), let un and
+un
+t be the computed solution at time tn and its time derivative, respectively. For each wave
+number ω = −N/2 + 1, . . . , N/2, we define
+R0 =
+� ˆeω
+un �
+,
+˜R0 =
+� ˆeω
+un
+t
+�
+,
+and then compute the QR factorizations
+R0 = X1B0,
+˜R0 = ˜X1 ˜B0.
+Block Lanczos iteration yields TK and ˜TK from X1 and ˜X1. Then, the solution and its time
+derivative are approximated by
+[ˆun+1]ω =
+�
+BH
+0 cos[T 1/2
+K
+∆t]1:2,1:2B0
+�
+12 +
+�
+˜BH
+0 ( ˜T −1/2
+K
+sin[ ˜T 1/2
+K
+∆t])1:2,1:2 ˜B0
+�
+12 ,
+[ˆun+1
+t
+]ω = −
+�
+BH
+0 (T 1/2
+K
+sin[T 1/2
+K
+∆t])1:2,1:2B0
+�
+12 +
+�
+˜BH
+0 cos[ ˜T 1/2
+K
+∆t]1:2,1:2 ˜B0
+�
+12 .
+Let u(x, ∆t) be the exact solution, and let ˜u(x, ∆t) be the approximate solution. If K
+iterations of block Lanczos are performed, then, for ω = −N/2 + 1, . . . , N/2, [14]
+|⟨ˆeω, u(·, ∆t) − ˜u(·, ∆t)⟩| = O(∆t4K),
+|⟨ˆeω, ut(·, ∆t) − ˜ut(·, ∆t)⟩| = O(∆t4K−1).
+The higher order of accuracy in time is due to the second derivative with respect to time in the
+PDE.
+2.3. Stability Results for KSS. The following has been proven about the stability of
+KSS methods:
+• Heat equation ut = puxx + q(x)u, where p is constant, q(x) is bandlimited: a
+first-order KSS method is unconditionally stable [16],
+• Wave equation utt = puxx + q(x)u, where p is constant, q(x) is bandlimited: a
+second-order KSS method is unconditionally stable [14],
+• Reaction-diffusion system of the form vt = Lv: a first-order KSS method with con-
+stant diffusion coefficient and bandlimited reaction term coefficient is unconditionally
+stable [20],
+• Wave equation utt = puxx + q(x)u, where p is constant, q(x) is bandlimited: a
+second-order block KSS method is unconditionally stable [14], and
+• Heat equation ut = (p(x)ux)x + q(x)u where p(x) and q(x) are bandlimited: a
+first-order block KSS method is unconditionally stable [20].
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+6
+B. RESTER AND J. V. LAMBERS
+3. Convergence Analysis. We will now analyze convergence of a second-order KSS
+method, with K = 1, for the IVP (2.4), (2.5), (2.6), (2.7), under the assumptions that the
+Fourier coefficients ˆp(ω), ˆq(ω) of p(x) and q(x), respectively, satisfy ˆp(ω) = q(ω) = 0 when
+|ω| > ωmax for some threshold ωmax. That is, we assume that p(x) and q(x) are bandlimited.
+We first carry out spatial discretization. We use a uniform grid, with spacing ∆x = 2π/N,
+where N is assumed to be even. Then, we let xN be an N-vector of grid points
+xj = j∆x,
+j = 0, 1, 2, . . . , N − 1.
+We denote by ωj the corresponding wave numbers
+ωj = j − N/2,
+j = 1, 2, . . . , N.
+We then denote by DN an N × N matrix that discretizes the second derivative operator using
+the discrete Fourier transform:
+DN = F −1
+N ΛNFN,
+where
+[FN]jk = 1
+N e−iωjxk,
+[ΛN]jj = −ω2
+j .
+We also let IN denote the N × N identity matrix, whereas I is the identity operator on
+functions of x.
+Let u1 = u and u2 = ut. We then rewrite (2.4) as the first-order system
+∂u1
+∂t = u2,
+(3.1)
+∂u2
+∂t = −Lu1,
+(3.2)
+which, for convenience, we write as
+(3.3)
+vt = ˜Lv,
+v =
+� u1
+u2
+�
+,
+˜L =
+�
+0
+I
+−L
+0
+�
+.
+Spatial discretization of (3.3) yields a system of ODEs
+v′
+N(t) = ˜LNv(t),
+where
+vN(t) =
+� u1,N(t)
+u2,N(t)
+�
+is the spatial discretization of the vector field v, and ˜LN is a 2N × 2N matrix which has the
+2 × 2 block structure
+˜LN =
+�
+0
+IN
+−LN
+0
+�
+.
+We define the exact solution operator of (3.1), (3.2) as
+(3.4)
+S(t) = exp[˜Lt] =
+� S11(t)
+S12(t)
+S12(t)
+S22(t)
+�
+=
+�
+R0(t)
+R1(t)
+−L R1(t)
+R0(t)
+�
+,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+7
+where, as before, R1(t) = L−1/2 sin(L1/2t) and R0(t) = cos(L1/2t). Then we let
+SN(∆t) =
+� SN,11(∆t)
+SN,12(∆t)
+SN,12(∆t)
+SN,22(∆t)
+�
+,
+where each SN,ij(∆t) is the approximation of Sij(∆t) by the KSS method.
+A KSS method applied to (2.4) with K = 1 uses two block Gauss quadrature nodes for
+each Fourier coefficient. Using an approach described in [18], we estimate these nodes, rather
+than using block Lanczos iteration explicitly. This significantly improves the efficiency of KSS
+methods, but it will also simplify the convergence analysis to be carried out in this section.
+The quadrature nodes will be prescribed as follows:
+(3.5)
+l1,ω = 0,
+l2,ω = pω2 + q,
+ω = −N/2 + 1, . . . , N/2,
+where p and q are the average values of p(x) and q(x), respectively, on [0, 2π]. To interpolate
+the functions
+f11(λ) = f22(λ) = cos(λ1/2∆t),
+f12(λ) = λ−1/2 sin(λ1/2∆t),
+f21(λ) = −λf12(λ)
+at the nodes l1,ω, l2,ω, we compute the slopes
+Mij,ω = fij(l2,ω) − fij(l1,ω)
+l2,ω − l1,ω
+,
+ω = −N/2 + 1, . . . , N/2,
+i, j = 1, 2.
+We then describe the computed solution at time tn+1 by
+un+1 = z11 + z12,
+un+1
+t
+= z21 + z22,
+(3.6)
+where
+zi1 = SN,i1(∆t)un,
+zi2 = SN,i2(∆t)un
+t ,
+i = 1, 2.
+We also define ˜p = p − ¯p, ˜q = q − ¯q, and let PN, QN, ˜PN, and ˜QN be diagonal matrices with
+the values of the coefficients p(x), q(x), ˜p(x), and ˜q(x), respectively, at the grid points on the
+main diagonal.
+Let Iω = {k ∈ Z|0 < |k − ω| ≤ ωmax}. The discrete Fourier coefficients of zij,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+8
+B. RESTER AND J. V. LAMBERS
+i, j = 1, 2, are then given by
+ˆz11(ω) = S11(l2,ω)(ˆeH
+ω un) + M11,ωˆeH
+ω (LN − l2,ωI)un
+= S11(l2,ω)ˆu(ω) − iωM11,ω
+�
+k∈Iω
+ˆp(ω − k)i(k)ˆu(k) +
+M11,ω
+�
+k∈Iω
+ˆq(ω − k)ˆu(k),
+(3.7)
+ˆz21(ω) = S21(l2,ω)ˆu(ω) − iωM21,ω
+�
+k∈Iω
+ˆp(ω − k)i(k)ˆu(k) +
+M21,ω
+�
+k∈Iω
+ˆq(ω − k)ˆu(k),
+(3.8)
+ˆz12(ω) = S12(l2,ω)ˆut(ω) − iωM12,ω
+�
+k∈Iω
+ˆp(ω − k)i(k)ˆut(k) +
+M12,ω
+�
+k∈Iω
+ˆq(ω − k)ˆut(k),
+(3.9)
+ˆz22(ω) = S22(l2,ω)ˆut(ω) − iωM22,ω
+�
+k∈Iω
+ˆp(ω − k)i(k)ˆut(k) +
+M22,ω
+�
+k∈Iω
+ˆq(ω − k)ˆut(k),
+(3.10)
+where i = √−1 and ω = −N/2 + 1, . . . , N/2. To obtain these formulas, we used the
+simplification
+ˆeH
+ω (LN − l2,ωI)un = ˆeH
+ω [−DNPNDN + QN]un − l2,ω(ˆeH
+ω un)
+= −iωˆeH
+ω PNDNun + ˆeH
+ω QNun − l2,ω(ˆeH
+ω un)
+= (¯pω2 + ¯q)(ˆeH
+ω un) − iωˆeH
+ω ˜PNDNun + ˆeH
+ω ˜QNun − l2,ω(ˆeH
+ω un)
+= −iωˆeH
+ω ˜PNDNun + ˆeH
+ω ˜QNun.
+To bound error, we need to establish an upper bound of a norm of the approximate solution
+operator SN(∆t). We elect to use the C-norm, defined by
+∥ (u, v) ∥2
+C = ⟨u, Cu⟩ + ⟨v, v⟩,
+where, as before, ⟨·, ·⟩ is the standard inner product on (0, 2π), and its discrete counterpart,
+the CN-norm, defined by
+∥ (u, v) ∥2
+CN = uT CNu + ∥v∥2
+2.
+The N × N matrix CN discretizes the constant-coefficient differential operator C = −¯p∂xx +
+¯qI, where u and v are N-vectors. We choose to bound the CN-norm of the solution operator
+for convenience, because the operator CN has a very simple expression in Fourier space due
+to its constant coefficients, which simplifies the analysis.
+3.1. Stability. We wish to express ∥SN(∆t)∥CN as the 2-norm of some matrix, since
+that will be easier to bound. We define ∥SN(∆t)∥CN by
+∥SN(∆t)∥2
+CN =
+sup
+w=(u,v)̸=0
+∥SN(∆t)w∥2
+CN
+∥w∥2
+CN
+.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+9
+Then
+(3.11)
+∥SN(∆t)∥2
+CN =
+sup
+(u,v)̸=0
+˜uT CN ˜u + ∥˜v2∥2
+uT CNu + ∥v2∥2 ,
+where
+�˜u
+˜v
+�
+= SN(∆t)
+�u
+v
+�
+= SN(∆t)w. In matrix form, we have
+∥SN(∆t)∥2
+CN = sup wT SN(∆t)T ˜CNSN(∆t)w
+wT ˜CNw
+= sup wT SN(∆t)T ˜CNSN(∆t)w
+( ˜C1/2
+N w)T ( ˜C1/2
+N w)
+,
+where ˜CN =
+�CN
+0
+0
+I
+�
+. Let z = ˜C1/2
+N w. Then
+∥SN(∆t)∥2
+CN = sup zT ( ˜C1/2
+N SN(∆t) ˜C−1/2
+N
+)T ( ˜C1/2
+N SN(∆t) ˜C−1/2
+N
+)z
+zT z
+.
+Therefore,
+∥SN(∆t)∥CN = ∥B∥2 =
+�
+ρ (BT B) ≤
+�
+∥G∥∞,
+where B = ˜C1/2
+N SN(∆t) ˜C−1/2
+N
+, and
+G = BT B =
+�G11
+G12
+G21
+G22
+�
+,
+(3.12)
+with
+G11 = C−1/2
+N
+SN,11(∆t)T CNSN,11(∆t)C−1/2
+N
++
+C−1/2
+N
+SN,21(∆t)T SN,21(∆t)C−1/2
+N
+,
+(3.13)
+G12 = C−1/2
+N
+SN,11(∆t)T CNSN,12(∆t) + C−1/2
+N
+SN,21(∆t)T SN,22(∆t),
+(3.14)
+G21 = SN,12(∆t)T CNSN,11(∆t)C−1/2
+N
++ SN,22(∆t)T SN,21(∆t)C−1/2
+N
+,
+(3.15)
+G22 = SN,12(∆t)T CNSN,12(∆t) + SN,22(∆t)T SN,22(∆t).
+(3.16)
+To obtain a bound for the CN-norm of the overall approximate solution operator SN(∆t), we
+will proceed by bounding ∥G∥∞ through bounding ∥Gij∥∞ for i, j = 1, 2.
+To bound the norm of each such block, we use expressions for the computed solution
+un+1, un+1
+t
+at time tn+1, in terms of un and un
+t . We begin with
+∥(un+1, un+1
+t
+)∥2
+CN = (un+1)T CN(un+1) + (un+1
+t
+)T (un+1
+t
+)
+= [un]T ¯G11un + [un]T ¯G12un
+t + [un
+t ]T ¯G21un + [un
+t ]T ¯G22un
+t
+where
+¯G11 = SN,11(∆t)T CNSN,11(∆t) + SN,21(∆t)T SN,21(∆t),
+¯G12 = SN,11(∆t)T CNSN,12(∆t) + SN,21(∆t)T SN,22(∆t),
+¯G21 = SN,12(∆t)T CNSN,12(∆t) + SN,22(∆t)T SN,21(∆t),
+¯G22 = SN,12(∆t)T CNSN,12(∆t) + SN,22(∆t)T SN,22(∆t).
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+10
+B. RESTER AND J. V. LAMBERS
+We note that
+G11 = C−1/2
+N
+¯G11C−1/2
+N
+,
+G12 = C−1/2
+N
+¯G12,
+G21 = ¯G21C−1/2
+N
+,
+G22 = ¯G22.
+Therefore, we can proceed by bounding the entries of each ¯Gij, for i, j = 1, 2.
+LEMMA 3.1. Assume ˆp(ω) = 0 and ˆq(ω) = 0 for |ω| > ωmax. Then the matrix G11
+defined in (3.12) satisfies
+∥G11∥∞ ≤ 1 + C11,p∥˜p∥∞∆t2N 2 + C11,q∥˜q∥∞∆t2 + C11,p2∥˜p∥2
+∞∆t2N 2 +
+C11,pq∥˜p∥∞∥˜q∥∞∆t2 + C11,q2∥˜q∥2
+∞∆t2
+(3.17)
+where the constants C11,p, C11,q, C11,p2, C11,pq, and C11,q2 are independent of N and ∆t.
+Proof. Let ˆIN = {−N/2 + 1, . . . , N/2}. From (3.6) we have
+[un]T ¯G11un = zT
+11CNz11 + zT
+21z21
+=
+�
+ω∈ˆIN
+ˆz11(ω)ˆz11(ω)(¯pω2 + ¯q) +
+�
+ω∈ˆIN
+ˆz21(ω)ˆz21(ω)
+=
+�
+j∈ˆIN
+�
+k∈ˆIN
+ˆu(−j)ˆu(k)
+� ¯A + ¯B + ¯C + ¯D
+�
+jk
+where, by (3.7) and (3.8), we have, for j, k ∈ ˆIN,
+¯Ajj = (S11(l2,j))2 (pj2 + q) + (S21(l2,j))2
+= cos2 ��
+pj2 + q∆t
+� �
+pj2 + q
+�
++
+�
+−
+�
+pj2 + q
+�1/2 sin
+��
+pj2 + q∆t
+��2
+= pj2 + q,
+¯Bjk = i(j)S11(l2,k)ikM11,k ˆp(j − k)(pk2 + q) + S11(l2,k)M11,kˆq(j − k)(pk2 + q) +
+i(j)S21(l2,k)ikM21,k ˆp(j − k) + S21(l2,k)M21,kˆq(j − k),
+j ̸= k,
+¯Cjk = i(k)S11(l2,j)ijM11,j ˆp(j − k)(pj2 + q) + S11(l2,j)M11,j ˆq(j − k)(pj2 + q) +
+ijM21,j ˆp(j − k)i(k)S21(l2,j) + S21(l2,j)M21,j ˆq(j − k),
+j ̸= k,
+and
+¯Djk = i(k)i(j)
+�
+ω∈ˆIN\{k,j}
+ω2ˆp(ω − k)ˆp(j − ω)
+�
+M 2
+11,ω(pω2 + q) + M 2
+21,ω
+�
++
+i(k)i
+�
+ω∈ˆIN\{k,j}
+ωˆp(ω − k)ˆq(j − ω)
+�
+M 2
+11,ω(pω2 + q) + M 2
+21,ω
+�
++
+i(j)i
+�
+ω∈ˆIN\{k,j}
+ωˆq(ω − k)ˆp(j − ω)
+�
+M 2
+11,ω(pω2 + q) + M 2
+21,ω
+�
++
+�
+ω∈ˆIN\{k,j}
+ˆq(ω − k)ˆq(j − ω)
+�
+M 2
+11,ω(pω2 + q) + M 2
+21,ω
+�
+,
+with ¯Ajk = 0 for j ̸= k, and ¯Bjj = ¯Cjj = 0 for j ∈ ˆIN.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+11
+To obtain an upper bound for ∥G11∥∞, we use the following bounds on Sij(l2,ω) and
+Mij,ω, which are the coefficients in the linear approximations of the various components of
+the solution operator:
+|S11(l2,ω)| ≤
+���cos
+�
+l1/2
+2,ω∆t
+���� ≤ 1,
+|S21(l2,ω)| ≤
+���−l1/2
+2,ω sin
+�
+l1/2
+2,ω∆t
+���� ≤
+���l1/2
+2,ω
+��� l1/2
+2,ω∆t = l2,ω∆t,
+|M11,ω| ≤ ∆t2
+2 ,
+|M11,ω| ≤
+∆t
+(pω2 + q)1/2 ,
+|M21,ω| ≤ ∆t.
+We have multiple bounds for M11 so that different terms will have the same order of magnitude
+in terms of N and ∆t. Then, for j ̸= k, we have
+�� ¯Bjk
+�� ≤
+����jkˆp(j − k)(pk2 + q)∆t2
+2
+���� +
+����
+∆t2
+2 ˆq(j − k)(pk2 + q)
+���� +
+��jkˆp(j − k)(pk2 + q)∆t2�� +
+��ˆq(j − k)(pk2 + q)∆t2��
+≤ 3
+2∆t2(pk2 + q) (|jkˆp(j − k)| + |ˆq(j − k)|) ,
+| ¯Cjk| ≤
+����jkˆp(j − k)(pj2 + q)∆t2
+2
+���� +
+����ˆq(j − k)(pj2 + q)∆t2
+2
+���� +
+��jkˆp(j − k)(pj2 + q)∆t2�� +
+��ˆq(j − k)(pj2 + q)∆t2��
+≤ 3
+2∆t2(pj2 + q) (|jkˆp(j − k)| + |ˆq(j − k)|) ,
+| ¯Djk| ≤
+������
+i(k)i(j)
+�
+ω∈ˆIN\{k,j}
+ω2|ˆp(ω − k)ˆp(j − ω)|
+�
+2∆t2�
++
+i(k)i
+�
+ω∈ˆIN\{k,j}
+ω|ˆp(ω − k)ˆq(j − ω)|
+�
+2∆t2�
++
+i(j)i
+�
+ω∈ˆIN\{k,j}
+ω|ˆq(ω − k)ˆp(j − ω)|
+�
+2∆t2�
++
+�
+ω∈ˆIN\{k,j}
+|ˆq(ω − k)ˆq(j − ω)|
+�
+2∆t2�
+������
+.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+12
+B. RESTER AND J. V. LAMBERS
+From these bounds, we obtain
+∥G11∥∞ ≤ max
+1≤j≤N
+�
+k∈ˆIN\j
+(¯pj2 + ¯q)−1/2 �� ¯Ajk + ¯Bjk + ¯Cjk + ¯Djk
+�� (¯pk2 + ¯q)−1/2
+≤ max
+1≤j≤N(¯pj2 + ¯q)−1/2(¯pj2 + ¯q)−1/2(pj2 + q) +
+3
+2
+�
+k∈ˆIN\j
+���jkˆp(j − k)(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)1/2∆t2��� +
+3
+2
+�
+k∈ˆIN\j
+���ˆq(j − k)(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)1/2∆t2��� +
+3
+2
+�
+k∈ˆIN\j
+���jkˆp(j − k)(¯pj2 + ¯q)1/2(¯pk2 + ¯q)−1/2∆t2��� +
+3
+2
+�
+k∈ˆIN\j
+���ˆq(j − k)(¯pj2 + ¯q)1/2(¯pk2 + ¯q)−1/2∆t2��� +
+�
+k∈ˆIN
+������
+i(k)i(j)(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2
+�
+ω∈ˆIN\{k,j}
+ω2|ˆp(ω − k)ˆp(j − ω)|∆t2
+������
++
+�
+k∈ˆIN
+������
+i(k)i(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2
+�
+ω∈ˆIN\{k,j}
+ω|ˆp(ω − k)ˆq(j − ω)|∆t2
+������
++
+�
+k∈ˆIN
+������
+i(j)i(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2
+�
+ω∈ˆIN\{k,j}
+ω|ˆq(ω − k)ˆp(j − ω)|∆t2
+������
++
+�
+k∈ˆIN
+������
+(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2
+�
+ω∈ˆIN\{k,j}
+|ˆq(ω − k)ˆq(j − ω)|∆t2
+������
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+13
+≤ max
+1≤j≤N 1 + 3
+2j∆t2∥˜p∥∞(pj2 + q)−1/2 �
+k∈Ij
+���k(pk2 + q)1/2��� +
+3
+2∆t2∥˜q∥∞(pj2 + q)−1/2 �
+k∈Ij
+���(pk2 + q)1/2��� +
+3
+2j∆t2∥˜p∥∞(pj2 + q)1/2 �
+k∈Ij
+���k(pk2 + q)−1/2��� +
+3
+2∆t2∥˜q∥∞(pj2 + q)1/2 �
+k∈Ij
+���(pk2 + q)−1/2��� +
+j∆t2∥˜p∥2
+∞(pj2 + q)−1/2 �
+k∈ˆIN
+������
+k(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω2
+������
++
+∆t2∥˜p∥∞∥˜q∥∞(pj2 + q)−1/2 �
+k∈ˆIN
+������
+k(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω
+������
++
+j∆t2∥˜p∥∞∥˜q∥∞(pj2 + q)−1/2 �
+k∈ˆIN
+������
+(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω
+������
++
+∆t2∥˜q∥2
+∞(pj2 + q)−1/2 �
+k∈ˆIN
+������
+(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+1
+������
+.
+(3.18)
+We can bound each of the summations in (3.18) as in the following examples.
+• We first derive a bound for
+(3.19)
+(pj2 + q)1/2 �
+k∈Ij
+���(pk2 + q)−1/2��� .
+For |j| > ωmax, we have
+(pj2 + q)1/2 �
+k∈Ij
+���(pk2 + q)−1/2��� ≤ (pj2 + q)1/2 �
+k∈Ij
+1
+(p(|j| − ωmax)2 + q)1/2
+≤ (pj2 + q)1/2
+2ωmax
+(p(|j| − ωmax)2 + q)1/2 .
+As |j| → ∞, we obtain
+lim
+j→∞(pj2 + q)1/2 �
+k∈Ij
+���(pk2 + q)−1/2��� ≤ 2ωmax.
+Therefore, the expression (3.19) can be bounded independently of N.
+• Next, we derive a bound for
+(3.20)
+(pj2 + q)−1/2 �
+k∈ˆIN
+������
+k(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω
+������
+.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+14
+B. RESTER AND J. V. LAMBERS
+We have
+�
+k∈ˆIN
+������
+k(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω
+������
+≤
+�
+k∈ˆIN
+�
+ω∈Ij∩Ik
+���k(pk2 + q)−1/2ω
+���
+≤
+�
+ω∈Ij
+|ω|
+�
+k∈Iω
+���k(pk2 + q)−1/2���
+≤
+�
+ω∈Ij
+|ω|
+�
+η∈I0
+���(ω + η)(p(ω + η)2 + q)−1/2��� .
+If j > 2ωmax, then ω > ωmax, and
+�
+k∈ˆIN
+������
+k(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω
+������
+≤
+�
+ω∈Ij
+|j + ωmax|
+�
+η∈I0
+����
+j + 2ωmax
+(p(j − 2ωmax)2 + q)1/2
+���� .
+We then have
+lim
+j→∞(pj2 + q)−1/2 �
+k∈ˆIN
+������
+k(pk2 + q)−1/2
+�
+ω∈Ij∩Ik
+ω
+������
+≤ 1
+¯p.
+We conclude that the expression (3.20) can also be bounded independently of N.
+Using a similar approach to bound the remaining summations in (3.18), we obtain
+∥G11∥∞ ≤ 1 + C11,p∥˜p∥∞∆t2N 2 + C11,q∥˜q∥∞∆t2 + C11,p2∥˜p∥2
+∞∆t2N 2 +
+C11,pq∥˜p∥∞∥˜q∥∞∆t2 + C11,q2∥˜q∥2
+∞∆t2,
+for constants C11,p, C11,q, C11,p2, C11,pq, C11,q2 that are independent of N and ∆t, which
+completes the proof.
+Using the same approach, we find that the matrix G22 defined in
+(3.12) satisfies a bound of the same form as that of G11, with appropriate constant factors.
+LEMMA 3.2. Assume ˆp(ω) = 0 and ˆq(ω) = 0 for |ω| > ωmax. Then the matrix G12
+defined in (3.12) satisfies
+∥G12∥∞ ≤ C12,p∥˜p∥∞∆tN + C12,q∥˜q∥∞∆t + C12,p2∥˜p∥2
+∞∆tN +
+C12,pq∥˜p∥∞∥˜q∥∞∆t + C12,q2∥˜q∥2
+∞∆t,
+(3.21)
+where each constant is independent of N and ∆t.
+Proof. From (3.6), we have
+[un]T ¯G12un
+t =
+�
+ω∈ˆIN
+ˆz11(ω)ˆz12(ω)(¯pω2 + ¯q) +
+�
+ω∈ˆIN
+ˆz21(ω)ˆz22(ω)
+=
+�
+j∈ˆIN
+�
+k∈ˆIN
+ˆu(−j)ˆu(k)
+� ¯A + ¯B + ¯C + ¯D
+�
+jk ,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+15
+where, for j, k ∈ ˆIN,
+¯Ajj = S11(l2,j)S12(l2,j)(¯pj2 + ¯q) + S21(l2,j)S22(l2,j)
+= cos
+�
+(pj2 + q)1/2∆t
+�
+(pj2 + q)−1/2 sin
+��
+pj2 + q∆t
+�
+(¯pj2 + ¯q) +
+�
+−(pj2 + q)1/2 sin
+�
+(pj2 + q)1/2∆t
+�
+cos
+��
+pj2 + q∆t
+��
+= cos
+�
+(pj2 + q)1/2∆t
+�
+(pj2 + q)1/2 sin
+��
+pj2 + q∆t
+�
+−
+(pj2 + q)1/2 sin
+�
+(pj2 + q)1/2∆t
+�
+cos
+��
+pj2 + q∆t
+�
+= 0,
+¯Bjk = i(j)ikM11,k ˆp(j − k)S12(l2,k)(¯pk2 + ¯q) + M11,kˆq(j − k)S12(l2,k)(¯pk2 + ¯q) +
+i(j)ikM21,k ˆp(j − k)S22(l2,k) + M21,kˆq(j − k)S22(l2,k),
+j ̸= k,
+¯Cjk = ijS11(l2,j)M12,j ˆp(j − k)i(k)(¯pj2 + ¯q) + S11(l2,j)M12,j ˆq(j − k)(¯pj2 + ¯q) +
+S21(l2,j)ijM22,j ˆp(j − k)i(k) + S21(l2,j)M22,j ˆq(j − k),
+j ̸= k,
+and
+¯Djk = i(j)i(k)
+�
+ω∈ˆIN\{k,j}
+ω2ˆp(j − ω)ˆp(ω − k)
+�
+M11,ωM12,ω(¯pω2 + ¯q) + M21,ωM22,ω
+�
++
+i(j)i
+�
+ω∈ˆIN\{k,j}
+ωˆp(j − ω)ˆq(ω − k)
+�
+M11,ωM12,ω(¯pω2 + ¯q) + M21,ωM22,ω
+�
++
+i(k)i
+�
+ω∈ˆIN\{k,j}
+ωˆq(j − ω)ˆp(ω − k)
+�
+M11,ωM12,ω(¯pω2 + ¯q) + M21,ωM22,ω
+�
++
+�
+ω∈ˆIN\{k,j}
+ˆq(j − ω)ˆq(ω − k)
+�
+M11,ωM12,ω(¯pω2 + ¯q) + M21,ωM22,ω
+�
+,
+with ¯Ajk = 0 for j ̸= k, and ¯Bjj = ¯Cjj = 0 for j ∈ ˆIN.
+To obtain an upper bound for ∥G12∥∞, we use the following bounds on Sij(l2,ω) and
+Mij,ω :
+|S12(l2,ω)| ≤
+���l−1/2
+2,ω
+sin(l1/2
+2,ω∆t)
+��� ≤
+���l−1/2
+2,ω
+��� l1/2
+2,ω∆t = ∆t,
+|M11,ω| = |M22,ω| ≤
+2
+pω2 + q ,
+|M12,ω| ≤
+∆t
+pω2 + q .
+Then we have
+| ¯Bjk| ≤ 3 (|jkˆp(j − k)∆t| + |ˆq(j − k)∆t|) ,
+| ¯Cjk| ≤ 3 (|jkˆp(j − k)∆t| + |ˆq(j − k)∆t|) ,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+16
+B. RESTER AND J. V. LAMBERS
+and
+| ¯Djk| ≤
+������
+i(j)i(k)
+�
+ω∈ˆIN\{k,j}
+ω2|ˆp(j − ω)ˆp(ω − k)|
+4∆t
+pω2 + q +
+i(j)i
+�
+ω∈ˆIN\{k,j}
+ω|ˆp(j − ω)ˆq(ω − k)|
+4∆t
+pω2 + q +
+i(k)i
+�
+ω∈ˆIN\{k,j}
+ω|ˆq(j − ω)ˆp(ω − k)|
+4∆t
+pω2 + q +
+�
+ω∈ˆIN\{k,j}
+|ˆq(j − ω)ˆq(ω − k)|
+4∆t
+pω2 + q
+������
+.
+From these bounds, we obtain
+∥G12∥∞ ≤ max
+1≤j≤N
+�
+k∈ˆIN\j
+���
+� ¯Ajk + ¯Bjk + ¯Cjk + ¯Djk
+�
+(¯pj2 + ¯q)−1/2���
+≤ max
+1≤j≤N 6
+�
+k∈ˆIN\j
+���jkˆp(j − k)(¯pj2 + ¯q)−1/2∆t
+��� +
+6
+�
+k∈ˆIN\j
+���ˆq(j − k)(¯pj2 + ¯q)−1/2∆t
+��� +
+�
+k∈ˆIN\j
+������
+i(j)i(k)
+�
+ω∈ˆIN\{k,j}
+ω2|ˆp(j − ω)ˆp(ω − k)|(¯pj2 + ¯q)−1/2
+4∆t
+pω2 + q
+������
++
+�
+k∈ˆIN\j
+������
+i(j)i
+�
+ω∈ˆIN\{k,j}
+ω|ˆp(j − ω)ˆq(ω − k)|(¯pj2 + ¯q)−1/2
+4∆t
+pω2 + q
+������
++
+�
+k∈ˆIN\j
+������
+i(k)i
+�
+ω∈ˆIN\{k,j}
+ω|ˆq(j − ω)ˆp(ω − k)|(¯pj2 + ¯q)−1/2
+4∆t
+pω2 + q
+������
++
+�
+k∈ˆIN\j
+������
+�
+ω∈ˆIN\{k,j}
+|ˆq(j − ω)ˆq(ω − k)|(¯pj2 + ¯q)−1/2
+4∆t
+pω2 + q
+������
+≤ max
+1≤j≤N 6j∥˜p∥∞(¯pj2 + ¯q)−1/2∆t
+�
+k∈Ij
+|k| +
+6∥˜q∥∞(¯pj2 + ¯q)−1/2∆t(2ωmax) +
+j∥˜p∥2
+∞(¯pj2 + ¯q)−1/2∆t4
+p
+�
+k∈ˆIN
+�
+ω∈Ij∩Ik
+|k| +
+∥˜p∥∞∥˜q∥∞(¯pj2 + ¯q)−1/2∆t4
+p
+�
+ω∈Ij
+�
+2jωmax +
+�
+k∈Iω
+|k|
+�
++
+∥˜q∥2
+∞(¯pj2 + ¯q)−1/2∆t4
+q (4ω2
+max).
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+17
+These summations can be bounded as in the proof of Lemma 3.1. As an example, we
+obtain a bound for
+(3.22)
+(¯pj2 + ¯q)−1/2 �
+k∈Ij
+|k| .
+If |j| > ωmax, then
+(3.23)
+�
+k∈Ij
+|k| =
+�
+η∈I0
+|j − η| =
+ωmax
+�
+η=1
+|j − η| + |j + η| =
+ωmax
+�
+η=1
+2|j| = 2|j|ωmax.
+It follows that
+lim
+j→∞(¯pj2 + ¯q)−1/2 �
+k∈Ij
+|k| = lim
+j→∞(¯pj2 + ¯q)−1/22|j|ωmax = 2ωmax
+¯p
+.
+That is, the expression (3.22) is bounded independently of N, whereas it would be O(N) if
+p(x) was not bandlimited.
+Proceeding in a similar manner for the remaining summations, we conclude that there
+exist constants C12,p, C12,q, C12,p2, C12,pq and C12,q2 such that
+∥G12∥∞ ≤ C12,p∥˜p∥∞∆tN + C12,q∥˜q∥∞∆t + C12,p2∥˜p∥2
+∞∆tN +
+C12,pq∥˜p∥∞∥˜q∥∞∆t + C12,q2∥˜q∥2
+∞∆t,
+which completes the proof.
+Using the same approach, it can be shown that the matrix
+G21 in (3.12) satisfies a bound of the same form as that of G12.
+We now prove a result that suggests the second-order KSS method applied to (2.4), (2.5),
+(2.6) is unstable.
+THEOREM 3.3. Assume ˆp(ω) = 0 and ˆq(ω) = 0 for |ω| > ωmax. Then the solution
+operator SN(∆t) satisfies
+∥SN(∆t)∥CN ≤ 1 + (Cp∥˜p∥∞N + Cq∥˜q∥∞)∆t,
+(3.24)
+where the constants Cp and Cq are independent of N and ∆t.
+Proof. From
+(3.25)
+����
+�G11
+G12
+G21
+G22
+�����
+∞
+≤ max{∥G11∥∞ + ∥G12∥∞, ∥G21∥∞ + ∥G22∥∞},
+we have, for some k ∈ {1, 2},
+∥SN(∆t)∥CN = ∥B∥2 ≤
+�
+∥G∥∞ ≤
+�
+∥Gk1∥∞ + ∥Gk2∥∞.
+From Lemma 3.1 and Lemma 3.2, we have
+∥G∥∞ ≤ 1 + ∆tN
+�
+Ck2,p∥˜p∥∞ + Ck2,p2∥˜p∥2
+∞
+�
++ ∆t2N 2 �
+Ck1,p∥˜p∥∞ + Ck1,p2∥˜p∥2
+∞
+�
++
+∆t
+�
+Ck2,q∥˜q∥∞ + Ck2,pq∥˜p∥∞∥˜q∥∞ + Ck2,q2∥˜q∥2
+∞
+�
++
+∆t2 �
+Ck1,q∥˜q∥∞ + Ck1,pq∥˜p∥∞∥˜q∥∞ + Ck1,q2∥˜q∥2
+∞
+�
+.
+(3.26)
+Let
+R1 = N
+�
+Ck2,p∥˜p∥∞ + Ck2,p2∥˜p∥2
+∞
+�
++ Ck2,q∥˜q∥∞ + Ck2,pq∥˜p∥∞∥˜q∥∞ + Ck2,q2∥˜q∥2
+∞,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+18
+B. RESTER AND J. V. LAMBERS
+R2 = N 2 �
+Ck1,p∥˜p∥∞ + Ck1,p2∥˜p∥2
+∞
+�
++ Ck1,q∥˜q∥∞ + Ck1,pq∥˜p∥∞∥˜q∥∞ + Ck1,q2∥˜q∥2
+∞,
+and R = max{R1/2
+2
+, R1/2}. We then have
+∥SN(∆t)∥CN ≤
+�
+∥G∥∞ ≤ 1 + R∆t ≤ 1 + (Cp∥˜p∥∞N + Cq∥˜q∥∞)∆t,
+from which the result follows.
+COROLLARY 3.4. Assume the leading coefficient p(x) is constant. Then, under the
+assumptions of Theorem 3.3,
+∥SN(∆t)∥CN ≤ 1 + Cq∥˜q∥∞∆t.
+Proof. Because the leading coefficient p(x) is constant, we have ˜p(x) = p(x) − ¯p ≡ 0.
+The result follows immediately from the last line of the proof of Theorem 3.3.
+Therefore,
+a second-order KSS method applied to (2.4), (2.5), (2.6), (2.7), under the assumptions that
+p(x) is constant and q(x) is bandlimited, is unconditionally stable.
+3.2. Consistency. For the remainder of this convergence analysis, we assume the coeffi-
+cient p(x) from (2.7) is constant, since stability has been proved only for this case.
+Before we obtain an estimate of the local truncation error, we introduce additional notation.
+We first define the restriction operator
+RNf(x) = f(xN),
+and interpolation operator
+TNg = TN
+� g1
+g2
+�
+=
+�
+��������
+N/2
+�
+ω=−N/2+1
+eiωx˜g1(ω)
+N/2
+�
+ω=−N/2+1
+eiωx˜g2(ω)
+�
+��������
+,
+where, for i = 1, 2,
+˜gi(ω) = 1
+N
+N
+�
+j=1
+e−iωxj[gi]j.
+Then, the operator IN = TNRN on L2([0, 2π])×L2([0, 2π]) computes the Fourier interpolant
+of each component function. By contrast, if we define
+ˆRNf(x) = ˆRN
+� f1(x)
+f2(x)
+�
+=
+�
+��������
+N/2
+�
+ω=−N/2+1
+eiωxN ˆf1(ω)
+N/2
+�
+ω=−N/2+1
+eiωxN ˆf2(ω)
+�
+��������
+,
+where, for i = 1, 2,
+ˆfi(ω) = 1
+2π
+� 2π
+0
+e−iωxfi(x) dx,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+19
+then the operator PN = TN ˆRN on L2([0, 2π]) × L2([0, 2π]) is the orthogonal projection
+operator onto span{eiωx}N/2
+ω=−N/2+1. Finally, the continuous approximate solution operator
+˜SN(∆t) : L2([0, 2π]) → L2([0, 2π]) is defined by ˜SN(∆t) = TNSN(∆t)RN.
+THEOREM 3.5. Let f ∈ Hm+1
+p
+([0, 2π]) × Hm
+p ([0, 2π]) for m ≥ 4. Then, for the
+problem (2.4), (2.5), (2.6), (2.7) on the domain (0, 2π) × (0, T), with p(x) constant and q(x)
+bandlimited, the two-node block KSS method with prescribed nodes (3.5) is consistent. That is,
+the local truncation error satisfies
+1
+∆t
+���IN exp[˜L∆t]f − ˜SN(∆t)f
+���
+C ≤ C1∆xm−1 + C2∆t2,
+where the constants C1 and C2 are independent of ∆x and ∆t.
+Proof: We split the local truncation error into two components:
+E1(∆t, ∆x) = IN exp(˜L∆t)f(x) − TN exp(˜LN∆t)RNf(x)
+E2(∆t, ∆x) = TN exp(˜LN∆t)RNf(x) − ˜SN(∆t)f(x)
+= TN[exp(˜LN∆t) − SN(∆t)]RNf(x).
+First, we bound E1(∆t, ∆x). Because of the regularity of f, we have
+(3.27)
+∥f − fN∥C ≤ C0∆xm
+for some constant C0 (see [11, Theorem 2.16]). Next, we note that the exact solution v(x, t) =
+exp[˜Lt]f(x) has the spectral decomposition
+v(x, t) =
+∞
+�
+k=1
+eµktvk(x)⟨vk, f⟩
+where {µk}∞
+k=1 are the purely imaginary eigenvalues of ˜L, and {vk(x)}∞
+k=1 are the cor-
+responding orthonormal eigenfunctions, each of which belongs to C∞
+p [0, 2π]. Using this
+spectral decomposition, it can be shown using an approach similar to that used in [4, Section
+7.2, Theorem 6] for other hyperbolic PDEs that if f ∈ Hm+1
+p
+([0, 2π]) × Hm
+p ([0, 2π]), then
+v(x, t) ∈ L2(0, T, Hm+1
+p
+([0, 2π])×Hm
+p ([0, 2π])). That is, the regularity of f(x) is preserved
+in v(x, ∆t) for each fixed ∆t > 0. Therefore, there exists a constant CT such that
+(3.28)
+∥(I − IN)v(·, ∆t)∥C ≤ CT ∆xm,
+0 < ∆t ≤ T.
+Using an approach based on [2] and applied in [20], we write E1(∆t, ∆x) as
+eN(x, t) = INv(x, t) − TN exp(˜LNt)RNf(x)
+= INv(x, t) − exp(IN ˜Lt)INf(x)
+Then, eN(x, t) solves the IVP
+∂
+∂teN = IN ˜LeN + IN ˜L(I − IN)v,
+eN(x, 0) = 0,
+and therefore
+eN(x, ∆t) =
+� ∆t
+0
+eIN ˜L(∆t−τ)IN ˜L(I − IN)v(x, τ) dτ.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+20
+B. RESTER AND J. V. LAMBERS
+From (3.28), and applying [4, Section 7.2, Theorem 6], it follows that
+∥E1(∆t, ∆x)∥C = ∥eN(x, ∆t)∥C
+≤ ∆t max
+0≤τ≤∆t
+���IN ˜LeIN ˜L(∆t−τ)(I − IN)v(·, τ)
+���
+C
+≤ C1∆t∆xm−1,
+where the constant C1 is independent of ∆x and ∆t. Here we note that because the coefficients
+of L are assumed to be constant or bandlimited, E1(∆t, ∆x) does not include aliasing error.
+Now, we examine E2(∆t, ∆x). If we let
+RNf =
+� fN,1
+fN,2
+�
+,
+E2(∆t, ∆x) = TN
+� eN,1
+eN,2
+�
+,
+then we have
+eN,1 = [cos(L1/2
+N ∆t) − SN,11(∆t)]fN,1 + [L−1/2
+N
+sin(L1/2
+N ∆t) − SN,12(∆t)]fN,2,
+eN,2 = [−L1/2
+N sin(L1/2
+N ∆t) − SN,21(∆t)]fN,1 + [cos(L1/2
+N ∆t) − SN,22(∆t)]fN,2.
+We have, by Parseval’s identity,
+∥E2(∆t, ∆x)∥2
+C = 2π
+N/2
+�
+ω=−N/2+1
+(pω2 + q)
+��ˆeH
+ω eN,1
+��2 +
+��ˆeH
+ω eN,2
+��2 .
+For each ω = −N/2 + 1, . . . , N/2, and i, j = 1, 2, we use the polynomial interpolation error
+in SN,ij to obtain
+ˆeH
+ω eN,1 = 1
+2
+∂2
+∂λ2
+�
+cos(λ1/2∆t)
+�����
+λ=ξω,11
+ˆeH
+ω (LN − l1,ωI)(LN − l2,ωI)fN,1 +
+1
+2
+∂2
+∂λ2
+�
+λ−1/2 sin(λ1/2∆t)
+�����
+λ=ξω,12
+ˆeH
+ω (LN − l1,ωI)(LN − l2,ωI)fN,2,
+ˆeH
+ω eN,2, = 1
+2
+∂2
+∂λ2
+�
+−λ1/2 sin(λ1/2∆t)
+�����
+λ=ξω,21
+ˆeH
+ω (LN − l1,ωI)(LN − l2,ωI)fN,1 +
+1
+2
+∂2
+∂λ2
+�
+cos(λ1/2∆t)
+�����
+λ=ξω,22
+ˆeH
+ω (LN − l1,ωI)(LN − l2,ωI)fN,2,
+where ξω,ij ∈ [l1,ω, l2,ω] for i, j = 1, 2. In view of the regularity of f(x), and the fact that LN
+is a discretization of a second-order differential operator with bandlimited coefficients, and a
+constant leading coefficient, we have
+��ˆeH
+0 (LN − l1,0I)(LN − l2,0I)fN,j
+�� =
+����
+1
+N (LN ˜qN)T fN,j
+���� ≤ ∥LN ˜qN∥∞∥fN,j∥∞.
+It follows that there exist constants Cij, for i, j = 1, 2, independent of N and ∆t, such that
+¯q1/2 ��ˆeH
+0 eN,1
+�� ≤ ∆t4C11 + ∆t5C12,
+��ˆeH
+0 eN,2
+�� ≤ ∆t3C21 + ∆t4C22,
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+21
+and for ω = −N/2 + 1, . . . , −1, 1, . . . , N/2, by Taylor expansion of the sines and cosines in
+ˆeH
+ω eN,1 and ˆeH
+ω eN,2, we have
+(¯pω2 + ¯q)1/2 ��ˆeH
+ω eN,1
+�� ≤ ∆t4
+C11
+|ω|m−2 + ∆t5
+C12
+|ω|m−3 ,
+��ˆeH
+ω eN,2
+�� ≤ ∆t3
+C21
+|ω|m−1 + ∆t4
+C22
+|ω|m−2 .
+It is important to note that because the leading coefficient p(x) of L is constant, (LN −
+l2,ωI)ˆeω = ˜qN ◦ ˆeω, where ◦ denotes componentwise multiplication. Therefore, this expres-
+sion is bounded independently of ω.
+Finally, we obtain
+∥E2(∆t, ∆x)∥C ≤ ∆t3
+√
+2π
+�
+C2
+11∆t2
+�
+1 + 2
+∞
+�
+ω=1
+ω4−2m
+�
++ 2C11C12∆t3
+�
+1 + 2
+∞
+�
+ω=1
+ω5−2m
+�
++
+C2
+12∆t4
+�
+1 + 2
+∞
+�
+ω=1
+ω6−2m
+�
++ C2
+21
+�
+1 + 2
+∞
+�
+ω=1
+ω2−2m
+�
++
+2C21C22∆t
+�
+1 + 2
+∞
+�
+ω=1
+ω3−2m
+�
++ C2
+22∆t2
+�
+1 + 2
+∞
+�
+ω=1
+ω4−2m
+��1/2
+≤ C2∆t3.
+Since m ≥ 4, it follows that all of the summations over ω converge to a sum that can be
+bounded independently of N. We conclude that the constant C2 is independent of ∆x and ∆t.
+□
+3.3. Convergence. Now we can prove that the second-order KSS method converges for
+the problem (2.4), (2.5), (2.6), (2.7) in the case of p(x) being constant and q(x) bandlimited.
+We say that a method is convergent of order (m, n) if there exist constants Ct and Cx,
+independent of the time step ∆t and grid spacing ∆x = 2π/N, such that
+∥u(·, t) − uN(·, t)∥C ≤ Ct∆tm + Cx∆xn,
+0 ≤ t ≤ T,
+where u(x, t) is the exact solution and uN(x, t) is the approximate solution computed using
+an N-point grid.
+THEOREM 3.6. Under the assumptions of Theorem 3.5, the two-node block KSS method
+with prescribed nodes (3.5) is convergent of order (2, m − 1).
+Proof: We recall that S(∆t) from (3.4) is the exact solution operator for the problem
+(2.4), (2.5), (2.6), (2.7). For any nonnegative integer n and fixed grid size N, we define
+En = ∥INS(∆t)nf − ˜SN(∆t)nINf∥C.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+22
+B. RESTER AND J. V. LAMBERS
+Then, by Theorem 3.5 and Corollary 3.4, there exist constants C1, C2 and Cq such that
+En+1 = ∥INS(∆t)n+1f − ˜SN(∆t)n+1INf∥C
+= ∥INS(∆t)S(∆t)nf − ˜SN(∆t) ˜SN(∆t)nINf∥C
+= ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf + ˜SN(∆t)S(∆t)nf − ˜SN(∆t) ˜SN(∆t)nINf∥C
+≤ ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf∥C + ∥ ˜SN(∆t)[INS(∆t)nf − ˜SN(∆t)nINf]∥C
+≤ ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf∥C + ∥TNSN(∆t)RN[INS(∆t)nf − ˜SN(∆t)nINf]∥C
+≤ ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf∥C + ∥TNSN(∆t) ˆRN[INS(∆t)nf − ˜SN(∆t)nINf]∥C
+≤ ∥INS(∆t)u(·, tn) − ˜SN(∆t)INu(·, tn)∥C + ∥SN(∆t)∥CN En
+≤ C1∆t3 + C2∆t∆xm−1 + (1 + Cq∥˜q∥∞∆t)En.
+It follows that
+En ≤
+eCq∥˜q∥∞T − 1
+1 + Cq∥˜q∥∞∆t − 1(C1∆t3 + C2∆t∆xm−1) ≤ ˜C1∆t2 + ˜C2∆xm−1
+for some constants ˜C1 and ˜C2 that depend only on T. We conclude that
+∥u(·, tn) − uN(·, tn)∥C ≤ ∥INu(·, tn) − uN(·, tn)∥C + ∥(I − IN)u(·, tn)∥C
+≤ ˜C1∆t2 + ˜C2∆xm−1 + ˜C3∆xm.
+□
+4. Numerical Experiments. We now perform some numerical experiments to corrobo-
+rate the theory presented in Section 3. For each test case, relative error was estimated using
+the ℓ2 norm, in comparison to a reference solution computed by the MATLAB ODE solver
+ode15s, witih absolute and relative tolerances set to 10−12.
+4.1. Constant Leading Coefficient. We consider the initial value problem
+(4.1)
+utt + Lu = 0,
+0 < x < 2π,
+t > 0,
+where L is of the form (2.7), with
+p(x) = 1,
+(4.2)
+q(x) = 1 + 1
+2 sin x + 1
+4 cos 2x + 1
+8 sin 3x.
+(4.3)
+The initial conditions are
+u(x, 0) =
+�
+1 − 2
+π|x − π|
+π/2 ≤ x ≤ 3π/2,
+0
+0 ≤ x < π/2.
+3π/2 < x < 2π,
+(4.4)
+ut(x, 0) = 0,
+0 < x < 2π,
+(4.5)
+and we impose periodic boundary conditions. We note that the initial data belongs to
+Hm+1
+p
+([0, 2π]) × Hm
+p ([0, 2π]) for m = 1, which is not sufficiently regular to satisfy the
+assumptions of Theorem 3.5.
+The results are shown in Table 4.1. As predicted by Theorem 3.5 and Corollary 3.4, we
+observe second-order accuracy in time, in spite of the low regularity of the initial data, even
+when the CFL limit is exceeded by using the same time step as the spatial resolution increases.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+23
+TABLE 4.1
+Relative errors in the solution of (4.1), (4.2), (4.3), (4.4), (4.5) with periodic boundary conditions on the domain
+(0, 2π) × (0, 10), using the second-order KSS method described in Section 3, with N grid points and time step ∆t.
+∆t
+N = 256
+N = 512
+N = 1024
+N = 2048
+π/128
+1.38e-04
+1.33e-04
+1.30e-04
+1.29e-04
+π/256
+3.32e-05
+3.24e-05
+3.25e-05
+3.20e-05
+π/512
+8.04e-06
+8.49e-06
+8.27e-06
+8.08e-06
+4.2. Variable Leading Coefficient. We now solve the problem (4.1), with
+(4.6)
+p(x) = 1 − 1
+2 sin x + 1
+4 cos 2x
+and initial conditions
+(4.7)
+u(x, 0) = e−(x−π)2,
+ut(x, 0) = 0,
+0 < x < 2π.
+The results are shown in Figures 4.1 and 4.2. We see that when the CFL number is greater
+than one, the method is unstable, as high-frequency components quickly become the dominant
+terms of the solution, and their amplitudes grow without bound. On the other hand, when the
+CFL number is less than one, the solution is well-behaved.
+FIG. 4.1. Solutions of (4.1), (4.6), (4.3), (4.7) on the domain (0, 2π) × (0, 1), using the second-order KSS
+method described in Section 3, with N = 256 grid points and time step CFL number ≈ 1.74.
+4.3. Generalizations. In this paper, we have limited our analysis to the wave equation in
+one space dimension, with periodic boundary conditions, and spatial differentiation performed
+
+t=0.2439
+0.8
+0.7
+0.6
+0.5
+(t'x)n
+0.4
+0.3
+0.2
+0.1
+0
+1
+2
+3
+4
+5
+9
+xt=0.4878
+0.6
+0.5
+0.4
+0.3
+x)n
+0.2
+0.1
+0
+0
+2
+3
+4
+5
+9
+xt=0.73171
+100
+80
+60
+40
+20
+(x,t)
+0
+-20
+-40
+-60
+-80
+-100
+0
+2
+3
+4
+5
+6
+XX105
+t=0.97561
+1.5
+0.5
+(t*x)
+-0.5
+-1
+-1.5
+0
+2
+3
+4
+5
+6
+xETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+24
+B. RESTER AND J. V. LAMBERS
+FIG. 4.2. Solutions of (4.1), (4.6), (4.3), (4.7) on the domain (0, 2π) × (0, 1), using the second-order KSS
+method described in Section 3, with N = 256 grid points and time step CFL number ≈ 0.87.
+using the FFT. We now consider some variations of this problem, to investigate whether our
+conclusions may apply more broadly.
+4.3.1. Finite Differencing in Space. We solve the problem (4.1), (4.2), (4.3), (4.7), on
+the domain (0, 2π) × (0, 10), with periodic boundary conditions, and using centered differ-
+encing in space. Because of the change of spatial discretization, we modify the interpolation
+points from (3.5) by prescribing
+(4.8)
+l2,ω = p2 − 2 cos(ω∆x)
+∆x2
++ q,
+ω = −N/2 + 1, . . . , N/2.
+The results are shown in Table 4.2. It can be seen that the same unconditional stability that was
+established for spectral differentiation applies in the case of finite differencing, as an accurate
+solution is obtained even when the CFL number is as large as eight.
+TABLE 4.2
+Relative errors in the solution of (4.1), (4.2), (4.3), (4.7) with periodic boundary conditions on the domain
+(0, 2π) × (0, 10), using the second-order KSS method described in Section 3, with N grid points, time step ∆t,
+central differencing in space, and interpolation points (4.8).
+∆t
+N = 256
+N = 512
+N = 1024
+N = 2048
+π/128
+1.00e-04
+1.00e-04
+1.00e-04
+1.00e-04
+π/256
+2.47e-05
+2.48e-05
+2.48e-05
+2.47e-05
+π/512
+6.15e-06
+6.16e-06
+6.15e-06
+6.15e-06
+
+t=0.2439
+0.8
+0.7
+0.6
+0.5
+(t'x)n
+0.4
+0.3
+0.2
+0.1
+0
+1
+2
+3
+4
+5
+9
+xt=0.4878
+0.6
+0.5
+0.4
+(x,t)
+0.3
+0.2
+0.1
+0
+1
+2
+3
+4
+5
+6
+xt=0.73171
+0.45
+0.4
+0.35
+0.3
+0.25
+(t'x)n
+0.2
+0.15
+0.1
+0.05
+0
+1
+2
+3
+4
+5
+9
+xt=0.97561
+0.4
+0.35
+0.3
+0.25
+(tx)n
+0.2
+0.15
+0.1
+0.05
+0
+2
+3
+4
+5
+9
+xETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+25
+4.3.2. Other Boundary Conditions. We repeat the problem from Section 4.3.1, except
+with homogeneous Dirichlet boundary conditions. The interpolation points from (3.5) are
+modified as follows:
+(4.9)
+l2,ω = p2 − 2 cos(ω∆x/2)
+∆x2
++ q,
+ω = 0, 1, 2, . . . , N − 1.
+As in the case of periodic boundary conditions, unconditional stability is indicated by the
+results, shown in Table 4.3.
+TABLE 4.3
+Relative errors in the solution of (4.1), (4.2), (4.3), (4.7) with homogeneous Dirichlet boundary conditions on
+the domain (0, 2π) × (0, 10), using the second-order KSS method described in Section 3, with N grid points, time
+step ∆t, central differencing in space, and interpolation points (4.9).
+∆t
+N = 256
+N = 512
+N = 1024
+N = 2048
+π/128
+1.53e-04
+1.53e-04
+1.53e-04
+1.53e-04
+π/256
+3.85e-05
+3.85e-05
+3.85e-05
+3.86e-05
+π/512
+9.67e-06
+9.67e-06
+9.67e-06
+9.68e-06
+4.3.3. Higher Space Dimension. We solve a two-dimensional wave equation
+(4.10)
+utt + Lu = 0,
+0 < x, y < 2π,
+0 < t < 10,
+where the differential operator L is defined by
+(4.11)
+Lu = −∆u + q(x, y)u,
+where
+(4.12)
+q(x, y) = 1 + 1
+2 sin x cos y + 1
+4 cos 2y + 1
+8 sin 3x.
+Our initial conditions are
+(4.13)
+u(x, y, 0) = e−(x−π)2+(y−π)2,
+ut(x, y, 0) = 0,
+0 < x, y < 2π.
+and we impose periodic boundary conditions. For spatial discretization, we use a N × N grid
+with spacing ∆x = ∆y = 2π/N, and centered differencing in space. This leads to the choice
+of interpolation points
+(4.14)
+l1,ω1,ω2 = 0,
+l2,ω1,ω2 =
+1
+∆x2 [4 − 2 cos(ω1∆x) − 2 cos(ω2∆y)] + q,
+for ω1, ω2 = −N/2 + 1, . . . , N/2, where q is the average value of q(x, y) on (0, 2π)2. The
+results, shown in Table 4.4, indicate that unconditional stability again holds, as the CFL
+number exceeds one without loss of accuracy or stability.
+4.4. Modeling an Acceleration Wave in a Dusty Gas. Finally, we consider the appli-
+cation of a KSS method to the PDE [13]
+(4.15) λ0wttt + (1 + κ0)wtt − (1 − z/H1)wzz + kwz + λ0[kwtz − (1 − z/H1)wtzz] = 0,
+on the domain 0 < z < ℓ < H1, t > 0, with initial conditions
+(4.16)
+w(z, 0) = wt(z, 0) = wtt(z, 0) = 0,
+0 < z < ℓ.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+26
+B. RESTER AND J. V. LAMBERS
+TABLE 4.4
+Relative errors in the solution of (4.10), (4.12), (4.13) with periodic boundary conditions on the domain
+(0, 2π)2 × (0, 10), using the second-order KSS method described in Section 3, with N grid points per dimension,
+time step ∆t, central differencing in space, and interpolation points (4.14).
+∆t
+N = 16
+N = 32
+N = 64
+N = 128
+π/8
+3.83e-02
+4.14e-02
+3.95e-02
+3.84e-02
+π/16
+8.69e-03
+8.80e-03
+8.30e-03
+8.06e-05
+π/32
+2.01e-03
+1.95e-03
+1.84e-03
+1.78e-03
+We impose the boundary conditions
+(4.17)
+w(0, t) = sin(Ωt),
+w(ℓ, t) = 0.
+This problem arises from the modeling of particle-laden flow formulation (i.e., a “dusty gas”
+model) [12].
+4.4.1. Transforming the Differential Operator. Using techniques from [19], we trans-
+form this problem so that it can be solved effectively using a Fourier spectral method. For
+convenience, we define the differential operator
+Lu(z) = −a2(z)uzz(z) + a1(z)uz(z),
+a2(z) = 1 − z/H1,
+a1(z) = k
+Our goal is to find similarity transformations V1, V2 so that the transformed operator
+¯L = V −1
+2
+V −1
+1
+LV1V2
+has these properties: (1) the coefficient of wzz constant (through V1), and (2) there is no wz
+term (through V2).
+We first perform the transformation
+V1f(z) = f(φ(z)),
+where
+y = φ(z) = ℓ
+� z
+0
+a2(s)−1/2 ds
+� ℓ
+0
+a2(s)−1/2 ds
+.
+This yields the transformed operator
+V −1
+1
+LV1u(y) = −(a2 ◦ φ−1)uyy + (a1 ◦ φ−1)uy,
+a2 = C2,
+a1(z) = (a1 + a′
+2/2)φz.
+Next, we need to eliminate the first derivative term in the differential operator. Let
+˜a2 = a2 ◦ φ−1, ˜a1 = a1 ◦ φ−1. Then
+V −1
+1
+LV1 = −˜a2uyy + ˜a1uy.
+We now apply the transformation
+V2f(y) = ψ(y)f(y),
+ψ(y) = exp
+�� ˜a1(y)
+2˜a2
+dy
+�
+.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+27
+This yields the transformed operator ¯L defined by
+(4.18)
+¯Lu = V −1
+2
+(V −1
+1
+LV1)V2u = −˜a2uyy + a0(y)u
+where
+a0 = −1
+2˜a′
+1 + ˜a2
+1
+4˜a2
+.
+Our transformed PDE is now
+λ0 ¯wttt + (1 + κ0) ¯wtt + ¯L ¯w + λ0 ¯L ¯wt = 0,
+0 < y < ℓ,
+t > 0,
+where w = V1V2 ¯w. The initial and boundary conditions for ¯w are the same as those for w.
+4.4.2. Homogenizing the Boundary Conditions. Due to the Dirichlet boundary condi-
+tions, a KSS method will represent the solution as a Fourier sine series. We therefore need to
+homogenize the boundary conditions, by writing the solution ¯w(y, t) as
+¯w(y, t) = ¯u(y, t) + F(y, t)
+where ¯u satisfies homogeneous boundary conditions, and F satisfies the original inhomoge-
+neous boundary conditions. Because this transformation will introduce a source term to the
+PDE, which we will denote by G(y, t), we will also require that G satisfies homogeneous
+boundary conditions.
+To that end, we prescribe
+F(y, t) = f0(t) + f1(t)y + f2(t)y2 + f3(t)y3 + f4(t)y4
+The source term introduced into the PDE is given by
+G(y, t) = λFttt + (1 + κ0)Ftt − a2Fyy + a0F + λ0[−a2Ftyy + a0Ft]
+The functions F and G must satisfy the condidtions
+F(0, t) = f0(t) = sin(Ωt),
+F(ℓ, t) = f0(t) + f1(t)ℓ + f2(t)ℓ2 + f3(t)ℓ3 + f4(t)ℓ4 = 0,
+G(0, t) = λ0f ′′′
+0 (t) + (1 + κ0)f ′′
+0 (t) − 2a2f2(t) − a0(0)f0(t) − λ0[a0(0)f ′
+0(t) + 2a2f ′
+2(t)] = 0
+G(ℓ, t) = −a2[2f2(t) + 6f3(t)ℓ + 12f4(t)ℓ2] = 0
+Fy(ℓ, t) = f1(t) + 2f2(t)ℓ + 3f3(t)ℓ2 + 4f4(t)ℓ3 = 0.
+Solving the ODE G(0, t) = 0 for f2(t) yields
+f2(t) = A2 sin Ωt + B2 cos Ωt,
+A2 = bΩ − ac
+Ω2 + a2 ,
+B2 = −ab + cΩ
+Ω2 + a2 ,
+where
+a = − 1
+λ0
+,
+b = −
+� Ω3
+2a2
++ Ωa−
+0
+2a2
+�
+,
+c = −
+�Ω2(1 + κ0)
+2a2λ0
++
+a−
+0
+2a2λ0
+�
+,
+a−
+0 = a0(0).
+We can then solve a system of three linear equations for the remaining three unknowns f1(t),
+f3(t) and f4(t). In addition to the above conditions, we also have Fyy(ℓ, t) = 0.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+28
+B. RESTER AND J. V. LAMBERS
+4.4.3. Transformation to a Wave Equation. The PDE that will actually be solved via
+KSS is
+(4.19)
+utt = −¯Lu − κ0¯utt − G,
+where
+u = ¯u + λ0¯ut.
+We note that the equation relating u and ¯u is a first-order linear ODE, which can be solved for
+¯u to obtain
+¯u(y, tn+1) = e−∆t/λ0 ¯u(y, tn) + 1
+λ0
+� ∆t
+0
+e−(∆t−s)/λ0u(y, tn + s) ds.
+We replace u(y, tn + s) within the integral by a Taylor expansion centered at tn+1 to obtain
+the approximation
+¯u(y, tn+1) ≈ e−∆t/λ0 ¯u(y, tn) + (1 − e−∆t/λ0)[u(y, tn+1) − ∆tut(y, tn+1)] +
+ut(y, tn+1)
+�
+∆t − λ0 + λ0e−∆t/λ0�
+.
+Having obtained ¯u, we can use the equation u = ¯u + λ0¯ut to obtain
+¯ut = (u − ¯u)/λ0,
+¯utt = (ut − ¯ut)/λ0.
+4.4.4. Solution Operator. We approximate our PDE with a PDE of the form
+utt = −¯Lu + b,
+where b is a source term, set equal to −κ0wtt − G, evaluated at time tn for the time step to
+tn+1. We rewrite this PDE as a first-order system
+� u
+ut
+�
+t
+= J
+� u
+ut
+�
++
+� 0
+b
+�
+.
+J =
+�
+0
+I
+−L
+0
+�
+.
+If we define
+v =
+� u
+ut
+�
+,
+c =
+� 0
+b
+�
+then the solution of our first-order system is
+v(tn+1) = φ0(Jt)v(tn) + tφ1(Jt)c(tn).
+where
+φ0(Jt) = eJt =
+�
+cos(L1/2t)
+L−1/2 sin(L1/2t)
+−L1/2 sin(L1/2t)
+cos(L1/2t)
+�
+and
+tφ1(Jt) = (eJt − I)J−1 =
+� L−1/2 sin(L1/2t)
+L−1 − L−1 cos(L1/2t)
+cos(L1/2t) − I
+L−1/2 sin(L1/2t)
+�
+.
+
+ETNA
+Kent State University and
+Johann Radon Institute (RICAM)
+CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION
+29
+4.4.5. Numerical Results. We now solve the problem (4.15), (4.16), (4.17), with param-
+eters
+(4.20)
+ℓ = 2,
+λ0 ≈ 0.01336,
+κ0 = 0.05,
+H1 ≈ 4.7278,
+k ≈ 1.4806,
+Ω = π,
+by transforming the PDE to (4.19), applying the second-order KSS method described in
+Section 4.3.2, and transforming back. The computed solution is shown in Figure 4.3. We
+see that the solution obtained via KSS matches the solution obtained via numerical Laplace
+inversion, and the analytical location of the wave front, given by
+(4.21)
+σ(t) = t −
+t2
+4H1
+.
+Although the differential operator ¯L from (4.18) has a coefficient a0(y) that is not bandlimited,
+it is not necessary to perform regularization to mitigate aliasing error that can lead to weak
+instability [8]. However, such regularization may be necessary when applying a KSS method
+to other wave propagation problems [17, 19].
+FIG. 4.3. Solution of dusty gas problem (4.15), (4.16), (4.17), (4.20) computed using a second-order KSS
+method (solid blue curve) with N = 16, 384 grid points and CFL number 10, and numerical Laplace transform
+inversion (black circles). The red dashed line indicates the location of the wavefront given by (4.21).
+5. Conclusion. We have established an upper bound for the approximate solution op-
+erator of a second-order KSS method applied to the 1-D wave equation with bandlimited
+coefficients and periodic boundary conditions. Unfortunately, the bound is not independent of
+the grid size, which indicates that the unconditional stability proved for the heat equation for
+the same kind of spatial differential operator does not extend to the wave equation. Numerical
+experiments support this assertion, while also suggesting that conditional stability may still
+hold.
+We have also proved that the same KSS method applied to the wave equation with
+periodic boundary conditions is unconditionally stable in the case of a constant wave speed
+and bandlimited lower-order coefficient. This is the first result proving unconditional stability
+for a KSS method with prescribed nodes applied to the wave equation, as opposed to nodes of
+Gauss quadrature rules. Numerical experiments suggest that this unconditional stability also
+holds for related problems.
+Ideas for future work include: 1) to investigate whether the same result holds if coefficients
+are not bandlimited but sufficiently smooth, 2) to generalize the analysis to higher-order KSS
+methods or higher space dimensions with different boundary conditions, and 3) to attempt to
+modify the KSS method to ensure at least conditional stability, without having to transform
+the spatial differential operator as in Section 4.4.1.
+
+t=1.0596
+3
+2.5
+2
+1.5
+w(z,t)
+1
+0.5
+0
+-0.5
+-1
+-1.5
+0
+0.5
+1
+1.5
+2
+zt=1.8921
+3
+2.5
+2
+1.5
+w(z,t)
+1
+0.5
+0
+-0.5
+-1
+-1.5
+0
+0.5
+1
+1.5
+2
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+Kent State University and
+Johann Radon Institute (RICAM)
+30
+B. RESTER AND J. V. LAMBERS
+REFERENCES
+[1] K. Atkinson, An Introduction to Numerical Analysis, 2nd Ed. Wiley (1989).
+[2] C. Bardos and E. Tadmor, “Stability and spectral convergence of Fourier method for nonlinear problems. On
+the shortcomings of the 2/3 de-aliasing method”, Numerische Mathematik 129 (2014), p. 749-782.
+[3] A. Cibotarica, J. V. Lambers and E. M. Palchak, “Solution of Nonlinear Time-Dependent PDE Through
+Componentwise Approximation of Matrix Functions”, Journal of Computational Physics 321 (2016), p.
+1120-1143.
+[4] L. C. Evans, Partial Differential Equations. American Mathematical Society (1998)
+[5] S. J. Farlow, Partial differential equations for scientists and engineers. Dover Publications, Inc., New York
+(1993)
+[6] G. H. Golub and G. Meurant, “Matrices, moments and quadrature”. In: Proceedings of the 15th Dundee
+Conference, June-July 1993. Longman Scientific and Technical (1994)
+[7] G. H. Golub and R. Underwood, “The block Lanczos method for computing eigenvalues”, Mathematical
+Software III, J. Rice Ed., (1977), p. 361-377.
+[8] J. Goodman, T. Hou, E. Tadmor, “On the stability of the unsmoothed Fourier method for hyperbolic equations”,
+Numerische Mathematik 67(1) (1994), p. 93-129.
+[9] P. Guidotti, J. V. Lambers and K. Sølna, “Analysis of the 1D Wave Equation in Inhomogeneous Media”,
+Numerical Functional Analysis and Optimization 27 (2006), p. 25-55.
+[10] B. Gustafsson, H.-O. Kreiss and J. Oliger, Time dependent problems and difference methods. Wiley, Amsterdam
+(1995)
+[11] J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-Dependent Problems. Cambridge
+University Press (2007)
+[12] P. M. Jordan, “Acoustic propagation in inhomogeneous fluids: regularization via the introduction of fine
+particles”, Archives of Mechanics 72 (2020), p. 59-73.
+[13] P. M. Jordan, “Test case: One-layer problem based on the dusty gas model for homentropic gas”, unpublished
+communication, August 29, 2021.
+[14] J. V. Lambers, “An Explicit, Stable, High-Order Spectral Method for the Wave Equation Based on Block
+Gaussian Quadrature", IAENG Journal of Applied Mathematics 38 (2008), p. 233-248.
+[15] J. V. Lambers, “Derivation of High-Order Spectral Methods for Time-Dependent PDE Using Modified
+Moments", Electronic Transactions on Numerical Analysis 28 (2008), p. 114-135.
+[16] J. V. Lambers, “Enhancement of Krylov Subspace Spectral Methods by Block Lanczos Iteration", Electronic
+Transactions on Numerical Analysis 31 (2008), p. 86-109.
+[17] J. V. Lambers and P. M. Jordan, “On the application of a Krylov subspace spectral method to poroacoustic
+shocks in inhomogeneous gases”, Numerical Methods for Partial Differential Equations 37(6) (2021), p.
+2955-2972.
+[18] E. M. Palchak, A. Cibotarica and J. V. Lambers, “Solution of Time-Dependent PDE Through Rapid Estimation
+of Block Gaussian Quadrature Nodes", Linear Algebra and its Applications 468 (2015), p. 233-259.
+[19] B. Rester, J. V. Lambers and P. M. Jordan, “Acoustic singular surfaces in an exponential class of inhomogeneous
+gases: A new numerical approach based on Krylov subspace spectral methodologies”, submitted.
+[20] S. Sheikholeslami, J. V. Lambers and C. Walker, “Convergence Analysis of Krylov Subspace Spectral Methods
+for Reaction-Diffusion Equations", Journal of Scientific Computing 78(3) (2019), p. 1768-1789.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf,len=1200
+page_content='ETNA  Kent State University and  Johann Radon Institute (RICAM)  Electronic Transactions on Numerical Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Volume 46, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 1–xx, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Copyright © 2023, Kent State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ISSN 1068–9613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  CONVERGENCE ANALYSIS OF A KRYLOV SUBSPACE SPECTRAL METHOD  FOR THE 1-D WAVE EQUATION IN AN INHOMOGENEOUS MEDIUM∗  BAILEY RESTER† AND JAMES V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS†  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for  partial differential equations (PDEs) that possess stability characteristic of implicit methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' KSS methods compute  each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' As  a result, KSS methods scale effectively to higher spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This paper will present a convergence analysis of  a second-order KSS method applied to a 1-D wave equation in an inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Numerical experiments  that corroborate the established theory are included, along with a discussion of generalizations, such as to higher  space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' spectral methods, wave equation, convergence analysis, variable coefficients  AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 65M70, 65M12, 65F60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Consider the 1-D wave equation in an inhomogeneous medium,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1)  utt = (p(x)ux)x + q(x)u,  on a bounded domain, with appropriate initial and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Analytical methods  are not practical to use for this problem, since the coefficients are not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For instance,  applying separation of variables [5] would result in a spatial ODE that cannot be solved  analytically, and therefore numerical methods are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' However, standard time-stepping  methods, such as Runge-Kutta methods or multistep methods, suffer from a lack of scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  As the number of grid points increases, a smaller time step would be needed due to the CFL  condition [10] for explicit methods, or an increasingly ill-conditioned system must be solved  for implicit methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' It follows that increasing the number of grid points significantly increases  the computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Therefore, a more practical numerical method for solving this  kind of variable-coefficient PDE is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping  methods that possess stability characteristic of implicit methods [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' By contrast with other  time-stepping methods, KSS methods employ a componentwise approach, in which each  Fourier coefficient of the solution is computed using an approximation of the solution operator  of the PDE that is tailored to that coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This customization is based on techniques for  approximating bilinear forms involving matrix functions by treating them as Riemann-Stieltjes  integrals [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This componentwise approach allows KSS methods to circumvent difficulties  caused by stiffness, and thus scale effectively to higher spatial resolution [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  A first-order KSS method applied to the heat equation with a constant leading coefficient  was proven to be unconditionally stable [15, 16], as well as a second-order KSS method  applied to the wave equation with a constant leading coefficient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In all of these studies,  lower-order coefficients of the spatial differential operator were assumed to be bandlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' A  first-order KSS method applied to the heat equation with a bandlimited leading coefficient is  also unconditionally stable [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In this paper, we analyze stability of a KSS method applied  to the wave equation with bandlimited coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The outline of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Section 2 provides an overview of KSS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Section 3 presents a stability analysis of a second-order KSS method applied to the PDE  ∗Received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Published online on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Recommended by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='.  †School of Mathematics and Natural Sciences, The University of Southern Mississippi, 118 College Dr #5043,  Hattiesburg, MS 39406 USA  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='08316v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='NA]  19 Jan 2023  ETNA  Kent State University and  Johann Radon Institute (RICAM)  2  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1) with bandlimited coefficients p(x) and q(x), and periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In that  same section, a full convergence analysis in the case of p(x) ≡ constant is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Corroborating numerical experiments are given in Section 4, along with application of the  second-order KSS method to more general problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Conclusions and ideas for future work  are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Overview of KSS Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We begin with the derivation of KSS methods, which  are simpler to describe for a variable-coefficient diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Consider a linear parabolic  PDE of the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1)  ut + Lu = 0  on the domain (0, 2π) × (0, ∞), where L is a second-order, self-adjoint differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  If we impose periodic boundary conditions, then the solution u(x, t) can be expressed as a  Fourier series, in which each Fourier coefficient  ˆu(ω, t) =  � 1  2π eiωx, e−Ltu(x, 0)  �  = 1  2π  � 2π  0  e−iωxe−Ltu(x, 0) dx  is the standard inner product on (0, 2π) of a Fourier basis function and the solution operator  e−Lt applied to the initial data u(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Spatial discretization results in a bilinear form  involving a matrix function of the form  uHf(A)v,  where f(A) = e−At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  In [6] Golub and Meurant described a method for computing bilinear forms involving  matrix functions, that have the form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2)  uT f(A)v,  where u and v are N-vectors, A is an N × N symmetric positive definite matrix, and f is  a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Since the matrix A is symmetric positive definite, it has real, positive  eigenvalues  b = λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' ≥ λN = a > 0,  and orthogonal corresponding eigenvectors qj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Using the spectral decomposi-  tion of A, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2) can be rewritten as  uT f(A)v =  N  �  j=1  f(λj)uT qjqT  j v,  but computing this summation would be impractical for large values of N, because computing  all of the eigenvalues and eigenvectors of A would require O(N 3) floating point operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Instead, we let α(λ) be a step function defined in terms of the coefficients of u and v in  the basis of eigenvectors, given by  α(λ) =  �  �  �  �  �  0,  if λ < a  �N  j=i ujvj,  if λi ≤ λ < λi−1  �N  j=1 ujvj,  if b ≤ λ  ,  uj = uT qj,  vj = qT  j v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  3  Then, the bilinear form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2) can be viewed as a Riemann-Stieltjes integral over the spectral  domain,  uT f(A)v = I[f] =  � b  a  f(λ) dα(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  This integral can then be approximated using a Gauss quadrature rule, which yields  I[f] =  K  �  j=1  wjf(λj) + error,  where the nodes λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , K, and the weights wj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , K can be obtained from  the Lanczos algorithm applied to A with initial vectors u and v, which produces a tridiagonal  matrix TK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The eigenvalues of TK are the Gauss quadrature nodes, and the first components of  the eigenvectors, squared and scaled by uT v, are the corresponding weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' When the vectors  u and v are equal, α(λ) is positive and increasing, but if u ̸= v, this may not necessarily be  the case, which can lead to a numerically unstable quadrature rule due to negative weights [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  As an alternative, we use a block approach, also described in [6], in which we compute  � u  v �T f(A)  � u  v �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Computing this block quadratic form results in the 2×2 matrix integral  � b  a  f(λ) dα(λ) =  � uT f(A)u  uT f(A)v  vT f(A)u  vT f(A)v  �  =  2K  �  j=1  f(λj)ujuT  j + error,  where uj is a 2-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We can approximate this integral using a quadrature rule with twice  as many nodes as in the non-block case, in which each outer product ujuT  j , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , 2K,  plays the role of a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The nodes and (matrix-valued) weights are obtained by generalizing the Lanczos algorithm  to the block case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Applying the block Lanczos algorithm [7] to A with initial block X1 =  � u  v �  , where XT  1 X1 = I, results in the 2K × 2K block tridiagonal matrix  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3)  TK =  �  ������  M1  BT  1  B1  M2  BT  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  BK−2  MK−1  BT  K−1  BK−1  MK  �  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We then define the quadrature rule for  � u  v �T f(A)  � u  v �  as  � b  a  f(λ) dα(λ) ≈  2K  �  j=1  f(λj)ujuT  j = [f(TK)]1:2,1:2,  where λj an eigenvalue of TK, and uj contains the first two elements of the corresponding  normalized eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  A KSS method for the PDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1) proceeds as follows, during each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Let un be  the computed solution at time tn = n∆t, and let ˆeω be an N-vector of values of the Fourier  basis function ˆeω(x) =  1  2πeiωx on a uniform N-point grid on [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For each wave number  ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2, we define  R0 =  � ˆeω  un �  ,  and then compute the “economy size” QR factorization  R0 = X1B0,  where X1 is N × 2 and B0 is 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Block Lanczos iteration yields TK from X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then, the  solution is approximated by  [ˆun+1]ω =  �  BH  0 exp[−TK∆t]1:2,1:2B0  �  12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Let u(x, ∆t) be the exact solution, and let ˜u(x, ∆t) be the approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' If K  iterations of block Lanczos are performed, then, for ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2, [16]  |⟨ˆeω, u(·, ∆t) − ˜u(·, ∆t)⟩| = O(∆t2K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  In addition to high-order accuracy in time, KSS methods have been demonstrated to be  effective even when using a time step that is larger than what the CFL condition would allow,  even though KSS methods are explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In other words, KSS methods represent a “best-of-both  worlds” balance between the computational efficiency of explicit methods and the stability of  implicit methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This balance is achieved through their componentwise approach, in which  each Fourier coefficient is treated as a Riemann-Stieltjes integral with a frequency-dependent  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' KSS Methods for the Wave Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Now, we apply these ideas to the second-  order wave equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4)  utt + Lu = 0 on (0, 2π) × (0, ∞),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5)  u(x, 0) = f(x),  ut(x, 0) = g(x),  0 < x < 2π,  with periodic boundary conditions  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6)  u(0, t) = u(2π, t),  t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The spatial differential operator L is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7)  Lu = −(p(x)ux)x + q(x)u,  where we assume p(x) > 0 and q(x) ≥ 0, to guarantee that L is self-adjoint and positive  definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  A spectral representation of the operator L allows us to describe the solution operator, the  propagator, as a function of L [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We introduce  R1(t) = L−1/2 sin(L1/2t),  R0(t) = cos(L1/2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  5  Then, the solution can be written as  � u(x, t)  ut(x, t)  �  =  �  R0(t)  R1(t)  −L R1(t)  R0(t)  � � u(x, 0)  ut(x, 0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The entries of this matrix, as functions of L, indicate which functions are the integrands in the  Riemann-Stieltjes integrals used to compute the Fourier coefficients of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  By analogy with the description of a KSS method for the parabolic PDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), let un and  un  t be the computed solution at time tn and its time derivative, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For each wave  number ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2, we define  R0 =  � ˆeω  un �  ,  ˜R0 =  � ˆeω  un  t  �  ,  and then compute the QR factorizations  R0 = X1B0,  ˜R0 = ˜X1 ˜B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Block Lanczos iteration yields TK and ˜TK from X1 and ˜X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then, the solution and its time  derivative are approximated by  [ˆun+1]ω =  �  BH  0 cos[T 1/2  K  ∆t]1:2,1:2B0  �  12 +  �  ˜BH  0 ( ˜T −1/2  K  sin[ ˜T 1/2  K  ∆t])1:2,1:2 ˜B0  �  12 ,  [ˆun+1  t  ]ω = −  �  BH  0 (T 1/2  K  sin[T 1/2  K  ∆t])1:2,1:2B0  �  12 +  �  ˜BH  0 cos[ ˜T 1/2  K  ∆t]1:2,1:2 ˜B0  �  12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Let u(x, ∆t) be the exact solution, and let ˜u(x, ∆t) be the approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' If K  iterations of block Lanczos are performed, then, for ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2, [14]  |⟨ˆeω, u(·, ∆t) − ˜u(·, ∆t)⟩| = O(∆t4K),  |⟨ˆeω, ut(·, ∆t) − ˜ut(·, ∆t)⟩| = O(∆t4K−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The higher order of accuracy in time is due to the second derivative with respect to time in the  PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Stability Results for KSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The following has been proven about the stability of  KSS methods:  Heat equation ut = puxx + q(x)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' where p is constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' q(x) is bandlimited: a  first-order KSS method is unconditionally stable [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Wave equation utt = puxx + q(x)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' where p is constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' q(x) is bandlimited: a  second-order KSS method is unconditionally stable [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Reaction-diffusion system of the form vt = Lv: a first-order KSS method with con-  stant diffusion coefficient and bandlimited reaction term coefficient is unconditionally  stable [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Wave equation utt = puxx + q(x)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' where p is constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' q(x) is bandlimited: a  second-order block KSS method is unconditionally stable [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' and  Heat equation ut = (p(x)ux)x + q(x)u where p(x) and q(x) are bandlimited: a  first-order block KSS method is unconditionally stable [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Convergence Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We will now analyze convergence of a second-order KSS  method, with K = 1, for the IVP (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7), under the assumptions that the  Fourier coefficients ˆp(ω), ˆq(ω) of p(x) and q(x), respectively, satisfy ˆp(ω) = q(ω) = 0 when  |ω| > ωmax for some threshold ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' That is, we assume that p(x) and q(x) are bandlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We first carry out spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We use a uniform grid, with spacing ∆x = 2π/N,  where N is assumed to be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then, we let xN be an N-vector of grid points  xj = j∆x,  j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We denote by ωj the corresponding wave numbers  ωj = j − N/2,  j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We then denote by DN an N × N matrix that discretizes the second derivative operator using  the discrete Fourier transform:  DN = F −1  N ΛNFN,  where  [FN]jk = 1  N e−iωjxk,  [ΛN]jj = −ω2  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We also let IN denote the N × N identity matrix, whereas I is the identity operator on  functions of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Let u1 = u and u2 = ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We then rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4) as the first-order system  ∂u1  ∂t = u2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1)  ∂u2  ∂t = −Lu1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2)  which, for convenience, we write as  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3)  vt = ˜Lv,  v =  � u1  u2  �  ,  ˜L =  �  0  I  −L  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Spatial discretization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3) yields a system of ODEs  v′  N(t) = ˜LNv(t),  where  vN(t) =  � u1,N(t)  u2,N(t)  �  is the spatial discretization of the vector field v, and ˜LN is a 2N × 2N matrix which has the  2 × 2 block structure  ˜LN =  �  0  IN  −LN  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We define the exact solution operator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2) as  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4)  S(t) = exp[˜Lt] =  � S11(t)  S12(t)  S12(t)  S22(t)  �  =  �  R0(t)  R1(t)  −L R1(t)  R0(t)  �  ,  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  7  where, as before, R1(t) = L−1/2 sin(L1/2t) and R0(t) = cos(L1/2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then we let  SN(∆t) =  � SN,11(∆t)  SN,12(∆t)  SN,12(∆t)  SN,22(∆t)  �  ,  where each SN,ij(∆t) is the approximation of Sij(∆t) by the KSS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  A KSS method applied to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4) with K = 1 uses two block Gauss quadrature nodes for  each Fourier coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Using an approach described in [18], we estimate these nodes, rather  than using block Lanczos iteration explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This significantly improves the efficiency of KSS  methods, but it will also simplify the convergence analysis to be carried out in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The quadrature nodes will be prescribed as follows:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5)  l1,ω = 0,  l2,ω = pω2 + q,  ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2,  where p and q are the average values of p(x) and q(x), respectively, on [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' To interpolate  the functions  f11(λ) = f22(λ) = cos(λ1/2∆t),  f12(λ) = λ−1/2 sin(λ1/2∆t),  f21(λ) = −λf12(λ)  at the nodes l1,ω, l2,ω, we compute the slopes  Mij,ω = fij(l2,ω) − fij(l1,ω)  l2,ω − l1,ω  ,  ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2,  i, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We then describe the computed solution at time tn+1 by  un+1 = z11 + z12,  un+1  t  = z21 + z22,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6)  where  zi1 = SN,i1(∆t)un,  zi2 = SN,i2(∆t)un  t ,  i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We also define ˜p = p − ¯p, ˜q = q − ¯q, and let PN, QN, ˜PN, and ˜QN be diagonal matrices with  the values of the coefficients p(x), q(x), ˜p(x), and ˜q(x), respectively, at the grid points on the  main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Let Iω = {k ∈ Z|0 < |k − ω| ≤ ωmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The discrete Fourier coefficients of zij,  ETNA  Kent State University and  Johann Radon Institute (RICAM)  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  i, j = 1, 2, are then given by  ˆz11(ω) = S11(l2,ω)(ˆeH  ω un) + M11,ωˆeH  ω (LN − l2,ωI)un  = S11(l2,ω)ˆu(ω) − iωM11,ω  �  k∈Iω  ˆp(ω − k)i(k)ˆu(k) +  M11,ω  �  k∈Iω  ˆq(ω − k)ˆu(k),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7)  ˆz21(ω) = S21(l2,ω)ˆu(ω) − iωM21,ω  �  k∈Iω  ˆp(ω − k)i(k)ˆu(k) +  M21,ω  �  k∈Iω  ˆq(ω − k)ˆu(k),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8)  ˆz12(ω) = S12(l2,ω)ˆut(ω) − iωM12,ω  �  k∈Iω  ˆp(ω − k)i(k)ˆut(k) +  M12,ω  �  k∈Iω  ˆq(ω − k)ˆut(k),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='9)  ˆz22(ω) = S22(l2,ω)ˆut(ω) − iωM22,ω  �  k∈Iω  ˆp(ω − k)i(k)ˆut(k) +  M22,ω  �  k∈Iω  ˆq(ω − k)ˆut(k),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='10)  where i = √−1 and ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' To obtain these formulas, we used the  simplification  ˆeH  ω (LN − l2,ωI)un = ˆeH  ω [−DNPNDN + QN]un − l2,ω(ˆeH  ω un)  = −iωˆeH  ω PNDNun + ˆeH  ω QNun − l2,ω(ˆeH  ω un)  = (¯pω2 + ¯q)(ˆeH  ω un) − iωˆeH  ω ˜PNDNun + ˆeH  ω ˜QNun − l2,ω(ˆeH  ω un)  = −iωˆeH  ω ˜PNDNun + ˆeH  ω ˜QNun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  To bound error, we need to establish an upper bound of a norm of the approximate solution  operator SN(∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We elect to use the C-norm, defined by  ∥ (u, v) ∥2  C = ⟨u, Cu⟩ + ⟨v, v⟩,  where, as before, ⟨·, ·⟩ is the standard inner product on (0, 2π), and its discrete counterpart,  the CN-norm, defined by  ∥ (u, v) ∥2  CN = uT CNu + ∥v∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The N × N matrix CN discretizes the constant-coefficient differential operator C = −¯p∂xx +  ¯qI, where u and v are N-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We choose to bound the CN-norm of the solution operator  for convenience, because the operator CN has a very simple expression in Fourier space due  to its constant coefficients, which simplifies the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We wish to express ∥SN(∆t)∥CN as the 2-norm of some matrix, since  that will be easier to bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We define ∥SN(∆t)∥CN by  ∥SN(∆t)∥2  CN =  sup  w=(u,v)̸=0  ∥SN(∆t)w∥2  CN  ∥w∥2  CN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  9  Then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='11)  ∥SN(∆t)∥2  CN =  sup  (u,v)̸=0  ˜uT CN ˜u + ∥˜v2∥2  uT CNu + ∥v2∥2 ,  where  �˜u  ˜v  �  = SN(∆t)  �u  v  �  = SN(∆t)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In matrix form, we have  ∥SN(∆t)∥2  CN = sup wT SN(∆t)T ˜CNSN(∆t)w  wT ˜CNw  = sup wT SN(∆t)T ˜CNSN(∆t)w  ( ˜C1/2  N w)T ( ˜C1/2  N w)  ,  where ˜CN =  �CN  0  0  I  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Let z = ˜C1/2  N w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then  ∥SN(∆t)∥2  CN = sup zT ( ˜C1/2  N SN(∆t) ˜C−1/2  N  )T ( ˜C1/2  N SN(∆t) ˜C−1/2  N  )z  zT z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Therefore,  ∥SN(∆t)∥CN = ∥B∥2 =  �  ρ (BT B) ≤  �  ∥G∥∞,  where B = ˜C1/2  N SN(∆t) ˜C−1/2  N  , and  G = BT B =  �G11  G12  G21  G22  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12)  with  G11 = C−1/2  N  SN,11(∆t)T CNSN,11(∆t)C−1/2  N  +  C−1/2  N  SN,21(∆t)T SN,21(∆t)C−1/2  N  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='13)  G12 = C−1/2  N  SN,11(∆t)T CNSN,12(∆t) + C−1/2  N  SN,21(∆t)T SN,22(∆t),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='14)  G21 = SN,12(∆t)T CNSN,11(∆t)C−1/2  N  + SN,22(∆t)T SN,21(∆t)C−1/2  N  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15)  G22 = SN,12(∆t)T CNSN,12(∆t) + SN,22(∆t)T SN,22(∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='16)  To obtain a bound for the CN-norm of the overall approximate solution operator SN(∆t), we  will proceed by bounding ∥G∥∞ through bounding ∥Gij∥∞ for i, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  To bound the norm of each such block, we use expressions for the computed solution  un+1, un+1  t  at time tn+1, in terms of un and un  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We begin with  ∥(un+1, un+1  t  )∥2  CN = (un+1)T CN(un+1) + (un+1  t  )T (un+1  t  )  = [un]T ¯G11un + [un]T ¯G12un  t + [un  t ]T ¯G21un + [un  t ]T ¯G22un  t  where  ¯G11 = SN,11(∆t)T CNSN,11(∆t) + SN,21(∆t)T SN,21(∆t),  ¯G12 = SN,11(∆t)T CNSN,12(∆t) + SN,21(∆t)T SN,22(∆t),  ¯G21 = SN,12(∆t)T CNSN,12(∆t) + SN,22(∆t)T SN,21(∆t),  ¯G22 = SN,12(∆t)T CNSN,12(∆t) + SN,22(∆t)T SN,22(∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  10  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  We note that  G11 = C−1/2  N  ¯G11C−1/2  N  ,  G12 = C−1/2  N  ¯G12,  G21 = ¯G21C−1/2  N  ,  G22 = ¯G22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Therefore, we can proceed by bounding the entries of each ¯Gij, for i, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  LEMMA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Assume ˆp(ω) = 0 and ˆq(ω) = 0 for |ω| > ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then the matrix G11  defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12) satisfies  ∥G11∥∞ ≤ 1 + C11,p∥˜p∥∞∆t2N 2 + C11,q∥˜q∥∞∆t2 + C11,p2∥˜p∥2  ∞∆t2N 2 +  C11,pq∥˜p∥∞∥˜q∥∞∆t2 + C11,q2∥˜q∥2  ∞∆t2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='17)  where the constants C11,p, C11,q, C11,p2, C11,pq, and C11,q2 are independent of N and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Let ˆIN = {−N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6) we have  [un]T ¯G11un = zT  11CNz11 + zT  21z21  =  �  ω∈ˆIN  ˆz11(ω)ˆz11(ω)(¯pω2 + ¯q) +  �  ω∈ˆIN  ˆz21(ω)ˆz21(ω)  =  �  j∈ˆIN  �  k∈ˆIN  ˆu(−j)ˆu(k)  � ¯A + ¯B + ¯C + ¯D  �  jk  where, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' for j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' k ∈ ˆIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ¯Ajj = (S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j))2 (pj2 + q) + (S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j))2  = cos2 ��  pj2 + q∆t  � �  pj2 + q  �  +  �  −  �  pj2 + q  �1/2 sin  ��  pj2 + q∆t  ��2  = pj2 + q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ¯Bjk = i(j)S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k)ikM11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k ˆp(j − k)(pk2 + q) + S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k)M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='kˆq(j − k)(pk2 + q) +  i(j)S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k)ikM21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k ˆp(j − k) + S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k)M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='kˆq(j − k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ¯Cjk = i(k)S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)ijM11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆp(j − k)(pj2 + q) + S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆq(j − k)(pj2 + q) +  ijM21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆp(j − k)i(k)S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j) + S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆq(j − k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  and  ¯Djk = i(k)i(j)  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω2ˆp(ω − k)ˆp(j − ω)  �  M 2  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(pω2 + q) + M 2  21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  +  i(k)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ωˆp(ω − k)ˆq(j − ω)  �  M 2  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(pω2 + q) + M 2  21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  +  i(j)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ωˆq(ω − k)ˆp(j − ω)  �  M 2  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(pω2 + q) + M 2  21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  +  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ˆq(ω − k)ˆq(j − ω)  �  M 2  11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(pω2 + q) + M 2  21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  with ¯Ajk = 0 for j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' and ¯Bjj = ¯Cjj = 0 for j ∈ ˆIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  11  To obtain an upper bound for ∥G11∥∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we use the following bounds on Sij(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω) and  Mij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' which are the coefficients in the linear approximations of the various components of  the solution operator:  |S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω)| ≤  ���cos  �  l1/2  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∆t  ���� ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  |S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω)| ≤  ���−l1/2  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω sin  �  l1/2  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∆t  ���� ≤  ���l1/2  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  ��� l1/2  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∆t = l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  |M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω| ≤ ∆t2  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  |M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω| ≤  ∆t  (pω2 + q)1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  |M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω| ≤ ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We have multiple bounds for M11 so that different terms will have the same order of magnitude  in terms of N and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' for j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we have  �� ¯Bjk  �� ≤  ����jkˆp(j − k)(pk2 + q)∆t2  2  ���� +  ����  ∆t2  2 ˆq(j − k)(pk2 + q)  ���� +  ��jkˆp(j − k)(pk2 + q)∆t2�� +  ��ˆq(j − k)(pk2 + q)∆t2��  ≤ 3  2∆t2(pk2 + q) (|jkˆp(j − k)| + |ˆq(j − k)|) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  | ¯Cjk| ≤  ����jkˆp(j − k)(pj2 + q)∆t2  2  ���� +  ����ˆq(j − k)(pj2 + q)∆t2  2  ���� +  ��jkˆp(j − k)(pj2 + q)∆t2�� +  ��ˆq(j − k)(pj2 + q)∆t2��  ≤ 3  2∆t2(pj2 + q) (|jkˆp(j − k)| + |ˆq(j − k)|) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  | ¯Djk| ≤  ������  i(k)i(j)  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω2|ˆp(ω − k)ˆp(j − ω)|  �  2∆t2�  +  i(k)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω|ˆp(ω − k)ˆq(j − ω)|  �  2∆t2�  +  i(j)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω|ˆq(ω − k)ˆp(j − ω)|  �  2∆t2�  +  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  |ˆq(ω − k)ˆq(j − ω)|  �  2∆t2�  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  12  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  From these bounds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we obtain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='∥G11∥∞ ≤ max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1≤j≤N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN\\j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='(¯pj2 + ¯q)−1/2 �� ¯Ajk + ¯Bjk + ¯Cjk + ¯Djk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�� (¯pk2 + ¯q)−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='≤ max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1≤j≤N(¯pj2 + ¯q)−1/2(¯pj2 + ¯q)−1/2(pj2 + q) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN\\j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���jkˆp(j − k)(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)1/2∆t2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN\\j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���ˆq(j − k)(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)1/2∆t2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN\\j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���jkˆp(j − k)(¯pj2 + ¯q)1/2(¯pk2 + ¯q)−1/2∆t2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN\\j  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���ˆq(j − k)(¯pj2 + ¯q)1/2(¯pk2 + ¯q)−1/2∆t2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='i(k)i(j)(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω2|ˆp(ω − k)ˆp(j − ω)|∆t2  ������  +  �  k∈ˆIN  ������  i(k)i(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω|ˆp(ω − k)ˆq(j − ω)|∆t2  ������  +  �  k∈ˆIN  ������  i(j)i(¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω|ˆq(ω − k)ˆp(j − ω)|∆t2  ������  +  �  k∈ˆIN  ������  (¯pj2 + ¯q)−1/2(¯pk2 + ¯q)−1/2  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='|ˆq(ω − k)ˆq(j − ω)|∆t2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ETNA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='Kent State University and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='Johann Radon Institute (RICAM)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='≤ max  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1≤j≤N 1 + 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2j∆t2∥˜p∥∞(pj2 + q)−1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈Ij  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���k(pk2 + q)1/2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2∆t2∥˜q∥∞(pj2 + q)−1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈Ij  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���(pk2 + q)1/2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2j∆t2∥˜p∥∞(pj2 + q)1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈Ij  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���k(pk2 + q)−1/2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2∆t2∥˜q∥∞(pj2 + q)1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈Ij  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='���(pk2 + q)−1/2��� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j∆t2∥˜p∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='∞(pj2 + q)−1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k(pk2 + q)−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∈Ij∩Ik  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='∆t2∥˜p∥∞∥˜q∥∞(pj2 + q)−1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k(pk2 + q)−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∈Ij∩Ik  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j∆t2∥˜p∥∞∥˜q∥∞(pj2 + q)−1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='(pk2 + q)−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∈Ij∩Ik  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='∆t2∥˜q∥2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='∞(pj2 + q)−1/2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k∈ˆIN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='(pk2 + q)−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω∈Ij∩Ik  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='18)  We can bound each of the summations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='18) as in the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We first derive a bound for  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='19)  (pj2 + q)1/2 �  k∈Ij  ���(pk2 + q)−1/2��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  For |j| > ωmax, we have  (pj2 + q)1/2 �  k∈Ij  ���(pk2 + q)−1/2��� ≤ (pj2 + q)1/2 �  k∈Ij  1  (p(|j| − ωmax)2 + q)1/2  ≤ (pj2 + q)1/2  2ωmax  (p(|j| − ωmax)2 + q)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  As |j| → ∞, we obtain  lim  j→∞(pj2 + q)1/2 �  k∈Ij  ���(pk2 + q)−1/2��� ≤ 2ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Therefore, the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='19) can be bounded independently of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Next, we derive a bound for  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='20)  (pj2 + q)−1/2 �  k∈ˆIN  ������  k(pk2 + q)−1/2  �  ω∈Ij∩Ik  ω  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  14  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  We have  �  k∈ˆIN  ������  k(pk2 + q)−1/2  �  ω∈Ij∩Ik  ω  ������  ≤  �  k∈ˆIN  �  ω∈Ij∩Ik  ���k(pk2 + q)−1/2ω  ���  ≤  �  ω∈Ij  |ω|  �  k∈Iω  ���k(pk2 + q)−1/2���  ≤  �  ω∈Ij  |ω|  �  η∈I0  ���(ω + η)(p(ω + η)2 + q)−1/2��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  If j > 2ωmax, then ω > ωmax, and  �  k∈ˆIN  ������  k(pk2 + q)−1/2  �  ω∈Ij∩Ik  ω  ������  ≤  �  ω∈Ij  |j + ωmax|  �  η∈I0  ����  j + 2ωmax  (p(j − 2ωmax)2 + q)1/2  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We then have  lim  j→∞(pj2 + q)−1/2 �  k∈ˆIN  ������  k(pk2 + q)−1/2  �  ω∈Ij∩Ik  ω  ������  ≤ 1  ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We conclude that the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='20) can also be bounded independently of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Using a similar approach to bound the remaining summations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='18), we obtain  ∥G11∥∞ ≤ 1 + C11,p∥˜p∥∞∆t2N 2 + C11,q∥˜q∥∞∆t2 + C11,p2∥˜p∥2  ∞∆t2N 2 +  C11,pq∥˜p∥∞∥˜q∥∞∆t2 + C11,q2∥˜q∥2  ∞∆t2,  for constants C11,p, C11,q, C11,p2, C11,pq, C11,q2 that are independent of N and ∆t, which  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Using the same approach, we find that the matrix G22 defined in  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12) satisfies a bound of the same form as that of G11, with appropriate constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  LEMMA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Assume ˆp(ω) = 0 and ˆq(ω) = 0 for |ω| > ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then the matrix G12  defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12) satisfies  ∥G12∥∞ ≤ C12,p∥˜p∥∞∆tN + C12,q∥˜q∥∞∆t + C12,p2∥˜p∥2  ∞∆tN +  C12,pq∥˜p∥∞∥˜q∥∞∆t + C12,q2∥˜q∥2  ∞∆t,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='21)  where each constant is independent of N and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we have  [un]T ¯G12un  t =  �  ω∈ˆIN  ˆz11(ω)ˆz12(ω)(¯pω2 + ¯q) +  �  ω∈ˆIN  ˆz21(ω)ˆz22(ω)  =  �  j∈ˆIN  �  k∈ˆIN  ˆu(−j)ˆu(k)  � ¯A + ¯B + ¯C + ¯D  �  jk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  15  where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' for j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' k ∈ ˆIN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ¯Ajj = S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)S12(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)(¯pj2 + ¯q) + S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)S22(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)  = cos  �  (pj2 + q)1/2∆t  �  (pj2 + q)−1/2 sin  ��  pj2 + q∆t  �  (¯pj2 + ¯q) +  �  −(pj2 + q)1/2 sin  �  (pj2 + q)1/2∆t  �  cos  ��  pj2 + q∆t  ��  = cos  �  (pj2 + q)1/2∆t  �  (pj2 + q)1/2 sin  ��  pj2 + q∆t  �  −  (pj2 + q)1/2 sin  �  (pj2 + q)1/2∆t  �  cos  ��  pj2 + q∆t  �  = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ¯Bjk = i(j)ikM11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k ˆp(j − k)S12(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k)(¯pk2 + ¯q) + M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='kˆq(j − k)S12(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k)(¯pk2 + ¯q) +  i(j)ikM21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k ˆp(j − k)S22(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k) + M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='kˆq(j − k)S22(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ¯Cjk = ijS11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)M12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆp(j − k)i(k)(¯pj2 + ¯q) + S11(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)M12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆq(j − k)(¯pj2 + ¯q) +  S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)ijM22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆp(j − k)i(k) + S21(l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j)M22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j ˆq(j − k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  and  ¯Djk = i(j)i(k)  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω2ˆp(j − ω)ˆp(ω − k)  �  M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(¯pω2 + ¯q) + M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  +  i(j)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ωˆp(j − ω)ˆq(ω − k)  �  M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(¯pω2 + ¯q) + M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  +  i(k)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ωˆq(j − ω)ˆp(ω − k)  �  M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(¯pω2 + ¯q) + M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  +  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ˆq(j − ω)ˆq(ω − k)  �  M11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω(¯pω2 + ¯q) + M21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωM22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  with ¯Ajk = 0 for j ̸= k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' and ¯Bjj = ¯Cjj = 0 for j ∈ ˆIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  To obtain an upper bound for ∥G12∥∞, we use the following bounds on Sij(l2,ω) and  Mij,ω :  |S12(l2,ω)| ≤  ���l−1/2  2,ω  sin(l1/2  2,ω∆t)  ��� ≤  ���l−1/2  2,ω  ��� l1/2  2,ω∆t = ∆t,  |M11,ω| = |M22,ω| ≤  2  pω2 + q ,  |M12,ω| ≤  ∆t  pω2 + q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Then we have  | ¯Bjk| ≤ 3 (|jkˆp(j − k)∆t| + |ˆq(j − k)∆t|) ,  | ¯Cjk| ≤ 3 (|jkˆp(j − k)∆t| + |ˆq(j − k)∆t|) ,  ETNA  Kent State University and  Johann Radon Institute (RICAM)  16  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  and  | ¯Djk| ≤  ������  i(j)i(k)  �  ω∈ˆIN\\{k,j}  ω2|ˆp(j − ω)ˆp(ω − k)|  4∆t  pω2 + q +  i(j)i  �  ω∈ˆIN\\{k,j}  ω|ˆp(j − ω)ˆq(ω − k)|  4∆t  pω2 + q +  i(k)i  �  ω∈ˆIN\\{k,j}  ω|ˆq(j − ω)ˆp(ω − k)|  4∆t  pω2 + q +  �  ω∈ˆIN\\{k,j}  |ˆq(j − ω)ˆq(ω − k)|  4∆t  pω2 + q  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  From these bounds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we obtain  ∥G12∥∞ ≤ max  1≤j≤N  �  k∈ˆIN\\j  ���  � ¯Ajk + ¯Bjk + ¯Cjk + ¯Djk  �  (¯pj2 + ¯q)−1/2���  ≤ max  1≤j≤N 6  �  k∈ˆIN\\j  ���jkˆp(j − k)(¯pj2 + ¯q)−1/2∆t  ��� +  6  �  k∈ˆIN\\j  ���ˆq(j − k)(¯pj2 + ¯q)−1/2∆t  ��� +  �  k∈ˆIN\\j  ������  i(j)i(k)  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω2|ˆp(j − ω)ˆp(ω − k)|(¯pj2 + ¯q)−1/2  4∆t  pω2 + q  ������  +  �  k∈ˆIN\\j  ������  i(j)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω|ˆp(j − ω)ˆq(ω − k)|(¯pj2 + ¯q)−1/2  4∆t  pω2 + q  ������  +  �  k∈ˆIN\\j  ������  i(k)i  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  ω|ˆq(j − ω)ˆp(ω − k)|(¯pj2 + ¯q)−1/2  4∆t  pω2 + q  ������  +  �  k∈ˆIN\\j  ������  �  ω∈ˆIN\\{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='j}  |ˆq(j − ω)ˆq(ω − k)|(¯pj2 + ¯q)−1/2  4∆t  pω2 + q  ������  ≤ max  1≤j≤N 6j∥˜p∥∞(¯pj2 + ¯q)−1/2∆t  �  k∈Ij  |k| +  6∥˜q∥∞(¯pj2 + ¯q)−1/2∆t(2ωmax) +  j∥˜p∥2  ∞(¯pj2 + ¯q)−1/2∆t4  p  �  k∈ˆIN  �  ω∈Ij∩Ik  |k| +  ∥˜p∥∞∥˜q∥∞(¯pj2 + ¯q)−1/2∆t4  p  �  ω∈Ij  �  2jωmax +  �  k∈Iω  |k|  �  +  ∥˜q∥2  ∞(¯pj2 + ¯q)−1/2∆t4  q (4ω2  max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  17  These summations can be bounded as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' As an example, we  obtain a bound for  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='22)  (¯pj2 + ¯q)−1/2 �  k∈Ij  |k| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  If |j| > ωmax, then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='23)  �  k∈Ij  |k| =  �  η∈I0  |j − η| =  ωmax  �  η=1  |j − η| + |j + η| =  ωmax  �  η=1  2|j| = 2|j|ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  It follows that  lim  j→∞(¯pj2 + ¯q)−1/2 �  k∈Ij  |k| = lim  j→∞(¯pj2 + ¯q)−1/22|j|ωmax = 2ωmax  ¯p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  That is, the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='22) is bounded independently of N, whereas it would be O(N) if  p(x) was not bandlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proceeding in a similar manner for the remaining summations, we conclude that there  exist constants C12,p, C12,q, C12,p2, C12,pq and C12,q2 such that  ∥G12∥∞ ≤ C12,p∥˜p∥∞∆tN + C12,q∥˜q∥∞∆t + C12,p2∥˜p∥2  ∞∆tN +  C12,pq∥˜p∥∞∥˜q∥∞∆t + C12,q2∥˜q∥2  ∞∆t,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Using the same approach, it can be shown that the matrix  G21 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12) satisfies a bound of the same form as that of G12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We now prove a result that suggests the second-order KSS method applied to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6) is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Assume ˆp(ω) = 0 and ˆq(ω) = 0 for |ω| > ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then the solution  operator SN(∆t) satisfies  ∥SN(∆t)∥CN ≤ 1 + (Cp∥˜p∥∞N + Cq∥˜q∥∞)∆t,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='24)  where the constants Cp and Cq are independent of N and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' From  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='25)  ����  �G11  G12  G21  G22  �����  ∞  ≤ max{∥G11∥∞ + ∥G12∥∞, ∥G21∥∞ + ∥G22∥∞},  we have, for some k ∈ {1, 2},  ∥SN(∆t)∥CN = ∥B∥2 ≤  �  ∥G∥∞ ≤  �  ∥Gk1∥∞ + ∥Gk2∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2, we have  ∥G∥∞ ≤ 1 + ∆tN  �  Ck2,p∥˜p∥∞ + Ck2,p2∥˜p∥2  ∞  �  + ∆t2N 2 �  Ck1,p∥˜p∥∞ + Ck1,p2∥˜p∥2  ∞  �  +  ∆t  �  Ck2,q∥˜q∥∞ + Ck2,pq∥˜p∥∞∥˜q∥∞ + Ck2,q2∥˜q∥2  ∞  �  +  ∆t2 �  Ck1,q∥˜q∥∞ + Ck1,pq∥˜p∥∞∥˜q∥∞ + Ck1,q2∥˜q∥2  ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='26)  Let  R1 = N  �  Ck2,p∥˜p∥∞ + Ck2,p2∥˜p∥2  ∞  �  + Ck2,q∥˜q∥∞ + Ck2,pq∥˜p∥∞∥˜q∥∞ + Ck2,q2∥˜q∥2  ∞,  ETNA  Kent State University and  Johann Radon Institute (RICAM)  18  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  R2 = N 2 �  Ck1,p∥˜p∥∞ + Ck1,p2∥˜p∥2  ∞  �  + Ck1,q∥˜q∥∞ + Ck1,pq∥˜p∥∞∥˜q∥∞ + Ck1,q2∥˜q∥2  ∞,  and R = max{R1/2  2  , R1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We then have  ∥SN(∆t)∥CN ≤  �  ∥G∥∞ ≤ 1 + R∆t ≤ 1 + (Cp∥˜p∥∞N + Cq∥˜q∥∞)∆t,  from which the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  COROLLARY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Assume the leading coefficient p(x) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then, under the  assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3,  ∥SN(∆t)∥CN ≤ 1 + Cq∥˜q∥∞∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Because the leading coefficient p(x) is constant, we have ˜p(x) = p(x) − ¯p ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The result follows immediately from the last line of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Therefore,  a second-order KSS method applied to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7), under the assumptions that  p(x) is constant and q(x) is bandlimited, is unconditionally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For the remainder of this convergence analysis, we assume the coeffi-  cient p(x) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) is constant, since stability has been proved only for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Before we obtain an estimate of the local truncation error, we introduce additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We first define the restriction operator  RNf(x) = f(xN),  and interpolation operator  TNg = TN  � g1  g2  �  =  �  ��������  N/2  �  ω=−N/2+1  eiωx˜g1(ω)  N/2  �  ω=−N/2+1  eiωx˜g2(ω)  �  ��������  ,  where, for i = 1, 2,  ˜gi(ω) = 1  N  N  �  j=1  e−iωxj[gi]j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Then, the operator IN = TNRN on L2([0, 2π])×L2([0, 2π]) computes the Fourier interpolant  of each component function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' By contrast, if we define  ˆRNf(x) = ˆRN  � f1(x)  f2(x)  �  =  �  ��������  N/2  �  ω=−N/2+1  eiωxN ˆf1(ω)  N/2  �  ω=−N/2+1  eiωxN ˆf2(ω)  �  ��������  ,  where, for i = 1, 2,  ˆfi(ω) = 1  2π  � 2π  0  e−iωxfi(x) dx,  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  19  then the operator PN = TN ˆRN on L2([0, 2π]) × L2([0, 2π]) is the orthogonal projection  operator onto span{eiωx}N/2  ω=−N/2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Finally, the continuous approximate solution operator  ˜SN(∆t) : L2([0, 2π]) → L2([0, 2π]) is defined by ˜SN(∆t) = TNSN(∆t)RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Let f ∈ Hm+1  p  ([0, 2π]) × Hm  p ([0, 2π]) for m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then, for the  problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) on the domain (0, 2π) × (0, T), with p(x) constant and q(x)  bandlimited, the two-node block KSS method with prescribed nodes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5) is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' That is,  the local truncation error satisfies  1  ∆t  ���IN exp[˜L∆t]f − ˜SN(∆t)f  ���  C ≤ C1∆xm−1 + C2∆t2,  where the constants C1 and C2 are independent of ∆x and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proof: We split the local truncation error into two components:  E1(∆t, ∆x) = IN exp(˜L∆t)f(x) − TN exp(˜LN∆t)RNf(x)  E2(∆t, ∆x) = TN exp(˜LN∆t)RNf(x) − ˜SN(∆t)f(x)  = TN[exp(˜LN∆t) − SN(∆t)]RNf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  First, we bound E1(∆t, ∆x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Because of the regularity of f, we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='27)  ∥f − fN∥C ≤ C0∆xm  for some constant C0 (see [11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Next, we note that the exact solution v(x, t) =  exp[˜Lt]f(x) has the spectral decomposition  v(x, t) =  ∞  �  k=1  eµktvk(x)⟨vk, f⟩  where {µk}∞  k=1 are the purely imaginary eigenvalues of ˜L, and {vk(x)}∞  k=1 are the cor-  responding orthonormal eigenfunctions, each of which belongs to C∞  p [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Using this  spectral decomposition, it can be shown using an approach similar to that used in [4, Section  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2, Theorem 6] for other hyperbolic PDEs that if f ∈ Hm+1  p  ([0, 2π]) × Hm  p ([0, 2π]), then  v(x, t) ∈ L2(0, T, Hm+1  p  ([0, 2π])×Hm  p ([0, 2π])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' That is, the regularity of f(x) is preserved  in v(x, ∆t) for each fixed ∆t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Therefore, there exists a constant CT such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='28)  ∥(I − IN)v(·, ∆t)∥C ≤ CT ∆xm,  0 < ∆t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Using an approach based on [2] and applied in [20], we write E1(∆t, ∆x) as  eN(x, t) = INv(x, t) − TN exp(˜LNt)RNf(x)  = INv(x, t) − exp(IN ˜Lt)INf(x)  Then, eN(x, t) solves the IVP  ∂  ∂teN = IN ˜LeN + IN ˜L(I − IN)v,  eN(x, 0) = 0,  and therefore  eN(x, ∆t) =  � ∆t  0  eIN ˜L(∆t−τ)IN ˜L(I − IN)v(x, τ) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  20  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='28), and applying [4, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2, Theorem 6], it follows that  ∥E1(∆t, ∆x)∥C = ∥eN(x, ∆t)∥C  ≤ ∆t max  0≤τ≤∆t  ���IN ˜LeIN ˜L(∆t−τ)(I − IN)v(·, τ)  ���  C  ≤ C1∆t∆xm−1,  where the constant C1 is independent of ∆x and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Here we note that because the coefficients  of L are assumed to be constant or bandlimited, E1(∆t, ∆x) does not include aliasing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Now, we examine E2(∆t, ∆x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' If we let  RNf =  � fN,1  fN,2  �  ,  E2(∆t, ∆x) = TN  � eN,1  eN,2  �  ,  then we have  eN,1 = [cos(L1/2  N ∆t) − SN,11(∆t)]fN,1 + [L−1/2  N  sin(L1/2  N ∆t) − SN,12(∆t)]fN,2,  eN,2 = [−L1/2  N sin(L1/2  N ∆t) − SN,21(∆t)]fN,1 + [cos(L1/2  N ∆t) − SN,22(∆t)]fN,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We have, by Parseval’s identity,  ∥E2(∆t, ∆x)∥2  C = 2π  N/2  �  ω=−N/2+1  (pω2 + q)  ��ˆeH  ω eN,1  ��2 +  ��ˆeH  ω eN,2  ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  For each ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' N/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' and i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we use the polynomial interpolation error  in SN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ij to obtain  ˆeH  ω eN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1 = 1  2  ∂2  ∂λ2  �  cos(λ1/2∆t)  �����  λ=ξω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='11  ˆeH  ω (LN − l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)(LN − l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)fN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1 +  1  2  ∂2  ∂λ2  �  λ−1/2 sin(λ1/2∆t)  �����  λ=ξω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12  ˆeH  ω (LN − l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)(LN − l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)fN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ˆeH  ω eN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' = 1  2  ∂2  ∂λ2  �  −λ1/2 sin(λ1/2∆t)  �����  λ=ξω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='21  ˆeH  ω (LN − l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)(LN − l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)fN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1 +  1  2  ∂2  ∂λ2  �  cos(λ1/2∆t)  �����  λ=ξω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='22  ˆeH  ω (LN − l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)(LN − l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ωI)fN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  where ξω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ij ∈ [l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' l2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='ω] for i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In view of the regularity of f(x), and the fact that LN  is a discretization of a second-order differential operator with bandlimited coefficients, and a  constant leading coefficient, we have  ��ˆeH  0 (LN − l1,0I)(LN − l2,0I)fN,j  �� =  ����  1  N (LN ˜qN)T fN,j  ���� ≤ ∥LN ˜qN∥∞∥fN,j∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  It follows that there exist constants Cij, for i, j = 1, 2, independent of N and ∆t, such that  ¯q1/2 ��ˆeH  0 eN,1  �� ≤ ∆t4C11 + ∆t5C12,  ��ˆeH  0 eN,2  �� ≤ ∆t3C21 + ∆t4C22,  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  21  and for ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , −1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2, by Taylor expansion of the sines and cosines in  ˆeH  ω eN,1 and ˆeH  ω eN,2, we have  (¯pω2 + ¯q)1/2 ��ˆeH  ω eN,1  �� ≤ ∆t4  C11  |ω|m−2 + ∆t5  C12  |ω|m−3 ,  ��ˆeH  ω eN,2  �� ≤ ∆t3  C21  |ω|m−1 + ∆t4  C22  |ω|m−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  It is important to note that because the leading coefficient p(x) of L is constant, (LN −  l2,ωI)ˆeω = ˜qN ◦ ˆeω, where ◦ denotes componentwise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Therefore, this expres-  sion is bounded independently of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Finally, we obtain  ∥E2(∆t, ∆x)∥C ≤ ∆t3  √  2π  �  C2  11∆t2  �  1 + 2  ∞  �  ω=1  ω4−2m  �  + 2C11C12∆t3  �  1 + 2  ∞  �  ω=1  ω5−2m  �  +  C2  12∆t4  �  1 + 2  ∞  �  ω=1  ω6−2m  �  + C2  21  �  1 + 2  ∞  �  ω=1  ω2−2m  �  +  2C21C22∆t  �  1 + 2  ∞  �  ω=1  ω3−2m  �  + C2  22∆t2  �  1 + 2  ∞  �  ω=1  ω4−2m  ��1/2  ≤ C2∆t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Since m ≥ 4, it follows that all of the summations over ω converge to a sum that can be  bounded independently of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We conclude that the constant C2 is independent of ∆x and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Now we can prove that the second-order KSS method converges for  the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) in the case of p(x) being constant and q(x) bandlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We say that a method is convergent of order (m, n) if there exist constants Ct and Cx,  independent of the time step ∆t and grid spacing ∆x = 2π/N, such that  ∥u(·, t) − uN(·, t)∥C ≤ Ct∆tm + Cx∆xn,  0 ≤ t ≤ T,  where u(x, t) is the exact solution and uN(x, t) is the approximate solution computed using  an N-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5, the two-node block KSS method  with prescribed nodes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5) is convergent of order (2, m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Proof: We recall that S(∆t) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4) is the exact solution operator for the problem  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For any nonnegative integer n and fixed grid size N, we define  En = ∥INS(∆t)nf − ˜SN(∆t)nINf∥C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  22  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  Then, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' there exist constants C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' C2 and Cq such that  En+1 = ∥INS(∆t)n+1f − ˜SN(∆t)n+1INf∥C  = ∥INS(∆t)S(∆t)nf − ˜SN(∆t) ˜SN(∆t)nINf∥C  = ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf + ˜SN(∆t)S(∆t)nf − ˜SN(∆t) ˜SN(∆t)nINf∥C  ≤ ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf∥C + ∥ ˜SN(∆t)[INS(∆t)nf − ˜SN(∆t)nINf]∥C  ≤ ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf∥C + ∥TNSN(∆t)RN[INS(∆t)nf − ˜SN(∆t)nINf]∥C  ≤ ∥INS(∆t)S(∆t)nf − ˜SN(∆t)S(∆t)nf∥C + ∥TNSN(∆t) ˆRN[INS(∆t)nf − ˜SN(∆t)nINf]∥C  ≤ ∥INS(∆t)u(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' tn) − ˜SN(∆t)INu(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' tn)∥C + ∥SN(∆t)∥CN En  ≤ C1∆t3 + C2∆t∆xm−1 + (1 + Cq∥˜q∥∞∆t)En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  It follows that  En ≤  eCq∥˜q∥∞T − 1  1 + Cq∥˜q∥∞∆t − 1(C1∆t3 + C2∆t∆xm−1) ≤ ˜C1∆t2 + ˜C2∆xm−1  for some constants ˜C1 and ˜C2 that depend only on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We conclude that  ∥u(·, tn) − uN(·, tn)∥C ≤ ∥INu(·, tn) − uN(·, tn)∥C + ∥(I − IN)u(·, tn)∥C  ≤ ˜C1∆t2 + ˜C2∆xm−1 + ˜C3∆xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Numerical Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We now perform some numerical experiments to corrobo-  rate the theory presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For each test case, relative error was estimated using  the ℓ2 norm, in comparison to a reference solution computed by the MATLAB ODE solver  ode15s, witih absolute and relative tolerances set to 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Constant Leading Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We consider the initial value problem  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1)  utt + Lu = 0,  0 < x < 2π,  t > 0,  where L is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7), with  p(x) = 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2)  q(x) = 1 + 1  2 sin x + 1  4 cos 2x + 1  8 sin 3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3)  The initial conditions are  u(x, 0) =  �  1 − 2  π|x − π|  π/2 ≤ x ≤ 3π/2,  0  0 ≤ x < π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  3π/2 < x < 2π,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4)  ut(x, 0) = 0,  0 < x < 2π,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5)  and we impose periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We note that the initial data belongs to  Hm+1  p  ([0, 2π]) × Hm  p ([0, 2π]) for m = 1, which is not sufficiently regular to satisfy the  assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' As predicted by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4, we  observe second-order accuracy in time, in spite of the low regularity of the initial data, even  when the CFL limit is exceeded by using the same time step as the spatial resolution increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  23  TABLE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1  Relative errors in the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5) with periodic boundary conditions on the domain  (0, 2π) × (0, 10), using the second-order KSS method described in Section 3, with N grid points and time step ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ∆t  N = 256  N = 512  N = 1024  N = 2048  π/128  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='38e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='33e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='30e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='29e-04  π/256  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='32e-05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='24e-05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='25e-05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='20e-05  π/512  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='04e-06  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='49e-06  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='27e-06  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='08e-06  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Variable Leading Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We now solve the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), with  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6)  p(x) = 1 − 1  2 sin x + 1  4 cos 2x  and initial conditions  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7)  u(x, 0) = e−(x−π)2,  ut(x, 0) = 0,  0 < x < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The results are shown in Figures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We see that when the CFL number is greater  than one, the method is unstable, as high-frequency components quickly become the dominant  terms of the solution, and their amplitudes grow without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' On the other hand, when the  CFL number is less than one, the solution is well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Solutions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) on the domain (0, 2π) × (0, 1), using the second-order KSS  method described in Section 3, with N = 256 grid points and time step CFL number ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In this paper, we have limited our analysis to the wave equation in  one space dimension, with periodic boundary conditions, and spatial differentiation performed  t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2439  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content="5  (t'x)n  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1  0  1  2  3  4  5  9  xt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4878  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  x)n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1  0  0  2  3  4  5  9  xt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='73171  100  80  60  40  20  (x,t)  0  20  40  60  80  100  0  2  3  4  5  6  XX105  t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='97561  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  (t*x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0  2  3  4  5  6  xETNA  Kent State University and  Johann Radon Institute (RICAM)  24  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Solutions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) on the domain (0, 2π) × (0, 1), using the second-order KSS  method described in Section 3, with N = 256 grid points and time step CFL number ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  using the FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We now consider some variations of this problem, to investigate whether our  conclusions may apply more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Finite Differencing in Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We solve the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7), on  the domain (0, 2π) × (0, 10), with periodic boundary conditions, and using centered differ-  encing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Because of the change of spatial discretization, we modify the interpolation  points from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5) by prescribing  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8)  l2,ω = p2 − 2 cos(ω∆x)  ∆x2  + q,  ω = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  The results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' It can be seen that the same unconditional stability that was  established for spectral differentiation applies in the case of finite differencing, as an accurate  solution is obtained even when the CFL number is as large as eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  TABLE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  Relative errors in the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) with periodic boundary conditions on the domain  (0, 2π) × (0, 10), using the second-order KSS method described in Section 3, with N grid points, time step ∆t,  central differencing in space, and interpolation points (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ∆t  N = 256  N = 512  N = 1024  N = 2048  π/128  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='00e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='00e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='00e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='00e-04  π/256  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='47e-05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='48e-05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='48e-05  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='47e-05  π/512  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15e-06  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='16e-06  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15e-06  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15e-06  t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2439  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content="5  (t'x)n  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content='4878  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content='4  (x,t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content='73171  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content="25  (t'x)n  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='05  0  1  2  3  4  5  9  xt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='97561  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='25  (tx)n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='05  0  2  3  4  5  9  xETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Other Boundary Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We repeat the problem from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1, except  with homogeneous Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The interpolation points from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5) are  modified as follows:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='9)  l2,ω = p2 − 2 cos(ω∆x/2)  ∆x2  + q,  ω = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  As in the case of periodic boundary conditions, unconditional stability is indicated by the  results, shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  TABLE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3  Relative errors in the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7) with homogeneous Dirichlet boundary conditions on  the domain (0, 2π) × (0, 10), using the second-order KSS method described in Section 3, with N grid points, time  step ∆t, central differencing in space, and interpolation points (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ∆t  N = 256  N = 512  N = 1024  N = 2048  π/128  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='53e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='53e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='53e-04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='53e-04  π/256  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='85e-05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='85e-05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='85e-05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='86e-05  π/512  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='67e-06  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='67e-06  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='67e-06  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='68e-06  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Higher Space Dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We solve a two-dimensional wave equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='10)  utt + Lu = 0,  0 < x, y < 2π,  0 < t < 10,  where the differential operator L is defined by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='11)  Lu = −∆u + q(x, y)u,  where  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12)  q(x, y) = 1 + 1  2 sin x cos y + 1  4 cos 2y + 1  8 sin 3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Our initial conditions are  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='13)  u(x, y, 0) = e−(x−π)2+(y−π)2,  ut(x, y, 0) = 0,  0 < x, y < 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  and we impose periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For spatial discretization, we use a N × N grid  with spacing ∆x = ∆y = 2π/N, and centered differencing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This leads to the choice  of interpolation points  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='14)  l1,ω1,ω2 = 0,  l2,ω1,ω2 =  1  ∆x2 [4 − 2 cos(ω1∆x) − 2 cos(ω2∆y)] + q,  for ω1, ω2 = −N/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' , N/2, where q is the average value of q(x, y) on (0, 2π)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The  results, shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4, indicate that unconditional stability again holds, as the CFL  number exceeds one without loss of accuracy or stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Modeling an Acceleration Wave in a Dusty Gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Finally, we consider the appli-  cation of a KSS method to the PDE [13]  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15) λ0wttt + (1 + κ0)wtt − (1 − z/H1)wzz + kwz + λ0[kwtz − (1 − z/H1)wtzz] = 0,  on the domain 0 < z < ℓ < H1, t > 0, with initial conditions  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='16)  w(z, 0) = wt(z, 0) = wtt(z, 0) = 0,  0 < z < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  26  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  TABLE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4  Relative errors in the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='13) with periodic boundary conditions on the domain  (0, 2π)2 × (0, 10), using the second-order KSS method described in Section 3, with N grid points per dimension,  time step ∆t, central differencing in space, and interpolation points (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ∆t  N = 16  N = 32  N = 64  N = 128  π/8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='83e-02  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='14e-02  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='95e-02  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='84e-02  π/16  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='69e-03  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='80e-03  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='30e-03  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='06e-05  π/32  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='01e-03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='95e-03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='84e-03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='78e-03  We impose the boundary conditions  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='17)  w(0, t) = sin(Ωt),  w(ℓ, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  This problem arises from the modeling of particle-laden flow formulation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=', a “dusty gas”  model) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Transforming the Differential Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Using techniques from [19], we trans-  form this problem so that it can be solved effectively using a Fourier spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' For  convenience, we define the differential operator  Lu(z) = −a2(z)uzz(z) + a1(z)uz(z),  a2(z) = 1 − z/H1,  a1(z) = k  Our goal is to find similarity transformations V1, V2 so that the transformed operator  ¯L = V −1  2  V −1  1  LV1V2  has these properties: (1) the coefficient of wzz constant (through V1), and (2) there is no wz  term (through V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We first perform the transformation  V1f(z) = f(φ(z)),  where  y = φ(z) = ℓ  � z  0  a2(s)−1/2 ds  � ℓ  0  a2(s)−1/2 ds  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  This yields the transformed operator  V −1  1  LV1u(y) = −(a2 ◦ φ−1)uyy + (a1 ◦ φ−1)uy,  a2 = C2,  a1(z) = (a1 + a′  2/2)φz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Next, we need to eliminate the first derivative term in the differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Let  ˜a2 = a2 ◦ φ−1, ˜a1 = a1 ◦ φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Then  V −1  1  LV1 = −˜a2uyy + ˜a1uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We now apply the transformation  V2f(y) = ψ(y)f(y),  ψ(y) = exp  �� ˜a1(y)  2˜a2  dy  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  27  This yields the transformed operator ¯L defined by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='18)  ¯Lu = V −1  2  (V −1  1  LV1)V2u = −˜a2uyy + a0(y)u  where  a0 = −1  2˜a′  1 + ˜a2  1  4˜a2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Our transformed PDE is now  λ0 ¯wttt + (1 + κ0) ¯wtt + ¯L ¯w + λ0 ¯L ¯wt = 0,  0 < y < ℓ,  t > 0,  where w = V1V2 ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The initial and boundary conditions for ¯w are the same as those for w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Homogenizing the Boundary Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Due to the Dirichlet boundary condi-  tions, a KSS method will represent the solution as a Fourier sine series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We therefore need to  homogenize the boundary conditions, by writing the solution ¯w(y, t) as  ¯w(y, t) = ¯u(y, t) + F(y, t)  where ¯u satisfies homogeneous boundary conditions, and F satisfies the original inhomoge-  neous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Because this transformation will introduce a source term to the  PDE, which we will denote by G(y, t), we will also require that G satisfies homogeneous  boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  To that end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' we prescribe  F(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = f0(t) + f1(t)y + f2(t)y2 + f3(t)y3 + f4(t)y4  The source term introduced into the PDE is given by  G(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = λFttt + (1 + κ0)Ftt − a2Fyy + a0F + λ0[−a2Ftyy + a0Ft]  The functions F and G must satisfy the condidtions  F(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = f0(t) = sin(Ωt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  F(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = f0(t) + f1(t)ℓ + f2(t)ℓ2 + f3(t)ℓ3 + f4(t)ℓ4 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  G(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = λ0f ′′′  0 (t) + (1 + κ0)f ′′  0 (t) − 2a2f2(t) − a0(0)f0(t) − λ0[a0(0)f ′  0(t) + 2a2f ′  2(t)] = 0  G(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = −a2[2f2(t) + 6f3(t)ℓ + 12f4(t)ℓ2] = 0  Fy(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' t) = f1(t) + 2f2(t)ℓ + 3f3(t)ℓ2 + 4f4(t)ℓ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Solving the ODE G(0, t) = 0 for f2(t) yields  f2(t) = A2 sin Ωt + B2 cos Ωt,  A2 = bΩ − ac  Ω2 + a2 ,  B2 = −ab + cΩ  Ω2 + a2 ,  where  a = − 1  λ0  ,  b = −  � Ω3  2a2  + Ωa−  0  2a2  �  ,  c = −  �Ω2(1 + κ0)  2a2λ0  +  a−  0  2a2λ0  �  ,  a−  0 = a0(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We can then solve a system of three linear equations for the remaining three unknowns f1(t),  f3(t) and f4(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' In addition to the above conditions, we also have Fyy(ℓ, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  28  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Transformation to a Wave Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The PDE that will actually be solved via  KSS is  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='19)  utt = −¯Lu − κ0¯utt − G,  where  u = ¯u + λ0¯ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We note that the equation relating u and ¯u is a first-order linear ODE, which can be solved for  ¯u to obtain  ¯u(y, tn+1) = e−∆t/λ0 ¯u(y, tn) + 1  λ0  � ∆t  0  e−(∆t−s)/λ0u(y, tn + s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We replace u(y, tn + s) within the integral by a Taylor expansion centered at tn+1 to obtain  the approximation  ¯u(y, tn+1) ≈ e−∆t/λ0 ¯u(y, tn) + (1 − e−∆t/λ0)[u(y, tn+1) − ∆tut(y, tn+1)] +  ut(y, tn+1)  �  ∆t − λ0 + λ0e−∆t/λ0�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Having obtained ¯u, we can use the equation u = ¯u + λ0¯ut to obtain  ¯ut = (u − ¯u)/λ0,  ¯utt = (ut − ¯ut)/λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Solution Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We approximate our PDE with a PDE of the form  utt = −¯Lu + b,  where b is a source term, set equal to −κ0wtt − G, evaluated at time tn for the time step to  tn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We rewrite this PDE as a first-order system  � u  ut  �  t  = J  � u  ut  �  +  � 0  b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  J =  �  0  I  −L  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  If we define  v =  � u  ut  �  ,  c =  � 0  b  �  then the solution of our first-order system is  v(tn+1) = φ0(Jt)v(tn) + tφ1(Jt)c(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  where  φ0(Jt) = eJt =  �  cos(L1/2t)  L−1/2 sin(L1/2t)  −L1/2 sin(L1/2t)  cos(L1/2t)  �  and  tφ1(Jt) = (eJt − I)J−1 =  � L−1/2 sin(L1/2t)  L−1 − L−1 cos(L1/2t)  cos(L1/2t) − I  L−1/2 sin(L1/2t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  ETNA  Kent State University and  Johann Radon Institute (RICAM)  CONVERGENCE ANALYSIS OF KSS FOR THE 1-D WAVE EQUATION  29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Numerical Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We now solve the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='17), with param-  eters  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='20)  ℓ = 2,  λ0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='01336,  κ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='05,  H1 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='7278,  k ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4806,  Ω = π,  by transforming the PDE to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='19), applying the second-order KSS method described in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='2, and transforming back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The computed solution is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We  see that the solution obtained via KSS matches the solution obtained via numerical Laplace  inversion, and the analytical location of the wave front, given by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='21)  σ(t) = t −  t2  4H1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Although the differential operator ¯L from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='18) has a coefficient a0(y) that is not bandlimited,  it is not necessary to perform regularization to mitigate aliasing error that can lead to weak  instability [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' However, such regularization may be necessary when applying a KSS method  to other wave propagation problems [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Solution of dusty gas problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='15), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='17), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='20) computed using a second-order KSS  method (solid blue curve) with N = 16, 384 grid points and CFL number 10, and numerical Laplace transform  inversion (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' The red dashed line indicates the location of the wavefront given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' We have established an upper bound for the approximate solution op-  erator of a second-order KSS method applied to the 1-D wave equation with bandlimited  coefficients and periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Unfortunately, the bound is not independent of  the grid size, which indicates that the unconditional stability proved for the heat equation for  the same kind of spatial differential operator does not extend to the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Numerical  experiments support this assertion, while also suggesting that conditional stability may still  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  We have also proved that the same KSS method applied to the wave equation with  periodic boundary conditions is unconditionally stable in the case of a constant wave speed  and bandlimited lower-order coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' This is the first result proving unconditional stability  for a KSS method with prescribed nodes applied to the wave equation, as opposed to nodes of  Gauss quadrature rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Numerical experiments suggest that this unconditional stability also  holds for related problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  Ideas for future work include: 1) to investigate whether the same result holds if coefficients  are not bandlimited but sufficiently smooth, 2) to generalize the analysis to higher-order KSS  methods or higher space dimensions with different boundary conditions, and 3) to attempt to  modify the KSS method to ensure at least conditional stability, without having to transform  the spatial differential operator as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='0596  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  w(z,t)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  2  zt=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='8921  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  w(z,t)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  2  zt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='52979  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  w(z,t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  Q  Q  0  000000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='5  2  zETNA  Kent State University and  Johann Radon Institute (RICAM)  30  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' RESTER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' LAMBERS  REFERENCES  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Atkinson, An Introduction to Numerical Analysis, 2nd Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Wiley (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content=' Tadmor, “Stability and spectral convergence of Fourier method for nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Lambers and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Palchak, “Solution of Nonlinear Time-Dependent PDE Through  Componentwise Approximation of Matrix Functions”, Journal of Computational Physics 321 (2016), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  1120-1143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' Evans, Partial Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' American Mathematical Society (1998)  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E_T4oBgHgl3EQfyBzt/content/2301.08316v1.pdf'}
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+Database Management System Performance Comparisons: A Systematic Survey
+TONI TAIPALUS, University of Jyväskylä, Finland
+Efficiency has been a pivotal aspect of the software industry since its inception, as a system that serves the end-user fast, and the
+service provider cost-efficiently benefits all parties. A database management system (DBMS) is an integral part of effectively all
+software systems, and therefore it is logical that different studies have compared the performance of different DBMSs in hopes of
+finding the most efficient one. This survey systematically synthesizes the results and approaches of studies that compare DBMS
+performance and provides recommendations for industry and research. The results show that performance is usually tested in a way
+that does not reflect real-world use cases, and that tests are typically reported in insufficient detail for replication or for drawing
+conclusions from the stated results.
+CCS Concepts: • Information systems → Data management systems; Information retrieval query processing; • General and
+reference → Surveys and overviews.
+Additional Key Words and Phrases: performance, database management system, database, comparison, survey
+Toni Taipalus. 2023. Database Management System Performance Comparisons: A Systematic Survey. 1, 1 (January 2023), 32 pages.
+https://doi.org/XXXXXXX.XXXXXXX
+1
+INTRODUCTION
+Efficiency is important in effectively all software systems, whether efficiency is measured by response times, how many
+concurrent users the system can serve, or how energy-efficient the system is [79]. Despite its importance, many software
+systems suffer from efficiency problems [49], as optimization has been largely recognized as a complex task [27, 79]. The
+more a system holds and handles data, the more the system’s performance depends on the database, and the database
+is often one of the first suspect when a performance issue is detected. The domain of database management systems
+(DBMS) saw rapid advancements in performance especially in the 1980s and 1990s, as benchmarking competitions
+between DBMS and hardware vendors led to innovations in DBMS technology that significantly improved DBMS
+performance [24]. Performance improvements are related to DBMS aspects such as different supporting data structures
+[82], and algorithms for sorting [29, 33] and joining [61, 72]. Given that DBMSs are annually a multi-billion dollar
+industry, the performance of a DBMS is one of the most crucial aspects when a company chooses a DBMS for their
+product or service [26]. As different DBMS performance comparison studies and DBMS vendor white-papers highlight
+the performance gains of one DBMS over another, it may seem tempting to either consider choosing the fastest DBMS
+for a business domain or to migrate from one DBMS to another for performance gains. However, as we show and argue
+in this study, performance is typically tested in very specific contexts which are not necessarily generalizable, and there
+are other aspects besides performance to consider.
+This study was inspired by a study by Raasveldt et al. [69], which claimed that “[...] we will explore the common
+pitfalls in database performance benchmarking that are present in a large number of scientific works [...]” while consciously
+refraining from citing example studies. While we agree with their claim based on our personal experiences, we wanted
+to systematically explore whether this phenomenon is common among performance benchmarks, and whether such
+Author’s address: Toni Taipalus, toni.taipalus@jyu.fi, University of Jyväskylä, Agora, Mattilanniemi 2, P.O. Box 35, Jyväskylä, Finland, FI-40014.
+© 2023
+Manuscript preprint
+Manuscript
+1
+arXiv:2301.01095v1  [cs.PF]  3 Jan 2023
+
+2
+Taipalus
+studies show performance gains of one DBMS over another in a setting that can be replicated. This study is not an
+attempt to criticize studies comparing DBMS performance, as no scientific study (ours included) is without threats to
+validity. Rather, based on the survey of the literature, the primary goals of our study are to (i) propagate information
+on how DBMS performance has been tested, (ii) how performance has been recommended to be tested, (iii) how
+the performance comparison results should be interpreted, (iv) what other aspects besides performance should be
+considered, and (v) what other avenues might be fruitful for DBMS performance testing. Additionally, we provide (vi) a
+relatively accessible background on database system performance, followed by (vii) a systematic survey of literature on
+DBMS performance comparisons, (viii) describing which DBMSs and which types of DBMSs have been compared with
+each other, (ix) the outcomes of the performance comparisons, and (x) by which benchmarks the DBMSs have been
+compared.
+The rest of this study is structured as follows. In Sections 2 and 3, we provide theoretical background for understanding
+the results and discussion provided by this study. These background sections are deliberately presented by refraining
+from using unnecessary information technology-related terms, acronyms, algorithms, or mathematics, to cater to the
+needs of readers from various backgrounds. For readers more technically inclined or interested, we have provided
+further reading at the end of Sections 2 and 3. Section 4 details how we searched, selected, and categorized the DBMS
+performance comparison studies, and Section 5 presents a high-level overview of the results, which is complemented
+by Appendix A detailing the performance comparison outcomes. In Section 6, we discuss what these findings mean,
+how they are applicable in industry, and present our recommendations for industry and research based on the findings.
+Section 7 concludes the study.
+2
+DATABASE SYSTEMS
+2.1
+Database System Overview
+A database is a collection of interrelated data, typically stored according to a data model. Typically, the data is used by
+one or several software applications via a DBMS. Collectively, the database, the DBMS, and the software application are
+referred to as a database system [31, p.7][17, p.65]. The separation of the database and the DBMS, especially in the realm
+of relational databases, is typically impossible without exporting the database in another format. In these situations, the
+database is often unusable by the DBMS, unless the database is imported back to a format understood by the DBMS.
+Possibly due to this inseparability, both the DBMS and the underlying database are often colloquially referred to simply
+as database. It is worth noting, though, that the former is a piece of software that does, while the other is a collection of
+data that is.
+Fig.1 shows a simplified example of a system where the components crucial for a database system and the scope of this
+study are emphasized. We refer to the components in the figure throughout this study. Several things are worth noting
+in considering the figure, as we have traded technical precision and comprehensiveness for ease of presentation by
+depicting only a single end-user, a single software application (some parts typically reside on the end-user’s device, while
+others reside on a separate server), a single DBMS, single hardware components, and a single database. Furthermore,
+we have not illustrated other DBMS components such as access control, data structures such as metadata, or outputs
+such as query execution plans. The figure also adopts the view that the database resides in persistent storage — this is
+not always the case. Additionally, we have depicted merely a centralized database system in which neither the DBMS
+nor the database has been distributed across multiple nodes. These are willful omissions given the scope of this study.
+Manuscript
+
+Database Management System Performance Comparisons
+3
+Parser
+Optimizer
+Execution
+engine
+Transaction
+logs
+Locks
+DBMS
+Memory
+CPU
+Software
+application
+User
+Fig. 1. A simplified view of a database system and the end-user with the emphasis on components relevant to this study; the arrows
+represent the flow of information from the end-user’s device to the database residing in persistent storage; the flow of information
+back to the software application is not illustrated here; gray rectangles represent boundaries of physical devices
+2.2
+Data Models
+Databases follow one or several data models, i.e., definitions of how and what data can be stored, and sometimes,
+what operations are available for data retrieval and manipulation. Data models may be conceptual, logical, or physical.
+Conceptual models such as the Entity-Relationship model [11] do not dictate how data should be stored, but are rather
+used to describe the interrelations and characteristics of the data. Logical data models such as the relational model [14]
+are related to how data is stored and presented, but often without describing how the data is physically stored, e.g.,
+which computing node is responsible for storing the data, where the data is located on a disk, and what types of indices
+(i.e., redundant data structures which facilitate query performance) and physical data retrieval operations are available.
+One DBMS is not limited to using a single data model [34].
+There are several popular logical data models, some of which are inseparable from their underlying physical data
+models. One of the most prominent logical data models is the relational data model rooted in set theory [14]. Relational
+DBMSs (RDBMS) follow many of the concepts introduced in the relational model. Many of the popular RDBMSs such
+as PostgreSQL and Oracle Database have adopted data structures from other logical data models as well [57]. What is
+common for effectively all modern RDBMS is that they utilize Structured Query Language (SQL) [46, 47] to define data
+structures and to retrieve and manipulate data. Typically, RDBMSs also implement a strong data consistency model
+which dictates or allows that database operations grouped into a transaction must all succeed or all fail, data must follow
+defined business logic, successful transactions persist in storage, and concurrent transactions [cf. 5] must result in the
+same data as if the transactions were serial. At least the last rule can often be loosened in modern implementations to
+various degrees. These constraints are collectively referred to as the ACID consistency model [42].
+NoSQL is an umbrella term for several data models, typically developed or popularized in the first decade of the
+2000s [38]. Contrary to the relational model, the data models within NoSQL typically have no formal definitions, and
+different NoSQL DBMSs implement different data models such as key-value (e.g., Redis), document (e.g., MongoDB),
+Manuscript
+
+4
+Taipalus
+wide-column (e.g., Cassandra) and graph (e.g., Neo4J) [22, 71]. Furthermore, these DBMSs often have a distinct query
+language developed to cater to the particular data structures available in the DBMS’s implementation of a data model.
+While RDBMSs have favored data consistency [9] by eliminating redundant data through logical database design, and
+through a strong consistency model, NoSQL DBMSs have generally adopted the opposite approach. In several NoSQL
+data models such as key-value pairs and documents, redundant data is stored at the cost of storage space [43]. This
+approach enables query languages to be simple [25], avoiding complex and potentially slow queries. Furthermore,
+consistency models are typically less strict than in RDBMSs [73], which facilitates higher performance demanded by,
+e.g., web applications with a large number of concurrent users [70].
+Although NoSQL DBMSs popularized several database-related approaches such as non-strict database structures,
+data availability over data consistency, and relatively effortless database replication (i.e., data is copied over computing
+nodes) and sharding (i.e., data is divided between computing nodes) [38], some industry leaders such as Google deemed
+a strong consistency model and an expressive query language important enough to design a DBMS which incorporates
+features from both RDBMSs and NoSQL DBMSs [19]. These so-called NewSQL DBMSs use the relational model, often
+with extensions, SQL as their primary query language, and a distributed database architecture [64]. In addition to
+these three main categories of RDBMS, NoSQL, and NewSQL data models, others such as object stores [148] and
+GPU-intensive [190] systems are used in specific contexts.
+2.3
+Query Execution
+The word query typically refers to query language statements that retrieve some data from the database. However, in
+this study, we use the word query to refer to any data retrieval or manipulation statement for brevity. In times it is
+necessary to differentiate between data retrieval or manipulation, we use appropriate terms such as read operations for
+data retrieval, and write operations for data insertion, updates, and deletes. In this subsection, we describe how queries
+are executed, using mainly general (i.e., not specific to a single DBMS) literature from the domain of RDBMS query
+execution.
+When a user — were it a human actor directly using a terminal, a transaction processing software application, or a
+database benchmark software — submits a query to a DBMS, a multitude of events must take place before the user
+receives feedback. Illustrated in a general fashion in Fig. 1, the query parser checks, among other things, that the query is
+syntactically valid [44]. If the query passes these (and other) checks, the query is translated to a lower-level presentation
+and passed to the query optimizer. The optimizer generates one or several query execution plans. These plans consist
+of physical operators for implementing, e.g., which physical data structures will be utilized in executing the query, and
+in RDBMSs in particular, how tables are joined together [36]. If several plans are generated, the optimizer evaluates
+which of these plans is the most effective in regards to, e.g., query execution time [44]. The accuracy of the optimizer
+relies on aspects such as database metadata [12], statistics of previous query executions, and the indices available [10].
+Generating effective query execution plans is a complex effort and takes time [10, 36], but once formulated, the plans
+can be re-used to a degree.
+Next, the query execution engine implements the query execution plan, using the physical operators therein.
+Simplified, the data objects required by the query are typically first searched from a memory area called the buffer pool
+which is allocated and maintained by the DBMS. If some or all data is not found, the data is requested from disk. Before
+accessing the disk, many systems may additionally utilize other areas of memory to avoid disk access [86].
+Effectively all database systems function in an environment where multiple concurrent end-users use the database.
+This concurrency presents challenges particularly when the users execute write operations on the same database, e.g.,
+Manuscript
+
+Database Management System Performance Comparisons
+5
+when two or more users withdraw money from the same bank account, concurrently updating the balance [5]. To
+guarantee that the write operations do not interfere with each other in a way that would cause the data to not represent
+the real world, DBMSs typically implement concurrency control through locking or versioning data. Effectively, the
+simpler implementations of locking restrict data objects or larger data structures from being accessed by other operations
+while the data objects are being modified [44]. These locking mechanisms may be implemented to ensure that no
+anomalies happen, or with implementations that theoretically allow some anomalies [4]. Typically, the business domain
+dictates what types of anomalies are tolerated.
+Finally, as strong consistency models often require that transactions persist in the database and that either all or
+none of the operations in a transaction either succeed or fail, locking is typically complemented by transaction logs.
+These logs are written before write operations are committed to the database, and can be used in reversing earlier write
+operations if a later write operation in the same transaction fails. All these considerations discussed in this section play
+a significant role from a performance perspective, which is discussed in the next section.
+Further reading on database systems: for readers interested in the basics of database systems, either the un-
+dergraduate level textbook by Connolly and Begg [17], or Elmasri and Navathe [31] are excellent albeit lengthy
+introductions covering the topic from several points of view and with the focus on RDBMSs. For readers interested in
+query processing, we point to studies by Chaudhury [10], and Hellerstein, Stonebraker and Hamilton [44]. If you are
+interested in logical relational database design, the book by Date [21] is an in-depth resource covering both formal and
+informal approaches. For a survey of literature on NoSQL data models, the study by Davoudian, Chen and Liu [22] is an
+accessible starting point.
+3
+PERFORMANCE
+3.1
+Performance Measurement
+In general, performance is a measurement of how efficiently a software system completes its tasks. Performance is
+typically measured in response time, throughput [44], or in some cases, utilization of computing resources [20, p.4].
+Response time is the time taken for a call in the system to traverse to some other part of the system and back. This is also
+sometimes called latency [39, p.10], and in the context of database systems, the response time may be measured as the
+response time to the first or the last result item [36]. In a broad perspective described in Fig. 1, the response time might
+be the time taken after the end-user sends a request to the software application (e.g., an online store), which passes
+the request to a DBMS, which returns a set of data to the software application, which finally presents the data to the
+end-user’s device. In database benchmarking, however, response time might be measured by running the benchmark on
+the same device the DBMS and the database reside, effectively eliminating inter-device-induced performance drawbacks
+such as network latency [23, 62] and firewalls, and mitigating the effects of other software running on the devices.
+Although DBMSs perform other tasks besides querying, querying is typically what is measured in DBMS performance
+testing [26]. While response time is perhaps the least arduous performance metric to measure, it is not often enough for
+reliable measurement of transaction processing environments [26] (often dubbed online transaction processing, OLTP).
+That is, response time might be a metric better suited for long-running queries in decision support environments (often
+dubbed online analytical processing, OLAP), but as transaction processing environments often process a large number
+of concurrent transactions, response time alone might not reliably account for the effects of concurrent transactions,
+unless response time is measured as an average of multiple concurrent transactions.
+Manuscript
+
+6
+Taipalus
+Performance can also be measured by throughput, i.e., how many transactions the DBMS can execute in a given time
+frame. Throughput is often expressed as transactions per second [26] and requires a more sophisticated approach, e.g.,
+benchmarking software. Again, throughput may be measured either locally (i.e., using only the hardware the DBMS
+and the database reside on), or over a network in case the database is distributed. Alternatively, throughput may be
+measured by connecting the benchmarking software to the software application, which simulates the throughput of the
+whole database system by accounting for, e.g., network and the software application [e.g., 54, 75]. Such an approach
+arguably requires significantly more investment, but provides a holistic perspective on the performance of the whole
+system, also uncovering potential performance issues unrelated to the DBMS and the database. Finally, performance
+may be measured by resource utilization, either CPU time, I/O, memory allocation, or energy consumption [36] in
+systems striving for energy-efficiency due to, e.g., limited battery power, or due to environmental concerns [40].
+In summary, we might consider the measurement of throughput a process that typically requires a simulation of
+some level, and the measurement of response time as an exact or approximated mathematical method. The former
+approach requires relatively high investments into the development of such simulations [20, p.142], while the latter
+often relies on a set of assumptions that do not necessarily reflect real-world scenarios due to inaccuracies in predicting
+what the real-world scenario ultimately is and how it can change.
+3.2
+Factors Affecting Performance
+Hardware: An intuitive factor in performance is the power of hardware [60, p.1], and while it is true that most of the
+local response time is attributed to time taken by CPU processing, memory and disk access, and software waiting for
+other tasks to complete [20, p.5], first investing in software performance rather than hardware performance is often
+more cost-effective. That being said, it is generally accepted that memory access is at least four orders of magnitude
+faster than disk access [e.g., 39, p.42]. That is, if memory access takes minutes (nanoseconds), disk access takes months
+(milliseconds). These numbers are largely dependent on the speed of memory and the type of disk, but paint a picture of
+how zealously DBMS optimization strives to minimize disk access. Since memory is typically more expensive than disk
+storage, keeping the whole database in memory is often not feasible. Additionally, the underlying hardware is important,
+as, e.g., some DBMSs have been shown to utilize multi-processor or multi-core environments more effectively than
+others [81]. Intuitively, how well a DBMS can exploit parallelism affects the performance of query execution [76, 80].
+Ultimately, performance measurement is about gains or losses in percentages, not in, e.g., response times.
+Data models: Data models described in Section 2.2 have indirect effects on DBMS performance. Relational databases
+often follow design guidelines that strive to minimize redundancy to eliminate potential data anomalies caused by
+redundant data [15, 16], and to minimize the need for storage space, which in turn typically causes queries to run
+slower due to a larger number of table joins. In contrast, different NoSQL data models — especially key-value, document,
+and wide-column — follow design guidelines according to which data structures are designed to effectively satisfy
+predetermined business logic queries, with the elimination of redundant data being a secondary concern [22]. It follows
+that because many NoSQL data structures are designed to serve queries, queries are typically simple [25], and their
+execution requires less computational resources than complex queries in relational databases. As discussed in Section 2.3,
+locking data objects (both on disk and in memory, and both primary data structures as well as indices), logging write
+operations, and how memory is managed by the DBMS all play a significant role in DBMS performance [44, 73]. For
+example, preventing write operation-induced anomalies is a costly action, and the level of granularity of database locks
+presents significant considerations on write operation performance, which is largely dictated by the ratio of read and
+write operations.
+Manuscript
+
+Database Management System Performance Comparisons
+7
+Distribution: Write operations in distributed configurations pose non-trivial challenges to both performance and
+data consistency [23]. In distributed database systems, effectively all transactions must choose either data consistency
+or data availability [7, 35]. The former guarantees that the data the end-user receives are not stale, with the cost of
+performance, while the latter guarantees to a degree that the end-user receives data faster, but with no guarantees that
+the dataset received is the most recent. The preferred approach is largely dictated by business logic.
+DBMS and OS parameters: Moving from data models and database system distribution to lower levels of abstraction,
+operating system (OS) and DBMS parameters and their interrelationships (e.g., page size) can have direct or indirect
+effects on performance [26]. Additionally, DBMS parameters such as the amount of memory the DBMS is allowed
+to use for data processing is typically closely related to the amount of memory available. Furthermore, as a query is
+sent to the optimizer (cf. Fig. 1), it depends on the DBMS internals how efficiently the optimizer can select the most
+efficient physical operations to implement the query, and what physical operations are available to the optimizer in
+the first place [10]. For example, MySQL implemented only one physical operation for table joins until 20181, limiting
+the number of options the optimizer could choose from. Regarding query optimization, the optimizers of RDBMSs in
+particular are relatively mature and can spot some unnecessary complications in queries, while overlooking others [6].
+Despite the benefits brought by the optimizers, some queries are inherently slow and can only be optimized through
+query rewrites.
+Physical database design: Last, but definitely not least, physical database design plays a key role in DBMS performance.
+It has been argued that performance bottlenecks are difficult to find in large systems [2], and that efficiency is gained by
+focusing on the vital few areas instead of the trivial many [51, p.450]. One of the most vital areas in database systems is
+physical design. In relational databases, efficient physical design is largely achieved through indices, and in NoSQL
+databases, typically through database distribution over computing nodes. In contrast to a holistic system overview,
+performance bottlenecks may be easier to find in queries, since many DBMSs provide detailed information on query
+execution (Fig. 2). PostgreSQL (Fig. 2a) lists the physical operations used to execute the query, which of the operations
+took the most time units, and which indices, if any, were used. For example, it can be seen in Fig. 2a that the sequential
+scan on line 12 accounted for approximately 94% of the execution time of the whole query (178 time units out of 189
+ms), probably because the query fetched a large number of records from the database. The query could be optimized by,
+e.g., selecting a smaller number of records, and showing the results to the end-user by paging them, i.e., showing a
+subset of results first, and fetching more later if necessary. In NoSQL systems, the query optimizer plays a smaller role
+due to typically less expressive query languages (cf. Fig. 2b). Some NoSQL systems such as Cassandra do not permit the
+execution of queries that do not utilize the physical structures effectively.
+3.3
+Database Performance Benchmarks
+There are several database performance benchmarks available, each typically consisting of a sample database and a
+workload that simulates how the database could be used [27, 68]. The benchmarks usually measure the efficiency of
+querying while taking into account factors such as concurrency but disregarding other DBMS tasks such as efficiency
+in data structure definition or bulk loading [26].
+In the domain of relational databases, the Transaction Processing Council (TPC) benchmarks [e.g., 37] are perhaps
+the most utilized [30, 80], and test the throughput of the DBMS with various parameters. For example, the TPC-A
+benchmark simulates a database of a bank with four tables and with one transaction, the TPC-B benchmark a database
+1https://dev.mysql.com/doc/refman/5.6/en/explain-output.html
+Manuscript
+
+8
+Taipalus
+1
+QUERY PLAN
+2
+---------------------------------------------------------------------------------------------------
+3
+Hash Join (cost=3802.67..5996.69 rows=469 width=67) (actual time=183.354..189.409 rows=473 loops=1)
+4
+Hash Cond: (o.customerid = c.customerid)
+5
+-> Bitmap Heap Scan on orders o (cost=167.89..2341.57 rows=7753 width=52) (actual time=1.446..5.581 rows=7925 loops=1)
+6
+Recheck Cond: (((paymenttype)::text = 'OC'::text) AND (orderdate > to_date('20101010'::text, 'YYYYMMDD'::text)))
+7
+Heap Blocks: exact=1772
+8
+-> Bitmap Index Scan on ord_paymenttype_orderdate (cost=0.00..165.95 rows=7753 width=0) (actual time=1.182..1.182 rows=7925 loops=1)
+9
+Index Cond: (((paymenttype)::text = 'OC'::text) AND (orderdate > to_date('20101010'::text, 'YYYYMMDD'::text)))
+10
+-> Hash (cost=3491.49..3491.49 rows=11463 width=15) (actual time=181.848..181.848 rows=11464 loops=1)
+11
+Buckets: 16384 Batches: 1 Memory Usage: 672kB
+12
+-> Seq Scan on customers c (cost=0.00..3491.49 rows=11463 width=15) (actual time=0.031..178.834 rows=11464 loops=1)
+13
+Filter: ((firstname)::text ~~* 'ma%'::text)
+14
+Rows Removed by Filter: 178095
+15
+Planning Time: 1.964 ms
+16
+Execution Time: 189.543 ms
+17
+(14 rows)
+(a) PostgreSQL query execution plan shows the time units anticipated and taken by each phase of the query execution
+1
+Tracing session: 4e5c44b0-6714-11ed-97f4-0597e908e9d9
+2
+activity
+| source
+| source_elapsed
+3
+------------------------------------------------------------------------+-----------+---------------
+4
+Execute CQL3 query
+| 127.0.0.1 |
+0
+5
+Parsing SELECT * FROM movies_by_tag; [Native-Transport-Requests-1]
+| 127.0.0.1 |
+1795
+6
+Preparing statement [Native-Transport-Requests-1]
+| 127.0.0.1 |
+2003
+7
+Computing ranges to query [Native-Transport-Requests-1]
+| 127.0.0.1 |
+2308
+8
+Submitting range requests on 17 ranges with a concurrency
+|
+|
+9
+of 7 (14.4 rows per range expected) [Native-Transport-Requests-1]
+| 127.0.0.1 |
+2562
+10
+Executing seq scan across 1 sstables for
+|
+|
+11
+(min(-9223372036854775808), min(-9223372036854775808)] [ReadStage-2]
+| 127.0.0.1 |
+7768
+12
+Submitted 1 concurrent range requests [Native-Transport-Requests-1]
+| 127.0.0.1 |
+8006
+13
+Read 3 live rows and 0 tombstone cells [ReadStage-2]
+| 127.0.0.1 |
+21272
+14
+Request complete
+| 127.0.0.1 |
+21747
+(b) Cassandra query execution plan; some output has been omitted for brevity
+Fig. 2. Query execution plans illustrating the physical operators (some bolded) chosen by the optimizer
+of a wholesale supplier with nine tables and with five transactions, and the TPC-E benchmark a brokerage database with
+33 tables and 12 transactions. All these benchmarks have the option for simulating strong consistency, and while TPC-A
+and TPC-B have transactions typical for transaction processing, TPC-E includes also decision support transactions [80].
+TPC-A simulates human end-user thinking by waiting between transactions, as a human arguably would wait between
+clicks in an online bank. TPC-B, on the other hand, does not wait and can be used as a precursor for TPC-A in adjusting
+DBMS parameters [26]. Alternatively to transaction processing, TPC-H benchmark measures the performance of a
+DBMS in decision support [3, 30].
+In the more general DBMS domain, the Yahoo! Cloud Serving Benchmark (YCSB) is a framework for benchmarking
+transaction processing in systems with different data models and architectures [18]. Due to its extensibility, YCSB can be
+adapted to different NoSQL data models. YCSB contains different workloads, each with a different ratio of read and write
+operations. YCSB and its extensions such as YCSB+T typically utilize transactions which consist of single operations
+and do not enforce strong consistency [25, 68]. The benchmarks described above are by no means an exhaustive list but
+cover the most popular benchmarks (cf. Section 2.1). Other benchmarks include LUBM [41], OLTP-Bench [27], and
+JOB [55]. Regardless of the data model and DBMS, transaction processing benchmarks have typically been the de facto
+method of comparing different DBMSs and hardware [80].
+Further reading on performance: for readers interested in physical database operations and query execution
+from a performance perspective, Graefe [36] provides an in-depth, DBMS-independent survey. For more information on
+Manuscript
+
+Database Management System Performance Comparisons
+9
+ACM DL
+IEEE Xplore
+ScienceDirect
+Exclusion based on title
+Exclusion based on full text
+Backward snowballing
+Exclusion based on full text
+Backward snowballing
+Exclusion based on full text
+Complementary
+Google Scholar search
+Backward snowballing
+Final selection = 117
+1,545
+1,969
+2,534
+124
+87 (-37)
+125 (+38)
+113 (-12)
+119 (+6)
+113 (-6)
+117 (+4)
+117 (+0)
+Fig. 3. The study selection process; the numbers refer to the number of primary studies selected in each stage of the process
+Table 1. Search strings
+Database
+Search string
+ACM DL
+[Abstract: performance] AND [Abstract: comparison] AND [[Abstract: database] OR [Abstract:
+dbms]] AND [Publication Date: (01/01/2000 TO 03/31/2022)]
+IEEE Xplore
+("Abstract":performance AND "Abstract":comparison AND ("Abstract":database OR "Ab-
+stract":dbms))
+ScienceDirect
+Title, abstract, keywords: performance AND comparison AND (database OR dbms)
+Google Scholar
+database performance comparison
+physical database design, especially indices and how they work, the book by Lightstone, Teorey and Nadeau [56] is a
+detailed and descriptive source. For a practical and concise guide on SQL query optimization, we point readers towards
+Winand’s book [85]. Regarding NoSQL DBMS optimization, we suggest referring to the manual of the DBMS of your
+choice, and always making sure that the source of information is current, as NoSQL systems tend to evolve rapidly.
+4
+STUDY SELECTION
+4.1
+Process and Criteria
+The DBMSs in this study were selected based on the selected primary studies. That is, we did not choose, e.g., the
+most popular DBMSs to include, but reported the DBMSs yielded by the primary studies. The results herein may
+be considered the most popular DBMSs in terms of benchmarking reported in scientific studies. Fig. 3 describes the
+primary study selection process starting from ACM Digital Library, IEEE Xplore, and ScienceDirect, complemented
+by subsequent Google Scholar searches. The search strings are detailed in Table 1. To account for potentially missing
+relevant studies, we conducted three rounds of backward snowballing (i.e., following the lists of references in selected
+studies), until snowballing revealed no additional studies. A total of 117 primary studies comparing DBMS performance
+were selected.
+Manuscript
+
+10
+Taipalus
+Table 2. Primary study selection criteria
+#
+Inclusion criterion
+1
+Article is written in English.
+2
+Full article can be accessed.
+3
+Article is published in a scientific journal, or conference or workshop proceedings.
+4
+Article is published between 2000 and March 2022.
+5
+Article focus is on query language statement execution performance comparison.
+6
+Article focus is on comparing the performance of two or more different DBMSs.
+7
+Article is based on at least seemingly objective metrics.
+Table 3. DBMSs discussed in this study divided into four types
+DBMS type
+DBMSs
+RDBMS
+Access, Azure SQL, Interbase, DB/2, H2, Hive, MariaDB, MySQL Cluster, MySQL, Oracle Database,
+PostgreSQL, PostgresXL, SQL Server, SQLite
+NoSQL
+ArangoDB, Azure Document Database, Cassandra, Couchbase, CouchDB, Elasticsearch, Firebase,
+HBase, Hypertable, memcached, MongoDB, Neo4J, Oracle NoSQL, OrientDB, RavenDB, Redis,
+RethinkDB, Riak, Scalaris, Tarantool, Voldemort
+NewSQL
+CockroachDB, MemSQL (now known as SingleStoreDB), NuoDB, VoltDB
+Other
+BlazingSQL, Caché, Db4o, OmniSciDB, PG-Strom
+Table 2 describes our inclusion criteria applied in the primary study selection. The first four criteria are related to
+bibliographic details, while the last three criteria are concerned with article focus and content. Regarding criterion #3,
+we excluded academic theses and dissertations [e.g., 13] due to the fact that they are typically not peer-reviewed. We
+also excluded white and gray literature for the same reason, and because those studies are often written or published by
+partial parties, e.g., DBMS vendors.
+We only selected studies that compared query (i.e., retrieving or modifying data) execution performance, not regarding
+e.g., database replication performance [32] or performance of different join operations [53]. We also excluded studies
+that compared a single DBMS performance in different configurations such as hardware, replication strategy, database
+structure, or query language [45] and studies that compared a DBMS with different data-related platforms [65]. Studies
+that reported pseudonymized DBMS names were also excluded. Finally, we only included studies that reported results
+based on at least seemingly objective metrics and empirical results. That is, studies simply stating the opinions of the
+authors such as “based on our experiences, we believe MySQL is faster than SQL Server” were not considered.
+4.2
+Selected Studies
+The selected 117 primary studies compared the performance of a total of 44 different DBMSs. We categorized these
+DBMSs into three top-level types defined and discussed in Section 2.2: RDBMSs, NoSQL systems, and NewSQL systems.
+Five DBMSs not clearly pertaining to any of these three categories were categorized under other systems (Table 3). It
+is worth noting that these DBMS types are not always clear-cut due to the lack of specificity and changing nature of
+the definitions, and should be interpreted as merely means to compartmentalize the results of this study into a more
+readable form. Five selected primary studies did not report results implying the performance of one DBMS over another
+[118, 138, 151, 168, 182].
+Manuscript
+
+Database Management System Performance Comparisons
+11
+Fig. 4 shows the distribution of publication years and the types of DBMSs discussed in the selected studies. Although
+our criteria allowed for studies from the year 2000, the first studies selected were published in 2008. The figure shows
+that generally, there is a somewhat constant number of DBMS performance comparison studies each year. It is worth
+noting that one study may pertain to several types of DBMSs.
+2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022*
+0
+10
+20
+30
+RDBMS
+NoSQL
+NewSQL
+Other
+Fig. 4. The number of publications by publication year and DBMS type; the year 2022 was only considered until March
+5
+PERFORMANCE COMPARISON RESULTS
+The most popular DBMS performance comparisons compared one or several RDBMSs to one or several NoSQL systems,
+one NoSQL system to another NoSQL system, or one RDBMS to another RDBMS, respectively. A total of 48 studies
+compared solely read performance, while 6 studies compared solely write performance. The rest of the studies compared
+both read and write performance, with the exception of two studies [111, 164] which were unclear whether they
+compared write operations. All comparisons and their results per DBMS type are summarized in Fig. 5.
+Fig. 6 presents an overview of which DBMSs and DBMS types the primary studies compared. The figure perhaps
+conveys how both other and NewSQL systems are typically compared within their respective DBMS type groups, while
+RDBMS and NoSQL systems are both compared within their respective groups as well as with each other. Additionally,
+the size of the nodes such as MongoDB, Redis, Cassandra, and MySQL show that these DBMSs typically outperform
+the DBMSs they are compared to. Due to their length, the detailed results from the primary study comparisons are
+presented in Appendix A, which includes tables detailing which DBMSs outperformed which.
+Regarding the benchmarks defined in earlier scientific literature, the most popular was YCSB, which was utilized by
+15 primary studies (approximately 13%) [90–93, 102, 125, 134, 142, 147, 174, 183, 185, 191, 192, 201]. The second most
+popular benchmark was the TPC-H benchmark and its variations, utilized by five primary studies (4%) [122, 167, 190, 196?
+]. It is worth noting, though, that two of the studies [167, 196] seemed to have executed the queries of TPC-H, instead of
+running the benchmark and accounting for, e.g., the effects of concurrent transactions. One primary study utilized the
+OLTP-Bench benchmark [193], one the LUBM benchmark [124], and one, in addition to TPC-H, the JOB benchmark
+Manuscript
+
+12
+Taipalus
+Other
+RDBMS
+NoSQL
+NewSQL
+r:1
+r:1, w:3
+r:18, w:12
+r:2, w:1
+r:1, w:1
+r:15, w:9
+r:55, w:32
+r:1
+r:1, w:1
+r:37, w:30
+r:2, w:2
+r:2, w:2
+r:3, w:2
+Fig. 5. DBMS performance comparisons overview; a directed edge from node a to node b represents the number of studies according
+to which a system of type a outperformed a system or systems of type b in (r)ead and (w)rite operations, e.g., a NoSQL system
+outperformed a NewSQL system in read operations in one study, and in write operations in one study; thicker edges visualize the
+most popular comparisons
+[190]. Regarding the benchmarks formulated by the primary study authors, 25 primary studies (21%) reported using
+ad hoc queries instead of earlier defined benchmarks to compare the performance of DBMSs. These queries were
+defined verbatim in the primary studies. In contrast, 70 of the primary studies (59%) compared DBMS performance
+using undisclosed ad hoc queries, likely formulated by the study authors. In other words, 22 primary studies (19%) used
+some type of earlier defined database benchmarking suite. The performance tests of these 22 primary studies and what
+aspects of the environment they reported are detailed in Table 4.
+6
+DISCUSSION
+6.1
+General Discussion
+The difficulty of rigorous performance testing is perhaps one of the root causes of why optimization is difficult, and
+several studies have highlighted the complexity of performance testing due to, e.g., the effects of DBMS parameters [66],
+testing environment settings [83], and how well the data in the performance test database reflects the real application
+data [67]. Is it also important whether an impartial actor has carried out the performance test, or whether the test
+results are published e.g., by a DBMS vendor [24]. However, this is sometimes difficult to assess and can be mitigated by
+simply explicitly reporting the test so that it can be replicated and verified by others.
+Despite the fact that we were aware of some DBMS performance comparison studies as they have been touched on in
+previous works, we were surprised by the extent the few examples presented in the previous works [69, 83] generalize
+to so many studies on the subject. For example, in read operations, MongoDB outperforms Cassandra according to
+ten studies, Cassandra outperforms Redis according to four studies, and Redis outperforms MongoDB according to
+six studies (cf. Appendix A), leading to a situation of 𝑀𝑜 > 𝐶𝑎 > 𝑅𝑒 > 𝑀𝑜, where MongoDB is both the best and the
+worst performing DBMS. Furthermore, as discussed in Section 5, few of the selected studies reported the test setting in
+enough detail for replication. Unfortunately, without sufficient details for replicating an experiment, such experimental
+Manuscript
+
+Database Management System Performance Comparisons
+13
+Access
+ArangoDB
+Azure DDB
+Azure SQL
+BlazingSQL
+Interbase
+Cassandra
+CockroachDB
+Couchbase
+CouchDB
+DB/2
+DB4o
+Elasticsearch
+Firebase
+H2
+HBase
+Hive
+Hypertable
+MariaDB
+memcached
+MemSQL
+MongoDB
+MySQL Cl.
+MySQL
+Neo4J
+NuoDB
+OmniSciDB
+Oracle 
+NoSQL
+Oracle
+OrientDB
+PG-Strom
+PostgreSQL
+PostgresXL
+RavenDB
+Redis
+RethinkDB
+Riak
+Scalaris
+SQL Server
+SQLite
+Tarantool
+Voldemort
+VoltDB
+Caché
+Fig. 6. An overview of read operation performance comparisons between NoSQL systems (green, upper right), NewSQL systems
+(yellow, lower right), RDBMSs (red, lower left), and other systems (blue, upper left); a clockwise turning edge from node a to
+node b depicts node a outperforming node b, and the color of the edged corresponds to the type of the outperforming node, e.g.,
+Caché outperforms PostgreSQL according to one or several studies; the size of a node represents out-degree, i.e., larger nodes have
+outperformed more systems than smaller nodes
+Manuscript
+
+14
+Taipalus
+Table 4. An overview of primary studies using previously defined benchmark software and which aspects of the testing environment
+they explicitly disclosed; performance measurements abbreviated as ET (execution time) and TP (throughput); 1the YCSB benchmark
+defines a single-table with n columns (or loose equivalents in non-relational data models)
+Study
+Explicitly reported
+Benchmark
+Measurement
+Nodes
+DBMS versions
+Hardware
+DB structure
+DBMS parameters
+[90]
+yes
+yes
+no1
+no
+YCSB
+ET
+1
+[92]
+yes
+yes
+no1
+no
+YCSB
+ET
+1
+[91]
+yes
+yes
+no1
+no
+YCSB
+ET
+1
+[93]
+no
+no
+no1
+no
+YCSB
+ET
+1
+[99]
+no
+yes
+logical only
+no
+Star Schema Benchmark
+ET
+1
+[102]
+yes
+yes
+no1
+no
+YCSB
+ET, TP
+2
+[122]
+no
+yes
+logical only
+no
+TPC-H database with custom queries
+ET
+5
+[124]
+yes
+yes
+no
+no
+LUBM-based
+ET
+9
+[125]
+no
+yes
+no1
+no
+YCSB
+ET, TP
+2-9
+[134]
+yes
+yes
+no1
+no
+YCSB
+ET, TP
+8
+[142]
+yes
+yes
+no1
+no
+YCSB
+ET, TP
+up to 5
+[147]
+yes
+no
+no1
+no
+YCSB
+ET, TP
+9
+[167]
+no
+yes
+logical only
+no
+TPC-H
+ET
+1
+[174]
+yes
+yes
+no1
+no
+YCSB
+ET, TP
+16 and 24
+[183]
+no
+yes
+no1
+yes (default)
+YCSB, Voter
+ET, TP
+3
+[185]
+yes
+yes
+no1
+no
+YCSB
+ET
+1
+[190]
+yes
+yes
+no1
+yes (default)
+TPC-H
+ET
+3
+[191]
+yes
+yes
+no1
+no
+YCSB
+TP
+up to 14
+[192]
+yes
+yes
+no1
+no
+YCSB
+ET, TP
+4
+[193]
+yes
+yes
+logical only
+no
+Sysbench, OLTP-Bench
+TP
+1
+[196]
+no
+yes
+logical only
+no
+TPC-H
+ET
+1
+[201]
+yes
+yes
+no1
+no
+YCSB
+ET, TP
+1
+results can claim any outcome [69]. One aspect that was typically reported was some details about the hardware the
+test was run on, i.e., processor make and model, clock rate, memory size, and disk size. Without other details about the
+DBMS parameters, parallel execution, etc., these details are inconsequential. Despite the importance of the topic of
+DBMS performance comparisons, with the exception of one study [174], no primary studies were published in major
+data management fora such as ACM SIGMOD or VLDB.
+6.2
+Considerations for Industry
+6.2.1
+Consider the Environments in Performance Testing Studies. If the environment in which the performance testing
+was carried out does not provide sufficient details, whatever the study states, you may interpret the results as if they do
+not generalize to other environments. That is, if you are in the process of deciding on a DBMS for your application,
+or perhaps considering changing one DBMS to another, consider whether the performance comparison study you
+are reading presents a similar use case. Compare your business domain to that presented in performance comparison
+studies, remembering that a single, sometimes even a seemingly inconsequential parameter (cf. e.g., data types in
+SQLite [66]) may change the results. DeWitt and Levine [24] aptly describe performance comparisons as the maximum
+potential performance gain of one DBMS over another. The performance gain in your particular environment might be
+less, or it might be that the DBMS that performed better in the comparison performs worse in your environment.
+One important aspect of the environment is the physical setup. Different hardware has been shown to affect DBMS
+performance, as some DBMSs exploit parallelism more efficiently than others [48, 58], effectively meaning that if a test
+was performed on one single-core CPU, the results might not generalize to distributed environments. Additionally,
+different hardware aspects such as the relative sizes of different CPU memory caches may significantly affect DBMS
+performance, making performance comparisons between different hardware a complex task [1, 83]. In distributed
+Manuscript
+
+Database Management System Performance Comparisons
+15
+environments, which were rarely tested in the primary studies, it is worth considering whether data availability
+is prioritized over data consistency, as the latter setup is typically significantly slower. Benchmarks that simulate
+concurrent users should also be considered separately from performance tests that merely execute queries sequentially.
+Concurrency introduces several challenges, many of which severely affect performance [83]. For example, SQLite uses
+database locking on a level of granularity which makes concurrent writes slow, but this has no negative effects on
+single-user writes [59]. Unfortunately, some studies have shown that developers do not widely understand concurrency-
+related security aspects [84], and that concurrency-related performance problems are understudied [87]. Some have
+even stated that the research has not been focusing on relevant issues [63].
+Intuitively, different business domains have different databases and they are used in different ways. For example,
+in some domains, the end-users typically read data, while in others, write operations are more common. The ratio
+of read and write operations in a performance test plays a crucial role, as some DBMSs are specifically designed for
+specific workloads [18]. The credibility of testing results is also related to how well the test database and data therein
+represent the target environment [67]. Furthermore, in business domains such as online stores, there are typically
+popular products, and thus the data related to them are targets of a relatively large number of database operations.
+For generalizable benchmarking results, the benchmark must account for such skewness in database use, rather than,
+e.g., randomly querying data objects. It is also worth considering how the performance tests have tested performance.
+For example, is your application about inserting 10,000 rows in bulk, but one row at a time randomly generated by
+the application? If it is not, you should not consider this type of benchmark results as an indication of how well one
+DBMS performs compared to another in your particular business context. It is also worth considering that decision
+support benchmarks such as TPC-H test performance in environments that can be fundamentally different from
+transaction processing environments. Finally, even similar business domains can have a myriad of different technical
+implementations.
+We have discussed some of the particulars involved in database system design in this subsection, and in Sections 2
+and 3, from which one can infer what has often been repeated in database system research: the environments and their
+optimization is a task so complex [18, 36] that DBMS optimization is a whole profession [69]. It follows that there are
+several threats to rigorous DBMS benchmarking. Even though RDBMS optimization is widely and deeply studied in
+both academic and industry contexts, RDBMS optimization remains a complex task. In the domain of NoSQL DBMSs,
+there exist far fewer scientific studies simply due to the age of the NoSQL DBMSs, and the heterogeneity of NoSQL
+data models. Additionally, there are several querying anti-patterns to avoid, such as performing joins in the software
+application instead of the DBMS, or paging query results by utilizing ordering and limiting. All these points considered,
+a reader of a performance comparison study must trust that the performance comparison study writers have been able
+to optimize the database systems to a similar degree for the performance comparison results to be credible. This requires
+particularly specific, in-depth expertise when DBMSs with more than one data model are compared. Furthermore,
+decades of benchmarking software development by entire councils (e.g., TPC) cannot simply be skipped by writing a
+set of (often arbitrary) queries, running them on two or more DBMSs in a single-user environment, recording response
+times, and consequently stating that one DBMS is faster than another. Although this was the case in over 80% of the
+selected primary studies, we do not consider this sufficient.
+In summary, if it is possible that changing even one of the environmental aspects discussed above may affect the
+performance test results significantly, it seems reasonable to argue that, no matter how many DBMS performance
+comparison studies state that one DBMS outperforms another, these DBMSs were not tested in an environment that is
+Manuscript
+
+16
+Taipalus
+the same as your environment, and thus have little concern in the decision of which DBMS is performance-wise the
+best fit for your environment.
+6.2.2
+Consider Other Aspects Besides Performance. There are other aspects besides response time or throughput to
+consider when choosing a DBMS. Performance gains, such as those provided by many NoSQL systems, rely heavily
+on redundant data to minimize the complexity of queries, thus providing faster response times. Naturally, storing
+redundant data increases the cost of storage, and may lead to data inconsistencies. Another comparison perspective is
+related to the features provided by the DBMSs compared. Intuitively, a DBMS that is tailored for a specific purpose
+outperforms a general-purpose DBMS [69, 74]. For example, one primary study [104] noted that while MongoDB
+outperformed PostgreSQL/PostGIS in most of the tests, MongoDB provides only a subset of the geospatial operations
+provided by PostGIS. If the rest of the operations needed by the business domain need to be implemented in the software
+application, it is not realistic to assume that such task is either trivial to implement, nor trivial to implement in a way
+that outperforms the solutions offered by existing DBMS features.
+Another consideration is the availability of suitable workforce, which is closely related to the DBMS technology and
+its maturity. It is not surprising that as query languages such as SQL have been a topic of effectively all information
+technology-related curricula in higher education for several years [50, 78], there is a relatively large number of
+professionals fluent in SQL, as opposed to new query languages. Some studies have also shown that strong consistency
+models [19] and the SQL language [8] are desired as skills as well as features in a DBMS. That is, it is worth considering
+how feasible it is to implement a database system with each specific technology, and DBMS performance is only one of
+the important aspects to consider.
+Finally, as the primary studies typically considered performance in terms of response time or throughput, we have
+approached the topic from a similar viewpoint. However, as discussed in Section 3.1, performance may be measured by
+the usage of computing resources, which can be a goal conflicting with response time [10]. It is typical that increasing
+parallelism through multiple CPUs lowers response time, but increases the total amount of work due to the parallelism
+overhead [60, p.13]. Finally, it has been shown that migrating data from one DBMS to another is all but trivial, and
+prone to fail due to a lack of clear methodologies [77] — especially when the DBMSs differ in data models and query
+languages [52]. Therefore, migrations such as RDBMS ⇔ RDBMS or RDBMS ⇔ NewSQL are arguably less complex
+than migrations such as NoSQL ⇔ NoSQL, RDBMS ⇔ NoSQL or NewSQL ⇔ NoSQL.
+6.3
+Considerations for Research
+6.3.1
+Consider Using Existing Guidelines for Testing and Reporting. Database benchmarking guidelines are not a novel
+invention in database system research and have been described in detail [37] and in short [26] in the early 1990s, and
+as a reader-friendly checklist later [69]. Additionally, benchmarking pitfalls have been discussed in numerous studies
+in respected database systems fora [30, 83]. Based on the primary studies, however, neither of these lines of research
+has been widely applied in practice. Database benchmarking has been argued to be difficult [69], as environmental
+parameters such as the nature of data [67], DBMS parameters [83], and data types [66] can all have significant impacts
+on performance testing results. Furthermore, benchmarking tools have received critique [38, 71] despite the fact that
+some of the tools have been under development for decades. Therefore, we urge researchers, at the very least, to
+consider whether using a performance test suite of one’s ad hoc queries is credible when well-known performance
+benchmarks are freely available.
+Manuscript
+
+Database Management System Performance Comparisons
+17
+As for reporting, Raasveldt et al. [69] provide a 24-point checklist for fair benchmarking. Some of the points are
+concerned about how performance is tested, and others about how the testing is reported. A performance comparison
+that cannot be replicated may present whatever results [69]. Furthermore, an empirical study without reproducible
+evidence should be considered an opinion of the authors, rather than an empirical study. Indeed, at the start of the
+NoSQL movement, we have witnessed several studies with high praise for the strengths of different NoSQL products,
+yet with little or no critical notions addressing the acknowledged shortcoming of such DBMSs. Therefore, we would
+caution the reader from inferring from these results that one DBMS performs better than another. Rather, each such
+argument should be carefully scrutinized and interpreted in a specific context, like in the primary study assessing the
+performance of GPU DBMSs [190], in which performance between DBMSs was compared, but the comparison was
+merely one aspect of the study.
+6.3.2
+Consider a Different Approach to DBMS-DBMS Testing. Especially for a junior researcher, comparing the perfor-
+mance of one DBMS to another may seem like a relatively simple research setting to both carry out, and also justify
+based on the prevalence of the DBMS industry. We hope that the arguments presented in previous studies as well as here
+have highlighted that neither of these points are as clear-cut. Following the guidelines [e.g., 69] can make performance
+testing a time-consuming task, and in many cases, perhaps overly time-consuming, and given the considerations on the
+generalizability of the results, the results may not be of interest in other environments. Alternatively, not following
+guidelines introduces significant threats to validity. While generalizability is hardly an intrinsic value, concluding that,
+e.g., MySQL outperforms PostgreSQL in “my webstore” but not in others unless they have similar data, hardware,
+number of end-users, etc., does not carry the implication of being as a scientifically impactful result as saying that, e.g.,
+MySQL will always outperform PostgreSQL. Therefore, we must either perform the performance comparisons with
+rigor and accept that the results do not probably generalize, or perform the comparisons without scientific rigor and
+state sophisms. Since the latter is hardly ethically sound, DBMS performance comparisons should be limited to domains
+where the goal of a study is not the generalizability of the results, but the betterment of the very particular domain the
+study is concerned [e.g., 100].
+Given the arguments above, we propose that future studies, if inter-DBMS performance must be compared, consider
+taking a different approach to performance testing. First, using a wide range of database system optimization experts to
+ensure that all aspects of the system are fairly optimized, and avoiding situations where one system is optimized beyond
+diminishing returns, while the other is barely optimized. We challenge research teams to explicitly disclose which
+authors optimized which systems, for authors to further one’s intellectual investments in the performance comparison.
+These solutions should be benchmarked by a party independent of both optimization teams, and fair benchmarking
+guidelines should be utilized. Second, after the benchmarking has been carried out, we urge researchers to consider what
+causes the differences in performance, and critically compare those aspects as well, as gains in performance arguably
+have root causes such as loosened consistency or increased storage space. Nonetheless, performance comparisons of two
+or more DBMS with different data models should be considered particularly complex. Unfortunately, such comparisons
+seem to be the most popular (cf. Fig. 5).
+6.3.3
+Consider Other Use Cases Besides DBMS-DBMS Testing Altogether. It is worth noting that benchmarking software
+has other use cases besides inter-DBMS performance comparisons. Instead of comparing one DBMS to another,
+researchers might consider testing the performance effects of different hardware [28], DBMS parameters [83], operating
+system parameters, query languages [45], physical configurations such as database distribution, physical structures
+such as different indices, or different levels of data consistency.
+Manuscript
+
+18
+Taipalus
+6.4
+Limitations and Threats to Validity
+It might be that some relevant studies are missing from this survey. However, it was not our intention to select primary
+studies to quantitatively demonstrate that one DBMS outperforms another by the number of studies corroborating
+such an argument. Rather, the results verify previous observations [69] according to which many of such comparisons
+are problematic and should be interpreted with care, if at all. Nevertheless, we have strived to include at least most of
+the primary studies that fit our criteria (Table 2) by several rounds of snowballing (Fig. 3) as well as a complementary
+literature search. Furthermore, as the DBMS classification (Table 3) and the interpretation of the primary study results
+(Appendix A) involve human judgment, it is possible that another group of researchers may attain at least slightly
+different results.
+7
+CONCLUSION
+Several database management system performance comparisons have been conducted and published as both vendor
+white-papers as well as in scientific fora. The approaches and reporting in such studies have been criticized in previous
+literature. In this study, we systematically surveyed 117 DBMS performance comparison studies. What seemed to be
+common among the selected primary studies is that they lack sufficient detail for reproducibility. Scientific, peer-reviewed
+research of high external validity concerning database management performance comparison is effectively scarce. Based
+on the survey, we presented several considerations for the industry as well as database system researchers. Namely,
+we argued for considering (i) the environments (i.e., business domain, amount of data, amount of concurrent users,
+hardware, database distribution, read/write operation ratio, etc.) when interpreting the results of DBMS performance
+comparison tests, and for considering (ii) other aspects besides DBMS performance when choosing a DBMS or changing
+one DBMS to another, and for researchers to consider (iii) using existing guidelines in performance testing and reporting
+the testing environments transparently, to consider (iv), different approaches to performance testing when one DBMS
+is compared to another, and to consider (v) other use cases for performance testing besides comparing the performance
+of one DBMS to another. The results highlight how rarely benchmarking software is used in performance testing,
+how often different DBMSs with data models are compared with each other, how often performance testing results in
+different studies conflict with each other, and why. This study is not an attempt to argue the performance gains of one
+DBMS over another using primary studies. That is, please do not cite this study by consulting the Appendix and stating
+that DBMS1 outperforms DBMS2.
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+
+Database Management System Performance Comparisons
+27
+A
+DETAILED COMPARISON RESULTS
+Table 5. DBMS performance comparisons in read operations; a table cell shows that according to the study or studies cited, the
+DBMS in the row outperformed the DBMS in the column, e.g., CouchDB outperformed ArangoDB [162], or ArangoDB outperformed
+CouchDB [127]
+Access
+ArangoDB
+Azure DDB
+Azure SQL DB
+BlazingSQL
+Caché
+Interbase
+Cassandra
+CockroachDB
+CouchDB
+Couchbase
+DB/2
+Db4o
+Elasticsearch
+Firebase
+H2
+Access
+ArangoDB
+[127]
+Azure DDB
+[103]
+Azure SQL DB
+BlazingSQL
+Caché
+Interbase
+Cassandra
+[141]
+CockroachDB
+CouchDB
+[162]
+[155]
+Couchbase
+[155, 192]
+[155]
+DB/2
+[106]
+Db4o
+Elasticsearch
+[91, 157]
+Firebase
+H2
+HBase
+[125, 191,
+192]
+Hive
+Hypertable
+[155]
+[155]
+MariaDB
+memcached
+MemSQL
+[143, 183]
+MongoDB
+[162]
+[90, 102, 123,
+125, 155, 158,
+185, 192, 195,
+204]
+[127, 149,
+155, 162,
+194]
+[113, 171,
+194]
+[93]
+[141]
+MySQL Cluster
+MySQL
+[106, 146]
+[174]
+[129]
+[178]
+Neo4J
+[161]
+NuoDB
+[143, 183]
+OmniSciDB
+Oracle DB
+[106, 146]
+[131]
+Oracle NoSQL
+OrientDB
+[93]
+PG-Strom
+PostgreSQL
+[123, 195]
+PostgresXL
+RavenDB
+Redis
+[91, 92, 174,
+185, 192]
+[171]
+[91]
+[141]
+RethinkDB
+Riak
+Scalaris
+[91]
+[91]
+SQL Server
+[106]
+[155, 180]
+[155]
+SQLite
+Tarantool
+[91]
+[91]
+Voldemort
+[174]
+VoltDB
+[143, 183]
+Manuscript
+
+28
+Taipalus
+Table 5. (cont.)
+HBase
+Hive
+Hypertable
+MariaDB
+memcached
+MemSQL
+MongoDB
+MySQL Cluster
+MySQL
+Neo4J
+NuoDB
+OmniSciDB
+Access
+ArangoDB
+[127]
+Azure DDB
+Azure SQL DB
+BlazingSQL
+Caché
+Interbase
+[94]
+Cassandra
+[91, 92, 133, 134,
+142, 174]
+[141]
+[90, 92, 102, 105,
+109, 134, 141,
+147, 161, 180,
+191]
+[105, 109, 161,
+163, 204]
+CockroachDB
+CouchDB
+[194]
+Couchbase
+[192]
+[155]
+[155, 181]
+[194]
+DB/2
+[106]
+Db4o
+[148]
+Elasticsearch
+[91]
+[91]
+[184]
+Firebase
+[166]
+H2
+HBase
+[191, 201]
+[124]
+[116, 154, 177,
+201]
+Hive
+Hypertable
+MariaDB
+memcached
+MemSQL
+[183]
+MongoDB
+[91, 92, 125, 134,
+192]
+[110]
+[155]
+[141]
+[96, 100, 109,
+112, 114, 115,
+117, 119, 121,
+126, 128, 137,
+139, 150, 165,
+170, 176, 194,
+198, 201, 204]
+[121, 186, 199]
+MySQL Cluster
+MySQL
+[174]
+[96, 100, 115,
+161]
+[187]
+Neo4J
+[161]
+[98, 107, 121,
+136, 161, 197]
+NuoDB
+[143]
+OmniSciDB
+Oracle DB
+[146, 204]
+Oracle NoSQL
+[91]
+OrientDB
+[91, 92]
+PG-Strom
+PostgreSQL
+[130]
+[159, 160, 181,
+195]
+[163, 194, 196]
+[186, 199]
+PostgresXL
+[122]
+RavenDB
+Redis
+[91, 92, 174, 192]
+[141]
+[91, 92, 141, 158,
+185, 192]
+[161, 174, 204]
+[204]
+RethinkDB
+Riak
+[147]
+Scalaris
+[91]
+[91]
+SQL Server
+[180]
+[106, 179, 196,
+204? ]
+SQLite
+Tarantool
+[91]
+[91]
+Voldemort
+[91, 174]
+[174]
+VoltDB
+[174]
+[143]
+[119]
+[119]
+Manuscript
+
+Database Management System Performance Comparisons
+29
+Table 5. (cont.)
+Oracle DB
+Oracle NoSQL
+OrientDB
+PG-Strom
+PostgreSQL
+PostgresXL
+RavenDB
+Redis
+RethinkDB
+Riak
+Scalaris
+SQL Server
+SQLite
+Tarantool
+Voldemort
+VoltDB
+Access
+ArangoDB
+[156]
+Azure DDB
+Azure SQL DB
+BlazingSQL
+[190]
+Caché
+[131]
+Interbase
+Cassandra
+[105,
+204]
+[91, 92]
+[163]
+[155]
+[105,
+141, 158,
+204]
+[147]
+[105,
+204]
+[174]
+CockroachDB
+CouchDB
+[194]
+[155,
+164]
+[194]
+Couchbase
+[181,
+194]
+[155]
+[155,
+194]
+DB/2
+[106]
+[106]
+Db4o
+Elasticsearch
+[91]
+[91]
+[91]
+Firebase
+H2
+HBase
+Hive
+Hypertable
+[155]
+[155]
+MariaDB
+[173]
+memcached
+MemSQL
+[167,
+183]
+MongoDB
+[188,
+204]
+[91–93]
+[95, 104,
+111, 123,
+140, 152,
+181, 186,
+194,
+199]
+[155]
+[93, 158,
+204]
+[171]
+[89, 155,
+169, 194,
+200,
+204]
+[202]
+MySQL Cluster
+MySQL
+[172]
+[101,
+172, 175,
+189]
+[194]
+[174]
+Neo4J
+[144,
+145,
+153]
+[202]
+NuoDB
+[143]
+OmniSciDB
+[190]
+Oracle DB
+[172]
+Oracle NoSQL
+[91]
+OrientDB
+[93]
+PG-Strom
+PostgreSQL
+[203]
+[194,
+203]
+PostgresXL
+RavenDB
+Redis
+[91]
+[91, 92]
+[204]
+[91]
+[174]
+RethinkDB
+Riak
+Scalaris
+[91]
+[91]
+[91]
+SQL Server
+[106,
+135,
+204]
+[196]
+[155]
+SQLite
+[97]
+Tarantool
+[91]
+[91]
+[91]
+Voldemort
+[91]
+[174]
+VoltDB
+[202]
+Manuscript
+
+30
+Taipalus
+Table 6. DBMS performance comparisons in write operations; a table cell shows that according to the study or studies cited, the
+DBMS in the row outperformed the DBMS in the column; subscripts (d)elete, (i)nsert and (u)pdate refer to the operations tested;
+citations without subscripts refer either to all three write operations, or to undisclosed general write operations
+Access
+ArangoDB
+Azure DDB
+Azure SQL DB
+Caché
+Interbase
+Cassandra
+CockroachDB
+CouchDB
+Couchbase
+DB/2
+Db4o
+Elasticsearch
+Firebase
+H2
+Access
+ArangoDB
+[127, 162]
+Azure DDB
+[103]
+Azure SQL DB
+Caché
+Interbase
+Cassandra
+[155]di
+[91][157]i
+CockroachDB
+CouchDB
+[194]
+Couchbase
+[192][155]di
+[155]di
+DB/2
+[106]du
+Db4o
+Elasticsearch
+[157]du
+Firebase
+H2
+[141]
+HBase
+[125, 142,
+174, 192]
+[91]
+Hypertable
+MariaDB
+memcached
+[141]
+[141]
+MemSQL
+[183][143]iu
+MongoDB
+[127]du
+[125, 185,
+191, 192,
+195] [102,
+155, 204]di
+[204]u
+[127,
+149][155,
+162]di
+[192,
+194][171]i
+MySQL
+[106]du
+[94]i
+[174]
+[129]u
+[194]u
+[106]du
+[178]d
+Neo4J
+NuoDB
+[183][143]iu
+Oracle DB
+[106]du
+[106]du
+Oracle NoSQL
+OrientDB
+[93]iu
+PostgreSQL
+[123]
+[194]u
+RavenDB
+[155]i
+Redis
+[91, 92,
+141, 174,
+185, 192]
+[204]i
+[192]
+[91]
+[141]
+RethinkDB
+[171]i
+Riak
+Scalaris
+[91]
+SQL Server
+[106]du
+[155]d
+[155]di
+[194]du
+[106]du
+SQLite
+Tarantool
+[91]
+[91]
+Voldemort
+[174]
+VoltDB
+[183][143]iu
+Manuscript
+
+Database Management System Performance Comparisons
+31
+Table 6. (cont.)
+HBase
+Hypertable
+MariaDB
+memcached
+MemSQL
+MongoDB
+MySQL
+NuoDB
+Oracle DB
+Oracle NoSQL
+OrientDB
+Access
+ArangoDB
+[162]du
+Azure DDB
+Azure SQL DB
+Caché
+[131]
+Interbase
+[94]du
+Cassandra
+[132, 133,
+191][134]iu
+[155]di
+[90–92, 123, 132,
+141, 147,
+180][134]iu[102]u
+[163, 204]
+[204]
+[91]
+[91, 92]
+CockroachDB
+CouchDB
+[194]du[162]u
+[194]
+Couchbase
+[192]
+[155]di
+[155]di [171]du
+[129]d [113]i
+[194] di
+DB/2
+Db4o
+[148][178]i
+Elasticsearch
+[91]
+[91]
+[91]
+Firebase
+[166]
+H2
+[141]
+HBase
+[91, 92, 132,
+201][134]iu
+[116, 174, 201]
+[91]
+[91, 92]
+Hypertable
+MariaDB
+memcached
+[141]
+MemSQL
+[183][143]u
+MongoDB
+[191, 192]
+[155]di
+[88, 112, 114,
+115, 120, 126,
+128, 165, 176,
+204][96, 117,
+139, 170, 194,
+198, 201]i
+[108,
+204][188]di
+[91, 92]
+MySQL
+[125]
+[193]
+[119][101]di[202]i
+[204]
+Neo4J
+[153]
+NuoDB
+[143]i
+Oracle DB
+[188]u
+[106]du[172]iu
+Oracle NoSQL
+[91]
+[91]
+OrientDB
+PostgreSQL
+[123]
+[163][194]di
+[172]iu
+RavenDB
+Redis
+[192]
+[173]du
+[91, 92, 141, 185,
+192][171]u
+[174, 204]
+[204]
+[91]
+[91, 92]
+RethinkDB
+[174]
+Riak
+[147]
+Scalaris
+[91]
+[91]
+[91]
+SQL Server
+[155]d
+[180]
+[194]di[106,
+179]du
+[204][106]du
+SQLite
+Tarantool
+[91]
+[91]
+[91]
+Voldemort
+[174]
+[91]
+[174]
+[91]
+VoltDB
+[119][202]i
+[119]
+Manuscript
+
+32
+Taipalus
+Table 6. (cont.)
+PostgreSQL
+RavenDB
+Redis
+RethinkDB
+Riak
+Scalaris
+SQL Server
+SQLite
+Tarantool
+Voldemort
+VoltDB
+Access
+ArangoDB
+Azure DDB
+Azure SQL DB
+Caché
+Interbase
+Cassandra
+[163, 195]
+[155]di
+[204]di
+[147]
+[91]
+[204][155]i
+[91]
+[174]
+CockroachDB
+CouchDB
+[194]
+[194]
+Couchbase
+[194]d
+[155]di
+[171]du
+[155]di[194]i
+DB/2
+Db4o
+Elasticsearch
+[91]
+Firebase
+H2
+HBase
+[91]
+[91]
+[174]
+Hypertable
+[155]di
+[155]di
+MariaDB
+memcached
+[141]
+MemSQL
+[183][143]iu
+MongoDB
+[140, 194, 195]
+[155]di
+[204]
+[171]di
+[89, 194,
+204][155]di[169]u
+[202]
+MySQL
+[194]u
+[204][179]i[194]u
+[202]
+[174]
+Neo4J
+NuoDB
+[143]iu
+Oracle DB
+[172]iu
+Oracle NoSQL
+[91]
+OrientDB
+[93]iu
+PostgreSQL
+[194]i
+RavenDB
+Redis
+[91]
+[204]
+[91, 174]
+[174]
+RethinkDB
+Riak
+Scalaris
+[91]
+SQL Server
+[194]d
+[155]di
+SQLite
+Tarantool
+[91]
+[91]
+Voldemort
+[174]
+VoltDB
+[202]
+Manuscript
+
diff --git a/W9AzT4oBgHgl3EQfKPvT/content/tmp_files/load_file.txt b/W9AzT4oBgHgl3EQfKPvT/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fcdca6e37cc9acf322a364810ae1c4776565b752
--- /dev/null
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@@ -0,0 +1,2035 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf,len=2034
+page_content='Database Management System Performance Comparisons: A Systematic Survey  TONI TAIPALUS, University of Jyväskylä, Finland  Efficiency has been a pivotal aspect of the software industry since its inception, as a system that serves the end-user fast, and the  service provider cost-efficiently benefits all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' A database management system (DBMS) is an integral part of effectively all  software systems, and therefore it is logical that different studies have compared the performance of different DBMSs in hopes of  finding the most efficient one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' This survey systematically synthesizes the results and approaches of studies that compare DBMS  performance and provides recommendations for industry and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The results show that performance is usually tested in a way  that does not reflect real-world use cases, and that tests are typically reported in insufficient detail for replication or for drawing  conclusions from the stated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  CCS Concepts: • Information systems → Data management systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Information retrieval query processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' • General and  reference → Surveys and overviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Additional Key Words and Phrases: performance, database management system, database, comparison, survey  Toni Taipalus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Database Management System Performance Comparisons: A Systematic Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1, 1 (January 2023), 32 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='XXXXXXX  1  INTRODUCTION  Efficiency is important in effectively all software systems, whether efficiency is measured by response times, how many  concurrent users the system can serve, or how energy-efficient the system is [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Despite its importance, many software  systems suffer from efficiency problems [49], as optimization has been largely recognized as a complex task [27, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The  more a system holds and handles data, the more the system’s performance depends on the database, and the database  is often one of the first suspect when a performance issue is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The domain of database management systems  (DBMS) saw rapid advancements in performance especially in the 1980s and 1990s, as benchmarking competitions  between DBMS and hardware vendors led to innovations in DBMS technology that significantly improved DBMS  performance [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Performance improvements are related to DBMS aspects such as different supporting data structures  [82], and algorithms for sorting [29, 33] and joining [61, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Given that DBMSs are annually a multi-billion dollar  industry, the performance of a DBMS is one of the most crucial aspects when a company chooses a DBMS for their  product or service [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' As different DBMS performance comparison studies and DBMS vendor white-papers highlight  the performance gains of one DBMS over another, it may seem tempting to either consider choosing the fastest DBMS  for a business domain or to migrate from one DBMS to another for performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' However, as we show and argue  in this study, performance is typically tested in very specific contexts which are not necessarily generalizable, and there  are other aspects besides performance to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  This study was inspired by a study by Raasveldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' [69], which claimed that “[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='] we will explore the common  pitfalls in database performance benchmarking that are present in a large number of scientific works [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=']” while consciously  refraining from citing example studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' While we agree with their claim based on our personal experiences, we wanted  to systematically explore whether this phenomenon is common among performance benchmarks, and whether such  Author’s address: Toni Taipalus, toni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='taipalus@jyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='fi, University of Jyväskylä, Agora, Mattilanniemi 2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Box 35, Jyväskylä, Finland, FI-40014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  © 2023  Manuscript preprint  Manuscript  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='01095v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='PF]  3 Jan 2023  2  Taipalus  studies show performance gains of one DBMS over another in a setting that can be replicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' This study is not an  attempt to criticize studies comparing DBMS performance, as no scientific study (ours included) is without threats to  validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Rather, based on the survey of the literature, the primary goals of our study are to (i) propagate information  on how DBMS performance has been tested, (ii) how performance has been recommended to be tested, (iii) how  the performance comparison results should be interpreted, (iv) what other aspects besides performance should be  considered, and (v) what other avenues might be fruitful for DBMS performance testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally, we provide (vi) a  relatively accessible background on database system performance, followed by (vii) a systematic survey of literature on  DBMS performance comparisons, (viii) describing which DBMSs and which types of DBMSs have been compared with  each other, (ix) the outcomes of the performance comparisons, and (x) by which benchmarks the DBMSs have been  compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  The rest of this study is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Sections 2 and 3, we provide theoretical background for understanding  the results and discussion provided by this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These background sections are deliberately presented by refraining  from using unnecessary information technology-related terms, acronyms, algorithms, or mathematics, to cater to the  needs of readers from various backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For readers more technically inclined or interested, we have provided  further reading at the end of Sections 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Section 4 details how we searched, selected, and categorized the DBMS  performance comparison studies, and Section 5 presents a high-level overview of the results, which is complemented  by Appendix A detailing the performance comparison outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Section 6, we discuss what these findings mean,  how they are applicable in industry, and present our recommendations for industry and research based on the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Section 7 concludes the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  2  DATABASE SYSTEMS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1  Database System Overview  A database is a collection of interrelated data, typically stored according to a data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Typically, the data is used by  one or several software applications via a DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Collectively, the database, the DBMS, and the software application are  referred to as a database system [31, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='7][17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The separation of the database and the DBMS, especially in the realm  of relational databases, is typically impossible without exporting the database in another format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In these situations, the  database is often unusable by the DBMS, unless the database is imported back to a format understood by the DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Possibly due to this inseparability, both the DBMS and the underlying database are often colloquially referred to simply  as database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is worth noting, though, that the former is a piece of software that does, while the other is a collection of  data that is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1 shows a simplified example of a system where the components crucial for a database system and the scope of this  study are emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' We refer to the components in the figure throughout this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Several things are worth noting  in considering the figure, as we have traded technical precision and comprehensiveness for ease of presentation by  depicting only a single end-user, a single software application (some parts typically reside on the end-user’s device, while  others reside on a separate server), a single DBMS, single hardware components, and a single database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore,  we have not illustrated other DBMS components such as access control, data structures such as metadata, or outputs  such as query execution plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The figure also adopts the view that the database resides in persistent storage — this is  not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally, we have depicted merely a centralized database system in which neither the DBMS  nor the database has been distributed across multiple nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These are willful omissions given the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  Database Management System Performance Comparisons  3  Parser  Optimizer  Execution  engine  Transaction  logs  Locks  DBMS  Memory  CPU  Software  application  User  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' A simplified view of a database system and the end-user with the emphasis on components relevant to this study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' the arrows  represent the flow of information from the end-user’s device to the database residing in persistent storage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' the flow of information  back to the software application is not illustrated here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' gray rectangles represent boundaries of physical devices  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2  Data Models  Databases follow one or several data models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', definitions of how and what data can be stored, and sometimes,  what operations are available for data retrieval and manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Data models may be conceptual, logical, or physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Conceptual models such as the Entity-Relationship model [11] do not dictate how data should be stored, but are rather  used to describe the interrelations and characteristics of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Logical data models such as the relational model [14]  are related to how data is stored and presented, but often without describing how the data is physically stored, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=',  which computing node is responsible for storing the data, where the data is located on a disk, and what types of indices  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', redundant data structures which facilitate query performance) and physical data retrieval operations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  One DBMS is not limited to using a single data model [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  There are several popular logical data models, some of which are inseparable from their underlying physical data  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' One of the most prominent logical data models is the relational data model rooted in set theory [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Relational  DBMSs (RDBMS) follow many of the concepts introduced in the relational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Many of the popular RDBMSs such  as PostgreSQL and Oracle Database have adopted data structures from other logical data models as well [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' What is  common for effectively all modern RDBMS is that they utilize Structured Query Language (SQL) [46, 47] to define data  structures and to retrieve and manipulate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Typically, RDBMSs also implement a strong data consistency model  which dictates or allows that database operations grouped into a transaction must all succeed or all fail, data must follow  defined business logic, successful transactions persist in storage, and concurrent transactions [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 5] must result in the  same data as if the transactions were serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' At least the last rule can often be loosened in modern implementations to  various degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These constraints are collectively referred to as the ACID consistency model [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  NoSQL is an umbrella term for several data models, typically developed or popularized in the first decade of the  2000s [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Contrary to the relational model, the data models within NoSQL typically have no formal definitions, and  different NoSQL DBMSs implement different data models such as key-value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', Redis), document (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', MongoDB),  Manuscript  4  Taipalus  wide-column (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', Cassandra) and graph (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', Neo4J) [22, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, these DBMSs often have a distinct query  language developed to cater to the particular data structures available in the DBMS’s implementation of a data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  While RDBMSs have favored data consistency [9] by eliminating redundant data through logical database design, and  through a strong consistency model, NoSQL DBMSs have generally adopted the opposite approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In several NoSQL  data models such as key-value pairs and documents, redundant data is stored at the cost of storage space [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' This  approach enables query languages to be simple [25], avoiding complex and potentially slow queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore,  consistency models are typically less strict than in RDBMSs [73], which facilitates higher performance demanded by,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', web applications with a large number of concurrent users [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Although NoSQL DBMSs popularized several database-related approaches such as non-strict database structures,  data availability over data consistency, and relatively effortless database replication (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', data is copied over computing  nodes) and sharding (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', data is divided between computing nodes) [38], some industry leaders such as Google deemed  a strong consistency model and an expressive query language important enough to design a DBMS which incorporates  features from both RDBMSs and NoSQL DBMSs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These so-called NewSQL DBMSs use the relational model, often  with extensions, SQL as their primary query language, and a distributed database architecture [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In addition to  these three main categories of RDBMS, NoSQL, and NewSQL data models, others such as object stores [148] and  GPU-intensive [190] systems are used in specific contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3  Query Execution  The word query typically refers to query language statements that retrieve some data from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' However, in  this study, we use the word query to refer to any data retrieval or manipulation statement for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In times it is  necessary to differentiate between data retrieval or manipulation, we use appropriate terms such as read operations for  data retrieval, and write operations for data insertion, updates, and deletes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In this subsection, we describe how queries  are executed, using mainly general (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', not specific to a single DBMS) literature from the domain of RDBMS query  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  When a user — were it a human actor directly using a terminal, a transaction processing software application, or a  database benchmark software — submits a query to a DBMS, a multitude of events must take place before the user  receives feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Illustrated in a general fashion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1, the query parser checks, among other things, that the query is  syntactically valid [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If the query passes these (and other) checks, the query is translated to a lower-level presentation  and passed to the query optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The optimizer generates one or several query execution plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These plans consist  of physical operators for implementing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', which physical data structures will be utilized in executing the query, and  in RDBMSs in particular, how tables are joined together [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If several plans are generated, the optimizer evaluates  which of these plans is the most effective in regards to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', query execution time [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The accuracy of the optimizer  relies on aspects such as database metadata [12], statistics of previous query executions, and the indices available [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Generating effective query execution plans is a complex effort and takes time [10, 36], but once formulated, the plans  can be re-used to a degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Next, the query execution engine implements the query execution plan, using the physical operators therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Simplified, the data objects required by the query are typically first searched from a memory area called the buffer pool  which is allocated and maintained by the DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If some or all data is not found, the data is requested from disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Before  accessing the disk, many systems may additionally utilize other areas of memory to avoid disk access [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Effectively all database systems function in an environment where multiple concurrent end-users use the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  This concurrency presents challenges particularly when the users execute write operations on the same database, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=',  Manuscript  Database Management System Performance Comparisons  5  when two or more users withdraw money from the same bank account, concurrently updating the balance [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' To  guarantee that the write operations do not interfere with each other in a way that would cause the data to not represent  the real world, DBMSs typically implement concurrency control through locking or versioning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Effectively, the  simpler implementations of locking restrict data objects or larger data structures from being accessed by other operations  while the data objects are being modified [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These locking mechanisms may be implemented to ensure that no  anomalies happen, or with implementations that theoretically allow some anomalies [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Typically, the business domain  dictates what types of anomalies are tolerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Finally, as strong consistency models often require that transactions persist in the database and that either all or  none of the operations in a transaction either succeed or fail, locking is typically complemented by transaction logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  These logs are written before write operations are committed to the database, and can be used in reversing earlier write  operations if a later write operation in the same transaction fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' All these considerations discussed in this section play  a significant role from a performance perspective, which is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Further reading on database systems: for readers interested in the basics of database systems, either the un-  dergraduate level textbook by Connolly and Begg [17], or Elmasri and Navathe [31] are excellent albeit lengthy  introductions covering the topic from several points of view and with the focus on RDBMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For readers interested in  query processing, we point to studies by Chaudhury [10], and Hellerstein, Stonebraker and Hamilton [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If you are  interested in logical relational database design, the book by Date [21] is an in-depth resource covering both formal and  informal approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For a survey of literature on NoSQL data models, the study by Davoudian, Chen and Liu [22] is an  accessible starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  3  PERFORMANCE  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1  Performance Measurement  In general, performance is a measurement of how efficiently a software system completes its tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Performance is  typically measured in response time, throughput [44], or in some cases, utilization of computing resources [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Response time is the time taken for a call in the system to traverse to some other part of the system and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' This is also  sometimes called latency [39, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='10], and in the context of database systems, the response time may be measured as the  response time to the first or the last result item [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In a broad perspective described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1, the response time might  be the time taken after the end-user sends a request to the software application (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', an online store), which passes  the request to a DBMS, which returns a set of data to the software application, which finally presents the data to the  end-user’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In database benchmarking, however, response time might be measured by running the benchmark on  the same device the DBMS and the database reside, effectively eliminating inter-device-induced performance drawbacks  such as network latency [23, 62] and firewalls, and mitigating the effects of other software running on the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Although DBMSs perform other tasks besides querying, querying is typically what is measured in DBMS performance  testing [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' While response time is perhaps the least arduous performance metric to measure, it is not often enough for  reliable measurement of transaction processing environments [26] (often dubbed online transaction processing, OLTP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  That is, response time might be a metric better suited for long-running queries in decision support environments (often  dubbed online analytical processing, OLAP), but as transaction processing environments often process a large number  of concurrent transactions, response time alone might not reliably account for the effects of concurrent transactions,  unless response time is measured as an average of multiple concurrent transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  6  Taipalus  Performance can also be measured by throughput, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', how many transactions the DBMS can execute in a given time  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Throughput is often expressed as transactions per second [26] and requires a more sophisticated approach, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=',  benchmarking software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Again, throughput may be measured either locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', using only the hardware the DBMS  and the database reside on), or over a network in case the database is distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Alternatively, throughput may be  measured by connecting the benchmarking software to the software application, which simulates the throughput of the  whole database system by accounting for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', network and the software application [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', 54, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Such an approach  arguably requires significantly more investment, but provides a holistic perspective on the performance of the whole  system, also uncovering potential performance issues unrelated to the DBMS and the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Finally, performance  may be measured by resource utilization, either CPU time, I/O, memory allocation, or energy consumption [36] in  systems striving for energy-efficiency due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', limited battery power, or due to environmental concerns [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  In summary, we might consider the measurement of throughput a process that typically requires a simulation of  some level, and the measurement of response time as an exact or approximated mathematical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The former  approach requires relatively high investments into the development of such simulations [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='142], while the latter  often relies on a set of assumptions that do not necessarily reflect real-world scenarios due to inaccuracies in predicting  what the real-world scenario ultimately is and how it can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2  Factors Affecting Performance  Hardware: An intuitive factor in performance is the power of hardware [60, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1], and while it is true that most of the  local response time is attributed to time taken by CPU processing, memory and disk access, and software waiting for  other tasks to complete [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='5], first investing in software performance rather than hardware performance is often  more cost-effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That being said, it is generally accepted that memory access is at least four orders of magnitude  faster than disk access [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', 39, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That is, if memory access takes minutes (nanoseconds), disk access takes months  (milliseconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These numbers are largely dependent on the speed of memory and the type of disk, but paint a picture of  how zealously DBMS optimization strives to minimize disk access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Since memory is typically more expensive than disk  storage, keeping the whole database in memory is often not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally, the underlying hardware is important,  as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', some DBMSs have been shown to utilize multi-processor or multi-core environments more effectively than  others [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Intuitively, how well a DBMS can exploit parallelism affects the performance of query execution [76, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Ultimately, performance measurement is about gains or losses in percentages, not in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', response times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Data models: Data models described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2 have indirect effects on DBMS performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Relational databases  often follow design guidelines that strive to minimize redundancy to eliminate potential data anomalies caused by  redundant data [15, 16], and to minimize the need for storage space, which in turn typically causes queries to run  slower due to a larger number of table joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In contrast, different NoSQL data models — especially key-value, document,  and wide-column — follow design guidelines according to which data structures are designed to effectively satisfy  predetermined business logic queries, with the elimination of redundant data being a secondary concern [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It follows  that because many NoSQL data structures are designed to serve queries, queries are typically simple [25], and their  execution requires less computational resources than complex queries in relational databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3,  locking data objects (both on disk and in memory, and both primary data structures as well as indices), logging write  operations, and how memory is managed by the DBMS all play a significant role in DBMS performance [44, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For  example, preventing write operation-induced anomalies is a costly action, and the level of granularity of database locks  presents significant considerations on write operation performance, which is largely dictated by the ratio of read and  write operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  Database Management System Performance Comparisons  7  Distribution: Write operations in distributed configurations pose non-trivial challenges to both performance and  data consistency [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In distributed database systems, effectively all transactions must choose either data consistency  or data availability [7, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The former guarantees that the data the end-user receives are not stale, with the cost of  performance, while the latter guarantees to a degree that the end-user receives data faster, but with no guarantees that  the dataset received is the most recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The preferred approach is largely dictated by business logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  DBMS and OS parameters: Moving from data models and database system distribution to lower levels of abstraction,  operating system (OS) and DBMS parameters and their interrelationships (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', page size) can have direct or indirect  effects on performance [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally, DBMS parameters such as the amount of memory the DBMS is allowed  to use for data processing is typically closely related to the amount of memory available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, as a query is  sent to the optimizer (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1), it depends on the DBMS internals how efficiently the optimizer can select the most  efficient physical operations to implement the query, and what physical operations are available to the optimizer in  the first place [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example, MySQL implemented only one physical operation for table joins until 20181, limiting  the number of options the optimizer could choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Regarding query optimization, the optimizers of RDBMSs in  particular are relatively mature and can spot some unnecessary complications in queries, while overlooking others [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Despite the benefits brought by the optimizers, some queries are inherently slow and can only be optimized through  query rewrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Physical database design: Last, but definitely not least, physical database design plays a key role in DBMS performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  It has been argued that performance bottlenecks are difficult to find in large systems [2], and that efficiency is gained by  focusing on the vital few areas instead of the trivial many [51, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='450].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' One of the most vital areas in database systems is  physical design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In relational databases, efficient physical design is largely achieved through indices, and in NoSQL  databases, typically through database distribution over computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In contrast to a holistic system overview,  performance bottlenecks may be easier to find in queries, since many DBMSs provide detailed information on query  execution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' PostgreSQL (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2a) lists the physical operations used to execute the query, which of the operations  took the most time units, and which indices, if any, were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example, it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2a that the sequential  scan on line 12 accounted for approximately 94% of the execution time of the whole query (178 time units out of 189  ms), probably because the query fetched a large number of records from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The query could be optimized by,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', selecting a smaller number of records, and showing the results to the end-user by paging them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', showing a  subset of results first, and fetching more later if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In NoSQL systems, the query optimizer plays a smaller role  due to typically less expressive query languages (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Some NoSQL systems such as Cassandra do not permit the  execution of queries that do not utilize the physical structures effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3  Database Performance Benchmarks  There are several database performance benchmarks available, each typically consisting of a sample database and a  workload that simulates how the database could be used [27, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The benchmarks usually measure the efficiency of  querying while taking into account factors such as concurrency but disregarding other DBMS tasks such as efficiency  in data structure definition or bulk loading [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  In the domain of relational databases, the Transaction Processing Council (TPC) benchmarks [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', 37] are perhaps  the most utilized [30, 80], and test the throughput of the DBMS with various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example, the TPC-A  benchmark simulates a database of a bank with four tables and with one transaction, the TPC-B benchmark a database  1https://dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content='1 |  21747  (b) Cassandra query execution plan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' some output has been omitted for brevity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Query execution plans illustrating the physical operators (some bolded) chosen by the optimizer  of a wholesale supplier with nine tables and with five transactions, and the TPC-E benchmark a brokerage database with  33 tables and 12 transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' All these benchmarks have the option for simulating strong consistency, and while TPC-A  and TPC-B have transactions typical for transaction processing, TPC-E includes also decision support transactions [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  TPC-A simulates human end-user thinking by waiting between transactions, as a human arguably would wait between  clicks in an online bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TPC-B, on the other hand, does not wait and can be used as a precursor for TPC-A in adjusting  DBMS parameters [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Alternatively to transaction processing, TPC-H benchmark measures the performance of a  DBMS in decision support [3, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  In the more general DBMS domain, the Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Cloud Serving Benchmark (YCSB) is a framework for benchmarking  transaction processing in systems with different data models and architectures [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Due to its extensibility, YCSB can be  adapted to different NoSQL data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' YCSB contains different workloads, each with a different ratio of read and write  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' YCSB and its extensions such as YCSB+T typically utilize transactions which consist of single operations  and do not enforce strong consistency [25, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The benchmarks described above are by no means an exhaustive list but  cover the most popular benchmarks (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Other benchmarks include LUBM [41], OLTP-Bench [27], and  JOB [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Regardless of the data model and DBMS, transaction processing benchmarks have typically been the de facto  method of comparing different DBMSs and hardware [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Further reading on performance: for readers interested in physical database operations and query execution  from a performance perspective, Graefe [36] provides an in-depth, DBMS-independent survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For more information on  Manuscript  Database Management System Performance Comparisons  9  ACM DL  IEEE Xplore  ScienceDirect  Exclusion based on title  Exclusion based on full text  Backward snowballing  Exclusion based on full text  Backward snowballing  Exclusion based on full text  Complementary  Google Scholar search  Backward snowballing  Final selection = 117  1,545  1,969  2,534  124  87 (-37)  125 (+38)  113 (-12)  119 (+6)  113 (-6)  117 (+4)  117 (+0)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The study selection process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' the numbers refer to the number of primary studies selected in each stage of the process  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Search strings  Database  Search string  ACM DL  [Abstract: performance] AND [Abstract: comparison] AND [[Abstract: database] OR [Abstract:  dbms]] AND [Publication Date: (01/01/2000 TO 03/31/2022)]  IEEE Xplore  ("Abstract":performance AND "Abstract":comparison AND ("Abstract":database OR "Ab-  stract":dbms))  ScienceDirect  Title,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' abstract,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' keywords: performance AND comparison AND (database OR dbms)  Google Scholar  database performance comparison  physical database design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' especially indices and how they work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' the book by Lightstone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Teorey and Nadeau [56] is a  detailed and descriptive source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For a practical and concise guide on SQL query optimization, we point readers towards  Winand’s book [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Regarding NoSQL DBMS optimization, we suggest referring to the manual of the DBMS of your  choice, and always making sure that the source of information is current, as NoSQL systems tend to evolve rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  4  STUDY SELECTION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1  Process and Criteria  The DBMSs in this study were selected based on the selected primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That is, we did not choose, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', the  most popular DBMSs to include, but reported the DBMSs yielded by the primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The results herein may  be considered the most popular DBMSs in terms of benchmarking reported in scientific studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 3 describes the  primary study selection process starting from ACM Digital Library, IEEE Xplore, and ScienceDirect, complemented  by subsequent Google Scholar searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The search strings are detailed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' To account for potentially missing  relevant studies, we conducted three rounds of backward snowballing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', following the lists of references in selected  studies), until snowballing revealed no additional studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' A total of 117 primary studies comparing DBMS performance  were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  10  Taipalus  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Primary study selection criteria  #  Inclusion criterion  1  Article is written in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  2  Full article can be accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  3  Article is published in a scientific journal, or conference or workshop proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  4  Article is published between 2000 and March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  5  Article focus is on query language statement execution performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6  Article focus is on comparing the performance of two or more different DBMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  7  Article is based on at least seemingly objective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' DBMSs discussed in this study divided into four types  DBMS type  DBMSs  RDBMS  Access,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Azure SQL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Interbase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' DB/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' H2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Hive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' MariaDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' MySQL Cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' MySQL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Oracle Database,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  PostgreSQL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' PostgresXL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' SQL Server,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' SQLite  NoSQL  ArangoDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Azure Document Database,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Cassandra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Couchbase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' CouchDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Elasticsearch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Firebase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  HBase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Hypertable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' memcached,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' MongoDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Neo4J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Oracle NoSQL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' OrientDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' RavenDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Redis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  RethinkDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Riak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Scalaris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Tarantool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Voldemort  NewSQL  CockroachDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' MemSQL (now known as SingleStoreDB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' NuoDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' VoltDB  Other  BlazingSQL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Caché,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Db4o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' OmniSciDB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' PG-Strom  Table 2 describes our inclusion criteria applied in the primary study selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The first four criteria are related to  bibliographic details, while the last three criteria are concerned with article focus and content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Regarding criterion #3,  we excluded academic theses and dissertations [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', 13] due to the fact that they are typically not peer-reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' We  also excluded white and gray literature for the same reason, and because those studies are often written or published by  partial parties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', DBMS vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  We only selected studies that compared query (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', retrieving or modifying data) execution performance, not regarding  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', database replication performance [32] or performance of different join operations [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' We also excluded studies  that compared a single DBMS performance in different configurations such as hardware, replication strategy, database  structure, or query language [45] and studies that compared a DBMS with different data-related platforms [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Studies  that reported pseudonymized DBMS names were also excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Finally, we only included studies that reported results  based on at least seemingly objective metrics and empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That is, studies simply stating the opinions of the  authors such as “based on our experiences, we believe MySQL is faster than SQL Server” were not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2  Selected Studies  The selected 117 primary studies compared the performance of a total of 44 different DBMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' We categorized these  DBMSs into three top-level types defined and discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2: RDBMSs, NoSQL systems, and NewSQL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Five DBMSs not clearly pertaining to any of these three categories were categorized under other systems (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It  is worth noting that these DBMS types are not always clear-cut due to the lack of specificity and changing nature of  the definitions, and should be interpreted as merely means to compartmentalize the results of this study into a more  readable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Five selected primary studies did not report results implying the performance of one DBMS over another  [118, 138, 151, 168, 182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  Database Management System Performance Comparisons  11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 4 shows the distribution of publication years and the types of DBMSs discussed in the selected studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Although  our criteria allowed for studies from the year 2000, the first studies selected were published in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The figure shows  that generally, there is a somewhat constant number of DBMS performance comparison studies each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is worth  noting that one study may pertain to several types of DBMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022*  0  10  20  30  RDBMS  NoSQL  NewSQL  Other  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The number of publications by publication year and DBMS type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' the year 2022 was only considered until March  5  PERFORMANCE COMPARISON RESULTS  The most popular DBMS performance comparisons compared one or several RDBMSs to one or several NoSQL systems,  one NoSQL system to another NoSQL system, or one RDBMS to another RDBMS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' A total of 48 studies  compared solely read performance, while 6 studies compared solely write performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The rest of the studies compared  both read and write performance, with the exception of two studies [111, 164] which were unclear whether they  compared write operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' All comparisons and their results per DBMS type are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 6 presents an overview of which DBMSs and DBMS types the primary studies compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The figure perhaps  conveys how both other and NewSQL systems are typically compared within their respective DBMS type groups, while  RDBMS and NoSQL systems are both compared within their respective groups as well as with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally,  the size of the nodes such as MongoDB, Redis, Cassandra, and MySQL show that these DBMSs typically outperform  the DBMSs they are compared to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Due to their length, the detailed results from the primary study comparisons are  presented in Appendix A, which includes tables detailing which DBMSs outperformed which.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Regarding the benchmarks defined in earlier scientific literature, the most popular was YCSB, which was utilized by  15 primary studies (approximately 13%) [90–93, 102, 125, 134, 142, 147, 174, 183, 185, 191, 192, 201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The second most  popular benchmark was the TPC-H benchmark and its variations, utilized by five primary studies (4%) [122, 167, 190, 196?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is worth noting, though, that two of the studies [167, 196] seemed to have executed the queries of TPC-H, instead of  running the benchmark and accounting for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', the effects of concurrent transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' One primary study utilized the  OLTP-Bench benchmark [193], one the LUBM benchmark [124], and one, in addition to TPC-H, the JOB benchmark  Manuscript  12  Taipalus  Other  RDBMS  NoSQL  NewSQL  r:1  r:1, w:3  r:18, w:12  r:2, w:1  r:1, w:1  r:15, w:9  r:55, w:32  r:1  r:1, w:1  r:37, w:30  r:2, w:2  r:2, w:2  r:3, w:2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' DBMS performance comparisons overview;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' a directed edge from node a to node b represents the number of studies according  to which a system of type a outperformed a system or systems of type b in (r)ead and (w)rite operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', a NoSQL system  outperformed a NewSQL system in read operations in one study, and in write operations in one study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' thicker edges visualize the  most popular comparisons  [190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Regarding the benchmarks formulated by the primary study authors, 25 primary studies (21%) reported using  ad hoc queries instead of earlier defined benchmarks to compare the performance of DBMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' These queries were  defined verbatim in the primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In contrast, 70 of the primary studies (59%) compared DBMS performance  using undisclosed ad hoc queries, likely formulated by the study authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In other words, 22 primary studies (19%) used  some type of earlier defined database benchmarking suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The performance tests of these 22 primary studies and what  aspects of the environment they reported are detailed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6  DISCUSSION  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1  General Discussion  The difficulty of rigorous performance testing is perhaps one of the root causes of why optimization is difficult, and  several studies have highlighted the complexity of performance testing due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', the effects of DBMS parameters [66],  testing environment settings [83], and how well the data in the performance test database reflects the real application  data [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Is it also important whether an impartial actor has carried out the performance test, or whether the test  results are published e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', by a DBMS vendor [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' However, this is sometimes difficult to assess and can be mitigated by  simply explicitly reporting the test so that it can be replicated and verified by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Despite the fact that we were aware of some DBMS performance comparison studies as they have been touched on in  previous works, we were surprised by the extent the few examples presented in the previous works [69, 83] generalize  to so many studies on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example, in read operations, MongoDB outperforms Cassandra according to  ten studies, Cassandra outperforms Redis according to four studies, and Redis outperforms MongoDB according to  six studies (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Appendix A), leading to a situation of 𝑀𝑜 > 𝐶𝑎 > 𝑅𝑒 > 𝑀𝑜, where MongoDB is both the best and the  worst performing DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, as discussed in Section 5, few of the selected studies reported the test setting in  enough detail for replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Unfortunately, without sufficient details for replicating an experiment, such experimental  Manuscript  Database Management System Performance Comparisons  13  Access  ArangoDB  Azure DDB  Azure SQL  BlazingSQL  Interbase  Cassandra  CockroachDB  Couchbase  CouchDB  DB/2  DB4o  Elasticsearch  Firebase  H2  HBase  Hive  Hypertable  MariaDB  memcached  MemSQL  MongoDB  MySQL Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  MySQL  Neo4J  NuoDB  OmniSciDB  Oracle  NoSQL  Oracle  OrientDB  PG-Strom  PostgreSQL  PostgresXL  RavenDB  Redis  RethinkDB  Riak  Scalaris  SQL Server  SQLite  Tarantool  Voldemort  VoltDB  Caché  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' An overview of read operation performance comparisons between NoSQL systems (green, upper right), NewSQL systems  (yellow, lower right), RDBMSs (red, lower left), and other systems (blue, upper left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' a clockwise turning edge from node a to  node b depicts node a outperforming node b, and the color of the edged corresponds to the type of the outperforming node, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=',  Caché outperforms PostgreSQL according to one or several studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' the size of a node represents out-degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', larger nodes have  outperformed more systems than smaller nodes  Manuscript  14  Taipalus  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' An overview of primary studies using previously defined benchmark software and which aspects of the testing environment  they explicitly disclosed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' performance measurements abbreviated as ET (execution time) and TP (throughput);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1the YCSB benchmark  defines a single-table with n columns (or loose equivalents in non-relational data models)  Study  Explicitly reported  Benchmark  Measurement  Nodes  DBMS versions  Hardware  DB structure  DBMS parameters  [90]  yes  yes  no1  no  YCSB  ET  1  [92]  yes  yes  no1  no  YCSB  ET  1  [91]  yes  yes  no1  no  YCSB  ET  1  [93]  no  no  no1  no  YCSB  ET  1  [99]  no  yes  logical only  no  Star Schema Benchmark  ET  1  [102]  yes  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  2  [122]  no  yes  logical only  no  TPC-H database with custom queries  ET  5  [124]  yes  yes  no  no  LUBM-based  ET  9  [125]  no  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  2-9  [134]  yes  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  8  [142]  yes  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  up to 5  [147]  yes  no  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  9  [167]  no  yes  logical only  no  TPC-H  ET  1  [174]  yes  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  16 and 24  [183]  no  yes  no1  yes (default)  YCSB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Voter  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  3  [185]  yes  yes  no1  no  YCSB  ET  1  [190]  yes  yes  no1  yes (default)  TPC-H  ET  3  [191]  yes  yes  no1  no  YCSB  TP  up to 14  [192]  yes  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  4  [193]  yes  yes  logical only  no  Sysbench,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' OLTP-Bench  TP  1  [196]  no  yes  logical only  no  TPC-H  ET  1  [201]  yes  yes  no1  no  YCSB  ET,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' TP  1  results can claim any outcome [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' One aspect that was typically reported was some details about the hardware the  test was run on, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', processor make and model, clock rate, memory size, and disk size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Without other details about the  DBMS parameters, parallel execution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', these details are inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Despite the importance of the topic of  DBMS performance comparisons, with the exception of one study [174], no primary studies were published in major  data management fora such as ACM SIGMOD or VLDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2  Considerations for Industry  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1  Consider the Environments in Performance Testing Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If the environment in which the performance testing  was carried out does not provide sufficient details, whatever the study states, you may interpret the results as if they do  not generalize to other environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That is, if you are in the process of deciding on a DBMS for your application,  or perhaps considering changing one DBMS to another, consider whether the performance comparison study you  are reading presents a similar use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Compare your business domain to that presented in performance comparison  studies, remembering that a single, sometimes even a seemingly inconsequential parameter (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', data types in  SQLite [66]) may change the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' DeWitt and Levine [24] aptly describe performance comparisons as the maximum  potential performance gain of one DBMS over another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The performance gain in your particular environment might be  less, or it might be that the DBMS that performed better in the comparison performs worse in your environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  One important aspect of the environment is the physical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Different hardware has been shown to affect DBMS  performance, as some DBMSs exploit parallelism more efficiently than others [48, 58], effectively meaning that if a test  was performed on one single-core CPU, the results might not generalize to distributed environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally,  different hardware aspects such as the relative sizes of different CPU memory caches may significantly affect DBMS  performance, making performance comparisons between different hardware a complex task [1, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In distributed  Manuscript  Database Management System Performance Comparisons  15  environments, which were rarely tested in the primary studies, it is worth considering whether data availability  is prioritized over data consistency, as the latter setup is typically significantly slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Benchmarks that simulate  concurrent users should also be considered separately from performance tests that merely execute queries sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Concurrency introduces several challenges, many of which severely affect performance [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example, SQLite uses  database locking on a level of granularity which makes concurrent writes slow, but this has no negative effects on  single-user writes [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Unfortunately, some studies have shown that developers do not widely understand concurrency-  related security aspects [84], and that concurrency-related performance problems are understudied [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Some have  even stated that the research has not been focusing on relevant issues [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Intuitively, different business domains have different databases and they are used in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example,  in some domains, the end-users typically read data, while in others, write operations are more common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The ratio  of read and write operations in a performance test plays a crucial role, as some DBMSs are specifically designed for  specific workloads [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The credibility of testing results is also related to how well the test database and data therein  represent the target environment [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, in business domains such as online stores, there are typically  popular products, and thus the data related to them are targets of a relatively large number of database operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  For generalizable benchmarking results, the benchmark must account for such skewness in database use, rather than,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', randomly querying data objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is also worth considering how the performance tests have tested performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  For example, is your application about inserting 10,000 rows in bulk, but one row at a time randomly generated by  the application?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If it is not, you should not consider this type of benchmark results as an indication of how well one  DBMS performs compared to another in your particular business context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is also worth considering that decision  support benchmarks such as TPC-H test performance in environments that can be fundamentally different from  transaction processing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Finally, even similar business domains can have a myriad of different technical  implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  We have discussed some of the particulars involved in database system design in this subsection, and in Sections 2  and 3, from which one can infer what has often been repeated in database system research: the environments and their  optimization is a task so complex [18, 36] that DBMS optimization is a whole profession [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It follows that there are  several threats to rigorous DBMS benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Even though RDBMS optimization is widely and deeply studied in  both academic and industry contexts, RDBMS optimization remains a complex task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In the domain of NoSQL DBMSs,  there exist far fewer scientific studies simply due to the age of the NoSQL DBMSs, and the heterogeneity of NoSQL  data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally, there are several querying anti-patterns to avoid, such as performing joins in the software  application instead of the DBMS, or paging query results by utilizing ordering and limiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' All these points considered,  a reader of a performance comparison study must trust that the performance comparison study writers have been able  to optimize the database systems to a similar degree for the performance comparison results to be credible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' This requires  particularly specific, in-depth expertise when DBMSs with more than one data model are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore,  decades of benchmarking software development by entire councils (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', TPC) cannot simply be skipped by writing a  set of (often arbitrary) queries, running them on two or more DBMSs in a single-user environment, recording response  times, and consequently stating that one DBMS is faster than another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Although this was the case in over 80% of the  selected primary studies, we do not consider this sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' if it is possible that changing even one of the environmental aspects discussed above may affect the  performance test results significantly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' it seems reasonable to argue that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' no matter how many DBMS performance  comparison studies state that one DBMS outperforms another,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' these DBMSs were not tested in an environment that is  Manuscript  16  Taipalus  the same as your environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' and thus have little concern in the decision of which DBMS is performance-wise the  best fit for your environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2  Consider Other Aspects Besides Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' There are other aspects besides response time or throughput to  consider when choosing a DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Performance gains, such as those provided by many NoSQL systems, rely heavily  on redundant data to minimize the complexity of queries, thus providing faster response times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Naturally, storing  redundant data increases the cost of storage, and may lead to data inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Another comparison perspective is  related to the features provided by the DBMSs compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Intuitively, a DBMS that is tailored for a specific purpose  outperforms a general-purpose DBMS [69, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' For example, one primary study [104] noted that while MongoDB  outperformed PostgreSQL/PostGIS in most of the tests, MongoDB provides only a subset of the geospatial operations  provided by PostGIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' If the rest of the operations needed by the business domain need to be implemented in the software  application, it is not realistic to assume that such task is either trivial to implement, nor trivial to implement in a way  that outperforms the solutions offered by existing DBMS features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Another consideration is the availability of suitable workforce, which is closely related to the DBMS technology and  its maturity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is not surprising that as query languages such as SQL have been a topic of effectively all information  technology-related curricula in higher education for several years [50, 78], there is a relatively large number of  professionals fluent in SQL, as opposed to new query languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Some studies have also shown that strong consistency  models [19] and the SQL language [8] are desired as skills as well as features in a DBMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That is, it is worth considering  how feasible it is to implement a database system with each specific technology, and DBMS performance is only one of  the important aspects to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Finally, as the primary studies typically considered performance in terms of response time or throughput, we have  approached the topic from a similar viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' However, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1, performance may be measured by  the usage of computing resources, which can be a goal conflicting with response time [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is typical that increasing  parallelism through multiple CPUs lowers response time, but increases the total amount of work due to the parallelism  overhead [60, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Finally, it has been shown that migrating data from one DBMS to another is all but trivial, and  prone to fail due to a lack of clear methodologies [77] — especially when the DBMSs differ in data models and query  languages [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Therefore, migrations such as RDBMS ⇔ RDBMS or RDBMS ⇔ NewSQL are arguably less complex  than migrations such as NoSQL ⇔ NoSQL, RDBMS ⇔ NoSQL or NewSQL ⇔ NoSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3  Considerations for Research  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1  Consider Using Existing Guidelines for Testing and Reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Database benchmarking guidelines are not a novel  invention in database system research and have been described in detail [37] and in short [26] in the early 1990s, and  as a reader-friendly checklist later [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Additionally, benchmarking pitfalls have been discussed in numerous studies  in respected database systems fora [30, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Based on the primary studies, however, neither of these lines of research  has been widely applied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Database benchmarking has been argued to be difficult [69], as environmental  parameters such as the nature of data [67], DBMS parameters [83], and data types [66] can all have significant impacts  on performance testing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, benchmarking tools have received critique [38, 71] despite the fact that  some of the tools have been under development for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Therefore, we urge researchers, at the very least, to  consider whether using a performance test suite of one’s ad hoc queries is credible when well-known performance  benchmarks are freely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  Database Management System Performance Comparisons  17  As for reporting, Raasveldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' [69] provide a 24-point checklist for fair benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Some of the points are  concerned about how performance is tested, and others about how the testing is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' A performance comparison  that cannot be replicated may present whatever results [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, an empirical study without reproducible  evidence should be considered an opinion of the authors, rather than an empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Indeed, at the start of the  NoSQL movement, we have witnessed several studies with high praise for the strengths of different NoSQL products,  yet with little or no critical notions addressing the acknowledged shortcoming of such DBMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Therefore, we would  caution the reader from inferring from these results that one DBMS performs better than another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Rather, each such  argument should be carefully scrutinized and interpreted in a specific context, like in the primary study assessing the  performance of GPU DBMSs [190], in which performance between DBMSs was compared, but the comparison was  merely one aspect of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2  Consider a Different Approach to DBMS-DBMS Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Especially for a junior researcher, comparing the perfor-  mance of one DBMS to another may seem like a relatively simple research setting to both carry out, and also justify  based on the prevalence of the DBMS industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' We hope that the arguments presented in previous studies as well as here  have highlighted that neither of these points are as clear-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Following the guidelines [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', 69] can make performance  testing a time-consuming task, and in many cases, perhaps overly time-consuming, and given the considerations on the  generalizability of the results, the results may not be of interest in other environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Alternatively, not following  guidelines introduces significant threats to validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' While generalizability is hardly an intrinsic value, concluding that,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', MySQL outperforms PostgreSQL in “my webstore” but not in others unless they have similar data, hardware,  number of end-users, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', does not carry the implication of being as a scientifically impactful result as saying that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=',  MySQL will always outperform PostgreSQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Therefore, we must either perform the performance comparisons with  rigor and accept that the results do not probably generalize, or perform the comparisons without scientific rigor and  state sophisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Since the latter is hardly ethically sound, DBMS performance comparisons should be limited to domains  where the goal of a study is not the generalizability of the results, but the betterment of the very particular domain the  study is concerned [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Given the arguments above, we propose that future studies, if inter-DBMS performance must be compared, consider  taking a different approach to performance testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' First, using a wide range of database system optimization experts to  ensure that all aspects of the system are fairly optimized, and avoiding situations where one system is optimized beyond  diminishing returns, while the other is barely optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' We challenge research teams to explicitly disclose which  authors optimized which systems, for authors to further one’s intellectual investments in the performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  These solutions should be benchmarked by a party independent of both optimization teams, and fair benchmarking  guidelines should be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Second, after the benchmarking has been carried out, we urge researchers to consider what  causes the differences in performance, and critically compare those aspects as well, as gains in performance arguably  have root causes such as loosened consistency or increased storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Nonetheless, performance comparisons of two  or more DBMS with different data models should be considered particularly complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Unfortunately, such comparisons  seem to be the most popular (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='3  Consider Other Use Cases Besides DBMS-DBMS Testing Altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' It is worth noting that benchmarking software  has other use cases besides inter-DBMS performance comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Instead of comparing one DBMS to another,  researchers might consider testing the performance effects of different hardware [28], DBMS parameters [83], operating  system parameters, query languages [45], physical configurations such as database distribution, physical structures  such as different indices, or different levels of data consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  18  Taipalus  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='4  Limitations and Threats to Validity  It might be that some relevant studies are missing from this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' However, it was not our intention to select primary  studies to quantitatively demonstrate that one DBMS outperforms another by the number of studies corroborating  such an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Rather, the results verify previous observations [69] according to which many of such comparisons  are problematic and should be interpreted with care, if at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Nevertheless, we have strived to include at least most of  the primary studies that fit our criteria (Table 2) by several rounds of snowballing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 3) as well as a complementary  literature search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Furthermore, as the DBMS classification (Table 3) and the interpretation of the primary study results  (Appendix A) involve human judgment, it is possible that another group of researchers may attain at least slightly  different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  7  CONCLUSION  Several database management system performance comparisons have been conducted and published as both vendor  white-papers as well as in scientific fora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The approaches and reporting in such studies have been criticized in previous  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In this study, we systematically surveyed 117 DBMS performance comparison studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' What seemed to be  common among the selected primary studies is that they lack sufficient detail for reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Scientific, peer-reviewed  research of high external validity concerning database management performance comparison is effectively scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Based  on the survey, we presented several considerations for the industry as well as database system researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Namely,  we argued for considering (i) the environments (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=', business domain, amount of data, amount of concurrent users,  hardware, database distribution, read/write operation ratio, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=') when interpreting the results of DBMS performance  comparison tests,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' and for considering (ii) other aspects besides DBMS performance when choosing a DBMS or changing  one DBMS to another,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' and for researchers to consider (iii) using existing guidelines in performance testing and reporting  the testing environments transparently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' to consider (iv),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' different approaches to performance testing when one DBMS  is compared to another,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' and to consider (v) other use cases for performance testing besides comparing the performance  of one DBMS to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The results highlight how rarely benchmarking software is used in performance testing,  how often different DBMSs with data models are compared with each other, how often performance testing results in  different studies conflict with each other, and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' This study is not an attempt to argue the performance gains of one  DBMS over another using primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' That is, please do not cite this study by consulting the Appendix and stating  that DBMS1 outperforms DBMS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' DBMSs on a modern processor: Where does time go?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' In New  Contributions in Information Systems and Technologies, Alvaro Rocha, Ana Maria Correia, Sandra Costanzo, and Luis Paulo Reis (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' Association for  Computing Machinery, New York, NY, USA, 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Concurrency Control in Distributed Database Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content='356846  [6] Stefan Brass and Christian Goldberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Semantic errors in SQL queries: A quite complete list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Journal of Systems and Software 79, 5 (2006),  630–644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content='028 Quality Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  Database Management System Performance Comparisons  19  [7] Eric Brewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' CAP twelve years later: How the "rules" have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Computer 45, 2 (2012), 23–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content='1109/MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='37  [8] Stephen Cass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' SQL Should Be Your Second Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' IEEE Spectrum 59, 10 (2022), 20–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='1109/MSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='9915547  [9] Natalia Chaudhry and Muhammad Murtaza Yousaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' In Proceedings of the Seventeenth ACM SIGACT-SIGMOD-SIGART  Symposium on Principles of Database Systems (Seattle, Washington, USA) (PODS ’98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Association for Computing Machinery, New York, NY, USA,  34–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' Comparing the Performance of Open Source and Proprietary Relational Database Management Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' On the State of NoSQL Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Proceedings of the 8th  ACM/SPEC on International Conference on Performance Engineering Companion (L’Aquila, Italy) (ICPE ’17 Companion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Association for Computing  Machinery, New York, NY, USA, 107–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' DeWitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' A Performance Evaluation of Four Parallel Join Algorithms in a Shared-Nothing Multiprocessor  Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Proceedings of the 1989 ACM SIGMOD International Conference on Management of Data (Portland, Oregon, USA) (SIGMOD ’89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' ACM 53, 4 (2010), 10–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' One size fits all?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Part 2: Benchmarking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' CIDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' Mellor-Crummey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' Open Journal of Databases (OJDB),  1(2):17–24, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' In 2021  International Russian Automation Conference (RusAutoCon), pages 683–687, Sochi, Russian Federation, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content='  [197] Chad Vicknair, Michael Macias, Zhendong Zhao, Xiaofei Nan, Yixin Chen, and Dawn Wilkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' In Proceedings of the 48th Annual Southeast Regional Conference on - ACM SE ’10, page 1, Oxford, Mississippi,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' ACM Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [198] Zhu Wei-ping, Li Ming-xin, and Chen Huan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Using MongoDB to implement textbook management system instead of MySQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In 2011 IEEE 3rd  International Conference on Communication Software and Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' IEEE, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [199] Linggis Galih Wiseso, Mahmud Imrona, and Andry Alamsyah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Performance analysis of Neo4j, MongoDB, and PostgreSQL on 2019 national  election big data management database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In 2020 6th International Conference on Science in Information Technology (ICSITech).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [200] Wei Xu, Zhonghua Zhou, Hong Zhou, Wu Zhang, and Jiang Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' MongoDB improves big data analysis performance on electric health record  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Communications in Computer and Information Science, pages 350–357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Springer Berlin Heidelberg, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [201] Amal W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Yassien and Amr F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Desouky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' RDBMS, NoSQL, Hadoop: A Performance-Based Empirical Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In Proceedings of the 2nd Africa and  Middle East Conference on Software Engineering - AMECSE ’16, pages 52–59, Cairo, Egypt, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' ACM Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [202] Yinfeng Wang, Guiquan Zhong, Lin Kun, Longxiang Wang, Huang Kai, Fuliang Guo, Chengzhe Liu, and Xiaoshe Dong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' The Performance Survey  of in Memory Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In 2015 IEEE 21st International Conference on Parallel and Distributed Systems (ICPADS), pages 815–820, Melbourne, VIC,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [203] Zhonghai Zhou, Bin Zhou, Wenwen Li, Brian Griglak, Carmen Caiseda, and Qunying Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Evaluating query performance on object-relational  spatial databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' In 2009 2nd IEEE International Conference on Computer Science and Information Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' IEEE, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  [204] Roman Čerešňák and Michal Kvet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Comparison of query performance in relational a non-relation databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' Transportation Research Procedia,  40:170–177, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='  Manuscript  Database Management System Performance Comparisons  27  A  DETAILED COMPARISON RESULTS  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' DBMS performance comparisons in read operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' a table cell shows that according to the study or studies cited, the  DBMS in the row outperformed the DBMS in the column, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' CouchDB outperformed ArangoDB [162],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' or ArangoDB outperformed  CouchDB [127]  Access  ArangoDB  Azure DDB  Azure SQL DB  BlazingSQL  Caché  Interbase  Cassandra  CockroachDB  CouchDB  Couchbase  DB/2  Db4o  Elasticsearch  Firebase  H2  Access  ArangoDB  [127]  Azure DDB  [103]  Azure SQL DB  BlazingSQL  Caché  Interbase  Cassandra  [141]  CockroachDB  CouchDB  [162]  [155]  Couchbase  [155,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 192]  [155]  DB/2  [106]  Db4o  Elasticsearch  [91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' DBMS performance comparisons in write operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' a table cell shows that according to the study or studies cited, the  DBMS in the row outperformed the DBMS in the column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' subscripts (d)elete, (i)nsert and (u)pdate refer to the operations tested;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+page_content=' or to undisclosed general write operations  Access  ArangoDB  Azure DDB  Azure SQL DB  Caché  Interbase  Cassandra  CockroachDB  CouchDB  Couchbase  DB/2  Db4o  Elasticsearch  Firebase  H2  Access  ArangoDB  [127,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
+page_content=' 162]  Azure DDB  [103]  Azure SQL DB  Caché  Interbase  Cassandra  [155]di  [91][157]i  CockroachDB  CouchDB  [194]  Couchbase  [192][155]di  [155]di  DB/2  [106]du  Db4o  Elasticsearch  [157]du  Firebase  H2  [141]  HBase  [125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfKPvT/content/2301.01095v1.pdf'}
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+Robust web element identification for evolving
+applications by considering visual overlaps
+Michel Nass, Emil Al´egroth, Robert Feldt
+SERL
+Blekinge Institute of Technlogy
+Karlskrona, Sweden
+Riccardo Coppola
+DAUIN
+Politecnico di Torino
+Turin, Italy
+Abstract—Fragile test execution is a common challenge for
+automated GUI-based testing of web applications as they evolve.
+Despite recent progress, there is still room for improvement since
+test execution failures caused by technical limitations result in
+unnecessary maintenance costs that limit its effectiveness and
+efficiency. One of the most reported technical challenges for web-
+based tests concerns how to reliably locate a web element used
+by a test script.
+This paper proposes the novel concept of Visually Overlapping
+Nodes (VON) that reduces fragility by utilizing the phenomenon
+that visual web elements (observed by the user) are constructed
+from multiple web-elements in the Document Object Model
+(DOM) that overlaps visually.
+We demonstrate the approach in a tool, VON Similo, which
+extends the state-of-the-art multi-locator approach (Similo) that
+is also used as the baseline for an experiment. In the experiment,
+a ground truth set of 1163 manually collected web element pairs,
+from different releases of the 40 most popular websites on the
+internet, are used to compare the approaches’ precision, recall,
+and accuracy.
+Our results show that VON Similo provides 94.7% accuracy
+in identifying a web element in a new release of the same SUT.
+In comparison, Similo provides 83.8% accuracy.
+These results demonstrate the applicability of the visually
+overlapping nodes concept/tool for web element localization in
+evolving web applications and contribute a novel way of thinking
+about web element localization in future research on GUI-based
+testing.
+Index Terms—component, formatting, style, styling, insert
+I. INTRODUCTION
+In modern software engineering, test automation is a key
+activity, where automated tests are used to continuously mon-
+itor the software’s quality and provide frequent feedback to
+developers [1]. However, much of this automation has been
+restricted to lower-level testing such as unit and integration
+tests [2]. Higher level testing, particularly with graphical user
+interface (GUI) tests, are still mostly manual and therefore a
+costly activity in practice [3], [4].
+While GUI tests can be used to verify the correctness
+of the GUI’s appearance, the focus of many GUI tests is
+on verifying functional correctness of the system under test
+(SUT) [5], i.e. system testing through the SUT’s GUI [6].
+However, despite continued research since the 1980s, several
+key challenges remain and limit the widespread adoption of
+automated GUI testing [7]. One of these challenges is the
+robust identification of GUI elements. This issue has been
+described for many domains, and several approaches have
+been proposed to increase the robustness of GUI element
+localization [8]–[11]. Robustness is, in this instance, defined as
+the correct identification of a web element when it is available
+and reporting no match when a web element is unavailable.
+This property is particularly important as automated web appli-
+cation tests are typically used for regression testing as software
+systems, and their GUI elements, change and evolve [12].
+Despite its importance, research has had marginal success in
+solving the challenge of robust GUI element localization [7].
+Instead, much research has been focused on extending GUI
+testing technologies, only utilizing the already available web
+element localization solutions—example of such extensions
+are test generation and GUI ripping [13], [14]. Some research,
+however, has been made investigating new types of locators,
+e.g., image recognition [15], or multi-locators [9]. However,
+we still consider web element localization an unsolved chal-
+lenge [7], warranting more research to improve the general
+robustness and maintainability of available GUI testing tech-
+niques and tools.
+Nass et al. proposed an approach called similarity-based
+web element localization (Similo) [16], which calculates a
+weighted similarity score between the target web element
+in a previous version, and all web elements (candidates),
+in a revised version, of a web application. The target web
+element contains the desired properties (e.g., attributes) that
+are compared with each candidate. They compared the Similo
+approach with the multi-locator approach proposed by Leotta
+et al. [17] and found that Similo can correctly locate more
+target web elements than the multi-locator when evaluating
+web elements extracted from 40 commonly used homepages.
+This study presents the novel concept of Visually Over-
+lapping Nodes (VON). The concept makes use of the struc-
+ture of how modern web applications are constructed (i.e.,
+as hierarchies of components with specific attributes and
+characteristics) and how this structure is formalized in the
+document object model (DOM) [18]. Our investigations show
+that multiple DOM nodes often point to the same visual
+element in the rendered GUI, implying that any of these nodes,
+if found to be a match to the sought visual element, will
+constitute a valid match if used for a graphic validation of
+the SUT. DOM nodes (typically arranged in a hierarchy) that
+represent the same visual element (e.g., a button or menu item)
+arXiv:2301.03863v1  [cs.SE]  10 Jan 2023
+
+Fig. 1: Graphical representation of the GUI test case execution
+process, highlighting the step (web element identification) that
+is studied in this work.
+are, in this paper, referred to as visually overlapping nodes
+(VON). As an example, a menu item that is represented by
+an anchor tag (<a>) containing two span tags (<span>) in
+the DOM. All three DOM nodes visually appear as the same
+element (as observed by the user) even though they are three
+separate nodes. The benefit of this approach is that it increases
+the chance to find at least one good match for a provided target,
+thus leading to more robust web element identification.
+While many other essential activities need to be in place for
+robust GUI-based test automation, we here focus on the vital
+aspect of GUI web element identification. If the identification
+is not robust, later steps of the chain cannot compensate for
+this. This focus is visualized in Figure 1.
+As such, the main contributions of this work are:
+• Insights into the relative power of different web element
+attributes for web element localization;
+• A generally applicable, yet novel, concept called Visually
+Overlapping Nodes (VON);
+• An improved version of similarity-based web element lo-
+calization (Similo) that implements VON (VON Similo).
+The continuation of this paper is structured as follows. First,
+in Section II, we present related work and give more detailed
+descriptions of the technologies evaluated in this study. Section
+IV then explains the study’s experimental setup. We present
+the results in Section V-A, which describes the findings of our
+experiment. These findings are discussed in Section VI before
+the paper is concluded in Section VII.
+II. BACKGROUND AND RELATED WORK
+In the GUI-based web application testing discipline, a web-
+element locator is defined as a method, function, approach or
+algorithm that can locate a web element in a given web page
+according to a specific parameter. State-of-the-art techniques
+and tools typically make use of conditions on the attributes of
+the web elements in the HTML DOM tree (e.g., ids, attributes,
+or class names). Popular testing tools (e.g., Selenium) provide
+the possibility to use XPath or CSS expressions to locate
+elements on the web page [19]. We refer to these types of
+locators as single-locators.
+Single-locators are a reported source of fragility for test
+suites. Fragility is defined as the lack of robustness of the
+locators to changes in the GUI definition of the SUT. Test
+script fragility typically manifests through test cases that fail
+not because of functional misbehaviors in the SUT but because
+of the inability of the testing engine to find the web elements
+needed during the test sequences using existing locators [20].
+When a locator cannot be found in the DOM tree of the current
+web page, it is typically referred to as a Broken locator [17].
+Given the increased complexity and high pace of change of
+modern web applications, it is highly likely that the attributes
+or the XPaths of web elements are modified between two
+versions of the same web application. If single-locators are
+used, any such change would result in failed localization
+attempts and require additional effort by the testers to repair
+the broken test suites.
+Leotta et al. proposed the multi-locator approach [17] to
+limit the fragility of single-locators in web application testing.
+The approach evaluates multiple locators and uses a voting
+procedure between the single-locators to improve the accuracy
+of locating the correct web element in a web-page, thereby
+improving the robustness and reducing the script maintenance
+cost.
+Similo is another approach of multi-locators based on a
+weighted similarity score computed on the differences between
+the locator parameters of the web element to locate and each
+of the web elements on the current web page [16]. Unlike
+the multi-locator approach proposed by Leotta et al. [17],
+which uses five single-locators that uniquely identify precisely
+one (or none) web element each, Similo can take advantage
+of locators that pinpoint more than one web element. For
+example, Similo can use an absolute XPath that pinpoints
+exactly one web element in the DOM, but unlike the multi-
+locator proposed by Leotta et al., Similo can also use the
+CSS selector ”.name” to find all web elements that contain
+a specific class name even if the query results in several
+matching web elements.
+In the Similo study [16], currently available as a preprint
+at arXiv, the authors used 14 different locator parameters with
+corresponding comparison operators and weights summarized
+in Figure 2. The locator parameters Tag, Class, Name, Id,
+HRef, Alt, XPath and ID relative XPath, Location, Visible
+Text are collected directly from the DOM-tree. IsButton is
+a derivate boolean parameter, that was set to true or false
+according to the values of the attributes Tag, Type, and
+Class. Neighbor Texts contain a space-separated text of words
+collected from the visible text of nearby web elements. Spe-
+cific comparison operators that return a value between zero
+and one (or exactly zero or one) were selected for each
+locator parameter. Some locator parameters were compared
+by the Java method equalsIgnoreCase (e.g., Tag, Name, and
+Id). Others were compared using Levenshtein distance, word
+comparison, or Euclidean distance. Detailed information about
+the comparison operators can be found in the Similo paper
+[16].
+The weights for each locator parameter (and comparison
+operator) were assigned based on their respective stability and
+uniqueness found by the COLOR study by Kirinuki at al. that
+used a similar selection of locator parameters when evaluating
+four open-source web applications [21]. The COLOR approach
+considers various properties such as attributes, positions, texts,
+
+Test case execution
+Widget
+Test action(s)
+Synchronization
+Test result
+identification
+Input: Target widget
+Output: Candidate
+locator
+location/handle or nullFig. 2: Graphical representation of the computation of simi-
+larity score between two different sets of locator parameters.
+and images to propose a repair unlike Similo that approaches
+the same problem in a preventive way (i.e., before the script
+failure occurs). In Similo, the locator parameters were divided
+into two groups based on the corresponding weights from the
+COLOR study. Locator parameters with a higher stability and
+uniqueness were assigned the weight value 1.5 (bold lines in
+Figure 2) and the remaining were assigned the value 0.5 (thin
+lines in the graphical representation).
+III. VISUALLY OVERLAPPING NODES APPROACH
+In this section, we propose an approach to enhance the
+current state of the art in multi-locators for web application
+testing with an improved version of similarity-based web
+element localization (Similo) that implements visually over-
+lapping nodes (VON Similo).
+A characteristic of modern web applications is that they are
+built from elements that contain elements, e.g., a text label
+can be contained in a button that is itself contained in a
+div tag, which in turn may be placed in another container,
+such as a frame. This hierarchical structure is modeled within
+the DOM, which is used by web browsers and GUI testing
+tools to render, identify or interact with elements. From a
+DOM perspective, each element is considered a unique entity,
+as it is described by a DOM node identifiable through a
+unique absolute XPath. However, from a visual perspective,
+multiple nodes — due to size, placement, and content —
+represent the same visual element, e.g., a button. This feature
+of modern websites thereby implies the existence of a many-
+to-one connection between DOM nodes and visual elements.
+In this paper, all the DOM nodes that belong to the same visual
+element are referred to as visually overlapping nodes (VON).
+Fig. 3 visualizes this many-to-one correspondence, showing
+how web elements can be contained in other web elements
+yet, from a visual perspective, occupy the same area of the
+screen. In this case, while web elements W2 and W3 are
+represented with different XPaths in the DOM, both visually
+point to the same (or nearby) screen area or web element. This
+phenomenon implies that both web elements (W2 and W3) are
+equally correct candidate elements (i.e., elements available on
+the current web page) for a target element (i.e., the element
+that contains the properties that we are looking for) pointing
+to that screen area. We refer to this phenomenon as visually
+Fig. 3: A visualization of a hierarchy of web elements repre-
+sented both visually and from a DOM perspective. It shows
+that although W2 and W3 are unique entities, they appear to
+be the same visual component or, at least, overlap visually.
+overlapping nodes (VON), where property-based localization
+approaches, like Similo, can utilize VON to increase the
+number of correctly located candidate web elements. Hence,
+rather than relying on finding one specific DOM node, or
+absolute XPath, any located DOM node that belongs to the
+same visual element is deemed a correct match. This approach
+mitigates test execution fragility by, as mentioned, increasing
+the number of candidate DOM nodes that can be matched
+when identifying one and the same web element. Essentially,
+the approach will be more robust as long as only one or a
+subset of these nodes change between two revisions.
+Formally, we define the approach to identify equivalent web
+elements as:
+Given a web element W1, we define the set of equivalent
+web elements e0, ... eN as the set of web elements that satisfy
+the following properties:
+1) The ratio between the overlapping areas of the web
+elements on the screen, and the union of the areas of
+the two web elements, is higher than a set threshold
+value. Such ratio is computed as,
+∩(R1, R2)
+∪(R1, R2)
+where: R1 and R2 are the rectangles occupied on the
+screen by the two web elements; the set intersection
+symbol indicates the size (in pixels) of the common area
+occupied on screen by R1 and R2, and the set union
+symbol indicates the size (in pixels) of the union of R1
+and R2.
+By experimenting with different threshold values to
+identify visually overlapping nodes, we finally selected
+0.85 as a value for the threshold that allowed us to
+avoid a definition of visual overlap that was too loose
+(thereby considering visually separate GUI elements as
+overlapping) or too strict (thereby finding none or a few
+visually overlapping nodes).
+2) The center of the web element W2 is contained in the
+rectangle R1.
+The addition of the VON concept transforms the locator
+parameters (e.g., tag. id, or xpath) that are stored and utilized
+by the Similo algorithm. Each locator parameter value is no
+
+Tag - equals (0 or 1)
+Class -Levenshtein dist (0 -1)
+weight = 1.5
+Name -equals (0 or 1)
+Id - equals (0 or 1)
+HRef -Levenshtein dist (0 -1)
+Alt-Levenshteindist (0-1)
+AbsoluteXPath-Levenshtein dist (0-1)
+-Similarityscore
+ID relative XPath - Levenshtein dist (0 - 1)
+IsButton-equals(0or1)
+Location (x, y) - 2D dist (0 - 1)
+Area (width * height) - integer dist (0 - 1)
+Shape (width / height) - integer dist (0 - 1)
+weight = 0.5
+Visible Text - Levenshtein dist (0 - 1)
+NeighborTexts-Wordcomparison(o-1)ualview
+DOM view
+Visu
+W1 - .../div[3]/input[2]
+W3 - .../div[3]/input[3]
+W1
+W2
++
+W2 - .../div[3]
+W3Fig. 4: Visualization of how visually overlapping nodes are
+implemented in VON Similo.
+longer a single value but is instead substituted by a list of
+equivalent values collected from all the visually overlapping
+web elements of a DOM node.
+The VON Similo score substitutes the comparison functions
+of the original Similo, with the following function:
+Given the set ew1 (set of web elements equivalent to w1:
+e1.1, e1.2, ... e1.N) and ew2 (set of web elements equivalent to
+w2: e2.1, e2.2, ... e2:M), and the values for a specific attribute
+of the equivalent web elements, VON Similo computes the
+similarity between all the possible combinations of visually
+overlapping web elements for a specific attribute. The maxi-
+mum of these similarity values is the VON Similo similarity
+score for that attribute in the comparison, representing the
+best possible match to the target web element. However, while
+the match may not be the target web element, this approach
+ensures that the coordinates of the match overlap with the
+intended target. As such, from a visual perspective, the target
+element is identified.
+Figure 4 presents a visualization of the process of how
+visually overlapping nodes are utilized in VON Similo. In
+the first step, denoted A in the figure, a set of target web
+elements (denoted Tx∈TS) and candidate web elements (de-
+noted Cy∈CS) are available, where |TS| ≤ |CS|. In the second
+step, denoted B in the figure, a pre-analysis of TS and CS is
+performed, clustering all target and candidate web elements
+according to the visual web elements they are associated with,
+using the formula presented previously in this section. The
+outcome of the pre-analysis are clusters TS1-TSi and CS1-
+CSj containing components with overlapping target areas on
+the screen but otherwise with an unknown overlap in terms of
+locator properties. In step 3, denoted C in the figure, each
+target web element Tx∈ TSi is compared to every other
+candidate web element Cy∈ CSj, and a similarity score is
+calculated. After the comparison, the maximum similarity
+score of each cluster is kept and associated with all target
+web elements Tn-Tm∈TSi, resulting in a mapping between Ti
+and Cj as visualized in the last step of the figure, denoted D.
+This mapping implies that (from a DOM perspective) a given
+target web element Ti may not be mapped to the candidate
+component Ci=Ti that was initially used when the target was
+set in the previous version of the SUT. In Figure 4, T1 is
+equivalent to C9 but is mapped to its parent component C6.
+However, from a visual perspective, this is irrelevant since
+both C6 and C9 point to the same visual area, T1.
+The benefits of this approach is that the number of valid
+matching candidates increases, implying that for clusters
+where a target web element can not be mapped to a suitable
+candidate, a candidate can still be associated. This increases
+test robustness by reducing the number of failed test actions
+caused by minor changes to components that would otherwise
+be considered false positive test results. One could argue that
+this approach would increase the number of false positives,
+or false negatives, by mapping targets to incorrect candidates.
+However, as we will show, no such trends were identified.
+IV. EMPIRICAL EVALUATION
+In this section, we report the goal, the research questions,
+the methodology, and the results of the empirical evaluation
+we performed when comparing VON Similo to the baseline.
+Similo was selected as the baseline since a previous study
+[16] (currently only available as a preprint) showed that it
+was able to locate more web elements when compared to the
+multi-locator approach proposed by Leotta et al. [17].
+A. Goal
+The study’s high-level goal is to evaluate the efficiency
+of VON-based web element locators when applied to web
+applications. The results are interpreted from the perspective of
+software testing procedures needing methods to automatically
+locate web elements in evolving GUIs with high levels of
+accuracy.
+B. Research Questions and Metrics
+• RQ1: Which are the most frequently used element at-
+tributes in web applications?
+• RQ2: What is the effectiveness of similarity-based web
+element localization (Similo)?
+• RQ3: What is the effectiveness of VON-based web
+element localization (VON Similo)?
+• RQ4: Which type of multi-locator (Similo or VON Sim-
+ilo) performs better in terms of accuracy?
+The objective of RQ1 is to corroborate the selection of
+an optimal set of attributes for the multi-locator approach
+employed in Similo (attributes shown in figure 2). We chose
+these attributes based on related literature. Still, it is unknown
+how often the attributes are populated with data on a generic
+web page and, thereby, what value they give to the employed
+calculation of similarity. It is further unknown if any attribute,
+which could provide value, is not part of this set. To answer
+RQ1, we compute two different metrics: the Relative Non-null
+values for each attribute and the Variability of each attribute.
+The objective of RQ2 is to extend and complement the
+original study performed on Similo [16]. Our purpose is
+to compute Accuracy, Precision, and Recall for the original
+
+A
+B
+TS1
+T3
+CS1
+C1
+C4
+C8
+TS2
+T4
+T5
+CS2
+C3
+C5
+C
+C6
+T2
+CS3
+T2
+C8
+T1
+TS1
+3
+CS3
+CS1
+T3
+T4
+T5
+D
+csimilarity-based web element localization approach, to serve as
+a baseline for our further evaluations. Real-world websites are
+used, where we can observe minor and significant changes to
+the site’s appearance and functionality. The standard Accuracy,
+Precision, and Recall measures are calculated based on the use
+of a human oracle by mapping target web elements from the
+old version of the website to the new website.
+The objective of RQ3 is to evaluate the accuracy with which
+VON Similo can locate the correct web element (from a visual
+perspective ), given a target web element, on a new release
+of the same website. To answer RQ3, we resort again to
+measuring three standard measures, Accuracy, Precision, and
+Recall.
+Finally, the objective of RQ4, as a side-objective of the
+measurement of the Accuracy, Precision, and Recall provided
+by both types of locators, is to provide a comparison between
+them.
+C. Methodology
+This section presents the steps we took to evaluate the
+research questions in a controlled experiment.
+A replication package is available as an open-source repos-
+itory 1.
+We will present threats to the validity of the study design
+in Section VI-A.
+1) Application Collection: To gather a set of experimental
+subjects for our evaluations, we collected pairs of ver-
+sions of the same webpage inside the same website. The
+objective was to emulate the maintenance of websites by
+using target web elements from an old site version and
+identifying them on a new version.
+To avoid biased selections, we selected the 40 top-rated
+websites in the United States from the Alexa ranking
+website23. Since the list contained one case of mirrored
+sites (different URLs leading to the same website), we
+used only one of them. We also excluded one website
+with adult content due to ethical reasons. The final list
+of 38 websites can be found in the replication package.
+2) Application Version Selection: The Internet Archive
+website was used4, to acquire the old version of the web-
+sites chosen in the previous step. To further reduce po-
+tential biases, we choose to replicate a design employed
+by Leotta et al., selecting as the newer version (v2)
+of each website a version published in December 2020
+and as the older version (v1) a version dated R months
+backward in time (with R randomly varying between
+12 and 60 months, 36 months on average). This choice
+ensured enough time had elapsed between releases to
+see graphical and functional differences between the two
+versions of the site, enabling us to evaluate the web
+element finding ability of the approach over time for
+both minor and significant changes. We perceive minor
+1https://figshare.com/s/e63c3679e925730397ac
+2http://www.alexa.com
+3The website has since the experiment been discontinued.
+4http://web.archive.org
+changes to have higher construct validity for regular
+operations in practice. Still, we do not want to exclude
+more significant changes, which can occur when com-
+panies re-brand or make more extensive technological
+updates to their websites.
+3) Application scraping: We developed a Selenium-based
+web scraper to analyze the distribution of the most
+commonly used web elements and attributes and their
+distribution among websites. The scraper collects all
+attribute values for all web elements for a web page
+and stores them for further analysis. This scraping is
+achieved by a script that cycles all web elements in the
+DOM tree and extracts values defined in the W3C list
+of HTML attributes. The scraped information is stored
+in CSV files.
+Based on the scraped information, we were able to
+collect the following statistics to answer RQ1: (i) the
+number of occurrences of non-empty attributes, i.e.,
+different than ”” (empty) or null; (ii) the variability of
+the values, i.e., the ratio between the number of different
+values divided by the total number of non-null, non-
+empty values. We recognize that the latter metric has
+more or less inherent variability for specific attributes,
+e.g., labels. Still, since we did not look at the content for
+this evaluation, only the variation in content, we perceive
+this effect to be minor.
+4) Correspondent web element Selection: To acquire target
+web elements and oracles for the automated evaluation,
+we manually selected web elements from each of the
+40 website homepages. We only selected elements from
+the start page since the Internet archive only stores
+static pages, meaning that javascript, databases, etc., do
+not always work. We were only interested in the web
+element finding ability of the approach and perceived
+that it is unlikely that web elements on other pages on
+a website have a significantly different distribution than
+those in the home pages. The sample of web elements
+was also analyzed to verify that we had acquired a
+comprehensive set of different types of web elements.
+The web elements that were chosen for the evaluation
+had to comply with one or several of the following
+attributes: (1) were possible to perform actions on (e.g.,
+anchors, buttons, menu items, input fields, text fields,
+check-boxes, and radio-buttons); (2) can be used for
+assertions or synchronization (e.g., top-level headlines);
+(3) belong to the core functionality of the website
+homepage; (4) are present in both versions of the web-
+site homepage. The rationale for (4) is given by the
+study’s objective to look at the approach’s ability to find
+web elements over different versions of the websites.
+Through this manual selection, we devised a set of 442
+matches (couples of corresponding web elements) for
+the experiment.
+5) Equivalent web elements Selection: The number of
+manually-identified matches is then expanded by apply-
+ing the equivalence definition according to the formula
+
+in Section III of the present paper. By considering the
+set of equivalents of both web elements in each of the
+442 couples, we came up with an extended set of 1163
+matches.
+We finally define a balanced test set for applying the
+Similo and VON Similo algorithms. To build the test
+set, we include the aforementioned set of 1163 matches
+and 1163 randomly-selected non-matching web element
+couples. We take the same number of matching and non-
+matching web element couples for each considered web
+application.
+6) Match Definition: In the experiment step, for each couple
+of web elements (either matching or non-matching), we
+compute the Similo similarity score and the VON Similo
+score. The scores are normalized in the range [0, 1].
+No normalization step was performed in the original for-
+mulation of the Similo locators. The algorithm, instead,
+computed the similarity scores for all candidate web ele-
+ments and ranked them with no normalization steps. For
+our purposes, it was necessary to add a normalization
+step to compare the performance of Similo with that of
+VON-based locators.
+A web element of the new website is considered a match
+to the target web element if the Similo (or VON Similo)
+score is higher than a threshold. The threshold is pro-
+vided as a variable parameter to the algorithm, which is
+necessary since both algorithms would otherwise always
+return a web element (even with a meager similarity
+score).
+7) Analysis: The results were then analyzed using formal
+and descriptive statistics to identify the metrics used to
+answer the study’s research questions. For this study, we
+use the target web element (Wt) to describe the sought
+web element taken from the older website version. In
+turn, the correct candidate web element (Wc) is sought
+in the new version of the website such that Wc ∈ WS,
+where WS is the set of all web elements in the newer
+version of the website. Given these definitions, we get
+the following outcomes of a web element localization:
+• True positive (TP) - When there is a found match
+such that Wt = Wc when Wc ∈ WS.
+• False negative (FN) - When there is no match found
+such that Wt ̸= Wc despite Wc ∈ WS.
+• True negative (TN) - When there is no match found
+such that Wt ̸= Wc when Wc /∈ WS.
+• False positive (FP) - When there is a match found
+such that Wt = Wc despite Wc /∈ WS.
+Counting the occurrences and distributions of the above
+measurements provides us with the information required
+to answer all the study’s research questions. Based
+on such definitions we in fact compute the following
+derived metrics: Precision, as TP/(TP+FP); Recall, as
+TP/(TP+FN); Accuracy, as (TP+TN)/(TP+FP+TN+FN).
+Fig. 5: Distribution of absent, empty and valued attributes in
+the selected web pages
+Fig. 6: Top attributes for weighted variability in the selected
+web pages
+V. RESULTS
+In this section we report the results measured to answer the
+research questions defined for the study.
+A. RQ1 - Ideal set of attributes for VON Similo
+To analyze and justify the selection of attributes for the
+multi-locator-based algorithms, we computed statistics on the
+distribution of the attributes on all the included web applica-
+tions. We computed statistics for all the 170 standard HTML
+attributes listed by W3C5.
+In the considered web applications, we utilized only 35
+attributes (i.e., they were assigned at least a non-null value).
+In figure 5 we report the percentage of present, empty (””)
+or absent (null) values for each attribute. For the sake of
+readability, we only report the attributes for which the per-
+centage of times a value is present is above the median for
+all attributes (0.7%). We distinguish between null and empty
+values since for specific attributes (e.g., the textual content
+of a web element, identified by the attribute visible text), the
+5https://www.w3.org/wiki/Html/Attributes/ Global
+
+xpath 
+tag
+idxpath 
+spellcheck
+draggable
+translate 
+Value
+Attributes
+class
+Absent
+visible_text
+Empty
+href-
+Present
+id-
+title
+rel 
+type
+tabindex 
+target
+00'0
+0.25
+0.50
+0.75
+1.00
+Percentage of valuesxpath -
+idxpath 
+class -
+visible_text -
+href-
+id -
+tag"
+Attribute
+title
+draggable
+name
+type
+spellcheck
+rel -
+placeholder
+tabindex 
+target -
+translate
+0.00
+0.25
+0.50
+0.75
+1.00
+Variability * Ratio of valued occurrencesTABLE I: Mean (SD) values of Precision, Recall and Accuracy
+over the five test sets for Similo at varying thresholds
+Threshold
+Precision
+Recall
+Accuracy
+0.20
+0.688 (0.024)
+0.965 (0.056)
+0.766 (0.036)
+0.28
+0.796 (0.029)
+0.898 (0.077)
+0.823 (0.045)
+0.40
+0.892 (0.034)
+0.615 (0.121)
+0.770 (0.066)
+0.60
+1.000 (0.006)
+0.265 (0.105)
+0.632 (0.052)
+0.80
+1.000 (0.000)
+0.008 (0.012)
+0.503 (0.006)
+absence of a value is itself a piece of information that we can
+use to locate a web element.
+We compute the variability of each attribute as the ratio
+between the number of different values present in the set of
+considered web elements for a given attribute and the number
+of times the attribute is valued (i.e., other than null or empty).
+To assign a weight to the variability with the relevance of the
+web elements in identifying web elements, we multiply it with
+the percentage of valued occurrences measured in the previous
+step. In figure 6 we report the top attributes for variability in
+the considered web pages. This result validates the original
+selection of attributes performed in the original Similo paper
+since only the XPath, ID Xpath, class, text, href, id, and tag
+attributes were those with the highest variability.
+It is worth noting that the XPath and ID XPath attributes
+have a value equal to 1 for the variability multiplied by the
+valued occurrence ratio, meaning that they are the only two
+attributes valued for every web element and for which no
+couple of web elements can have equal values.
+Answer to RQ1: The analysis of Non-Null values and
+variability for attributes in web pages identify XPath, ID
+XPath, class, visible text, href, id, and tags as the attributes
+that are most likely to change when present with non-
+null values in elements of DOM models. The selection of
+attributes for the Similo algorithm is therefore confirmed as
+optimal for web elements multi-locators.
+B. RQ2 - Similo Performance
+Having finalized the selection of attributes to consider for
+comparing target and candidate web elements, we evaluated
+the original Similo over the set of 1163 matches as defined in
+section C.5.
+We performed 5-fold cross validation and optimized the
+treshold value, used to detect a match, on a training set
+and then evaluated at that treshold on the hold-out test set.
+The optimal treshold values were very stable over all folds;
+for simplicity we thus report results for the full set of web
+elements.
+Table I reports the mean and standard deviation of precision
+(P), recall (R), and accuracy (A) at varying thresholds. The
+optimal threshold, selected to maximize accuracy on the test
+sets, is shown in bold.
+With minimal variations between different folds, the optimal
+threshold for the algorithm is a rather low value of 0.28.
+TABLE II: Mean (SD) values of Precision, Recall and Ac-
+curacy over the five test sets for VON Similo at varying
+thresholds
+Threshold
+Precision
+Recall
+Accuracy
+0.20
+0.680 (0.011)
+0.991 (0.008)
+0.760 (0.011)
+0.40
+0.968 (0.007)
+0.922 (0.110)
+0.941 (0.052)
+0.60
+1.000 (0.004)
+0.475 (0.210)
+0.738 (0.105)
+0.80
+1.000 (0.000)
+0.030 (0.042)
+0.515 (0.021)
+Raising the threshold of the algorithm, in fact, lowers the
+number of comparisons that are signalled as matches, which
+reduces the number of false positives. This low value for
+the threshold may suggest that few attributes with stable
+values (over the selected set of 14) are sufficient to identify a
+match between the original and the new web element. Raising
+the threshold over 0.40 guarantees a high precision for the
+algorithm (over 90%) at the expense of a rapidly increasing
+number of false negatives (i.e., number of candidates that are
+not matched while being correspondent to the target).
+Answer to RQ2: applied on the test dataset, the original
+version of the Similo algorithm is capable of providing a
+mean 82.3% when a match threshold of 0.28 is used.
+C. RQ3 - VON Similo performance
+In table III, we report the results for VON Similo, in the
+same format as for Similo above. Also here, the optimal
+treshold value was stable over the five folds.
+By comparing the results in the two tables, it is clear that
+the optimal threshold is higher for VON Similo. This is likely
+since the comparison of multiple attribute values — instead
+of the one-to-one comparison of the original Similo algorithm
+— increases the likelihood that a candidate web element is
+deemed as matching with the target one. In other words, the
+increase of the size of the candidate space, as described in
+section III, requires a larger treshold or the number of false
+positives increases.
+Even in this case the number of true positives found decays
+rapidly with increasing threshold, however we can observe
+a precision of 100% at a threshold of 0.60, supported by a
+47.5% recall. We note that practitioners can choose to select
+a different treshold level that favors either precision or recall
+depending on the risks and costs they judge a false positive or
+a false negative to have. However, it is clear that VON Similo
+can achieve higher accuracy scores than the original Similo
+algorithm.
+Answer to RQ3: applied on the test dataset, the Visually
+Overlapping Nodes-based version of the Similo algorithm
+(VON Similo) is capable of providing 94.1% accuracy when
+a match threshold of 0.40 is used.
+
+Fig. 7: ROC comparison for Similo, VON Similo, and the
+baseline corresponding to a random classifier
+TABLE III: Comparison of the mean (std deviation) of pre-
+cision, recall and accuracy of Similo vs. VON Similo, per
+subject application
+Approach
+Precision
+Recall
+Accuracy
+Similo
+0.799 (0.130)
+0.772 (0.243)
+0.783 (0.157)
+VON Similo
+0.965 (0.061)
+0.790 (0.915)
+0.879 (0.153)
+D. Comparison between Similo and VON Similo
+In fig. 7 we report the comparison of the two ROC curves
+drawn for Similo (blue) and VON Similo (red). The ROC
+curves show the trade-off between the true positive rate (i.e.
+sensitivity) and the false positive rate (i.e. specificity) of an
+algorithm, plotted at varying thresholds. For simplicity, we
+report the ROC curves computed over the whole set of matches
+without applying the 5-fold split on the data set.
+We also report in the graph, as a baseline, a random clas-
+sifier, which is expected to provide points along the diagonal
+(i.e., number of true positives equal to the number of false
+positives). An ideal classifier instead would instead provide a
+single point where TPR = 1.00 and FPR = 0.00. In a ROC
+curve, the closer the curve is to the main diagonal of the ROC
+space, the less accurate the test.
+For our purposes, there is no benefit in obtaining a
+Precision-Recall curve over a ROC curve since, by construc-
+tion, the data are equally distributed between positives and
+negatives.
+By visual comparison, it is evident how both the algorithms,
+Similo and VON Similo, provide ROC curves that are signif-
+icantly distant from the main diagonal (corresponding to a
+Random classifier). At the same time, it is visually evident
+that the VON Similo algorithm is overall better than that of
+Similo. This advantage in using VON Similo can be quantified
+by computing the area below the ROC Curve (Area Under
+Curve, or AUC).
+As a final comparison, in table III, we report a comparison
+of the performance of the two approaches in terms of average
+accuracy, precision and recall over the set of 33 web pages
+used in the experiment. From the comparison we notice that
+VON Similo has globally better values for all the measures on
+the set of apps, with 88% accuracy against 78.3% accuracy
+for Similo. We also observe rather high std. deviation values
+for the computed measures, caused by a variabile number of
+matches and performance obtained with the tool on individual
+applications.
+One can also perform a Wilcoxon (paired) signed rank
+test [22] of the accuracy values per app which supports (p-
+value of 0.0001614) the hypothesis that von Similo (mean
+accuracy, over the apps, of 88.0%) performs differently than
+Similo (78.3%). Since the argument has been made that
+Bayesian statistical analysis can have benefits [23] we also
+compared the two approaches with a Bayesian signed rank
+test [24] which gave a probability of 99.9% that VON Similo
+has a higher accuracy than Similo6.
+Answer to RQ4: As demonstrated by the comparison of the
+respective ROC-curves and the statistical tests, VON Similo
+(AUC = 0.91, mean accuracy = 0.94, mean accuracy per app
+= 88.0) performs significantly better than Similo (AUC =
+0.88, mean accuracy = 0.82, mean accuracy per app = 78.3).
+VI. DISCUSSION
+This study has shown that multi-locator-based approaches,
+using web element attributes, can provide robust web element
+localization, in the real world, for evolving web applications.
+When coupled with visually overlapping node based (VON-
+based) locators, the method is robust with a 94 percent success
+rate in finding the correct web element.
+This result is significant, especially when put into context
+of how tests are maintained in industry. The study results are
+acquired from a random sample of websites with both larger
+and smaller amounts of change between versions, controlled
+by the time in the past a website was sampled (from 6 months
+to 5 years). In a real scenario, in industry, the probability of
+only more minor changes between versions is theoretically
+higher than the scenario presented here since test suites would
+be run daily or at least weekly. This leads us to the logical
+conclusion that; in an authentic setting, the average accuracy
+of the proposed approach would be higher than 94 percent. We
+base this discussion on the attribute-based locator technology,
+which precision is affected by change. For example, if only one
+attribute is changed, the accuracy will be higher than if two or
+more attributes have changed. For shorter iterations between
+test runs, the number of changes to the application will be
+less and, as such, the solution accuracy increases. However,
+although logically sound, future work is required to verify this
+conclusion, e.g., through empirical and industrial research.
+Regardless, given the result of the study, an interesting
+question becomes what industry is willing to accept in terms
+of false test results, since, as stated in Section I, robust
+6There were clear differences also for precision and recall. We used Python
+3.10 and the library baycomp for the Bayesian SignedRankTest and (base) R
+4.2 for the Wilcoxon test.
+
+1.00
+0.75
+Rate
+Classifier
+Similo
+0.50
+Similo-EML
+True
+Random
+0.25
+0.00
+0.00
+0.25
+0.50
+0.75
+1.00
+False Positives Rateweb element localization is one of the core challenges with
+automated GUI testing and perceived a root-cause to its lack of
+general use in practice. Hence, although the presented results
+are significant, compared to, for instance, state of the art
+research by Leotta et al., which reported a success rate of 33-
+93 percent (93 percent being a theoretical best case limit) [17],
+a philosophical question remains if 94 percent success rate
+is good enough? A 94 percent success rate still implies that
+six web elements will not be found every 100 test actions,
+or one faulty test behavior every 17 steps of script code run
+on modified parts of the SUT. For a test suite ordering in
+the hundreds of test cases, 94 percent success rate would still
+constitute a significant amount of root-cause analysis and test
+script maintenance. However, once more, the selected time
+elapsed between tests of the website versions were in the
+experiment abnormal when compared to the time between
+executions of a GUI-based regression test suite in industrial
+practice.
+However, as a speculative counterpoint, since any accuracy
+below 100 percent seems to be an issue for larger test
+suites, the question arises if 100 percent robustness is even
+achievable. Keeping in mind that web element localization
+implies searching for a target web element in a context where
+there is often not an exact match, only a partial match, due
+to changes in the attributes, appearance, or behavior of the
+sought web element. Under these circumstances, we also see
+humans having less than a 100 percent success rate, so the
+question is how far an automated approach can reach.
+Additionally, the attribute-based approach shows that with
+impartial data, we can still identify web elements with high
+probability. When one of these web elements is identified,
+the attributes that are no longer valid can be automatically
+repaired—Repair, in this case, implies updating attributes that
+are no longer valid. Thus, mitigating much of the maintenance
+costs associated with test scripts. Consequently, keeping the
+delta between test script versions and the SUT low implies a
+larger probability of web element identification success. Thus,
+once more speaking towards the potential of the approach’s
+ability to reach higher than 94 percent success rate in a real
+scenario in industrial practice.
+Conceptually, VON-based locators can work for any DOM-
+based multi-locator approach, i.e., on other platforms not
+used in this paper, such as Android or desktop. However,
+VON-based locators only work with multi-locator approaches
+and would not work for single locator (e.g., Selenium-based)
+scripts due to a lack of information. Consider a Selenium
+action based on an XPath locator, but said XPath is no longer
+available in a new version of the target website. All candidate
+information can be acquired from the new site, but if additional
+information is not available for the target web element, there
+would not be a way to map the target to any of the candidates
+reliably. This is a delimitation of the approach, and it is
+unknown what the minimal amount of information is to make
+it work in practice. This is a subject of future work that entails
+ranking the attributes based on their relative power to be used
+as locators.
+Furthermore, the analysis of the attributes used for this
+work shows that the 14 attributes that Similo used are the
+most frequently populated ones on modern web pages. Thus,
+implying that these provide the most value for web element
+localization. This result is significant, as previous works have
+presented lists of suitable attributes [21], [25], [26], but not
+provided empirical evidence to support these claims. The list
+of attributes provided in this work, although only contempo-
+rary, thereby provides insights into how to design more robust
+GUI testing tools for the foreseeable future. Future work could
+also investigate if there are notable dependencies, or even
+domain aspects, to how these attributes are populated and how,
+for further improvements into web element localization and
+automated repair. Such analysis should also look at the relative
+power of different attributes. Looking at the list of attributes in
+Figures 5 and 6 we note some interesting observations. ID is
+an often mentioned attribute for web element localization [17],
+[27], [28]. However, as shown by the results, ID is seldom used
+in practice, implying that reliance on this attribute would have
+detrimental effects on web element localization. Hence, from a
+practical perspective, the attributes need to be weighed in terms
+of power versus availability. One or several commonly avail-
+able attributes, e.g., XPath or tag, have a higher probability of
+being useful than a perceived more powerful attribute, which
+is seldom available, such as ID. Further research is, however,
+required to evaluate this relative power between attributes.
+Another finding concerns the contents of the attributes of
+the web elements. In Similo, locator parameters are compared
+with various comparators such as equals, Levenshtein distance,
+and integer distance. These are utilized in this research based
+on expert judgment, but other comparators, e.g., euclidean dis-
+tance, could also be used. In future work, a valuable analysis
+would investigate the types and variability of attribute data
+to assign the most suitable comparators. For instance, XPath
+strings are unique to each candidate and are represented as
+Strings of longer length. This paper uses Levenshtein distance
+for their comparison, but other approaches would likely work
+equally well or even better. This mapping between key web
+element attributes, the characteristics of the attribute data,
+and comparators could theoretically be done using machine
+learning, e.g., using learning to rank [29]. Similarly, we can
+also optimize the weighting of each comparator-attribute tuple.
+However, this is once more a subject of future research.
+A. Threats to validity
+In this section we discuss the threats to the study and its
+results pertaining to the internal, external and construct validity
+of the research.
+Generalizability: The paper provides two contributions;
+(1) analysis of web element attributes used in industrial
+practice and (2) VON-based locators. For (1), we perceive
+the results to be generally valid, as they are gathered from
+commonly used web pages from larger companies that should
+be perceived as adopters of best state-of-practice approaches.
+However, for (2), the results are delimited to web element
+localization with multi-locators as discussed in Section VI.
+
+This diminishes the result’s industrial applicability since most
+industrial frameworks, like Selenium, utilize a single locator
+approach. As such, adopting VON-based multi-locators would
+also require a technological shift in approaches or tooling for
+developing such test cases. Although scripts that use multi-
+locators can be developed manually, we perceive recording
+such tests as beneficial from an efficiency perspective.
+Subject selection: We took the subjects for this analysis
+from the Alexa.com website (as of writing, the site has been
+discontinued), which reduces bias in the sampling as the web-
+sites on the list are outside the researchers’ control. However,
+we chose only a subset of the list, i.e., the top 40 pages (33
+used after excluding, for instance, broken and duplicate sites).
+This set is still perceived as significant when compared to
+other studies in this area of research. However, it is unknown
+how representative the result is to any general website on the
+internet. Analysis of the tags used in the web elements of these
+web pages does, however, provide us with confidence that all
+types of web elements that can, theoretically, be encountered
+have been identified. We stress, however, that this research
+only focused on web element identification, and therefore
+detrimental effects that may arise during test execution were
+not covered. An example of such an effect is the acquisition
+of web element attribute data during a state transition. A lack
+of synchronization between the test script and AUT could
+result in missing attributes if pre-analysis of web element
+information begins before the AUT is fully loaded.
+Comprehensiveness: VON Similo was designed with flex-
+ibility in mind to optimize its potential use in practice. A
+drawback of this design is that the comparators and attributes
+used in this study are a subset of possible attributes and
+comparators, as discussed in Section VI. While not reducing
+the validity of the results presented in this work, this implies
+that we could achieve better results with other constellations
+of parameters. We defend this unorthodox statement by the
+presented results that indicate a considerable increase in pre-
+cision and recall compared to state-of-art. As such, although
+there is no threat to the validity of the results of the current
+implementation, given the set of analyzed web pages, we
+cannot state if this is the best implementation of the approach.
+Further research is thereby required with other constellations
+and contexts to optimize the approach.
+VII. CONCLUSIONS AND FUTURE WORK
+In this paper, we proposed a novel location algorithm for
+web elements in web applications, i.e., Visually Overlapping
+Nodes (VON). We applied this approach by extending an
+existing multi-locator approach (Similo), thereby obtaining the
+VON Similo approach.
+To investigate the accuracy of the algorithm, we performed
+an empirical investigation by manually identifying a set of
+matches between mutated widgets in different releases of
+the same application and then measuring the effectiveness
+of the original and extended approaches to determine correct
+matches. This analysis led us to measure a +9.9% increase
+in accuracy for VON Similo with respect to the original
+approach. It is worth underlining that the original Similo was
+already proven as better performing than state-of-the-art multi-
+locator algorithms (e.g., Leotta’s ROBULA+ [9]).
+This work first assessed the possible benefits introduced by
+the concept of Visually Overlapping Nodes when performing
+GUI testing of web applications. Therefore, fine-tuning the
+approach can be obtained by empirically finding the optimal
+values for several fixed coefficients and variables employed by
+the algorithm.
+The algorithms evaluated and compared in this study used
+only static weights defined in the original Similo algorithm
+and were compatible with state-of-the-art tools like COLOR
+[21]. Selecting a different set of weights for the attributes used
+in the comparison can lead to significantly different results in
+terms of precision and accuracy. The same reasoning applies
+to selecting the attributes involved in the score computation.
+In our immediate future work, we plan to employ ma-
+chine learning-based approaches to identify the optimal set
+of attributes, and related weights, to maximize the accuracy
+of Similo. As a further improvement of the algorithm, it is
+possible to consider returning as output not only a single
+matching web element but a ranked list of the most matching
+candidates. This evolution of the algorithm can also benefit
+from the application of machine-learning-ranking (MLR or
+Learning to Rank) techniques.
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+
diff --git a/ZdE2T4oBgHgl3EQfZQey/content/tmp_files/load_file.txt b/ZdE2T4oBgHgl3EQfZQey/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf,len=793
+page_content='Robust web element identification for evolving  applications by considering visual overlaps  Michel Nass, Emil Al´egroth, Robert Feldt  SERL  Blekinge Institute of Technlogy  Karlskrona, Sweden  Riccardo Coppola  DAUIN  Politecnico di Torino  Turin, Italy  Abstract—Fragile test execution is a common challenge for  automated GUI-based testing of web applications as they evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Despite recent progress, there is still room for improvement since  test execution failures caused by technical limitations result in  unnecessary maintenance costs that limit its effectiveness and  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' One of the most reported technical challenges for web-  based tests concerns how to reliably locate a web element used  by a test script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This paper proposes the novel concept of Visually Overlapping  Nodes (VON) that reduces fragility by utilizing the phenomenon  that visual web elements (observed by the user) are constructed  from multiple web-elements in the Document Object Model  (DOM) that overlaps visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  We demonstrate the approach in a tool, VON Similo, which  extends the state-of-the-art multi-locator approach (Similo) that  is also used as the baseline for an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In the experiment,  a ground truth set of 1163 manually collected web element pairs,  from different releases of the 40 most popular websites on the  internet, are used to compare the approaches’ precision, recall,  and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Our results show that VON Similo provides 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='7% accuracy  in identifying a web element in a new release of the same SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  In comparison, Similo provides 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='8% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  These results demonstrate the applicability of the visually  overlapping nodes concept/tool for web element localization in  evolving web applications and contribute a novel way of thinking  about web element localization in future research on GUI-based  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Index Terms—component, formatting, style, styling, insert  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' INTRODUCTION  In modern software engineering, test automation is a key  activity, where automated tests are used to continuously mon-  itor the software’s quality and provide frequent feedback to  developers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, much of this automation has been  restricted to lower-level testing such as unit and integration  tests [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Higher level testing, particularly with graphical user  interface (GUI) tests, are still mostly manual and therefore a  costly activity in practice [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  While GUI tests can be used to verify the correctness  of the GUI’s appearance, the focus of many GUI tests is  on verifying functional correctness of the system under test  (SUT) [5], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' system testing through the SUT’s GUI [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  However, despite continued research since the 1980s, several  key challenges remain and limit the widespread adoption of  automated GUI testing [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' One of these challenges is the  robust identification of GUI elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This issue has been  described for many domains, and several approaches have  been proposed to increase the robustness of GUI element  localization [8]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Robustness is, in this instance, defined as  the correct identification of a web element when it is available  and reporting no match when a web element is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This property is particularly important as automated web appli-  cation tests are typically used for regression testing as software  systems, and their GUI elements, change and evolve [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Despite its importance, research has had marginal success in  solving the challenge of robust GUI element localization [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Instead, much research has been focused on extending GUI  testing technologies, only utilizing the already available web  element localization solutions—example of such extensions  are test generation and GUI ripping [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Some research,  however, has been made investigating new types of locators,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', image recognition [15], or multi-locators [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However,  we still consider web element localization an unsolved chal-  lenge [7], warranting more research to improve the general  robustness and maintainability of available GUI testing tech-  niques and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Nass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' proposed an approach called similarity-based  web element localization (Similo) [16], which calculates a  weighted similarity score between the target web element  in a previous version, and all web elements (candidates),  in a revised version, of a web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The target web  element contains the desired properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', attributes) that  are compared with each candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' They compared the Similo  approach with the multi-locator approach proposed by Leotta  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' [17] and found that Similo can correctly locate more  target web elements than the multi-locator when evaluating  web elements extracted from 40 commonly used homepages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This study presents the novel concept of Visually Over-  lapping Nodes (VON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The concept makes use of the struc-  ture of how modern web applications are constructed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=',  as hierarchies of components with specific attributes and  characteristics) and how this structure is formalized in the  document object model (DOM) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Our investigations show  that multiple DOM nodes often point to the same visual  element in the rendered GUI, implying that any of these nodes,  if found to be a match to the sought visual element, will  constitute a valid match if used for a graphic validation of  the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' DOM nodes (typically arranged in a hierarchy) that  represent the same visual element (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', a button or menu item)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='03863v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='SE]  10 Jan 2023  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 1: Graphical representation of the GUI test case execution  process, highlighting the step (web element identification) that  is studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  are, in this paper, referred to as visually overlapping nodes  (VON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' As an example, a menu item that is represented by  an anchor tag (<a>) containing two span tags (<span>) in  the DOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' All three DOM nodes visually appear as the same  element (as observed by the user) even though they are three  separate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The benefit of this approach is that it increases  the chance to find at least one good match for a provided target,  thus leading to more robust web element identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  While many other essential activities need to be in place for  robust GUI-based test automation, we here focus on the vital  aspect of GUI web element identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' If the identification  is not robust, later steps of the chain cannot compensate for  this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This focus is visualized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  As such, the main contributions of this work are:  Insights into the relative power of different web element  attributes for web element localization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A generally applicable, yet novel, concept called Visually  Overlapping Nodes (VON);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  An improved version of similarity-based web element lo-  calization (Similo) that implements VON (VON Similo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The continuation of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' First,  in Section II, we present related work and give more detailed  descriptions of the technologies evaluated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Section  IV then explains the study’s experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We present  the results in Section V-A, which describes the findings of our  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' These findings are discussed in Section VI before  the paper is concluded in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' BACKGROUND AND RELATED WORK  In the GUI-based web application testing discipline, a web-  element locator is defined as a method, function, approach or  algorithm that can locate a web element in a given web page  according to a specific parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' State-of-the-art techniques  and tools typically make use of conditions on the attributes of  the web elements in the HTML DOM tree (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', ids, attributes,  or class names).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Popular testing tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', Selenium) provide  the possibility to use XPath or CSS expressions to locate  elements on the web page [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We refer to these types of  locators as single-locators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Single-locators are a reported source of fragility for test  suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Fragility is defined as the lack of robustness of the  locators to changes in the GUI definition of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Test  script fragility typically manifests through test cases that fail  not because of functional misbehaviors in the SUT but because  of the inability of the testing engine to find the web elements  needed during the test sequences using existing locators [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  When a locator cannot be found in the DOM tree of the current  web page, it is typically referred to as a Broken locator [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Given the increased complexity and high pace of change of  modern web applications, it is highly likely that the attributes  or the XPaths of web elements are modified between two  versions of the same web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' If single-locators are  used, any such change would result in failed localization  attempts and require additional effort by the testers to repair  the broken test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Leotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' proposed the multi-locator approach [17] to  limit the fragility of single-locators in web application testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The approach evaluates multiple locators and uses a voting  procedure between the single-locators to improve the accuracy  of locating the correct web element in a web-page, thereby  improving the robustness and reducing the script maintenance  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Similo is another approach of multi-locators based on a  weighted similarity score computed on the differences between  the locator parameters of the web element to locate and each  of the web elements on the current web page [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Unlike  the multi-locator approach proposed by Leotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' [17],  which uses five single-locators that uniquely identify precisely  one (or none) web element each, Similo can take advantage  of locators that pinpoint more than one web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For  example, Similo can use an absolute XPath that pinpoints  exactly one web element in the DOM, but unlike the multi-  locator proposed by Leotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', Similo can also use the  CSS selector ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='name” to find all web elements that contain  a specific class name even if the query results in several  matching web elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  In the Similo study [16], currently available as a preprint  at arXiv, the authors used 14 different locator parameters with  corresponding comparison operators and weights summarized  in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The locator parameters Tag, Class, Name, Id,  HRef, Alt, XPath and ID relative XPath, Location, Visible  Text are collected directly from the DOM-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' IsButton is  a derivate boolean parameter, that was set to true or false  according to the values of the attributes Tag, Type, and  Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Neighbor Texts contain a space-separated text of words  collected from the visible text of nearby web elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Spe-  cific comparison operators that return a value between zero  and one (or exactly zero or one) were selected for each  locator parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Some locator parameters were compared  by the Java method equalsIgnoreCase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', Tag, Name, and  Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Others were compared using Levenshtein distance, word  comparison, or Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Detailed information about  the comparison operators can be found in the Similo paper  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The weights for each locator parameter (and comparison  operator) were assigned based on their respective stability and  uniqueness found by the COLOR study by Kirinuki at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' that  used a similar selection of locator parameters when evaluating  four open-source web applications [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The COLOR approach  considers various properties such as attributes, positions, texts,  Test case execution  Widget  Test action(s)  Synchronization  Test result  identification  Input: Target widget  Output: Candidate  locator  location/handle or nullFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 2: Graphical representation of the computation of simi-  larity score between two different sets of locator parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  and images to propose a repair unlike Similo that approaches  the same problem in a preventive way (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', before the script  failure occurs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In Similo, the locator parameters were divided  into two groups based on the corresponding weights from the  COLOR study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Locator parameters with a higher stability and  uniqueness were assigned the weight value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='5 (bold lines in  Figure 2) and the remaining were assigned the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='5 (thin  lines in the graphical representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' VISUALLY OVERLAPPING NODES APPROACH  In this section, we propose an approach to enhance the  current state of the art in multi-locators for web application  testing with an improved version of similarity-based web  element localization (Similo) that implements visually over-  lapping nodes (VON Similo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A characteristic of modern web applications is that they are  built from elements that contain elements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', a text label  can be contained in a button that is itself contained in a  div tag, which in turn may be placed in another container,  such as a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This hierarchical structure is modeled within  the DOM, which is used by web browsers and GUI testing  tools to render, identify or interact with elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' From a  DOM perspective, each element is considered a unique entity,  as it is described by a DOM node identifiable through a  unique absolute XPath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, from a visual perspective,  multiple nodes — due to size, placement, and content —  represent the same visual element, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', a button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This feature  of modern websites thereby implies the existence of a many-  to-one connection between DOM nodes and visual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  In this paper, all the DOM nodes that belong to the same visual  element are referred to as visually overlapping nodes (VON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 3 visualizes this many-to-one correspondence, showing  how web elements can be contained in other web elements  yet, from a visual perspective, occupy the same area of the  screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In this case, while web elements W2 and W3 are  represented with different XPaths in the DOM, both visually  point to the same (or nearby) screen area or web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This  phenomenon implies that both web elements (W2 and W3) are  equally correct candidate elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', elements available on  the current web page) for a target element (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', the element  that contains the properties that we are looking for) pointing  to that screen area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We refer to this phenomenon as visually  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 3: A visualization of a hierarchy of web elements repre-  sented both visually and from a DOM perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' It shows  that although W2 and W3 are unique entities, they appear to  be the same visual component or, at least, overlap visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  overlapping nodes (VON), where property-based localization  approaches, like Similo, can utilize VON to increase the  number of correctly located candidate web elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Hence,  rather than relying on finding one specific DOM node, or  absolute XPath, any located DOM node that belongs to the  same visual element is deemed a correct match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This approach  mitigates test execution fragility by, as mentioned, increasing  the number of candidate DOM nodes that can be matched  when identifying one and the same web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Essentially,  the approach will be more robust as long as only one or a  subset of these nodes change between two revisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Formally, we define the approach to identify equivalent web  elements as:  Given a web element W1, we define the set of equivalent  web elements e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' eN as the set of web elements that satisfy  the following properties:  1) The ratio between the overlapping areas of the web  elements on the screen, and the union of the areas of  the two web elements, is higher than a set threshold  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Such ratio is computed as,  ∩(R1, R2)  ∪(R1, R2)  where: R1 and R2 are the rectangles occupied on the  screen by the two web elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' the set intersection  symbol indicates the size (in pixels) of the common area  occupied on screen by R1 and R2, and the set union  symbol indicates the size (in pixels) of the union of R1  and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  By experimenting with different threshold values to  identify visually overlapping nodes, we finally selected  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='85 as a value for the threshold that allowed us to  avoid a definition of visual overlap that was too loose  (thereby considering visually separate GUI elements as  overlapping) or too strict (thereby finding none or a few  visually overlapping nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  2) The center of the web element W2 is contained in the  rectangle R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The addition of the VON concept transforms the locator  parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' id, or xpath) that are stored and utilized  by the Similo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Each locator parameter value is no  Tag - equals (0 or 1)  Class -Levenshtein dist (0 -1)  weight = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='5  Name -equals (0 or 1)  Id - equals (0 or 1)  HRef -Levenshtein dist (0 -1)  Alt-Levenshteindist (0-1)  AbsoluteXPath-Levenshtein dist (0-1)  Similarityscore  ID relative XPath - Levenshtein dist (0 - 1)  IsButton-equals(0or1)  Location (x, y) - 2D dist (0 - 1)  Area (width * height) - integer dist (0 - 1)  Shape (width / height) - integer dist (0 - 1)  weight = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='5  Visible Text - Levenshtein dist (0 - 1)  NeighborTexts-Wordcomparison(o-1)ualview  DOM view  Visu  W1 - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='/div[3]/input[2]  W3 - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='/div[3]/input[3]  W1  W2  +  W2 - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='/div[3]  W3Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 4: Visualization of how visually overlapping nodes are  implemented in VON Similo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  longer a single value but is instead substituted by a list of  equivalent values collected from all the visually overlapping  web elements of a DOM node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The VON Similo score substitutes the comparison functions  of the original Similo, with the following function:  Given the set ew1 (set of web elements equivalent to w1:  e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='1, e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='N) and ew2 (set of web elements equivalent to  w2: e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='1, e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' e2:M), and the values for a specific attribute  of the equivalent web elements, VON Similo computes the  similarity between all the possible combinations of visually  overlapping web elements for a specific attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The maxi-  mum of these similarity values is the VON Similo similarity  score for that attribute in the comparison, representing the  best possible match to the target web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, while  the match may not be the target web element, this approach  ensures that the coordinates of the match overlap with the  intended target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' As such, from a visual perspective, the target  element is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Figure 4 presents a visualization of the process of how  visually overlapping nodes are utilized in VON Similo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In  the first step, denoted A in the figure, a set of target web  elements (denoted Tx∈TS) and candidate web elements (de-  noted Cy∈CS) are available, where |TS| ≤ |CS|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In the second  step, denoted B in the figure, a pre-analysis of TS and CS is  performed, clustering all target and candidate web elements  according to the visual web elements they are associated with,  using the formula presented previously in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The  outcome of the pre-analysis are clusters TS1-TSi and CS1-  CSj containing components with overlapping target areas on  the screen but otherwise with an unknown overlap in terms of  locator properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In step 3, denoted C in the figure, each  target web element Tx∈ TSi is compared to every other  candidate web element Cy∈ CSj, and a similarity score is  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' After the comparison, the maximum similarity  score of each cluster is kept and associated with all target  web elements Tn-Tm∈TSi, resulting in a mapping between Ti  and Cj as visualized in the last step of the figure, denoted D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This mapping implies that (from a DOM perspective) a given  target web element Ti may not be mapped to the candidate  component Ci=Ti that was initially used when the target was  set in the previous version of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In Figure 4, T1 is  equivalent to C9 but is mapped to its parent component C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  However, from a visual perspective, this is irrelevant since  both C6 and C9 point to the same visual area, T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The benefits of this approach is that the number of valid  matching candidates increases, implying that for clusters  where a target web element can not be mapped to a suitable  candidate, a candidate can still be associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This increases  test robustness by reducing the number of failed test actions  caused by minor changes to components that would otherwise  be considered false positive test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' One could argue that  this approach would increase the number of false positives,  or false negatives, by mapping targets to incorrect candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  However, as we will show, no such trends were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' EMPIRICAL EVALUATION  In this section, we report the goal, the research questions,  the methodology, and the results of the empirical evaluation  we performed when comparing VON Similo to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Similo was selected as the baseline since a previous study  [16] (currently only available as a preprint) showed that it  was able to locate more web elements when compared to the  multi-locator approach proposed by Leotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Goal  The study’s high-level goal is to evaluate the efficiency  of VON-based web element locators when applied to web  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The results are interpreted from the perspective of  software testing procedures needing methods to automatically  locate web elements in evolving GUIs with high levels of  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Research Questions and Metrics  RQ1: Which are the most frequently used element at-  tributes in web applications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  RQ2: What is the effectiveness of similarity-based web  element localization (Similo)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  RQ3: What is the effectiveness of VON-based web  element localization (VON Similo)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  RQ4: Which type of multi-locator (Similo or VON Sim-  ilo) performs better in terms of accuracy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The objective of RQ1 is to corroborate the selection of  an optimal set of attributes for the multi-locator approach  employed in Similo (attributes shown in figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We chose  these attributes based on related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Still, it is unknown  how often the attributes are populated with data on a generic  web page and, thereby, what value they give to the employed  calculation of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' It is further unknown if any attribute,  which could provide value, is not part of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' To answer  RQ1, we compute two different metrics: the Relative Non-null  values for each attribute and the Variability of each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The objective of RQ2 is to extend and complement the  original study performed on Similo [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Our purpose is  to compute Accuracy, Precision, and Recall for the original  A  B  TS1  T3  CS1  C1  C4  C8  TS2  T4  T5  CS2  C3  C5  C  C6  T2  CS3  T2  C8  T1  TS1  3  CS3  CS1  T3  T4  T5  D  csimilarity-based web element localization approach, to serve as  a baseline for our further evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Real-world websites are  used, where we can observe minor and significant changes to  the site’s appearance and functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The standard Accuracy,  Precision, and Recall measures are calculated based on the use  of a human oracle by mapping target web elements from the  old version of the website to the new website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The objective of RQ3 is to evaluate the accuracy with which  VON Similo can locate the correct web element (from a visual  perspective ), given a target web element, on a new release  of the same website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' To answer RQ3, we resort again to  measuring three standard measures, Accuracy, Precision, and  Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Finally, the objective of RQ4, as a side-objective of the  measurement of the Accuracy, Precision, and Recall provided  by both types of locators, is to provide a comparison between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Methodology  This section presents the steps we took to evaluate the  research questions in a controlled experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A replication package is available as an open-source repos-  itory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  We will present threats to the validity of the study design  in Section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  1) Application Collection: To gather a set of experimental  subjects for our evaluations, we collected pairs of ver-  sions of the same webpage inside the same website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The  objective was to emulate the maintenance of websites by  using target web elements from an old site version and  identifying them on a new version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  To avoid biased selections, we selected the 40 top-rated  websites in the United States from the Alexa ranking  website23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Since the list contained one case of mirrored  sites (different URLs leading to the same website), we  used only one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We also excluded one website  with adult content due to ethical reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The final list  of 38 websites can be found in the replication package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  2) Application Version Selection: The Internet Archive  website was used4, to acquire the old version of the web-  sites chosen in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' To further reduce po-  tential biases, we choose to replicate a design employed  by Leotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', selecting as the newer version (v2)  of each website a version published in December 2020  and as the older version (v1) a version dated R months  backward in time (with R randomly varying between  12 and 60 months, 36 months on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This choice  ensured enough time had elapsed between releases to  see graphical and functional differences between the two  versions of the site, enabling us to evaluate the web  element finding ability of the approach over time for  both minor and significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We perceive minor  1https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='com/s/e63c3679e925730397ac  2http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='alexa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='com  3The website has since the experiment been discontinued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  4http://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='org  changes to have higher construct validity for regular  operations in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Still, we do not want to exclude  more significant changes, which can occur when com-  panies re-brand or make more extensive technological  updates to their websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  3) Application scraping: We developed a Selenium-based  web scraper to analyze the distribution of the most  commonly used web elements and attributes and their  distribution among websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The scraper collects all  attribute values for all web elements for a web page  and stores them for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This scraping is  achieved by a script that cycles all web elements in the  DOM tree and extracts values defined in the W3C list  of HTML attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The scraped information is stored  in CSV files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Based on the scraped information, we were able to  collect the following statistics to answer RQ1: (i) the  number of occurrences of non-empty attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=',  different than ”” (empty) or null;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' (ii) the variability of  the values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', the ratio between the number of different  values divided by the total number of non-null, non-  empty values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We recognize that the latter metric has  more or less inherent variability for specific attributes,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Still, since we did not look at the content for  this evaluation, only the variation in content, we perceive  this effect to be minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  4) Correspondent web element Selection: To acquire target  web elements and oracles for the automated evaluation,  we manually selected web elements from each of the  40 website homepages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We only selected elements from  the start page since the Internet archive only stores  static pages, meaning that javascript, databases, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', do  not always work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We were only interested in the web  element finding ability of the approach and perceived  that it is unlikely that web elements on other pages on  a website have a significantly different distribution than  those in the home pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The sample of web elements  was also analyzed to verify that we had acquired a  comprehensive set of different types of web elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The web elements that were chosen for the evaluation  had to comply with one or several of the following  attributes: (1) were possible to perform actions on (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=',  anchors, buttons, menu items, input fields, text fields,  check-boxes, and radio-buttons);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' (2) can be used for  assertions or synchronization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', top-level headlines);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  (3) belong to the core functionality of the website  homepage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' (4) are present in both versions of the web-  site homepage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The rationale for (4) is given by the  study’s objective to look at the approach’s ability to find  web elements over different versions of the websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Through this manual selection, we devised a set of 442  matches (couples of corresponding web elements) for  the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  5) Equivalent web elements Selection: The number of  manually-identified matches is then expanded by apply-  ing the equivalence definition according to the formula  in Section III of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' By considering the  set of equivalents of both web elements in each of the  442 couples, we came up with an extended set of 1163  matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  We finally define a balanced test set for applying the  Similo and VON Similo algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' To build the test  set, we include the aforementioned set of 1163 matches  and 1163 randomly-selected non-matching web element  couples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We take the same number of matching and non-  matching web element couples for each considered web  application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  6) Match Definition: In the experiment step, for each couple  of web elements (either matching or non-matching), we  compute the Similo similarity score and the VON Similo  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The scores are normalized in the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  No normalization step was performed in the original for-  mulation of the Similo locators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The algorithm, instead,  computed the similarity scores for all candidate web ele-  ments and ranked them with no normalization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For  our purposes, it was necessary to add a normalization  step to compare the performance of Similo with that of  VON-based locators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A web element of the new website is considered a match  to the target web element if the Similo (or VON Similo)  score is higher than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The threshold is pro-  vided as a variable parameter to the algorithm, which is  necessary since both algorithms would otherwise always  return a web element (even with a meager similarity  score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  7) Analysis: The results were then analyzed using formal  and descriptive statistics to identify the metrics used to  answer the study’s research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For this study, we  use the target web element (Wt) to describe the sought  web element taken from the older website version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In  turn, the correct candidate web element (Wc) is sought  in the new version of the website such that Wc ∈ WS,  where WS is the set of all web elements in the newer  version of the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Given these definitions, we get  the following outcomes of a web element localization:  True positive (TP) - When there is a found match  such that Wt = Wc when Wc ∈ WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  False negative (FN) - When there is no match found  such that Wt ̸= Wc despite Wc ∈ WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  True negative (TN) - When there is no match found  such that Wt ̸= Wc when Wc /∈ WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  False positive (FP) - When there is a match found  such that Wt = Wc despite Wc /∈ WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Counting the occurrences and distributions of the above  measurements provides us with the information required  to answer all the study’s research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Based  on such definitions we in fact compute the following  derived metrics: Precision, as TP/(TP+FP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Recall, as  TP/(TP+FN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Accuracy, as (TP+TN)/(TP+FP+TN+FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 5: Distribution of absent, empty and valued attributes in  the selected web pages  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 6: Top attributes for weighted variability in the selected  web pages  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' RESULTS  In this section we report the results measured to answer the  research questions defined for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' RQ1 - Ideal set of attributes for VON Similo  To analyze and justify the selection of attributes for the  multi-locator-based algorithms, we computed statistics on the  distribution of the attributes on all the included web applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We computed statistics for all the 170 standard HTML  attributes listed by W3C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  In the considered web applications, we utilized only 35  attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', they were assigned at least a non-null value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  In figure 5 we report the percentage of present, empty (””)  or absent (null) values for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For the sake of  readability, we only report the attributes for which the per-  centage of times a value is present is above the median for  all attributes (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We distinguish between null and empty  values since for specific attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', the textual content  of a web element, identified by the attribute visible text), the  5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content="org/wiki/Html/Attributes/ Global  xpath  tag  idxpath  spellcheck  draggable  translate  Value  Attributes  class  Absent  visible_text  Empty  href-  Present  id-  title  rel  type  tabindex  target  00'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00  Percentage of valuesxpath -  idxpath  class -  visible_text -  href-  id -  tag"  Attribute  title  draggable  name  type  spellcheck  rel -  placeholder  tabindex  target -  translate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
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+page_content='00  Variability * Ratio of valued occurrencesTABLE I: Mean (SD) values of Precision, Recall and Accuracy  over the five test sets for Similo at varying thresholds  Threshold  Precision  Recall  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
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+page_content='006)  absence of a value is itself a piece of information that we can  use to locate a web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  We compute the variability of each attribute as the ratio  between the number of different values present in the set of  considered web elements for a given attribute and the number  of times the attribute is valued (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', other than null or empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  To assign a weight to the variability with the relevance of the  web elements in identifying web elements, we multiply it with  the percentage of valued occurrences measured in the previous  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In figure 6 we report the top attributes for variability in  the considered web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This result validates the original  selection of attributes performed in the original Similo paper  since only the XPath, ID Xpath, class, text, href, id, and tag  attributes were those with the highest variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  It is worth noting that the XPath and ID XPath attributes  have a value equal to 1 for the variability multiplied by the  valued occurrence ratio, meaning that they are the only two  attributes valued for every web element and for which no  couple of web elements can have equal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Answer to RQ1: The analysis of Non-Null values and  variability for attributes in web pages identify XPath, ID  XPath, class, visible text, href, id, and tags as the attributes  that are most likely to change when present with non-  null values in elements of DOM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The selection of  attributes for the Similo algorithm is therefore confirmed as  optimal for web elements multi-locators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' RQ2 - Similo Performance  Having finalized the selection of attributes to consider for  comparing target and candidate web elements, we evaluated  the original Similo over the set of 1163 matches as defined in  section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  We performed 5-fold cross validation and optimized the  treshold value, used to detect a match, on a training set  and then evaluated at that treshold on the hold-out test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The optimal treshold values were very stable over all folds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  for simplicity we thus report results for the full set of web  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Table I reports the mean and standard deviation of precision  (P), recall (R), and accuracy (A) at varying thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The  optimal threshold, selected to maximize accuracy on the test  sets, is shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  With minimal variations between different folds, the optimal  threshold for the algorithm is a rather low value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
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+page_content='  TABLE II: Mean (SD) values of Precision, Recall and Ac-  curacy over the five test sets for VON Similo at varying  thresholds  Threshold  Precision  Recall  Accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
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+page_content='042)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
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+page_content='021)  Raising the threshold of the algorithm, in fact, lowers the  number of comparisons that are signalled as matches, which  reduces the number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This low value for  the threshold may suggest that few attributes with stable  values (over the selected set of 14) are sufficient to identify a  match between the original and the new web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Raising  the threshold over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='40 guarantees a high precision for the  algorithm (over 90%) at the expense of a rapidly increasing  number of false negatives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', number of candidates that are  not matched while being correspondent to the target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Answer to RQ2: applied on the test dataset, the original  version of the Similo algorithm is capable of providing a  mean 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='3% when a match threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='28 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' RQ3 - VON Similo performance  In table III, we report the results for VON Similo, in the  same format as for Similo above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Also here, the optimal  treshold value was stable over the five folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  By comparing the results in the two tables, it is clear that  the optimal threshold is higher for VON Similo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This is likely  since the comparison of multiple attribute values — instead  of the one-to-one comparison of the original Similo algorithm  — increases the likelihood that a candidate web element is  deemed as matching with the target one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In other words, the  increase of the size of the candidate space, as described in  section III, requires a larger treshold or the number of false  positives increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Even in this case the number of true positives found decays  rapidly with increasing threshold, however we can observe  a precision of 100% at a threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='60, supported by a  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='5% recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We note that practitioners can choose to select  a different treshold level that favors either precision or recall  depending on the risks and costs they judge a false positive or  a false negative to have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, it is clear that VON Similo  can achieve higher accuracy scores than the original Similo  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Answer to RQ3: applied on the test dataset, the Visually  Overlapping Nodes-based version of the Similo algorithm  (VON Similo) is capable of providing 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='1% accuracy when  a match threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='40 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 7: ROC comparison for Similo, VON Similo, and the  baseline corresponding to a random classifier  TABLE III: Comparison of the mean (std deviation) of pre-  cision, recall and accuracy of Similo vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' VON Similo, per  subject application  Approach  Precision  Recall  Accuracy  Similo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='799 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='130)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='772 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='243)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='783 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='157)  VON Similo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='965 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='061)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='790 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='915)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='879 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='153)  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Comparison between Similo and VON Similo  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' 7 we report the comparison of the two ROC curves  drawn for Similo (blue) and VON Similo (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The ROC  curves show the trade-off between the true positive rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  sensitivity) and the false positive rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' specificity) of an  algorithm, plotted at varying thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For simplicity, we  report the ROC curves computed over the whole set of matches  without applying the 5-fold split on the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  We also report in the graph, as a baseline, a random clas-  sifier, which is expected to provide points along the diagonal  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', number of true positives equal to the number of false  positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' An ideal classifier instead would instead provide a  single point where TPR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00 and FPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In a ROC  curve, the closer the curve is to the main diagonal of the ROC  space, the less accurate the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  For our purposes, there is no benefit in obtaining a  Precision-Recall curve over a ROC curve since, by construc-  tion, the data are equally distributed between positives and  negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  By visual comparison, it is evident how both the algorithms,  Similo and VON Similo, provide ROC curves that are signif-  icantly distant from the main diagonal (corresponding to a  Random classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' At the same time, it is visually evident  that the VON Similo algorithm is overall better than that of  Similo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This advantage in using VON Similo can be quantified  by computing the area below the ROC Curve (Area Under  Curve, or AUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  As a final comparison, in table III, we report a comparison  of the performance of the two approaches in terms of average  accuracy, precision and recall over the set of 33 web pages  used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' From the comparison we notice that  VON Similo has globally better values for all the measures on  the set of apps, with 88% accuracy against 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='3% accuracy  for Similo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We also observe rather high std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' deviation values  for the computed measures, caused by a variabile number of  matches and performance obtained with the tool on individual  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  One can also perform a Wilcoxon (paired) signed rank  test [22] of the accuracy values per app which supports (p-  value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='0001614) the hypothesis that von Similo (mean  accuracy, over the apps, of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='0%) performs differently than  Similo (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Since the argument has been made that  Bayesian statistical analysis can have benefits [23] we also  compared the two approaches with a Bayesian signed rank  test [24] which gave a probability of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='9% that VON Similo  has a higher accuracy than Similo6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Answer to RQ4: As demonstrated by the comparison of the  respective ROC-curves and the statistical tests, VON Similo  (AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='91, mean accuracy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='94, mean accuracy per app  = 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='0) performs significantly better than Similo (AUC =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='88, mean accuracy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='82, mean accuracy per app = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' DISCUSSION  This study has shown that multi-locator-based approaches,  using web element attributes, can provide robust web element  localization, in the real world, for evolving web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  When coupled with visually overlapping node based (VON-  based) locators, the method is robust with a 94 percent success  rate in finding the correct web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This result is significant, especially when put into context  of how tests are maintained in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The study results are  acquired from a random sample of websites with both larger  and smaller amounts of change between versions, controlled  by the time in the past a website was sampled (from 6 months  to 5 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In a real scenario, in industry, the probability of  only more minor changes between versions is theoretically  higher than the scenario presented here since test suites would  be run daily or at least weekly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This leads us to the logical  conclusion that;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' in an authentic setting, the average accuracy  of the proposed approach would be higher than 94 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We  base this discussion on the attribute-based locator technology,  which precision is affected by change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For example, if only one  attribute is changed, the accuracy will be higher than if two or  more attributes have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For shorter iterations between  test runs, the number of changes to the application will be  less and, as such, the solution accuracy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However,  although logically sound, future work is required to verify this  conclusion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', through empirical and industrial research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Regardless, given the result of the study, an interesting  question becomes what industry is willing to accept in terms  of false test results, since, as stated in Section I, robust  6There were clear differences also for precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We used Python  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='10 and the library baycomp for the Bayesian SignedRankTest and (base) R  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='2 for the Wilcoxon test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='75  Rate  Classifier  Similo  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='50  Similo-EML  True  Random  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='00  False Positives Rateweb element localization is one of the core challenges with  automated GUI testing and perceived a root-cause to its lack of  general use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Hence, although the presented results  are significant, compared to, for instance, state of the art  research by Leotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', which reported a success rate of 33-  93 percent (93 percent being a theoretical best case limit) [17],  a philosophical question remains if 94 percent success rate  is good enough?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' A 94 percent success rate still implies that  six web elements will not be found every 100 test actions,  or one faulty test behavior every 17 steps of script code run  on modified parts of the SUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For a test suite ordering in  the hundreds of test cases, 94 percent success rate would still  constitute a significant amount of root-cause analysis and test  script maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, once more, the selected time  elapsed between tests of the website versions were in the  experiment abnormal when compared to the time between  executions of a GUI-based regression test suite in industrial  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  However, as a speculative counterpoint, since any accuracy  below 100 percent seems to be an issue for larger test  suites, the question arises if 100 percent robustness is even  achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Keeping in mind that web element localization  implies searching for a target web element in a context where  there is often not an exact match, only a partial match, due  to changes in the attributes, appearance, or behavior of the  sought web element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Under these circumstances, we also see  humans having less than a 100 percent success rate, so the  question is how far an automated approach can reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Additionally, the attribute-based approach shows that with  impartial data, we can still identify web elements with high  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' When one of these web elements is identified,  the attributes that are no longer valid can be automatically  repaired—Repair, in this case, implies updating attributes that  are no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Thus, mitigating much of the maintenance  costs associated with test scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Consequently, keeping the  delta between test script versions and the SUT low implies a  larger probability of web element identification success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Thus,  once more speaking towards the potential of the approach’s  ability to reach higher than 94 percent success rate in a real  scenario in industrial practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Conceptually, VON-based locators can work for any DOM-  based multi-locator approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', on other platforms not  used in this paper, such as Android or desktop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However,  VON-based locators only work with multi-locator approaches  and would not work for single locator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', Selenium-based)  scripts due to a lack of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Consider a Selenium  action based on an XPath locator, but said XPath is no longer  available in a new version of the target website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' All candidate  information can be acquired from the new site, but if additional  information is not available for the target web element, there  would not be a way to map the target to any of the candidates  reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This is a delimitation of the approach, and it is  unknown what the minimal amount of information is to make  it work in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This is a subject of future work that entails  ranking the attributes based on their relative power to be used  as locators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Furthermore, the analysis of the attributes used for this  work shows that the 14 attributes that Similo used are the  most frequently populated ones on modern web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Thus,  implying that these provide the most value for web element  localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This result is significant, as previous works have  presented lists of suitable attributes [21], [25], [26], but not  provided empirical evidence to support these claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The list  of attributes provided in this work, although only contempo-  rary, thereby provides insights into how to design more robust  GUI testing tools for the foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Future work could  also investigate if there are notable dependencies, or even  domain aspects, to how these attributes are populated and how,  for further improvements into web element localization and  automated repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Such analysis should also look at the relative  power of different attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Looking at the list of attributes in  Figures 5 and 6 we note some interesting observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' ID is  an often mentioned attribute for web element localization [17],  [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, as shown by the results, ID is seldom used  in practice, implying that reliance on this attribute would have  detrimental effects on web element localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Hence, from a  practical perspective, the attributes need to be weighed in terms  of power versus availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' One or several commonly avail-  able attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', XPath or tag, have a higher probability of  being useful than a perceived more powerful attribute, which  is seldom available, such as ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Further research is, however,  required to evaluate this relative power between attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Another finding concerns the contents of the attributes of  the web elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In Similo, locator parameters are compared  with various comparators such as equals, Levenshtein distance,  and integer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' These are utilized in this research based  on expert judgment, but other comparators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', euclidean dis-  tance, could also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' In future work, a valuable analysis  would investigate the types and variability of attribute data  to assign the most suitable comparators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For instance, XPath  strings are unique to each candidate and are represented as  Strings of longer length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This paper uses Levenshtein distance  for their comparison, but other approaches would likely work  equally well or even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This mapping between key web  element attributes, the characteristics of the attribute data,  and comparators could theoretically be done using machine  learning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', using learning to rank [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Similarly, we can  also optimize the weighting of each comparator-attribute tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  However, this is once more a subject of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Threats to validity  In this section we discuss the threats to the study and its  results pertaining to the internal, external and construct validity  of the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Generalizability: The paper provides two contributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  (1) analysis of web element attributes used in industrial  practice and (2) VON-based locators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' For (1), we perceive  the results to be generally valid, as they are gathered from  commonly used web pages from larger companies that should  be perceived as adopters of best state-of-practice approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  However, for (2), the results are delimited to web element  localization with multi-locators as discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This diminishes the result’s industrial applicability since most  industrial frameworks, like Selenium, utilize a single locator  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' As such, adopting VON-based multi-locators would  also require a technological shift in approaches or tooling for  developing such test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Although scripts that use multi-  locators can be developed manually, we perceive recording  such tests as beneficial from an efficiency perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Subject selection: We took the subjects for this analysis  from the Alexa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='com website (as of writing, the site has been  discontinued), which reduces bias in the sampling as the web-  sites on the list are outside the researchers’ control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However,  we chose only a subset of the list, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', the top 40 pages (33  used after excluding, for instance, broken and duplicate sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This set is still perceived as significant when compared to  other studies in this area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' However, it is unknown  how representative the result is to any general website on the  internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Analysis of the tags used in the web elements of these  web pages does, however, provide us with confidence that all  types of web elements that can, theoretically, be encountered  have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We stress, however, that this research  only focused on web element identification, and therefore  detrimental effects that may arise during test execution were  not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' An example of such an effect is the acquisition  of web element attribute data during a state transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' A lack  of synchronization between the test script and AUT could  result in missing attributes if pre-analysis of web element  information begins before the AUT is fully loaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Comprehensiveness: VON Similo was designed with flex-  ibility in mind to optimize its potential use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' A  drawback of this design is that the comparators and attributes  used in this study are a subset of possible attributes and  comparators, as discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' While not reducing  the validity of the results presented in this work, this implies  that we could achieve better results with other constellations  of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We defend this unorthodox statement by the  presented results that indicate a considerable increase in pre-  cision and recall compared to state-of-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' As such, although  there is no threat to the validity of the results of the current  implementation, given the set of analyzed web pages, we  cannot state if this is the best implementation of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  Further research is thereby required with other constellations  and contexts to optimize the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' CONCLUSIONS AND FUTURE WORK  In this paper, we proposed a novel location algorithm for  web elements in web applications, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', Visually Overlapping  Nodes (VON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' We applied this approach by extending an  existing multi-locator approach (Similo), thereby obtaining the  VON Similo approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  To investigate the accuracy of the algorithm, we performed  an empirical investigation by manually identifying a set of  matches between mutated widgets in different releases of  the same application and then measuring the effectiveness  of the original and extended approaches to determine correct  matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This analysis led us to measure a +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='9% increase  in accuracy for VON Similo with respect to the original  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' It is worth underlining that the original Similo was  already proven as better performing than state-of-the-art multi-  locator algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=', Leotta’s ROBULA+ [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  This work first assessed the possible benefits introduced by  the concept of Visually Overlapping Nodes when performing  GUI testing of web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Therefore, fine-tuning the  approach can be obtained by empirically finding the optimal  values for several fixed coefficients and variables employed by  the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  The algorithms evaluated and compared in this study used  only static weights defined in the original Similo algorithm  and were compatible with state-of-the-art tools like COLOR  [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' Selecting a different set of weights for the attributes used  in the comparison can lead to significantly different results in  terms of precision and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' The same reasoning applies  to selecting the attributes involved in the score computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content='  In our immediate future work, we plan to employ ma-  chine learning-based approaches to identify the optimal set  of attributes, and related weights, to maximize the accuracy  of Similo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' As a further improvement of the algorithm, it is  possible to consider returning as output not only a single  matching web element but a ranked list of the most matching  candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
+page_content=' This evolution of the algorithm can also benefit  from the application of machine-learning-ranking (MLR or  Learning to Rank) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE2T4oBgHgl3EQfZQey/content/2301.03863v1.pdf'}
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diff --git a/ZtAzT4oBgHgl3EQfZPw6/content/tmp_files/2301.01347v1.pdf.txt b/ZtAzT4oBgHgl3EQfZPw6/content/tmp_files/2301.01347v1.pdf.txt
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+A retrospective analysis of mid-infrared observations of the Comet D/Shoemaker-Levy 9
+and Wesley impacts on Jupiter
+J. A. Sinclaira,, C. M. Lisseb, G. S. Ortona, M. Krishnamoorthyc, L. N. Fletcherd, J. Horae, C. Palotaif, T. Haywardg
+aJet Propulsion Laboratory/California Institute of Technology, 48Oak Grove Dr, Pasadena, CA 91109, United States
+bApplied Physics Laboratory, Johns Hopkins University, Baltimore, MD, United States
+cUniversity of Michigan, Ann Arbor, MI, United States
+dSchool of Physics & Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom
+eCenter for Astrophysics, Harvard & Smithsonian, Cambridge, MA, United States
+fFlorida Institute of Technology, Melbourne, FL, United States
+gGemini Observatory, NOIRLab, La Serena, Chile
+Abstract
+We present a retrospective analysis of Earth-based mid-infrared observations of Jupiter capturing the aftermath of the impacts by
+Comet D/Shoemaker-Levy 9 (henceforth SL9) in July 1994 and the unknown object previously termed the “Wesley impactor”
+in July 2009. While the effects of both impacts on Jupiter’s atmosphere have been reported previously, we were motivated to
+re-examine both events using consistent data reduction and analysis methods to enable robust, quantitative comparisons. To study
+the aftermath of the SL9 impacts, we examined infrared (7.8 - 20.5 µm) spectrophotometry of Jupiter measured by the MIRAC
+(Mid-Infrared Camera) on NASA’s Infrared Telescope Facility on July 20, 21 1994 and low-resolution (R = 100) N-band (7 – 13 µm)
+spectroscopy measured by SpectroCam-10 on the Palomar Telescope on July 22 1994. To study the aftermath of the Wesley impact,
+we examined low-resolution (R = 100) N- and Q-band (R = 80, 17 – 25 µm) spectroscopy recorded by Gemini South Telescope’s
+T-ReCS (Thermal-Region Camera Spectrograph). We analyzed the observations with two independent analyses: 1) a least-squares
+search over a grid of candidate mineral species to determine the composition of impact residue and 2) a radiative transfer analysis to
+derive atmospheric information. We observe that the SL9 impact sites are enhanced in stratospheric CH4 emissions at 7.9 µm, due
+to shock heating and adiabatic compression from plume re-entry, and from 8.5 - 11.5 µm due to stratospheric NH3 emission and
+non-gaseous cometary material, in agreement with previous work. In the G impact site, we derive NH3 concentrations of 5.7+4.5
+−2.8
+ppmv at 30 mbar. In new findings, we find that the SL9 impact sites also exhibit a non-gaseous emission feature at 18 - 19 µm.
+The non-gaseous emission at 8.5 - 11.5 µm and 18 - 19 µm emission result is best reproduced by predominantly amorphous olivine
+and (obsidian) silica at similar abundances. The Wesley impact site exhibits enhanced emissions from 8.8 - 11.5 µm and 18 - 19
+µm. We found this could be reproduced by predominantly amorphous olivine and stratospheric gaseous NH3 at concentrations of
+150+338
+−121 ppbv retrieved at 30 mbar. Stratospheric abundances of NH3 are a factor of ∼40 higher in the SL9 impacts compared to
+the Wesley impact, which confirms the former reached deeper, NH3-richer altitudes of Jupiter’s atmosphere compared to the latter.
+The absence of silicas in the Wesley impact would place an upper limit of 10 km/s on the incident angle and ∼9◦ on the entry
+angle of the impactor such that shock heating associated with the impact did not reach temperatures required for silicates to be
+converted.
+Keywords:
+1. Introduction
+Jupiter has the second largest gravitational sink in our solar
+system (after the Sun) and therefore experiences a relatively
+high impact rate compared to the other planets.
+In recent
+decades, several impacts have been detected in Jupiter. Most
+were bolide flashes that left no detectable residual influence on
+Jupiter including the September 10, 2012 impact (Hueso et al.,
+2013) and the April 2020 impact (Giles et al., 2021) detected
+by Juno’s ultraviolet instrument (UVS, Gladstone et al. 2017).
+However, two impact events left prominent perturbations to
+Jupiter’s atmosphere.
+One was the series of impacts by
+Email address: james.sinclair@jpl.nasa.gov (J. A. Sinclair)
+fragments of Comet D/Shoemaker-Levy 9 (hereafter SL9) in
+July 1994 (c.f. the review by Harrington et al. 2004). The other
+was an impact by an unknown body into the nightside of Jupiter
+on July 19, 2009, the aftermath of which was first detected
+by amateur astronomer Anthony Wesley (hereafter the Wesley
+impact, c.f. S´anchez-Lavega et al. 2010). Using spectroscopic
+and imaging observations from Earth-based telescopes, the
+effects of both impact events on the atmosphere of Jupiter were
+investigated intensely.
+SL9 was tidally disrupted into at least 16 fragments, which
+then impacted Jupiter’s atmosphere at ∼40◦S between July
+16 and 22, 1994. The impacts occured at relative velocities
+similar to Jupiter’s escape velocity (∼65 km/sec), shock-heated
+gas to extreme temperatures (30000 - 40000 K, e.g. Zahnle
+Preprint submitted to Icarus
+January 5, 2023
+arXiv:2301.01347v1  [astro-ph.EP]  3 Jan 2023
+
+and Mac Low 1994) and vaporized the cometary material.
+The residue then rose in a plume along the initial entry path,
+outside of the atmosphere, before re-entering the atmosphere
+at near-horizontal trajectories (e.g.
+Griffith et al. (1997),
+Figure 14).
+Upon re-entry, the material re-compressed and
+began sinking in atmosphere, converting kinetic to thermal
+energy and thereby heating the atmosphere as deep as the
+lower stratosphere and enhancing hydrocarbon emissions at
+mid-infrared wavelengths (e.g. Orton et al. 1995; B´ezard et al.
+1997).
+Species such as NH3, normally cold-trapped below
+Jupiter’s tropopause, were excavated to stratospheric altitudes
+and observed in emission at mid-infrared wavelengths (e.g.
+Kostiuk et al. 1996; Griffith et al. 1997).
+Shock-induced
+chemistry and/or delivery from the comet itself enhanced the
+abundances of existing species, such as HCN (e.g. B´ezard et al.
+1997, Noll et al. 1995) and introduced new chemical species
+into the stratosphere including H2O (e.g. Bjoraker et al. 1996,
+Encrenaz et al. 1997) and CO (e.g. Lellouch et al. 1997, Kim
+et al. 1999). In particular, stratospheric H2O and CO continue to
+remain detectable in Jupiter’s stratosphere at the time of writing
+(e.g. Lellouch et al. 2002, Cavali´e et al. 2013, Benmahi et al.
+2020).
+The aftermath of the Wesley impact was first noticed on July
+19 2009 at ∼55◦S as a dark “bruise” at visible wavelengths
+(S´anchez-Lavega et al., 2010) but also bright at near-infrared
+wavelengths (Hammel et al., 2010; de Pater et al., 2010; Orton
+et al., 2011) indicating the presence of material at millibar
+pressure levels. The same region exhibited stratospheric NH3
+emission at mid-infrared wavelengths (e.g. Fletcher et al. 2011,
+de Pater et al. 2010), which indicated an impactor had excavated
+NH3 from below the tropopause to stratospheric altitudes,
+similar to the SL9 impacts. By dynamical arguments, impacts
+on Jupiter are orders of magnitude more likely to be comets
+than asteroids since Jupiter should have cleared the latter from
+its orbit ∼Gyr ago (Schenk et al., 2004). Nevertheless, there
+was a convergence of evidence suggesting the Wesley impactor
+was an asteroid.
+Several studies inferred the material from
+the Wesley impactor was more “rocky” and less icy than
+material from the previous SL9 impacts.
+Using ultraviolet
+observations measured by the Hubble Space Telescope (HST),
+Hammel et al. (2010) noted that the 2009 debris field was
+less diffuse and shrank faster than comparable-sized SL9 fields,
+which implied the Wesley impactor material was heavier and
+denser and therefore sank faster in the atmosphere than the
+SL9 material. While strong, stratospheric heating was observed
+for several days following the SL9 impacts (see above), no
+such heating occured in the aftermath of the Wesley impact,
+as evidenced by the lack of any enhanced, stratospheric CH4
+emissions (e.g. Fletcher et al. 2010, Orton et al. 2011). This
+was interpreted as the impactor material being sufficiently
+heavier/denser such that the plume did not rise as high as the
+upper stratosphere. Enhanced, stratospheric C2H6 emissions
+were observed (Fletcher et al., 2010), which indicated an
+increase in its abundance within the impact region.
+An
+enhancement of C2H6, in contrast to oxidized products such
+as H2O and CO, was more consistent with a “dry” impactor
+with a higher C/O ratio (Zahnle, 1996), as expected for an
+asteroid.
+From the discovery images of the 2009 impact,
+S´anchez-Lavega et al. (2010) inferred an impact trajectory
+elevation angle of 20◦ ± 5◦, much shallower than the 45◦
+angle of SL9. Assuming ballistic trajectories, similar to the
+approach of Pankine and Ingersoll (1999), and comparing the
+visible debris field with the “intermediate” SL9 fragments, they
+estimated the impactor size as 0.5-1 km. These measurements
+left few constraints on the trajectory of the impactor, but Orton
+et al. (2011) examined a suite of possible trajectories and
+determined that (a) an asteroidal origin for the impactor was
+possible, and (b) of the most likely sources of asteroids ejected
+into Jupiter-encountering orbits - Trojans (Levison et al. 1997,
+Duncan et al. (2004)) or Main-Belt Hildas (Di Sisto et al., 2005)
+- that the Hilda family was more likely.
+A further contrast between the two impact events was the
+detection of a 9.1-µm non-gaseous feature in the Wesley
+impact region, attributed to crystalline silicas such as quartz
+and cristobalite (Orton et al., 2011; Fletcher et al., 2011) and
+its (apparent) absence from the SL9 impact. Stringent upper
+limits of silica of 0.5% in abundance have been derived from
+high signal-to-noise ratio spectra of comets, including comets
+9P/Tempel 1, C/Hale-Bopp 1995 O1 (Lisse et al., 2007a)
+and 17P/Holmes (Reach et al., 2010). Similarly, there is no
+spectroscopic evidence of silica on asteroids or Trojans (e.g.
+Emery et al. 2006, Vernazza et al. 2010). The presence of silicas
+in the residue of the Wesley object was therefore interpreted
+to have been formed during the impact by the high-pressure
+(>104 bar) and/or high-temperature (> 1500 K) alteration of
+ferromagnesian silicates, as seen in terrestrial studies of tektites,
+silica magmas and rock fulgurites (Lisse et al., 2009). However,
+such temperatures and pressures were achieved during the
+SL9 impacts (e.g. Mac Low and Zahnle 1994, Deming and
+Harrington 2001, Korycansky et al. 2006). Amorphous olivine
+and amorphous pyroxene (typically found in equal abundances
+in comets) would have been readily converted to silicas and
+so the apparent absence of silicas from the SL9 impacts was
+puzzling.
+Overall, contrasts between the SL9 and Wesley impacts led
+to the inference that the Wesley impactor must be different
+in composition and/or origin from SL9.
+However, previous
+observations of the SL9 and Wesley impacts employed different
+methods for data reduction, calibration and radiative transfer
+calculations and so previous comparisons between the events
+were not immediately robust.
+We were therefore motivated
+to re-examine and quantitatively compare both impacts using
+consistent methods. In this work, we perform an analysis of
+Earth-based, mid-infrared observations recorded following the
+impacts of SL9’s fragments in July 1994 and the Wesley impact
+in July 2009.
+2
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+60
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+7.85 µm 02:18 UT 33
+oW
+a)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+8.57 µm 01:51 UT   17
+oW
+b)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+10.5 µm 01:15 UT 355
+oW
+c)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+10.74 µm 01:55 UT 19
+oW
+d)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+11.25 µm 01:18 UT 357
+oW
+e)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+12.2 µm 01:20 UT 358
+oW
+f)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+12.5 µm 01:22 UT 359
+oW
+g)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+13.0 µm 01:33 UT   6
+oW
+h)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+13.3 µm 01:26 UT   2
+oW
+i)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+10
+20
+30
+40
+50
+ 
+x (arcsec)
+13.6 µm 01:28 UT 3
+oW
+j)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+10
+20
+30
+40
+50
+ 
+x (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+17.2 µm 00:56 UT 344
+oW
+k)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+60
+x (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+17.9 µm 00:59 UT 345
+oW
+l)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+y (arcsec)
+0
+10
+20
+30
+40
+50
+ 
+x (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+20.80 µm 01:00 UT   346
+oW
+m)
+����µ�����
+Figure 1: IRTF-MIRAC images recorded on 1994 July 20. The filter wavelength, time of observation (UT) and central meridian longitude
+(System III) are included in each panel. The aftermath of SL9 impacts are evident as bright regions on the disk, particularly at 7.85, 10.50 –
+11.25 and 17.20 µm.
+2. Observations
+2.1. SL9 impact observations in 1994
+2.1.1. IRTF-MIRAC
+Spectrophotometric images from 7.85 to 20.5 µm were
+measured using the Mid-Infrared Array Camera, MIRAC
+(Hoffmann et al., 1998) on NASA’s 3-m Infrared Telescope
+Facility (IRTF). The data were reduced,
+using standard
+object-minus-sky (A-B) subtraction and flat-fielding steps.
+Absolute calibration was performed by scaling the images to
+convolved, mid-infrared spectra recorded by the Voyager IRIS
+and Cassini CIRS instruments, following the approach used by
+Fletcher et al. (2009b). The noise-equivalent radiances (NESR)
+of the images were calculated by determining the standard
+deviation of sky pixels more than 5” outside of Jupiter’s
+limb.
+We concentrated on observations of the impact sites of the
+G, K and L fragments, as they were bright and had the
+best wavelength coverage. Images recorded on 1994 July 20
+captured the G and L impacts approximately 41 - 43 hours and
+2 - 4 hours after their predicted times of impact, respectively.
+Images on July 21 captured the K impact 18 - 20 hours
+after the predicted time of impact.
+Figures 1 and 2 show
+the images recorded on July 20 and 21 and Table 1 further
+details each image and the time elapsed since the G, K and L
+impacts.
+From each image recorded on July 20, a broadband spectrum
+of the L impact was derived by calculating the mean radiance
+and emission angle between 35-45◦S (planetocentric) and
+330-343◦W (System III). The uncertainty on the mean radiance
+was calculated as the larger of: 1) the standard deviation on the
+mean or 2) the NESR scaled by n0.5
+pixel, where npixel is the number
+of diffraction-resolved pixels averaged. A similar broadband
+3
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+60
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+7.85 µm 06:06 UT 321
+oW
+a)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+8.0 µm 03:43 UT
+  235
+oW
+b)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+8.2 µm 03:47 UT
+237
+oW
+c)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+8.57 µm 02:07 UT 177
+oW
+d)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+9.0 µm 03:50 UT
+239
+oW
+e)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+9.3 µm 03:52 UT
+240
+oW
+f)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+9.7 µm 03:55 UT
+242
+oW
+g)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+10.5 µm 03:09 UT   215
+oW
+h)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+10.74 µm 01:48 UT   166
+oW
+i)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+11.25 µm 03:19 UT 221
+oW
+j)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+12.2 µm 05:27 UT 298
+oW
+k)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+12.5 µm 03:23 UT 223
+oW
+l)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+y (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+x (arcsec)
+13.0 µm 03:24 UT 223
+oW
+m)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+x (arcsec)
+13.3 µm 04:05 UT   248
+oW
+n)
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+50
+ 
+x (arcsec)
+13.6 µm 04:09 UT   251
+oW
+o)
+����µ�����
+Figure 2: As in Figure 1 but using images recorded on 1994 July 21.
+spectrum of the G impact site was extracted from the July 20
+images at same latitude range but longitudes from 23 – 32◦W.
+Using the images recorded on July 21, a broadband spectrum
+of the K impact was similarly extracted using longitudes from
+260-275◦W in the same latitude range.
+We considered several methods for calculating the radiance of
+the unperturbed “background” atmosphere from each image
+in order to quantify the enhancement in emission due to the
+impacts.
+Ultimately, we chose to derive this background
+“off-impact” radiance by calculating the mean radiance within
+the same longitude range of the impacts (see above) but
+over a latitude range of 28-38◦N. The chosen latitude range
+was determined such that the radiances of the impact site
+and the “background” unperturbed site were as similar as
+possible in emission angle, such that both observations sound
+a similar altitude level in Jupiter’s atmosphere and such that
+foreshortening effects are removed when computing a ratio
+between the two. We chose not to calculate the background
+atmosphere using unperturbed areas within the same 35-45◦S
+latitude band.
+By July 20 and July 21, over 9 individual
+fragments of SL9 had impacted the atmosphere and it was
+near-impossible to extract radiances of an unperturbed region
+at a similar emission angle as the impact site. While MIRAC
+images were recorded on July 14 before the SL9 impacts, we
+chose not to use these images to calculate the background
+atmosphere for the following reasons. Firstly, the images on
+July 14, July 20 and 21 captured different longitude ranges
+of Jupiter and thus the sites of the future impacts were either
+not sampled on July 14 or recorded at very different emission
+angles compared to the July 20 - 21 images. Secondly, images
+on July 19/20 were recorded in several CVF (circular-variable
+filter) positions between 8 and 12 µm, which were not recorded
+4
+
+Date
+Instrument
+Time
+λ
+Airmass
+Longitude
+∆tG
+∆tK
+∆tL
+1994-Jul-20
+IRTF/MIRAC
+(UT)
+(µm)
+range
+(hrs)
+(hrs)
+(hrs)
+00:56
+17.2
+2.34
+264 - 64
+41.34
+-
+2.57
+00:59
+17.9
+2.29
+265 - 65
+41.39
+-
+2.62
+01:00
+20.80
+2.27
+266 - 66
+41.41
+-
+2.63
+01:15
+10.5
+2.05
+275 - 75
+41.66
+-
+2.88
+01:18
+11.25
+2.
+277 - 77
+41.71
+-
+2.93
+01:20
+12.2
+1.98
+278 - 78
+41.74
+-
+2.97
+01:22
+12.5
+1.96
+279 - 79
+41.78
+-
+3.
+01:26
+13.3
+1.90
+78 - 278
+41.84
+-
+3.07
+01:28
+13.6
+1.89
+77 - 277
+41.88
+-
+3.10
+01:33
+13.0
+2.15
+74 - 274
+41.96
+-
+3.18
+01:51
+8.57
+2.11
+63 - 263
+42.26
+-
+3.48
+01:55
+10.74
+2.02
+61 - 261
+42.33
+-
+3.55
+02:18
+7.85
+2.19
+47 - 247
+42.71
+-
+3.93
+1994-Jul-21
+IRTF/MIRAC
+01:48
+10.74
+2.44
+86 - 246
+-
+15.28
+-
+02:07
+8.57
+2.26
+97 - 257
+-
+15.6
+-
+03:09
+10.50
+1.30
+135 - 295
+-
+16.63
+-
+03:19
+11.25
+1.26
+141 - 301
+-
+16.8
+-
+03:23
+12.50
+1.25
+143 - 303
+-
+16.86
+-
+03:24
+13.
+1.57
+143 - 303
+-
+16.88
+-
+03:43
+8.00
+1.22
+155 - 315
+-
+17.2
+-
+03:47
+8.20
+1.21
+157 - 317
+-
+17.26
+-
+03:50
+9.00
+1.21
+159 - 319
+-
+17.31
+-
+03:52
+9.30
+1.20
+160 - 320
+-
+17.35
+-
+03:55
+9.70
+1.20
+162 - 322
+-
+17.4
+-
+03:58
+10.20
+1.20
+164 - 324
+-
+17.45
+-
+04:01
+11.70
+1.20
+166 - 326
+-
+17.5
+-
+04:05
+13.30
+1.19
+168 - 328
+-
+17.56
+-
+04:09
+13.60
+1.19
+171 - 331
+-
+17.63
+-
+05:27
+12.20
+1.19
+218 - 378
+-
+18.93
+-
+05:31
+10.30
+1.20
+220 - 380
+-
+19.0
+-
+06:06
+7.85
+1.28
+241 - 401
+-
+19.58
+-
+1994-Jul-22
+Palomar/SC-10
+01:49
+8 - 13.5
+1.64
+308 - 35
+90.23
+-
+-
+Table 1: Details of the observations examining the SL9 G, K and L impact sites. ∆tG, ∆tK and ∆tL denote the time elapsed (in hours) since
+impacts G, K and L.
+on July 14 images. Limiting the study of the K and L impact
+sites in July 19-20 images only to the filters used on July
+14 would remove measurements at key wavelengths where
+non-gaseous species have spectral features.
+In order to quantify the enhancement in emission due to
+impact, we calculated the fractional residual, fr, between
+radiances measured over the impact region, Ri, and the
+off-impact/background region, Rb, as defined in Equation
+1.
+fr = Ri − Rb
+Rb
+(1)
+Figure 3 shows the broadband spectra of the G, L and K
+impact sites, the background atmosphere as detailed above,
+and the fractional residual between them (Equation 1).
+As
+noted in previous work, the site of the L impact is enhanced
+at 7.85 µm due to CH4 emissions enhanced by stratospheric
+fallback heating. Marginal enhancements between 12 and 13
+µm are noted and are attributed to emissions of C2H6 and C2H2,
+which are also enhanced by stratospheric heating and possibly
+by enhancements to their abundances. In both the K and L
+impacts, the 8.5 – 11.5 µm range is enhanced due, in part,
+to emissions from gaseous NH3 lofted into the stratosphere as
+well as non-gaseous compounds produced from the cometary
+debris upon impact.
+Significant enhancements in emission
+are also observed at
+18 - 19 µm for the G and L impacts,
+whereas impact and background radiances at 20.5 µm agree
+within uncertainty. As far as we are aware, there are no gases
+in Jupiter’s atmosphere that have significant spectral features at
+this wavelength and so we attribute this feature to non-gaseous
+compounds produced from the cometary debris. For the G and
+K impact, radiances at 7.85 µm and 12 – 14 µm agree with
+background radiances within uncertainty, which indicates less
+stratospheric heating or chemical alteration relative to the L
+impact site. This suggests that the G and K fragments were
+smaller than the L impactor fragment and therefore produced
+a smaller atmospheric response. Alternatively, the G and K
+impact sites had evolved and cooled in the 18-20 and 41-43
+hours between the predicted times of their impacts and their
+respective measurements. In contrast, the site of the L impact
+was observed only 2 – 4 hours after the predicted impact
+time.
+2.1.2. Palomar/SpectroCam-10
+We also examined spectroscopic measurements recorded on
+1994 July 22 by the SpectroCam-10 (henceforth ‘SC-10’)
+instrument (McGhee, 2000) at the 200-inch (5.1 m) Hale
+telescope at the Palomar Observatory. These spectra capture
+the site of the SL9 G impact approximately 90 hours after the
+5
+
+a) SL9 Impact L: 1994-07-20
+9
+12
+15
+18
+21
+Wavelength (µm)
+0.1
+1
+10
+Radiance (µW cm-2 sr-1 µm-1)
+Impact L
+Off impact
+b) SL9 Impact L: 1994-07-20
+9
+12
+15
+18
+21
+Wavelength (µm)
+0.0
+0.5
+1.0
+1.5
+Fractional Residual
+c) SL9 Impact G: 1994-07-20
+9
+12
+15
+18
+21
+Wavelength (µm)
+0.1
+1
+10
+Radiance (µW cm-2 sr-1 µm-1)
+Impact G
+Off impact
+d) SL9 Impact G: 1994-07-20
+9
+12
+15
+18
+21
+Wavelength (µm)
+0.0
+0.5
+1.0
+1.5
+Fractional Residual
+e) SL9 Impact K: 1994-07-21
+7
+8
+9
+10
+11
+12
+13
+14
+Wavelength (µm)
+0.1
+1
+10
+Radiance (µW cm-2 sr-1 µm-1)
+Impact K
+Off impact
+f) SL9 Impact K: 1994-07-21
+7
+8
+9
+10
+11
+12
+13
+14
+Wavelength (µm)
+0.0
+0.5
+1.0
+1.5
+Fractional Residual
+ηιϕκλµνο�π
+Figure 3: a) IRTF-MIRAC spectrophotometry of the SL9 impact
+site L (red points) and background atmosphere (blue points) on
+1994 July 20, b) the fractional residual between the impact site and
+background (Equation 1). Similar results are shown for SL9 impact
+G on the same date (c, d) and SL9 impact K on 1994 July 21 (e,
+f). Horizontal error bars represent the full width at half maximum
+(FWHM) of the given filter. No Q band images were recorded on
+1994 July 21.
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+10
+20
+30
+40
+50
+60
+70
+Spatial pixel
+ηιϕκλµνο�π
+Figure 4:
+The spectral-spatial image of the Palomar/SC-10
+observation recorded on 1994 July 22. Spatial pixels 0 – 50 capture
+the unperturbed atmosphere west-to-east in the ∼44◦S latitude band
+, the enhanced emissions of SL9 impact G are evident near the limb
+at spatial pixels 55-60 and dark sky off planet is captured from
+spatial pixels 62 – 70.
+predicted time of impact. Table 1 provides further details of the
+Palomar/SC-10 observations recorded.
+We did not examine spectra from the R-impact site, as they
+were largely dominated by blackbody emission with fewer
+identifiable spectral features and have already been presented
+by Nicholson et al. (1995). Spectra were obtained using the
+1x15” slit with the slit length oriented parallel to Jupiter’s
+equator and covering the region of peak brightness due to
+the impact. The slit position also captured 5 – 6 sky pixels
+off Jupiter from which the standard deviation was calculated
+to derive the noise-equivalent radiance.
+Each exposure was
+reduced by subtracting an off-Jupiter exposure. Bad pixels in
+the image were removed by interpolation. A slight tilt of the
+spectral lines in the images was corrected such that they were
+aligned vertically in each image.
+The wavelength grid was
+derived using the known wavelength of telluric ozone. Figure 4
+shows the resulting spectral-spatial image.
+Spatial registration and viewing geometries of each pixel were
+calculated by identifying the location of Jupiter’s limb in the
+spectral-spatial image (Figure 4), the known latitude of the G
+impact (44◦S), the 0.25” pixel scale of the instrument and the
+sub-observer latitude and longitude at the time of observation
+determined by JPL Horizons.
+Initially, the absolute calibration was performed using Callisto,
+just as for the R-impact observations reported by Nicholson
+et al. (1995). However, this resulted in radiances (even away
+from the impact) that were factors of 3 – 4 higher than typical
+values for Jupiter.
+We suggest differences in atmospheric
+seeing between the measurements of Jupiter and Callisto could
+account for these unphysically-large radiances.
+In order to
+calibrate the Palomar/SC-10 data in absolute radiance, we
+instead scaled the observed radiances to CIRS (Composite
+Infrared Spectrometer, Kunde et al. 1996) spectra of Jupiter
+6
+
+Palomar/SC-10: 1994 July 22
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+10
+20
+30
+Radiance (µW cm-2 sr-1 µm-1)
+Impact G
+µ=0.292494
+Background
+µ=0.489952
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+-2
+0
+2
+4
+6
+8
+10
+Fractional residual
+ηιϕκλµνο�π
+Figure 5: The top panel shows Palomar/SpectroCam-10 spectra of
+the SL9 G impact site (red) and an unperturbed region in the same
+latitude band (blue). The lower panel shows the fractional residual
+between the two spectra. Note the difference in emission angles of
+the two observations.
+recorded during the 2000-2001 flyby.
+Using the 2.5 cm−1
+resolution (0.025 µm resolution at 10 µm) CIRS data, we
+computed a mean spectrum between 40-45◦S over all sampled
+longitudes. The CIRS spectrum was then convolved down to
+the coarser 0.1 µm (10 cm−1) resolution of Palomar/SC-10. We
+coadded Palomar/SC-10 spectra over 10 spatial pixels centered
+over the central meridian, and away from the impact region,
+which resulted in a mean emission angle similar to the CIRS
+spectrum. A calibration scale factor was derived by computing
+a ratio of the mean CIRS radiance to the mean Palomar count
+from 8.9 to 13 µm, and subsequently applied to all Palomar
+radiances.
+A coadded spectrum of the G impact site was computed
+by averaging the spectra obtained in spatial pixels 53 – 56
+(approximately longitudes of 23-32◦W), which resulted in a
+mean emission angle of 73◦ (µ ∼ 0.3, where µ is the cosine
+of the emission angle).
+The 15” slit of SC-10 samples
+dark sky off the eastern limb and the 44◦S latitude band
+east of the central meridian. Unfortunately, no spectra were
+measured at µ ∼ 0.3 west of the central meridian, which
+would allow a spectrum of the background atmosphere to be
+computed at a similar viewing geometry as the G impact site
+for comparison.
+We coadded spectra over spatial pixels 40
+– 50 (or longitudes of 0 – 18◦W) as a compromise between
+averaging a sufficient number of spectra to increase the effective
+signal-to-noise ratio and achieving a mean emission angle
+(µ ∼ 0.5) as similar as possible to the impact spectrum.
+We anticipate the difference in emission angle between the
+impact and non-impact region may introduce an offset in
+computing the fractional residual since different emission
+angles sound slightly different levels of the atmosphere (at
+different temperatures). Nevertheless, we continued to analyze
+the spectra such that a spectroscopic analysis of both SL9 and
+Wesley impacts could be performed.
+Figure 5 shows the Palomar spectra of the SL9 G impact site,
+background atmosphere and the fractional residual (Equation
+1) between the two locations.
+As observed by MIRAC at
+other SL9 sites (Figure 3), the G impact site is enhanced in
+CH4 emission at ∼8 due to stratospheric heating and between
+9 – 11 µm due to NH3 gas lofted in to the stratosphere
+and non-gaseous compounds produced from the cometary
+material.
+2.2. Wesley impact observations in 2009
+Low-resolution (R = 100) N-band (7.5-13 µm) and (R =
+80) Q-band (17 – 25 µm) spectra were measured using the
+Thermal-Region Camera Spectrograph, T-ReCS, on the 8-m
+Gemini South Telescope on 2009 July 24, approximately four
+(Earth) days or ∼10 Jupiter rotations after the effects of the
+impact were first observed.
+The slit of the spectrograph
+was aligned parallel to Jupiter’s equator, centered over the
+latitude of the Wesley impact (∼60◦S), and measurements
+were repeated while Jupiter rotated. 7 N-band and 6 Q-band
+measurements were recorded over the course of the night,
+providing multiple emission angle spectra of the impact site
+(with the core at ∼304◦W) as it rotated across Jupiter’s disk
+as well spectra of the unperturbed atmosphere in the same
+latitude band.
+The measurements are detailed in Table 2.
+Spectra were also measured of the Cohen standard HD216032
+(Cohen et al., 1999), which were used to perform the nominal,
+absolute calibration of the Jupiter spectra.
+These spectra
+were previously analyzed and presented by Fletcher et al.
+(2011) and Orton et al. (2011).
+Initially, they found that
+the N- and Q-band spectra could not be fit simultaneously
+with a sensible atmospheric model, which was attributed to
+the wavelength-dependent effects of telluric contamination. In
+order to resolve this, Fletcher et al. (2011) first calculated the
+ratio between spectra of the impact and background atmosphere
+such that telluric contamination is removed.
+Second, they
+computed a series of synthetic spectra from 7 - 25 µm over
+a range of emission angles using an atmospheric model derived
+from Cassini-CIRS (Composite Infrared Spectrometer, Kunde
+et al. 1996) measurements during the 2001 flyby (Fletcher et al.,
+2009a). The impact-to-background ratios were then applied to
+synthetic spectra (at the appropriate emission angle) to compute
+absolute spectra in radiance units without telluric artefacts.
+Further details of this technique are provided in Section 4.1 of
+Fletcher et al. (2011). We likewise adopt and re-analyze this
+7
+
+Date
+Instrument
+Time
+Spectrum Airmass
+CML
+2009-Jul-24
+Gemini-S/T-ReCS
+04:25
+N1
+1.176
+278
+04:31
+N2
+1.160
+281
+04:38
+N3
+1.145
+285
+04:45
+N4
+1.132
+290
+04:51
+N5
+1.119
+293
+04:57
+N6
+1.108
+297
+05:05
+N7
+1.097
+302
+06:35
+Q1
+1.041
+356
+06:41
+Q2
+1.042
+360
+06:48
+Q3
+1.045
+4
+06:55
+Q4
+1.048
+8
+07:01
+Q5
+1.052
+12
+07:08
+Q6
+1.057
+16
+Table 2: Details of the observations capturing the aftermath of the Wesley impact. Spectra beginning with ‘N’ and ‘Q’ are N-band (7.5 - 13
+µm) and Q-band (17 - 25 µm) spectra, respectively. ‘CML’ denotes the central meridian longitude at the time of observation in System III.
+(a) T-ReCS N band
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+2
+4
+6
+8
+Radiance (µW cm-2 sr-1 µm-1)
+Impact
+µ=0.592346
+Background
+µ=0.531442
+(b) T-ReCS Q band
+18
+20
+22
+24
+Wavelength (µm)
+0
+2
+4
+6
+8
+10
+12
+Radiance (µW cm-2 sr-1 µm-1)
+Impact
+µ=0.292949
+Background
+µ=0.437701
+(c) T-ReCS N band: similar µ
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+2
+4
+6
+8
+Radiance (µW cm-2 sr-1 µm-1)
+Impact
+µ=0.592346
+Background
+µ=0.576769
+(d) T-ReCS Q band: similar µ
+18
+20
+22
+24
+Wavelength (µm)
+0
+2
+4
+6
+8
+10
+12
+Radiance (µW cm-2 sr-1 µm-1)
+Impact
+µ=0.391369
+Background
+µ=0.425395
+(e) T-ReCS N band residual
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+Fractional residual
+(f) T-ReCS Q band residual
+18
+20
+22
+24
+Wavelength (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+Fractional residual
+ηιϕκλµνο�π
+Figure 6: T-ReCS N band (a) and Q band (b) coadded spectra of the Wesley impact (red spectra) and a background region for comparison
+(blue) as recorded on 2009 Jul 24. The cosine of emission angle of the spectra (µ) are shown. Panels (c) and (d) show similar spectra except
+where individual observations were omitted such that the coadded spectra of the impact and background were similar as possible in emission
+angle. Panels (e) and (f) show the fractional residual (Equation 1) between the spectra shown in panels (c) and (d).
+8
+
+10
+15
+20
+25
+Wavelength (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Normalized fractional residual
+Wesley: Gemini-S/T-ReCS
+SL9 G: Palomar/SC-10
+SL9 K: IRTF/MIRAC
+SL9 L: IRTF/MIRAC
+SL9 G: IRTF/MIRAC
+ηιϕκλµνο�π
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Normalized fractional residual
+ηιϕκλµνο�π
+Figure 7: Comparison of the fractional residuals of the SL9 and Wesley impact sites from all datasets. The top panels shows the 7.5 - 25 µm
+range, the bottom panel focuses on the 7.5 - 13 µm range. The fractional residuals have been normalized to unity for ease of comparison.
+corrected version of the T-ReCS spectra in this work.
+In inverting the T-ReCS spectra, Fletcher et al. (2011) used
+the multi emission angle observations of the same location
+to improve the vertical sensitivity of retrieved atmospheric
+parameters. In this work, we instead adopt a single, average
+spectrum such that the treatment of spectra for the SL9 and
+Wesley impacts are similar as possible.
+Nominally, spectra
+of the impact and background atmosphere were computed by
+coadding individual spectra recorded at respective longitude
+ranges of 299 -307◦W and 316-325◦W, as shown in Figure
+6a-b. While using all available spectra within these longitude
+ranges results in a higher signal-to-noise ratio, they capture
+the impact site and background atmosphere at very different
+mean emission angles. For a direct comparison of the spectra
+between impact site and background atmosphere, we repeated
+the coaddition of spectra over the same longitude ranges but
+omitted a subset of individual spectra such that the impact and
+background spectra were similar in mean emission angle. These
+are shown in Figure 6c-d and the fractional residual between
+them are shown in Figure 6e-f.
+The atmosphere at the site of the Wesley impact is enhanced in
+emission from 8.8 to 12 µm and 17 – 19 µm in comparison to
+the unperturbed atmosphere. Again, the 8.8-to-12 µm feature
+was nominally attributed to some mixture of stratospheric NH3
+gas (in emission) and non-gaseous material.
+The 17 – 19
+µm feature was also attributed to non-gaseous material.
+In
+Section 3.1, we adopt the fractional residual computed from
+spectra similar in emission angle (Figure 6c-f) as an emissivity
+spectrum of the impact region.
+For the purpose of comparing relative wavelength dependence
+between the impacts, the fractional residuals for each dataset
+were normalized, as shown in Figure 7.
+This higlights two
+main contrasts between the SL9 impacts and the Wesley impact.
+First, the SL9 impact sites exhibited enhanced 7 – 8 µm
+stratospheric emissions whereas the Wesley impact site was not
+enhanced in CH4 emissions outside of uncertainty, which is in
+agreement with previous work (e.g. Orton et al. 2011,Fletcher
+et al. 2011). Second, the broad N-band feature extends from 8.8
+to 13 µm whereas for SL9, the broad feature extends to shorter
+wavelengths of ∼8.5 µm. As we demonstrate in Section 3.1,
+the N-band feature extends to shorter wavelengths in the SL9
+impact due to the presence of obsidian (glassy silica) and its
+absence from the Wesley impact spectrum.
+3. Analysis & Results
+We performed two separate analyses to determine the species
+responsible for the observed emission features in the SL9
+and Wesley impact sites. Each analysis has advantages and
+9
+
+disadvantages over the other.
+First, the fractional residual
+between the spectra of the impact region and a region away
+from the impact was calculated and then normalized.
+The
+normalized fractional residual spectrum was then modelled
+by performing a least-squares search over a grid of candidate
+mineral species with varying abundances.
+In this part of
+the analysis, gaseous NH3 in Jupiter’s stratosphere was also
+treated as a “mineral” and included in the model grid search.
+The advantage of this method is that gaseous NH3 and
+non-gaseous mineral species are constrained simultaneously.
+The disadvantage is that the calculation of the fractional
+residual assumes the temperature of the impact region and
+off-impact region are similar and were measured at a similar
+emission angle. In the Wesley impact, negligible stratospheric
+heating was observed over the impact region (Figure 6, Fletcher
+et al. 2011) and both regions were sampled at a range of
+emission angles, which allowed spectra similar in emission
+angle to be calculated. However, for SL9, strong stratospheric
+heating was observed over the impact regions and it was
+impossible to sample the impact and off-impact region at
+similar emission angles.
+Thus, the assumption of similar
+temperature and emission angle is valid for the Wesley impact
+but is a poor assumption for the SL9 impact. This analysis is
+detailed further in Section 3.1.
+In the second approach, we inverted the spectra using the
+NEMESIS radiative transfer code (Irwin et al., 2008).
+The
+vertical profiles of temperature and stratospheric hydrocarbons
+were allowed to vary and retrievals were performed over
+a grid of NH3 vertical profiles.
+The combination of
+parameters producing a synthetic spectrum that minimized
+the goodness-of-fit was interpreted to represent the gaseous
+component of the spectrum. Wavelengths where the synthetic
+spectra did not adequately fit the observed spectra were
+interpreted to result from non-gaseous emission from impactor
+material and the wavelength dependence of the non-gaseous
+emission was compared with the absorptance spectra of several
+candidate mineral species. Unlike Fletcher et al. (2011), we did
+not attempt to include mineral species in the radiative transfer
+inversion to avoid introducing degenerate parameters such as
+aerosol size distribution and vertical profiles to an already large
+parameter space.
+The advantage of this method is that the
+effects of temperature and emission angle on the planet are
+fully characterized, a range of vertical profiles of NH3 can
+be tested, and absolute temperatures and NH3 concentrations
+are constrained. The disadvantage is that the inversion could
+misinterpret non-gaseous emission as gaseous emission. This
+analysis is detailed further in Section 3.2.
+3.1. Mineralogical dust modelling
+Mineralogical spectral modeling was performed by adopting a
+collection of candidate mineralogical species, performing a grid
+search to find a combination of species that produced a model
+emissivity spectrum that minimized the goodness-of-fit (χ2/n,
+where n is the number of degrees of freedom) to the emergent
+emissivity spectra of both impacts (Figure 5 and 6). For an
+optically thin refractory residue, a composite spectrum can be
+written as the combination of the contributions from individual
+species (Lisse et al., 2009), as in Equation 2
+Fv = Bν[Tatm]
+∆2
+�
+i
+� amax
+amin
+Qabs,i(a, λ)πa2 dni(r)
+da da
+(2)
+where ∆ is the distance between the observer and the dust,
+Bν is the Planck function at wavelength λ and atmospheric
+temperature Tatm.
+a is the particle radius, r is the distance
+from the Sun, Qabs,i is the emission efficiency of the ith
+species and dn/da is the differential particle size distribution
+as a function of radius, a. The emitted flux depends on the
+composition (location of spectral features) and the particle size
+(feature to continuum contrast). The particle size distribution
+was assumed to be a power law or power law reduced
+at small particle sizes to model radiation-pressure blowout
+effects. For the candidate mineralogical species, we nominally
+considered ∼100 species that have previously been used to fit
+the mid-infrared spectra of dust produced by comets, asteroids
+and jovian trojans in our solar system and by exocomets,
+exoasteroids, and aggregating planetesimals in other nearby
+star systems (Lisse et al., 2006, 2007a,b; Lisse et al., 2008;
+Lisse et al., 2009, 2012, 2017), including silica-rich hyper
+velocity impact systems like HD172555 (Lisse et al., 2009;
+Lisse et al., 2020). These species include multiple amorphous
+silicates with olivine-like and pyroxene-like composition;
+multiple ferromagnesian silicates (forsterite, fayalite, clino-
+and ortho-pyroxene, augite, anorthite, bronzite, diopside, and
+ferrosilite); silicas, both amorphous and crystalline (obsidian,
+tektites, quartz, cristobalite, tridymite); phyllosilicates (such as
+saponite, serpentine, smectite, montmorillonite, and chlorite);
+sulfates (such as gypsum,
+ferrosulfate,
+and magnesium
+sulfate); oxides (including various aluminas, spinels, hibonite,
+magnetite, and hematite); Mg/ Fe sulfides (including pyrrohtite,
+troilite, pyrite, and ningerite); carbonate minerals (including
+calcite, aragonite, dolomite, magnesite, and siderite); water-ice,
+clean and with carbon dioxide, carbon monoxide, methane,
+and ammonia clathrates; carbon dioxide ice; graphitic and
+amorphous carbon; and the neutral and ionized polycyclic
+aromatic hydrocarbon (PAH) emission models of Draine and
+Li (2007). The Supplementary material of Lisse et al. (2006)
+details the sources of absorption spectra of the aforementioned
+species.
+However, for both impacts in this study, the
+wavelengths at which non-gaseous spectral features were and
+were not observed (Figure 7) ruled out the large majority of
+these nominal species.
+Ultimately, we only considered the
+mineralogical species listed in Table 3.
+Since gaseous NH3 lofted into the stratosphere has a significant
+contribution to the observed spectral features, we also included
+and treated gaseous NH3 as a “mineral” in our spectral fitting.
+In order to compute a emissivity spectra of NH3, we forward
+modelled spectra of Jupiter with a range of vertical NH3 profiles
+and computed their fractional difference. Section 3.2 provides
+further details of the radiative transfer model and atmospheric
+model of Jupiter used to compute forward model spectra. The
+nominal, vertical NH3 profile was retrieved from Cassini-CIRS
+10
+
+a) NH3 vertical profiles
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+Volume Mixing Ratio
+103
+102
+10
+1
+10-1
+10-2
+10-3
+Pressure (mbar)
+f=0.2
+f=0.25
+f=0.3
+f=0.35
+f=0.4
+f=0.45
+f=0.5
+f=0.55
+f=0.6
+f=0.65
+f=0.7
+f=0.75
+f=0.8
+f=0.85
+f=0.9
+f=0.95
+f=1.0
+b) Forward models
+7
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+2
+4
+6
+8
+Radiance (µW cm-2 sr-1 µm-1)
+c) Emissivity spectra
+7
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Fractional residual
+v1: 0.9 - 0.2
+v2: 1.0 - 0.55
+v3: 0.7 - 0.2
+v4: 1.0 - 0.6
+v5: 0.65 - 0.2
+v6: 1.0 - 0.65
+v7: 0.55 - 0.2
+ηιϕκλµνο�π
+Figure 8:
+(a) A family of NH3 vertical profiles with varying
+fractional scale heights, as indicated in the legend, to parameterize
+stratospheric NH3.
+Panel (b) shows the corresponding forward
+model spectra at T-ReCS/SC-10 spectral resolution using the same
+color conversion. Panel (c) shows seven NH3 emissivity spectra
+computed by finding the fractional residual between two forward
+model spectra, as detailed by the legend. For example, spectrum
+‘v1’ was computed by finding the residual between the forward
+model spectra using fractional scale heights of 0.9 and 0.2.
+spectra at 45◦S on Jupiter and assumes NH3 is well mixed with
+a volume mixing ratio of 360 ppmv at pressures higher than
+890 mbar, and then decreases in abundance at lower pressures
+according to a fractional scale height of 0.21 (see Appendix A
+for further details). In order to parameterize NH3 gas lofted
+into the stratosphere, we adopted the approach presented by
+Fletcher et al. (2011), where the NH3 profile, q, is set to the
+CIRS-derived profile, qCIRS (p) at pressures higher than a cutoff
+pressure, p0, and then a decrease in abundance with altitude
+quantified by a fractional scale height, f. This is quantified in
+Equation 3 below.
+q(p ≥ p0) = qCIRS (p)
+q(p < p0) = q0
+� p
+p0
+� 1−f
+f
+(3)
+q0 is the abundance of NH3 at pressure, p0, f is a fractional
+scale height (between 0 and 1). For the purposes of computing
+emissivity spectra of NH3, we assumed p0 = 330 mbar and
+varied f between 0.2 and 1.0 in increments of 0.05. However,
+we note to readers that the vertical profile of NH3 in the SL9
+impact regions may be more complex than the power law
+parameterization we adopted here.
+Figure 8a, b show the resulting vertical profiles of NH3
+and corresponding forward model spectra at T-ReCS/SC-10
+spectral resolution. We computed a range of NH3 emissivity
+spectra by calculating the fractional residual between different
+combinations of forward model spectra, as shown in Figure
+8c. The goal was to produce a range of spectra with varying
+strengths of the stronger, 10.3- and 10.8-µm lines with respect
+to the weaker lines between 9 - 10 µm and 11 - 12 µm. In
+modeling the emission spectra of the SL9 and Wesley impacts,
+each emissivity spectrum was adopted in turn, and scaled by a
+factor at all wavelengths. We found that emissivity spectrum
+v2 (the fractional residual between the forward models using
+fractional scale heights of 1.0 and 0.55) yielded the best fits
+to the Palomar/SC-10 spectra of the SL9 G impact, whereas v6
+yielded the best fits to the Gemini/T-ReCS spectra of the Wesley
+impact.
+After testing the spectral mineralogical fitting with several
+different size distributions, we found that a single power law
+of dn/da ∼ a−4.20 optimized the fit to the Palomar/SC-10
+spectra of the SL9 G impact whereas a power law index
+of -4.30 best reproduced the Gemini-S/T-ReCS spectra of
+the Wesley impact.
+Both are extremely steep, as expected
+for impact-produced dust populations dominated by small,
+sub-micron- to micron-sized particles (Lisse et al., 2009;
+Takasawa et al., 2011; Johnson et al., 2012). While T-ReCS
+spectra of the Wesley impact recorded both the N and Q
+band, Palomar spectra of the SL9 impact only recorded the
+N band.
+In order for the analysis to be as consistent as
+possible for both impacts, we included the IRTF-MIRAC
+broadband 17.2-, 17.9 and 20.8-µm measurements of the SL9
+G impact (Figures 3) in the fitting of the Palomar spectra.
+The MIRAC measurements were scaled by a factor of 4 such
+that the difference in fractional residual between the Palomar
+N-band and MIRAC Q-band measurements was similar as the
+difference in fractional residual between the MIRAC N- and
+Q-band measurements.
+Finally, for both impacts, we omitted wavelengths 9.7 - 9.9
+µm from the spectral fitting since these wavelengths are
+obscured by telluric ozone. For Palomar/SC-10 spectra of SL9,
+wavelengths shorter than ∼8.5 µm contain strong stratospheric
+CH4 emission, which we did not want to be interpreted
+11
+
+Species
+SL9 G (1994)
+Wesley (2009)
+%
+∆χ2
+%
+∆χ2
+Amorphous Olivine
+0.3
+16.6
+0.65
+27.3
+Amorphous Pyroxene
+0
+0
+0
+0
+Obsidian
+0.32
+1.3
+0
+0
+Fosterite
+0.03
+0.04
+0.01
+0
+Fayalite
+0
+0
+0
+0
+Diopside
+0.08
+0.1
+0
+0
+Ferrosite
+0
+0
+0.07
+0.06
+Ortho Enstatite
+0
+0
+0.02
+0.01
+Australite Tektite
+0
+0
+0
+0
+Bediasite Tektite
+0.03
+0.2
+0
+0
+MgFeS
+0
+0
+0
+0
+PAH
+0
+0
+0
+0
+NH3 (g)
+0.2
+1.0
+0.24
+0.7
+Amorphous Carbon
+0
+0
+0
+0
+H2O (s)
+0.02
+0.1
+0
+0
+χ2/n
+1.1
+0.91
+Table 3: The list of species included in the spectral fitting, the derived proportion of each mineral (%) and the change in the reduced χ2 (∆χ2)
+when that species is omitted from the spectral fitting. Results for the SL9 G impact site are shown in the middle column and for the Wesley
+impact in the right-hand column.
+Palomar/SC-10, 1994-Jul-22
+9
+12
+15
+18
+Wavelength  (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional residual
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized absorptance
+Amorphous Olivine
+Obsidian
+Fosterite
+Diopsene
+Bedias. Tektite
+NH3 (g)
+H2O (s)
+Total
+χ2/n = 0.437
+Palomar/SC-10, 1994-Jul-22
+8
+9
+10
+11
+12
+13
+Wavelength  (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional residual
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized absorptance
+ηιϕκλµνο�π
+Figure 9: The observed fractional residual of SL9 Impact G using Palomar/SC-10 spectroscopy and the best-fitting total mineralogical spectrum
+(solid grey). The top panel shows the full wavelength range, the lower panel focuses on the 8 - 13 µm range. The individual species that
+contribute to the total mineralogical curve (see Table 3) are shown and colored according to the legend. Areas covered by 45◦-lines denote
+the wavelengths omitted in performing the mineralogical fitting due to contamination by telluric O3 or jovian, stratospheric (gaseous) CH4
+emission).
+12
+
+Gemini/T-ReCS, 2009-Jul-24
+9
+12
+15
+18
+21
+24
+Wavelength  (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional residual
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized absorptance
+Amorphous Olivine
+Fosterite
+Ferrosite
+Ortho Enstatite
+NH3 (g)
+Total
+χ2/n = 0.870
+Gemini/T-ReCS, 2009-Jul-24
+8
+9
+10
+11
+12
+13
+Wavelength  (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional residual
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized absorptance
+ηιϕκλµνο�π
+Figure 10: The observed fractional residual of the Wesley impact using Gemini-S/T-ReCS spectroscopy and the best-fitting total mineralogical
+spectrum (solid grey). The top panel shows the full wavelength range, the lower panel focuses on the 8 - 13 µm range. The individual species
+that contribute to the total mineralogical curve (see Table 3) are shown and colored according to the legend. Areas covered by 45◦-lines denote
+the wavelengths omitted in performing the mineralogical fitting due to contamination by telluric O3 or jovian, stratospheric (gaseous) CH4
+emission).
+as non-gaseous emission by the spectral fitting.
+For the
+Gemini/T-ReCS spectrum of the Wesley impact, we found
+that including the 8.0 - 8.7 µm drove unphysical solutions
+for mineralogical compositions.
+In order to analyze both
+impacts/observations as consistently as possible, we chose to
+omit wavelengths shorter than 8.7 µm in the fitting of both
+observations.
+Table 3 details the best-fitting compositions of minerals for
+each impact and Figures 9 and 10 compare the fractional
+residuals of both impacts with the best-fitting mineralogical
+composition. For the SL9 G impact, we found that the main
+components producing significant emissivities were amorphous
+olivine, silica (in the form of obsidian), and gaseous NH3 in the
+stratosphere. A small amount of the water ice, diopside, and
+forsterite typically found in comets was present in the D/SL9
+case, but their absence from the synthetic spectrum had only a
+small, statistically-insignificant effect on the reduced χ2. We
+found no evidence of amorphous pyroxene in the D/SL9 case,
+which was initially unexpected since comets typically have a
+∼1:1 ratio of pyroxenaceous to olivinaceous species (e.g. Lisse
+et al. 2006, 2007b). As we discuss in Section 4, we suggest
+that all the cometary pyroxene was converted into silicas by
+the strong heating (> 1000 K) associated with the impact. A
+similar finding was seen in the HD172555 exodisk system by
+Lisse et al. (2009).
+By contrast, the Wesley impact emission appears composed
+of almost pure amorphous olivine, with some ferromagnesian
+pyroxenes mixed in at an ∼8:1 ratio, together with gaseous
+NH3. No emission due to silica is apparent in our new reduction
+and spectral analysis, a marked difference from the findings
+of Fletcher et al. (2011). Without any evidence of silica, it is
+impossible to understand this mixture of materials being formed
+at high temperatures. In order to explain the conversion of the
+usually dominant crystalline olivine into amorphous olivine, we
+suggest that the Wesley residue was formed at low temperatures
+and high pressures, as discussed further in Section 4.
+Figures 9 and 10 also demonstrate the a posteriori fit of the
+best-fitting mineralogical composition to the 8.0 - 8.7 µm
+spectral region that was omitted from the least-squares search.
+For SL9, the fit to the Palomar/SC-10 spectrum between 8.3 -
+8.7 µm is adequate (within uncertainty). The fit is expectedly
+poorer from 8 - 8.3 µm, which captures stratospheric CH4
+emission that was not included in the grid search.
+For
+the Wesley impact, the fit of the best-fitting mineralogical
+composition does not adequately fit the 8.3 - 8.7 µm region.
+This suggests a missing (and currently unknown) source of
+opacity from 8.3 - 8.7 µm in the impact region, which is in
+13
+
+absorption rather than in emission.
+3.2. Radiative transfer inversions
+We adopt the NEMESIS (Non-linear optimal Estimator for
+MultivariatE spectral analysis, Irwin et al. (2008)) radiative
+transfer code to perform inversions of the data. This code has
+been used extensively in previous investigations (Orton et al.,
+2011; Fletcher et al., 2009b, 2010, 2011). Upper-tropospheric
+temperatures were sensed using H2-related collision-induced
+absorption (CIA) continuum that dominates wavelengths in the
+17-25 µm range, with coefficients derived by ab initio models
+for H2-H2 (Fletcher et al., 2018a), H2-He (Birnbaum et al.,
+1996), and H2-CH4 (Borysow and Frommhold, 1986).
+The
+spectroscopic line information for CH4 and its isotopologues,
+C2H2, C2H4, C2H6, NH3 and PH3, which are the relevant and
+dominant gaseous species at the wavelengths of this study,
+were adopted from Supplementary Table 1 of Fletcher et al.
+(2018b).
+For computational efficiency, we chose to perform forward
+models
+and
+inversions
+using
+the
+correlated-k
+method,
+where absorption coefficients of all relevant gases within a
+wavenumber are sorted in order of line strength and the
+distributions of line strengths, or k distribution, is computed
+(Lacis and Oinas, 1991; Rodgers, 2000).
+In order to
+model both Gemini/ T-ReCS and Palomar/SC-10 spectra, the
+aforementioned spectroscopic line information was convolved
+with a triangular line function with a FWHM of ∆ν= 10 cm−1
+and ∆ν = 5.55 cm−1 to model the N-band (7 – 13 µm) and
+Q-band (17 – 25 µm), respectively. K-distributions in each band
+were calculated and concatenated such that the N- and Q-band
+spectra could be modelled simultaneously.
+3.2.1. Modeling Palomar/SC-10 spectra
+Adopting a initial guess or a priori atmosphere derived
+from
+Cassini-CIRS
+(Composite
+Infrared
+Spectrometer)
+measurements of 40◦S in 2001 (see Appendix
+A), we fit
+the observed Palomar/SC-10 8–13 µm spectra of the G
+impact site (Figure 5) by allowing the vertical profiles of
+temperature, NH3, the abundance of tropospheric aerosol and
+C2H4 to vary.
+The tropospheric aerosol and stratospheric
+C2H4 abundances were parameterized by retrieving a scale
+factor applied to all altitudes of their respective CIRS-derived
+profiles. Temperature was allowed to vary continuously at all
+altitudes with wavelengths 8.2 – 8.5 µm providing sensitivity
+to the upper troposphere (1– 5 mbar) and CH4 (8 - 8.1 µm)
+and C2H6 emission (11.3 – 13.0 µm) providing sensitivity to
+the lower stratosphere (50 mbar < p < 0.1 mbar).
+Outside
+of the range of sensitivity, retrieved temperatures tend back
+to a priori (CIRS-derived) values.
+As in Section 3.1, we
+parameterized the vertical profile of NH3 using the approach
+presented in Fletcher et al. (2011).
+A 2-dimensional model
+grid was computed with 13 p0 values ranging from 50 mbar to
+7 mbar in increments of 50 mbar and fractional scale heights
+from 0.2 to 1 in steps of 0.05. The vertical profiles of NH3
+of the model grid were adopted in turn, fixed and the SC-10
+spectrum of the SL9 G impact was inverted.
+In previous
+work analyzing observations of the SL9 impacts, a range of
+parameterizations of the stratospheric NH3 vertical profile were
+adopted, including a step profile (e.g. Fast et al. 2002) and a
+double-peaked profile (e.g. Griffith et al. 1997). However, the
+observations analyzed in those studies were of high spectral
+resolving powers (∼104 to 107) and therefore better resolve the
+vertical structure in the impact regions compared to the ∼102
+resolving powers analyzed in this work. We therefore chose
+to parameterize the vertical profile of NH3 using a simple
+power law index (two parameters) to avoid a highly degenerate
+parameter space and to be consistent with our parameterization
+of the NH3 vertical profile in analyzing T-ReCS spectra of the
+Wesley impact.
+At mid-infrared wavelengths, the magnitude of stratospheric
+hydrocarbon emissions is modulated both by the vertical
+temperature profile and the abundance of the emitting species.
+In previous work (e.g.
+Nixon et al. 2007; Nixon et al.
+2010; Fletcher et al. 2016; Sinclair et al. 2020), stratospheric
+temperatures are normally derived from fitting CH4 emission
+at 7.2 – 8.5 µm since the vertical profile of CH4 is
+generally assumed to be horizontally homogeneous in the lower
+stratosphere.
+However, in this work, there are only a few
+spectral points capturing CH4 emission in this wavelength
+range and with a poorer signal-to-noise ratio (SNR) due to
+telluric obscuration.
+As a result, the vertical temperature
+profile is largely constrained from fitting C2H6 emission, since
+it is sampled by a greater number of spectral points and
+at a higher SNR. Our analysis assumes the CIRS-derived
+C2H6 profile at 45◦S derived from Cassini-CIRS measurements
+recorded during the 2001 Jupiter flyby, however, stratospheric
+abundances of C2H6 do vary temporally due radiative forcing
+and dynamics (e.g. Melin et al. 2018, Hue et al. 2018). Thus,
+absolute stratospheric temperatures and abundances should be
+interpreted with caution.
+Figure 11a-d shows the reduced χ2 fits to the Palomar spectrum
+at 8.0 – 8.5 µm (capturing continuum and stratospheric
+CH4 emission), 8.5 – 11.5 µm (predominantly NH3 gaseous
+emission and non-gaseous impact residue ), 11.5 – 13.0 µm
+(continuum and stratospheric C2H6 emission) and over all
+wavelengths.
+We find there is no model atmosphere within
+the described model grid that fits all wavelengths adequately,
+i.e. with χ2/n ∼ 1. While the 8.5 – 11.5 µm spectral range is
+relatively better fit (χ2/n ∼ 20) with models assuming higher
+values of p0 and f (i.e. higher NH3 abundances over a larger
+vertical range of atmosphere), the same models produce too
+much NH3 emission and therefore a poorer fit in the 11.5 –
+13.0 µm spectral range. This is also demonstrated in Figure 12.
+We attribute the inability to fit all wavelengths with the same
+temperature, aerosol and ammonia profile due to the presence
+of significant non-gaseous emission predominantly in the 8.5 –
+11.5 µm spectral range.
+Given the inability to fit all wavelengths of the spectrum due
+to non-gaseous emission, we instead modelled the spectrum in
+the following way. The retrievals of temperature and aerosol
+abundance were repeated across the same model grid of NH3
+14
+
+a) Fit from 8.0 - 8.5 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.163
+0.369
+0.575
+0.781
+0.987
+1.193
+1.399
+χ2/n
+b) Fit from 8.5 - 11.5 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+19.8
+28.6
+37.5
+46.4
+55.2
+64.1
+73.0
+χ2/n
+c) Fit from 11.5 - 13 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+1.06
+4.16
+7.27
+10.38
+13.49
+16.60
+19.71
+χ2/n
+d) All wavelengths
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+15.5
+21.3
+27.0
+32.7
+38.5
+44.2
+50.0
+χ2/n
+e) Fit from 8 - 8.5 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.163
+0.369
+0.575
+0.781
+0.987
+1.193
+1.399
+χ2/n
+f) Fit from 8.5 - 11.5 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+19.8
+28.6
+37.5
+46.4
+55.2
+64.1
+73.0
+χ2/n
+g) Fit from 11.5 - 13 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+1.06
+4.16
+7.27
+10.38
+13.49
+16.60
+19.71
+χ2/n
+h) All wavelengths
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+15.5
+21.3
+27.0
+32.7
+38.5
+44.2
+50.0
+χ2/n
+ηιϕκλµνο�π
+Figure 11: Reduced χ2 values in fitting the Palomar/SC-10 spectra as a function of deviation pressure, p0 and fractional scale height of the
+NH3 (gas) profile. The left column shows results derived from inverting the full 8 – 13 µm spectral range, the right column shows results
+derived from inverting the 8 – 8.5 and 11.5 – 13 µm spectral range (and radiances in the 8.5 – 11.5 µm range were forward modelled). are
+shown in the left column, Results are shown at wavelengths of 8 – 8.5 µm (R branch of CH4 emission, continuum), 8.5 – 11.5 µm (NH3 and
+non-gaseous emission), 11.5 – 13 µm (C2H6 emission) and all wavelengths. White lines denote χ2/n increments of 2. White crosses denote
+model results that fit the spectra within absolute χ2
+min + 2.3 (the 1-σ confidence level) and reproduce the observed strength of NH3 emission at
+10.36 and 10.75 µm within uncertainty.
+profiles (see above) but using only the 8 – 8.5 and 11.5 – 13
+µm spectral ranges. These ranges were chosen because they
+capture CH4 and C2H6 emission features, and also correspond
+to wavelength ranges where the candidate mineralogical species
+(see Section 3.1) have significantly weaker absorption features.
+The retrieved atmosphere was then forward-modelled across
+the entire 8 – 13 µm spectral range and the reduced χ2/n values
+were calculated. The goodness-of-fit values for these results are
+shown in 11e-h. In omitting the 8.5 – 11.5 µm spectral range,
+the retrieval is able to adequately fit (χ2/n ∼ 1) the 8 – 8.5 and
+15
+
+Palomar/SC-10, 1994-Jul-22
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+10
+20
+30
+40
+Radiance (µW cm-2 sr-1 µm-1)
+Best-fitting 8 - 13 µm retrieval
+Best-fitting 8 - 8.5, 11.5 - 13 µm retrieval
+100
+120
+140
+160
+180
+Retrieved Temperature (K)
+103
+102
+10
+1
+10-1
+10-2
+10-3
+Pressure (mbar)
+10-12
+10-10
+10-8
+10-6
+10-4
+10-2
+NH3 VMR
+ηιϕκλµνο�π
+Figure 12:
+The top panel shows the observed Palomar/SC-10
+spectrum (black points with error bars) and model spectra are
+shown as solid, colored lines. The red spectrum corresponds to
+the “best-fitting” result when inverting the full 8 – 13 µm range,
+which produces too much NH3 emission at wavelengths longer than
+11 µm. The blue spectrum is the best-fitting result when inverting
+only the 8 – 8.5 and 11.5 – 13 µm spectral range and that produces
+NH3 emission features of a similar strength as observed within
+uncertainty - see the text for further details. The blue synthetic
+spectrum is considered to represent the component of the observed
+spectrum formed by gaseous emission, as detailed further in the text.
+The bottom panel shows the corresponding temperature and NH3
+profiles (solid lines) associated with the synthetic spectra. Dotted
+lines represent 1-σ uncertainty. The black, solid lines denote the
+a priori (CIRS-derived) temperature and NH3 profiles adopted in
+inverting the spectra. Hashed regions at pressures lower than 0.1
+mbar indicate where the observations have little/no sensitivity.
+11.5 – 13 µm spectral range with the same atmosphere.
+In order to calculate the component of the spectrum produced
+by gaseous emission, we searched the model grid results for
+results that satisfied the following criteria. First, the synthetic
+spectrum must fit the observed spectrum in the 8.0 – 8.5
+and 11.5 – 13 µm ranges with an absolute χ2 value of less
+than χ2
+min + 2.3, which denotes the 1-σ confidence level
+(Press et al., 1992) when varying two parameters.
+Second,
+the synthetic spectrum must reproduce the line-to-continuum
+radiance difference of the strong NH3 emission features at
+10.36 and 10.75 µm within uncertainty on the radiance. The
+models that satisfy these criteria are shown as white crosses
+in Figure 11e-h. A model atmosphere with the NH3 profile
+deviating from the CIRS-derived profile at 650 mbar and
+with a fractional scale height of 0.75 yields the best-fitting
+spectrum that satisfy these criteria. The spectrum associated
+with this model was adopted as the gaseous emission spectrum.
+The uncertainty of the gaseous component of the spectrum
+was determined by calculating the standard deviation of the
+synthetic spectra that satisfied the aforementioned criteria.
+Similarly, at each atmospheric level, the standard deviation
+in NH3 values from the range of the model atmosphere
+that satisfied the aforementioned criteria were calculated and
+adopted as their respective uncertainties.
+We performed a
+similar calculation to determine the uncertainty on temperature
+but found that the standard deviation was smaller than the
+uncertainty on temperature from the best-fitting retrieval and so
+the latter was adopted as the uncertainty on temperature. The
+gaseous emission spectrum and the associated vertical profiles
+of temperature and NH3 and their uncertainties are shown in
+Figure 12.
+Relative to the CIRS-derived a priori temperature profile,
+heating of the atmosphere is evident from ∼100 mbar to ∼0.1
+mbar (with no sensitivity to temperature at lower pressures).
+The warmest temperatures of 174.3 ± 0.2 K were retrieved
+at the 2-mbar level. The observations have limited sensitivity
+to temperature at pressures lower than 0.1 mbar and so we
+cannot rule out heating of the atmosphere at lower pressures.
+At 30 mbar and 0.1 mbar, we derive NH3 concentrations of
+38+7
+−19 ppmv and 5.7+4.5
+−2.8 ppmv, respectively. The non-gaseous
+emission spectrum was derived by calculating the fractional
+residual (Equation 1) between the observed spectrum and the
+best-fitting gaseous emission spectrum (Figure 12 in blue).
+Figure 13 compares the non-gaseous fractional residual with
+the features of candidate mineralogical species.
+The derived non-gaseous emission exhibits a broad feature
+from wavelengths of 8.5 µm to 11.5 µm.
+As in Section
+3.1, the dominant opacity sources responsible for the broad
+feature are a mixture of amorphous olivine, obsidian, and tekt2.
+Superimposed on this broad feature are narrower features at
+wavelengths of 9.6, 9.8, 10.0, 10.2, 10.6 and 10.9 µm with
+(normalized) fractional residuals of 0.56 ± 0.12, 0.74 ± 0.16,
+0.57 ± 0.12, 1.± 0.18, 0.29 ± 0.08 and 0.30 ± 0.15. We do
+not consider the feature at 9.8 µm real due to its proximity
+to telluric O3. Although the observed feature at 10.0 µm is
+consistent in wavelength with forsterite, the mineral exhibits
+a much broader feature (FWHM ∼0.4 µm) compared to what
+is observed (FWHM ∼ 0.2 µm). In addition, forsterite has a
+feature at 11.25 µm, which is twice as strong as its feature
+at 10.0 µm, where no significant residual is observed.
+In
+taking into account the 0.1 µm (R = λ/∆λ = 100) accuracy
+of the wavelength grid, the features at 10.6 and 11.0 µm
+are near-coincident with those of fayalite at 10.5 and 10.9
+µm. However, in acknowledging the 0.1-µm accuracy of the
+wavelength grid, we also note that the features at 9.6, 9.8
+10.0 and 10.2 µm are within 0.1 µm of NH3 lines.
+Thus,
+it remains inconclusive whether the identified narrow features
+16
+
+Impact G, 1994-Jul-22, Palomar/SC-10
+8
+9
+10
+11
+12
+13
+Wavelength  (µm)
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional Residual
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized Absorptance
+Amorphous Olivine
+Amorphous Pyroxene
+Obsidian
+Fosterite
+Fayalite
+Diopsene
+Ferrosite
+Ortho Enstatite
+Austr. Tektite
+Bedias. Tektite
+NH3 (g)
+Amorphous Carbon
+H2O (s)
+ηιϕκλµνο�π
+Figure 13: Comparison of the fractional residual between observed and modelled SC-10 spectra of the SL9 G impact site, which denotes the
+non-gaseous component of the spectrum, and the absorptance spectra of candidate mineralogical species according to the legend. The fractional
+residual was normalized to 1 and all mineral absorptance spectra have been normalized to 0.6 for ease of comparison. The 9.65 – 9.9 µm range
+is hatched due to telluric ozone and so we do not consider any apparent features in this range.
+result from non-gaseous species or poor fitting of the NH3
+emission features.
+3.2.2. Modeling Gemini/T-ReCS spectra
+The atmosphere derived from Cassini-CIRS measurements of
+60◦S in 2001 (see Appendix A) was adopted as the background
+atmosphere and a priori.
+The T-ReCS spectra were then
+inverted over the same model grid of NH3 profiles adopted
+in inverting the SC-10 spectra (see Section 3.2.1).
+In order
+to optimize the signal-to-noise ratio, we adopted the coadded
+spectrum in Figure 6, where all individual spectra sampling the
+impact region were coadded. Initially, we inverted the entire
+T-ReCS spectral range from 7.5 to 13 µm and 17 to 25 µm.
+Figure 14a-d shows the reduced χ2 distributions of the model
+grid in fitting the observations.
+Overall, the fit to the observations is optimized when the
+vertical profile of NH3 deviates from the CIRS profile at 100
+– 200 mbar and decreases in abundance with a high fractional
+scale height. While we can adequately fit the 7.5 – 13 µm
+spectrum with variations in temperature, NH3 and tropospheric
+aerosol, we find that the 17 – 19 µm feature cannot be
+adequately fit with a physically-sensible atmospheric model
+and cannot be adequately fit using an atmospheric model that
+simultaneously fits the 7.5 – 13 µm range.
+This is further
+suggestive the 17 – 19 µm enhancement of the Wesley impact
+site, with respect to the background atmosphere, results from a
+non-gaseous species in the atmosphere.
+As shown in Figure 14b-c, when the 17 – 19 µm was included
+in the spectral inversion, we found that the overall fit to 17 –
+25 µm spectral range was poor. We performed a second set
+of retrievals where the 17 – 19 µm spectral range was omitted
+from the spectral inversion and found that the fit to the 19 – 25
+µm range significantly improved (comparing Figures 14c and
+g). In the remainder of this section, we will therefore discuss
+results from retrievals omitting the 17 – 19 µm range.
+The best-fitting (χ2/n = 0.629) NH3 profile deviates from
+the CIRS-derived profile at p0 = 200 mbar, and decreases
+with a fractional scale height of 1.
+However, we note that
+a range of solutions with p0 ranging from 150 to 350 mbar
+and fractional scale heights from 0.75 to 1.0 also fit the
+observed spectrum with (absolute) χ2 values less than χ2
+min
++ 2.3, which corresponds to the 1-σ confidence level when
+varying two parameters (Press et al., 1992).
+These models
+are marked in Figure 14e – h. A fractional scale height of 1
+imposes a constant abundance of NH3 with altitude from the
+200-mbar level to the top of the atmosphere, which is physically
+unrealistic.
+Instead, the vertical profile of NH3 is likely to
+be approximately constant with altitude from p0 = 2 mbar
+to ∼ 0.1 mbar, over which the observations have sensitivity,
+but then decreases significantly at higher altitudes outside the
+vertical range of sensitivity of the observations. We adopted
+the best-fitting spectrum (see above) as the gaseous component
+of the impact region. The uncertainty of the gaseous spectrum
+was determined by calculating the the standard deviation of
+the synthetic spectra that had (absolute) χ2 less than χ2
+min +
+2.3. At each atmospheric level, the standard deviation in NH3
+values from the range of the model atmosphere that satisfied the
+aforementioned criteria were calculated and adopted as their
+respective uncertainties. We performed a similar calculation
+to determine the temperature uncertainty but found that the
+17
+
+a) Fit from 7.5 - 13 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.58
+5.30
+10.01
+14.72
+19.44
+24.15
+28.87
+χ2/n
+b) Fit from 17 - 19 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+1.04
+1.37
+1.70
+2.03
+2.35
+2.68
+3.01
+χ2/n
+c) Fit from 19 - 25 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.194
+0.519
+0.845
+1.170
+1.496
+1.821
+2.147
+χ2/n
+d) All wavelengths
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.63
+4.41
+8.20
+11.98
+15.77
+19.55
+23.34
+χ2/n
+e) Fit from 7.5 - 13 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.58
+5.30
+10.01
+14.72
+19.44
+24.15
+28.87
+χ2/n
+g) Fit from 19 - 25 µm
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.194
+0.519
+0.845
+1.170
+1.496
+1.821
+2.147
+χ2/n
+h) All wavelengths
+100
+200
+300
+400
+500
+600
+700
+Deviation pressure (mbar)
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional scale height
+0.63
+4.41
+8.20
+11.98
+15.77
+19.55
+23.34
+χ2/n
+ηιϕκλµνο�π
+Figure 14: Reduced χ2 distributions calculated from the observed and modelled T-ReCS spectra. Results in the left column are for retrievals
+where the entire spectral range available (7.5 - 13, 17 - 25 µm) was included in the inversion. χ2/n are shown at wavelengths from a) 7.5 – 13
+µm, b) 17 – 19 µm, c) 19 – 25 µm and d) all wavelengths. Similar results are shown in panels e) – h) but where the 17 – 19 µm range was
+excluded from the inversion. White contours denote increments of χ/n of 0.5. White crosses denote model combinations that allow the 7.5 -
+13 µm and 19 - 25 µm spectral ranges to be fit within (absolute) χ2
+min + 2.3, which corresponds to the 1-σ confidence level.
+standard deviation was smaller than the temperature uncertainty
+from the best-fitting retrieval and so the latter was adopted as
+the real temperature uncertainty.
+Figure 15 compares the observed T-ReCS spectra with the
+gaseous spectrum noted above.
+This result was forward
+modelled at 17 – 19 µm (omitted from the inversion as
+noted above) so this spectral range could be included in the
+comparison. At 30 mbar, we derive NH3 abundances of 150+338
+−121
+ppbv, which is in agreement (within uncertainty) of the results
+derived by Fletcher et al. (2011). As also noted in previous
+work (e.g. Orton et al. 2011, Fletcher et al. 2011), there is no
+evidence of stratospheric heating associated with the Wesley
+impact. In fact, stratospheric temperatures are ∼10 K cooler
+than those derived from Cassini measurements in 2001. In the
+18
+
+2009-Jul-24, Wesley Impact, N band
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+0
+1
+2
+3
+4
+5
+6
+7
+Radiance (µW cm-2 sr-1 µm-1)
+2009-Jul-24, Wesley Impact, Q band
+18
+20
+22
+24
+Wavelength (µm)
+6
+7
+8
+9
+10
+11
+Radiance (µW cm-2 sr-1 µm-1)
+8
+9
+10
+11
+12
+13
+Wavelength (µm)
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Fractional Residual
+18
+20
+22
+24
+Wavelength (µm)
+-0.2
+-0.1
+0.0
+0.1
+0.2
+Fractional Residual
+100
+120
+140
+160
+180
+Retrieved Temperature (K)
+103
+102
+10
+1
+10-1
+10-2
+10-3
+Pressure (mbar)
+10-14
+10-12
+10-10
+10-8
+10-6
+10-4
+10-2
+NH3 VMR
+ηιϕκλµνο�π
+Figure 15: The top row compares observed T-ReCS spectra (black points with error bars) and the best-fitting synthetic spectra (solid, red lines)
+in the N (left) and Q (right) band. The middle row shows the fractional residual between the observed and modelled spectra . The bottom row
+shows the a priori (solid, back) and retrieved temperature profile (solid red, with 1-σ uncertainty shown as dotted lines) according the lower
+axis. The best-fitting NH3 profile (solid black) is shown according the upper axis. Hashed regions at pressures lower than 0.1 mbar indicate
+where the observations have little/no sensitivity.
+Q band, the model spectra cannot adequately fit the observed
+spectra from 17.5 – 19.5 µm. For example, at 18.5 µm, the
+fractional residual is 15.4 ± 4.3% and therefore greater than
+3σ. However, we find we can adequately fit the majority of the
+spectrum in the N band by varying only the vertical profiles of
+temperature, NH3 and scaling the abundance of C2H4.
+4. Discussion
+Using methods as consistently as possible, our analysis of
+mid-infrared data capturing the Jovian atmosphere in the
+aftermath of the SL9 and Wesley impacts have revealed
+similarities and differences between the two events.
+In both impacts, NH3, normally concentrated below Jupiter’s
+tropopause, was transported to stratospheric altitudes by
+ballistic splashback, which produced strong NH3 emission
+features in the N band with the strongest lines at ∼10.4 and
+∼10.8 µm (see Figure 7, as seen in previous work (Griffith et al.,
+1997; Fast et al., 2002; Orton et al., 2011; Fletcher et al., 2011).
+In inverting the spectra of both impacts, we parameterized the
+NH3 vertical profile by using the CIRS-derived profile (see
+Appendix
+A) at pressures higher than a deviation pressure,
+p0 and a decrease in abundance with altitude according to
+a fractional scale height, f.
+The fit to the continua and
+the strength of the NH3 emission lines in the Palomar/SC-10
+spectra of SL9’s G impact was optimized using p0 = 650
+mbar and a fractional scale height of 0.65.
+This results in
+NH3 concentrations of 38.4+6.8
+−18.7 ppmv and 5.66+4.54
+−2.82 ppmv at
+30 and 0.1 mbar, respectively. These are broadly-consistent
+with the findings of Griffith et al. (1997), who derived ∼1
+ppmv NH3 concentrations at 0.1 mbar using data recorded
+∼6 days after K impact, and Fast et al. (2002), who derived
+∼3 ppmv NH3 concentrations at pressures lower than 1 mbar
+using data recorded ∼4 days after the G impact.
+The fit
+to Gemini-South/T-ReCS spectra of the Wesley impact was
+optimized using p0 = 200 mbar. At 30 mbar, we derive NH3
+abundances of 150+338
+−121 ppbv, which is in agreement within
+uncertainty of the results derived by Fletcher et al. (2011).
+However, we note to the reader the our inversion of the spectra
+did not include the opacities of non-gaseous features and so
+19
+
+these concentrations represent an upper limit.
+Nevertheless,
+the higher values of p0 and higher concentrations of NH3
+required to fit the spectra of the SL9 impact, compared to the
+Wesley impact, is consistent with the SL9 fragments entering
+the atmosphere at angles closer to the local zenith, and reaching
+deeper, NH3-richer altitudes.
+In both events, material introduced by the impactors into the
+atmosphere was evident as non-gaseous emission features in
+the spectra of the impact. In order to determine the mineral
+species responsible for these non-gaseous emission features
+and their relative abundances, we performed a least-squares
+search of the mineral species listed in Table 3. In the impact
+sites of SL9, we would have expected a preponderance of
+amorphous olivine and pyroxene, plus water ice and amorphous
+carbon as well as NH3 gas lofted to stratospheric altitudes.
+The results of our mineralogical least-squares fitting indicate
+that amorphous olivine and NH3 gas were significant opacity
+sources in the impact site of SL9’s G fragment. However, our
+analysis detected silica (in the form of obsidian) and an absence
+of amorphous pyroxene.
+This was initially surprising since
+comets are expected to have similar abundances of olivinaceous
+to pyroxenacous species (e.g. Lisse et al. 2006, 2007b). We
+suggest that high temperatures and pressures produced by the
+cometary impacts converted all the cometary pyroxene into
+silicas. While amorphous pyroxene is stable up to temperatures
+of ∼1000 K, amorphous olivine is stable up to ∼1500 K
+and so presumably, the material was heated to a temperature
+between 1000 - 1500 K such that amorphous olivine remained
+unaltered. A similar finding was seen in the HD172555 exodisk
+system by Lisse et al. (2009). As far as we are aware, silica
+was not detected or reported in previous studies of the SL9
+impact residue though this is unsurprising given the majority of
+mid-infrared studies focused on high-resolution spectroscopy
+of NH3 lines between 10 - 11 µm (e.g. Griffith et al. 1997, Fast
+et al. 2002) and did not sample the 9 - 9.5 µm region. Nicholson
+et al. (1995) presented Palomar/SC-10 spectra (the same dataset
+used in this paper) of the R impact, also observing a broad
+N-band feature, but this was interpreted as silicate emission,
+not silica. However, their study did not attempt to derive an
+emissivity spectrum of the impact (by computing a residual
+between the impact and a non-impact region) or model the
+features using a least-squares search of mineral species.
+If the Wesley impactor was a comet, we would also have
+expected a composition of predominantly amorphous olivine
+and pyroxene, plus water ice and amorphous carbon. If the
+Wesley impactor was an asteroid, as suggested previously (e.g.
+Hammel et al. 2010; Fletcher et al. 2010; Orton et al. 2011, we
+would have expected a preponderance of crystalline olivine and
+phyllosilicates plus impact-alteration-produced silicas. Instead,
+our analysis found that the Wesley impact emission appears
+composed of almost pure amorphous olivine, with some
+ferromagnesian pyroxenes mixed in at an ∼8:1 ratio.
+No
+emission due to silicas (expected from 9.1 - 9.4 µm) is apparent
+in our new reduction and spectral analysis, a marked difference
+from the findings of Fletcher et al. (2011) who used the same
+dataset. We believe our results differ due to differences in how
+the spectra were coadded and modelled. Fletcher et al. (2011)
+inverted spectra of the impact obtained over a range of emission
+angles and found the 9.1-µm signature was most evident in the
+higher emission-angle observations, whereas, in this work, we
+averaged individual T-ReCS spectra over a range of emission
+angles to compute a single, coadded spectrum.
+Given the presence of silicas is analysis-dependent, we consider
+its non-detection in this work a tentative result. If we assume
+the non-detection of silica is correct, we must therefore assume
+residue from the Wesley impactor did not reach the required
+temperatures of ∼1000-1500K in order to convert rocky
+silicates into silicas. Instead, we must conclude from our new
+study that the Wesley residue was formed at low temperatures
+and high pressures such that the usually dominant crystalline
+olivine was readily converted into amorphous olivine. Whether
+these pressures were achieved due to the shock of impact,
+or due to the pressures achieved at depth inside a large
+differentiated body, is not clear. We currently favor the impact
+induced shock amorphization explanation, as all reports have
+the Wesley impactor as being relatively small in size (<10 km
+radius), which would be too small to thoroughly self-pressure
+alter its olivine throughout. In addition, such an explanation
+would require removal of the body’s low pressure, near-surface
+crystalline olivine without heating the high pressure phase
+enough to produce silica. This still belabors the question of
+how a body impinging on Jupiter with ∼65 km/sec (Jupiter’s
+escape speed) relative velocity and a huge specific energy
+per kg of material can manage to NOT melt and transform
+its matter.
+As noted above, SL9 was seen to create large
+amounts of glassy silica (a material not found in comets) from
+amorphous pyroxene. A remote possibility is that the Wesley
+2009 impactor entered the atmosphere at an extremely shallow
+angle, which would allow material to spall off the impactor at
+relatively low temperatures. If we take the velocity of impact
+required to convert silicates into silicas as ∼10 km/sec (Lisse
+et al. (2009) and references therein), an impact angle of less
+than tan−1(10/65) ∼ 9◦ with respect to the local horizontal
+would be required in order to impart the impactors kinetic
+energy slow enough to the body that it could have a chance to
+retain all its silicates. This is outside the 20◦ ± 5◦ estimated by
+S´anchez-Lavega et al. (2010) and Pond et al. (2012) in order to
+explain the lack of stratospheric heating (Fletcher et al., 2010)
+and the observed debris deposition between 3- 10 mbar (de
+Pater et al., 2010; Hammel et al., 2010).
+An impact on Jupiter by 0.5- to 1.5-km objects, similar
+in size to the fragments of SL9 and the Wesley impactor,
+is likely in the near future.
+While Schenk et al. (2004)
+inferred from the cratering record on the Galilean moons that
+SL9/Wesley-sized impacts on Jupiter will occur once every
+∼100 years, S´anchez-Lavega et al. (2010) instead calculated
+a rate of once every 10 - 20 years from the observed, recent
+rate.
+For future impact events, we believe a mid-infrared
+characterization of the impact site would allow the impactor
+and its effect on the atmosphere to be characterized in the
+following ways. First, the relative proportions of amorphous
+pyroxanaceous and olivinaceous species would be constrained
+20
+
+from their features at 9, 10, and 19 µm.
+Second, the
+detection or non-detection of silica would indicate whether
+temperatures and pressures produced by the impact reached
+the ∼1500 K or 4 - 6 GPa levels required to temperature
+alter silicate species. Third, if stratospheric NH3 emission is
+detected, concentrations of NH3 could be derived, which in turn
+would constrain the terminal depth of the impactor material.
+While broadband imaging and low-resolution spectroscopy
+readily allowed non-gaseous features from the impactor to
+be identified, the coarse spectral resolving power (R = 100)
+presented in this work introduced uncertainty in identifying
+gaseous vs.
+non-gaseous spectral features and allowed
+only simple (two variable) parameterizations for the vertical
+profile of NH3.
+Characterization of future impacts would
+therefore benefit from N- and Q-band spectroscopy performed
+near-simultaneously and at higher spectral resolving powers
+(1000 < 15000).
+For Earth-based observations, R > 1000
+would also better allow CH4 emissions from the planet to
+be disentangled relative to telluric CH4 features (assuming
+the Earth-Jupiter relative velocity exceeds ∼ 15 km/s). This
+would improve the retrieval of the stratospheric temperature
+field since the CH4 is generally considered to be horizontally
+homogeneous in the lower stratosphere.
+5. Conclusions
+We
+performed
+a
+retrospective
+analysis
+of
+mid-infrared
+Earth-based observations of Jupiter recorded in July 1994
+and July 2009 in order to compare the effects on Jupiter’s
+atmosphere
+by
+Comet
+D/Shoemaker-Levy
+9
+(SL9)
+and
+the Wesley impactor quantitatively.
+Spectrophotometry
+by IRTF/MIRAC and Palomar/Spectrocam-10 in 1994 and
+Gemini/T-ReCS in 2009 were reduced, calibrated and analyzed
+using methods as consistently as possible in order to facilitate
+robust comparisons between the two impacts.
+As presented
+in previous work, we find the 8 – 11.5 µm spectral range of
+both impact sites is significantly enhanced due to stratospheric
+emission from NH3, having been lofted from the upper
+troposphere to the stratosphere by ballistic “blowout”, as well
+as non-gaseous impact material (see below). Also, in agreement
+with previous work, we find that the SL9 impacts exhibited
+enhanced stratospheric CH4 emissions whereas no significant
+enhancement was evident in the site above the Wesley impact.
+In new findings, we determine that the sites of the SL9
+and Wesley impacts both exhibit enhanced emissions at 18
+– 19 µm.
+For example, at 17.9 µm, the SL9 L impact
+site is enhanced by 41.2 ± 11.5% and the Wesley impact
+site is by 16.4 ± 3.7%. In order to detect and quantify the
+species responsible for these observed spectral features, we
+performed two separate but complementary analyses.
+First,
+we performed mineralogical spectral modeling by adopting a
+collection of candidate mineralogical species and performing
+a grid search to find a combination of species that produced a
+model emissivity spectrum that minimized the goodness-of-fit
+to the observations. For the impact site of SL9’s G fragment,
+we find that stratospheric NH3 emission, amorphous olivine
+and silica (in the form of obsidian) are the dominant opacity
+sources responsible for the observed spectral features at 8.5 -
+11 µm and 18 - 19 µm. We find no evidence of amorphous
+pyroxene in the site of SL9’s G fragment, which was initially
+unexpected since comets typically exhibit a ∼1:1 ratio of
+pyroxenaceous species to olivinaceous species. We suggest that
+the high pressures and temperatures produced by the impact
+readily converted cometary pyroxene into silicas.
+For the
+Wesley impact, we find that stratospheric NH3 and amorphous
+olivine are the dominant opacity sources responsible for the
+observed spectral features. We find no evidence of silicas in
+the aftermath of the Wesley impacts.
+This is in contrast to
+previous work (Fletcher et al., 2011) who used the same data but
+processed/analyzed differently, which suggests the detection or
+non-detection of silica is analyssis-dependent.
+We therefore
+consider the non-detection of silica a tentative finding. If the
+non-detection of silica is correct, it would require that material
+from the Wesley impactor was modified at relatively lower
+temperatures such that species retain all their silicates. This
+would require that the Wesley impactor entered the atmosphere
+at a very shallow angle of 9◦, which is shallower than the
+20 ± 5◦ estimated previously (S´anchez-Lavega et al., 2010;
+Pond et al., 2012).
+Our second analysis involved inverting
+the spectra of both impacts using radiative transfer software in
+order to constrain NH3 abundances. For SL9’s G impact, we
+constrain NH3 concentrations of 38.4+6.8
+−18.7 and 5.66+4.54
+−2.82 ppmv at
+30 and 0.1 mbar, respectively, which is broadly consistent with
+previous work (Griffith et al., 1997; Fast et al., 2002) though
+our parameterization of NH3’s vertical profile differed. For the
+Wesley impact, we derive an NH3 concentration of 150 +338
+−121
+ppbv at 30 mbar, which is consistent (within uncertainty) with
+Fletcher et al. (2011). Higher concentrations of NH3 in the
+SL9 impacts is consistent with the fragments reaching a deeper
+terminal depth compared to the Wesley impact.
+6. Acknowledgements
+Some of this research was carried out at the Jet Propulsion
+Laboratory, California Institute of Technology, under a contract
+with the National Aeronautics and Space Administration
+(80NM0018D0004). During the course of this research, Meera
+Krishnamorthy was a student at the California Institute of
+Technology and worked at JPL as an intern in the Caltech
+Summer Undergraduate Research Fellowship (SURF) program,
+supported by these funds.
+Fletcher was supported by a
+European Research Council Consolidator Grant (under the
+European Union’s Horizon 2020 research and innovation
+programme, grant agreement No 723890) at the University of
+Leicester. We thank Mike Chaffin and Sonal Jain for creating
+IDL-compatible color tables1 that are perceptually-uniform,
+colorblind-friendly, and print correctly in black and white.
+Emma Dahl was supported by an appointment to the
+NASA Postdoctoral Program at the Jet Propulsion Laboratory,
+administered by Oak Ridge Associated Universities under
+contract with NASA.
+1https://github.com/planetarymike/IDL-Colorbars
+21
+
+7. Data Availability
+Archiving
+of
+the
+IRTF/MIRAC
+images
+and
+the
+Gemini-South/T-ReCS spectra at the Planetary Data System
+(PDS) is in progress (at the time of writing). Until they are
+available on the PDS, the data will be made available upon
+request. The calibrated spatial-spectral Palomar/SC-10 image
+of Jupiter is available at the Mendeley data archive2. Raw data
+files and the software used to process the Palomar data would
+also be made available upon request.
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+Letters 733, L39. doi:10.1088/2041-8205/733/2/L39.
+Vernazza, P., Carry, B., Emery, J., Hora, J.L., Cruikshank, D., Binzel, R.P.,
+Jackson, J., Helbert, J., Maturilli, A., 2010. Mid-infrared spectral variability
+for compositionally similar asteroids: Implications for asteroid particle size
+distributions. Icarus 207, 800–809. doi:10.1016/j.icarus.2010.01.
+011.
+Zahnle, K., 1996. Dynamics and chemistry of SL9 plumes, in: Noll, K.S.,
+Weaver, H.A., Feldman, P.D. (Eds.), IAU Colloq. 156: The Collision of
+Comet Shoemaker-Levy 9 and Jupiter, pp. 183–212.
+Zahnle, K., Mac Low, M.M., 1994.
+The Collision of Jupiter and Comet
+Shoemaker-Levy 9. Icarus 108, 1–17. doi:10.1006/icar.1994.1038.
+Appendix A. Cassini-CIRS retrievals
+During Cassini’s 2000/2001 flyby of Jupiter, four maps of 2.5
+cm−1 ‘MIRMAP’ observations - ‘ATMOS02A’, ‘ATMOS02B’,
+‘ATMOS02C’ and ‘ATMOS02D’ - were acquired from January
+1st to Jan 11th.
+Spectral from all four maps were sorted
+into 2◦ latitude bins, nyquist-sampled by 1◦S and coadded.
+The effective noise on these spatially-binned observations was
+assumed to be the largest of either: 1) the standard deviation of
+the mean or 2) the noise-equivalent spectral radiance (NESR)
+spectrum of CIRS by a factor of (0.5/∆˜ν)(1/(
+√
+N) + 1/(
+√
+M)),
+where ∆˜ν is the spectral resolution of the target spectra, N is
+the number of averaged target spectra and M is the number of
+deep-space spectra.
+The spectra at 40◦S (the latitude of the SL9 impacts) and
+60◦S (the latitude of the Wesley impact) were inverted by
+varying temperature, NH3, tropospheric aerosol, C2H2, C2H4
+and C2H6.
+Inversions were performed using the NEMESIS
+radiative transfer code (Irwin et al., 2008). The sources of line
+data, the atmospheric model are identical to those described in
+Section 3.2.
+The
+vertical
+temperature
+profile
+was
+allowed
+to
+vary
+continuously at all altitudes. The vertical profile of NH3 was
+parameterized with three variables: 1) a deep volume mixing
+ratio (qdeep) at pressures higher than a knee pressure, 2) the
+knee pressure (pknee) and 3) a fractional scale height ( fNH3).
+For tropospheric aerosol, a grey absorber was assumed with an
+a priori base pressure (pbase) of 0.7 bar, an optical depth (τ1) of
+1 and a fractional scale height ( fcloud) of 0.2, were assumed, and
+all three parameters were varied. The vertical profiles of C2H2,
+C2H4 and C2H6 were adopted from a photochemical model of
+Jupiter (Moses and Poppe, 2017) and varied by applying a scale
+factor to all altitudes.
+Comparisons of the observed and synthetic spectra and the
+corresponding retrieval parameters are shown in Figures
+A.16 and A.17.
+The retrieved atmospheric parameters
+at 40◦S and 60◦S are adopted as a priori values for
+inversions
+of
+the
+Palomar/SC-10
+and
+Gemini-S/T-ReCS
+spectra, respectively.
+24
+
+Cassini-CIRS (January 2001), 40oS
+7
+8
+9
+10
+11
+Wavelength (µm)
+0
+2
+4
+6
+8
+Radiance (µW cm-2 sr-1 µm-1)
+11
+12
+13
+14
+15
+16
+17
+Wavelength (µm)
+0
+2
+4
+6
+8
+10
+Radiance (µW cm-2 sr-1 µm-1)
+100
+120
+140
+160
+180
+Retrieved Temperature (K)
+103
+102
+10
+1
+10-1
+10-2
+10-3
+Pressure (mbar)
+10-12
+10-10
+10-8
+10-6
+10-4
+10-2
+NH3 VMR
+τ1 = 2.1 ± 0.3
+p1 = 700 mbar
+f1 = 0.29 ± 0.1
+qNH3 = 357.8 ± 32.5 ppmv
+pknee = 885 mbar
+fNH3 = 0.21 ± 0.01
+C2H2  x (0.60 ± 0.05)
+C2H4  x (1.00 ± 0.05)
+C2H6  x (0.78 ± 0.03)
+χ2/n = 1.1726409
+ηιϕκλµνο�π
+Figure A.16: Inversions of Cassini-CIRS spectra recorded during the 2001 Jupiter flyby at 40◦S. The top and middle panels show observed
+spectra (black circles with error bars) from 7 - 11 µm and 11 - 17 µm, and synthetic spectra are shown as solid lines. The retrieved atmospheric
+parameters corresponding to the synthetic spectra are shown in the bottom panel and the legend. Black, solid lines represent a priori values,
+solid lines denote retrieved values and dotted lines outline the 1-σ uncertainty.
+25
+
+Cassini-CIRS (January 2001), 60oS
+7
+8
+9
+10
+11
+Wavelength (µm)
+0
+1
+2
+3
+4
+5
+Radiance (µW cm-2 sr-1 µm-1)
+11
+12
+13
+14
+15
+16
+17
+Wavelength (µm)
+0
+1
+2
+3
+4
+5
+6
+7
+Radiance (µW cm-2 sr-1 µm-1)
+100
+120
+140
+160
+180
+Retrieved Temperature (K)
+103
+102
+10
+1
+10-1
+10-2
+10-3
+Pressure (mbar)
+10-12
+10-10
+10-8
+10-6
+10-4
+10-2
+NH3 VMR
+τ1 = 2.36 ± 0.2
+p1 = 700 mbar
+f1 = 0.53 ± 0.2
+qNH3 = 219.5 ± 21.1 ppmv
+pknee = 700 mbar
+fNH3 = 0.15 ± 0.04
+C2H2  x (0.28 ± 0.03)
+C2H4  x (1.00 ± 0.05)
+C2H6  x (0.93 ± 0.03)
+χ2/n = 0.76112759
+ηιϕκλµνο�π
+Figure A.17: Inversions of Cassini-CIRS spectra recorded during the 2001 Jupiter flyby at 60◦S. The top and middle panels show observed
+spectra (black circles with error bars) from 7 - 11 µm and 11 - 17 µm, and synthetic spectra are shown as solid lines. The retrieved atmospheric
+parameters corresponding to the synthetic spectra are shown in the bottom panel and the legend. Black, solid lines represent a priori values,
+solid lines denote retrieved values and dotted lines outline the 1-σ uncertainty.
+26
+
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new file mode 100644
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+page_content=' Krishnamoorthyc, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content=' Horae, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Palotaif, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Haywardg aJet Propulsion Laboratory/California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content=' United States bApplied Physics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' United States dSchool of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' University of Leicester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' University Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Leicester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' LE1 7RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' United Kingdom eCenter for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Harvard & Smithsonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' United States fFlorida Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Melbourne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' FL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' United States gGemini Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' La Serena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Chile Abstract We present a retrospective analysis of Earth-based mid-infrared observations of Jupiter capturing the aftermath of the impacts by Comet D/Shoemaker-Levy 9 (henceforth SL9) in July 1994 and the unknown object previously termed the “Wesley impactor” in July 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While the effects of both impacts on Jupiter’s atmosphere have been reported previously, we were motivated to re-examine both events using consistent data reduction and analysis methods to enable robust, quantitative comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  To study the aftermath of the SL9 impacts, we examined infrared (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 - 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm) spectrophotometry of Jupiter measured by the MIRAC (Mid-Infrared Camera) on NASA’s Infrared Telescope Facility on July 20, 21 1994 and low-resolution (R = 100) N-band (7 – 13 µm) spectroscopy measured by SpectroCam-10 on the Palomar Telescope on July 22 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' To study the aftermath of the Wesley impact, we examined low-resolution (R = 100) N- and Q-band (R = 80, 17 – 25 µm) spectroscopy recorded by Gemini South Telescope’s T-ReCS (Thermal-Region Camera Spectrograph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We analyzed the observations with two independent analyses: 1) a least-squares search over a grid of candidate mineral species to determine the composition of impact residue and 2) a radiative transfer analysis to derive atmospheric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We observe that the SL9 impact sites are enhanced in stratospheric CH4 emissions at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm, due to shock heating and adiabatic compression from plume re-entry, and from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm due to stratospheric NH3 emission and non-gaseous cometary material, in agreement with previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In the G impact site, we derive NH3 concentrations of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 ppmv at 30 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In new findings, we find that the SL9 impact sites also exhibit a non-gaseous emission feature at 18 - 19 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The non-gaseous emission at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm and 18 - 19 µm emission result is best reproduced by predominantly amorphous olivine and (obsidian) silica at similar abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The Wesley impact site exhibits enhanced emissions from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm and 18 - 19 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We found this could be reproduced by predominantly amorphous olivine and stratospheric gaseous NH3 at concentrations of 150+338 −121 ppbv retrieved at 30 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Stratospheric abundances of NH3 are a factor of ∼40 higher in the SL9 impacts compared to the Wesley impact, which confirms the former reached deeper, NH3-richer altitudes of Jupiter’s atmosphere compared to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The absence of silicas in the Wesley impact would place an upper limit of 10 km/s on the incident angle and ∼9◦ on the entry angle of the impactor such that shock heating associated with the impact did not reach temperatures required for silicates to be converted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Keywords: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Introduction Jupiter has the second largest gravitational sink in our solar system (after the Sun) and therefore experiences a relatively high impact rate compared to the other planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In recent decades, several impacts have been detected in Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Most were bolide flashes that left no detectable residual influence on Jupiter including the September 10, 2012 impact (Hueso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2013) and the April 2020 impact (Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2021) detected by Juno’s ultraviolet instrument (UVS, Gladstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, two impact events left prominent perturbations to Jupiter’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' One was the series of impacts by Email address: james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='sinclair@jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='gov (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Sinclair) fragments of Comet D/Shoemaker-Levy 9 (hereafter SL9) in July 1994 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  the review by Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The other was an impact by an unknown body into the nightside of Jupiter on July 19, 2009, the aftermath of which was first detected by amateur astronomer Anthony Wesley (hereafter the Wesley impact, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' S´anchez-Lavega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Using spectroscopic and imaging observations from Earth-based telescopes, the effects of both impact events on the atmosphere of Jupiter were investigated intensely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' SL9 was tidally disrupted into at least 16 fragments, which then impacted Jupiter’s atmosphere at ∼40◦S between July 16 and 22, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The impacts occured at relative velocities similar to Jupiter’s escape velocity (∼65 km/sec), shock-heated gas to extreme temperatures (30000 - 40000 K, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Zahnle Preprint submitted to Icarus January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='01347v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='EP]  3 Jan 2023  and Mac Low 1994) and vaporized the cometary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The residue then rose in a plume along the initial entry path, outside of the atmosphere, before re-entering the atmosphere at near-horizontal trajectories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (1997), Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Upon re-entry, the material re-compressed and began sinking in atmosphere, converting kinetic to thermal energy and thereby heating the atmosphere as deep as the lower stratosphere and enhancing hydrocarbon emissions at mid-infrared wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' B´ezard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Species such as NH3, normally cold-trapped below Jupiter’s tropopause, were excavated to stratospheric altitudes and observed in emission at mid-infrared wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Kostiuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Shock-induced chemistry and/or delivery from the comet itself enhanced the abundances of existing species, such as HCN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' B´ezard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997, Noll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1995) and introduced new chemical species into the stratosphere including H2O (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Bjoraker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1996, Encrenaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997) and CO (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lellouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In particular, stratospheric H2O and CO continue to remain detectable in Jupiter’s stratosphere at the time of writing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lellouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2002, Cavali´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2013, Benmahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The aftermath of the Wesley impact was first noticed on July 19 2009 at ∼55◦S as a dark “bruise” at visible wavelengths (S´anchez-Lavega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010) but also bright at near-infrared wavelengths (Hammel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' de Pater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011) indicating the presence of material at millibar pressure levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The same region exhibited stratospheric NH3 emission at mid-infrared wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011, de Pater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010), which indicated an impactor had excavated NH3 from below the tropopause to stratospheric altitudes, similar to the SL9 impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' By dynamical arguments, impacts on Jupiter are orders of magnitude more likely to be comets than asteroids since Jupiter should have cleared the latter from its orbit ∼Gyr ago (Schenk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nevertheless, there was a convergence of evidence suggesting the Wesley impactor was an asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Several studies inferred the material from the Wesley impactor was more “rocky” and less icy than material from the previous SL9 impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Using ultraviolet observations measured by the Hubble Space Telescope (HST), Hammel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2010) noted that the 2009 debris field was less diffuse and shrank faster than comparable-sized SL9 fields, which implied the Wesley impactor material was heavier and denser and therefore sank faster in the atmosphere than the SL9 material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  While strong, stratospheric heating was observed for several days following the SL9 impacts (see above), no such heating occured in the aftermath of the Wesley impact, as evidenced by the lack of any enhanced, stratospheric CH4 emissions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010, Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This was interpreted as the impactor material being sufficiently heavier/denser such that the plume did not rise as high as the upper stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Enhanced, stratospheric C2H6 emissions were observed (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010), which indicated an increase in its abundance within the impact region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' An enhancement of C2H6, in contrast to oxidized products such as H2O and CO, was more consistent with a “dry” impactor with a higher C/O ratio (Zahnle, 1996), as expected for an asteroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' From the discovery images of the 2009 impact, S´anchez-Lavega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2010) inferred an impact trajectory elevation angle of 20◦ ± 5◦, much shallower than the 45◦ angle of SL9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Assuming ballistic trajectories, similar to the approach of Pankine and Ingersoll (1999), and comparing the visible debris field with the “intermediate” SL9 fragments, they estimated the impactor size as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5-1 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These measurements left few constraints on the trajectory of the impactor, but Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011) examined a suite of possible trajectories and determined that (a) an asteroidal origin for the impactor was possible, and (b) of the most likely sources of asteroids ejected into Jupiter-encountering orbits - Trojans (Levison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  1997, Duncan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2004)) or Main-Belt Hildas (Di Sisto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2005) - that the Hilda family was more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A further contrast between the two impact events was the detection of a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1-µm non-gaseous feature in the Wesley impact region, attributed to crystalline silicas such as quartz and cristobalite (Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011) and its (apparent) absence from the SL9 impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Stringent upper limits of silica of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5% in abundance have been derived from high signal-to-noise ratio spectra of comets, including comets 9P/Tempel 1, C/Hale-Bopp 1995 O1 (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2007a) and 17P/Holmes (Reach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Similarly, there is no spectroscopic evidence of silica on asteroids or Trojans (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Emery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2006, Vernazza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The presence of silicas in the residue of the Wesley object was therefore interpreted to have been formed during the impact by the high-pressure (>104 bar) and/or high-temperature (> 1500 K) alteration of ferromagnesian silicates, as seen in terrestrial studies of tektites, silica magmas and rock fulgurites (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, such temperatures and pressures were achieved during the SL9 impacts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Mac Low and Zahnle 1994, Deming and Harrington 2001, Korycansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Amorphous olivine and amorphous pyroxene (typically found in equal abundances in comets) would have been readily converted to silicas and so the apparent absence of silicas from the SL9 impacts was puzzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Overall, contrasts between the SL9 and Wesley impacts led to the inference that the Wesley impactor must be different in composition and/or origin from SL9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, previous observations of the SL9 and Wesley impacts employed different methods for data reduction, calibration and radiative transfer calculations and so previous comparisons between the events were not immediately robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We were therefore motivated to re-examine and quantitatively compare both impacts using consistent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In this work, we perform an analysis of Earth-based, mid-infrared observations recorded following the impacts of SL9’s fragments in July 1994 and the Wesley impact in July 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2  0  10  20  30  40  50  60  y (arcsec)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 µm 02:18 UT 33 oW a)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='57 µm 01:51 UT   17 oW b)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm 01:15 UT 355 oW c)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='74 µm 01:55 UT 19 oW d)  0  10  20  30  40  50  y (arcsec)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25 µm 01:18 UT 357 oW e)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 µm 01:20 UT 358 oW f)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm 01:22 UT 359 oW g)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm 01:33 UT   6 oW h)  0  10  20  30  40  50  y (arcsec)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 µm 01:26 UT   2 oW i)  10  20  30  40  50  x (arcsec) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 µm 01:28 UT 3 oW j)  10  20  30  40  50  x (arcsec)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 µm 00:56 UT 344 oW k)  0  10  20  30  40  50  60  x (arcsec)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm 00:59 UT 345 oW l)  0  10  20  30  40  50  y (arcsec)  0  10  20  30  40  50  x (arcsec)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='80 µm 01:00 UT   346 oW m) ����µ����� Figure 1: IRTF-MIRAC images recorded on 1994 July 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The filter wavelength, time of observation (UT) and central meridian longitude (System III) are included in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The aftermath of SL9 impacts are evident as bright regions on the disk, particularly at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='50 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Observations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' SL9 impact observations in 1994 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' IRTF-MIRAC Spectrophotometric images from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm were measured using the Mid-Infrared Array Camera, MIRAC (Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1998) on NASA’s 3-m Infrared Telescope Facility (IRTF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The data were reduced, using standard object-minus-sky (A-B) subtraction and flat-fielding steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Absolute calibration was performed by scaling the images to convolved, mid-infrared spectra recorded by the Voyager IRIS and Cassini CIRS instruments, following the approach used by Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2009b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The noise-equivalent radiances (NESR) of the images were calculated by determining the standard deviation of sky pixels more than 5” outside of Jupiter’s limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We concentrated on observations of the impact sites of the G, K and L fragments, as they were bright and had the best wavelength coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Images recorded on 1994 July 20 captured the G and L impacts approximately 41 - 43 hours and 2 - 4 hours after their predicted times of impact, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Images on July 21 captured the K impact 18 - 20 hours after the predicted time of impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Figures 1 and 2 show the images recorded on July 20 and 21 and Table 1 further details each image and the time elapsed since the G, K and L impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' From each image recorded on July 20, a broadband spectrum of the L impact was derived by calculating the mean radiance and emission angle between 35-45◦S (planetocentric) and 330-343◦W (System III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The uncertainty on the mean radiance was calculated as the larger of: 1) the standard deviation on the mean or 2) the NESR scaled by n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 pixel, where npixel is the number of diffraction-resolved pixels averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A similar broadband 3  0  10  20  30  40  50  60  y (arcsec)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 µm 06:06 UT 321 oW a)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm 03:43 UT  235  oW  b)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 µm 03:47 UT  237  oW  c)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='57 µm 02:07 UT 177 oW d)  0  10  20  30  40  50  y (arcsec)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm 03:50 UT  239  oW  e)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 µm 03:52 UT  240  oW  f)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm 03:55 UT  242  oW  g)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm 03:09 UT   215 oW h)  0  10  20  30  40  50  y (arcsec)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='74 µm 01:48 UT   166 oW i)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25 µm 03:19 UT 221 oW j)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 µm 05:27 UT 298 oW k)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm 03:23 UT 223 oW l)  0  10  20  30  40  50  y (arcsec)  0  10  20  30  40  50  x (arcsec) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm 03:24 UT 223 oW m)  0  10  20  30  40  50  x (arcsec) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 µm 04:05 UT   248 oW n)  0  10  20  30  40  50  x (arcsec) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 µm 04:09 UT   251 oW o) ����µ����� Figure 2: As in Figure 1 but using images recorded on 1994 July 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' spectrum of the G impact site was extracted from the July 20 images at same latitude range but longitudes from 23 – 32◦W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Using the images recorded on July 21, a broadband spectrum of the K impact was similarly extracted using longitudes from 260-275◦W in the same latitude range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We considered several methods for calculating the radiance of the unperturbed “background” atmosphere from each image in order to quantify the enhancement in emission due to the impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Ultimately, we chose to derive this background “off-impact” radiance by calculating the mean radiance within the same longitude range of the impacts (see above) but over a latitude range of 28-38◦N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The chosen latitude range was determined such that the radiances of the impact site and the “background” unperturbed site were as similar as possible in emission angle, such that both observations sound a similar altitude level in Jupiter’s atmosphere and such that foreshortening effects are removed when computing a ratio between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We chose not to calculate the background atmosphere using unperturbed areas within the same 35-45◦S latitude band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' By July 20 and July 21, over 9 individual fragments of SL9 had impacted the atmosphere and it was near-impossible to extract radiances of an unperturbed region at a similar emission angle as the impact site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  While MIRAC images were recorded on July 14 before the SL9 impacts, we chose not to use these images to calculate the background atmosphere for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Firstly, the images on July 14, July 20 and 21 captured different longitude ranges of Jupiter and thus the sites of the future impacts were either not sampled on July 14 or recorded at very different emission angles compared to the July 20 - 21 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Secondly, images on July 19/20 were recorded in several CVF (circular-variable filter) positions between 8 and 12 µm, which were not recorded 4  Date Instrument Time λ Airmass Longitude ∆tG ∆tK ∆tL 1994-Jul-20 IRTF/MIRAC (UT) (µm) range (hrs) (hrs) (hrs) 00:56 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='34 264 - 64 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='34 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='57 00:59 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='29 265 - 65 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='39 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='62 01:00 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='27 266 - 66 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='41 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='63 01:15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='05 275 - 75 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='66 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='88 01:18 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 277 - 77 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='71 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='93 01:20 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='98 278 - 78 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='74 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='97 01:22 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='96 279 - 79 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='78 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='63 - 05:27 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='19 218 - 378 - 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='93 - 05:31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='20 220 - 380 - 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 - 06:06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='28 241 - 401 - 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='58 - 1994-Jul-22 Palomar/SC-10 01:49 8 - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='64 308 - 35 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='23 - - Table 1: Details of the observations examining the SL9 G, K and L impact sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  ∆tG, ∆tK and ∆tL denote the time elapsed (in hours) since impacts G, K and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' on July 14 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Limiting the study of the K and L impact sites in July 19-20 images only to the filters used on July 14 would remove measurements at key wavelengths where non-gaseous species have spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to quantify the enhancement in emission due to impact, we calculated the fractional residual, fr, between radiances measured over the impact region, Ri, and the off-impact/background region, Rb, as defined in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' fr = Ri − Rb Rb (1) Figure 3 shows the broadband spectra of the G, L and K impact sites, the background atmosphere as detailed above, and the fractional residual between them (Equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As noted in previous work, the site of the L impact is enhanced at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 µm due to CH4 emissions enhanced by stratospheric fallback heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Marginal enhancements between 12 and 13 µm are noted and are attributed to emissions of C2H6 and C2H2, which are also enhanced by stratospheric heating and possibly by enhancements to their abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In both the K and L impacts, the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm range is enhanced due, in part, to emissions from gaseous NH3 lofted into the stratosphere as well as non-gaseous compounds produced from the cometary debris upon impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Significant enhancements in emission are also observed at 18 - 19 µm for the G and L impacts, whereas impact and background radiances at 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm agree within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  As far as we are aware, there are no gases in Jupiter’s atmosphere that have significant spectral features at this wavelength and so we attribute this feature to non-gaseous compounds produced from the cometary debris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the G and K impact, radiances at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 µm and 12 – 14 µm agree with background radiances within uncertainty, which indicates less stratospheric heating or chemical alteration relative to the L impact site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This suggests that the G and K fragments were smaller than the L impactor fragment and therefore produced a smaller atmospheric response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Alternatively, the G and K impact sites had evolved and cooled in the 18-20 and 41-43 hours between the predicted times of their impacts and their respective measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In contrast, the site of the L impact was observed only 2 – 4 hours after the predicted impact time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Palomar/SpectroCam-10 We also examined spectroscopic measurements recorded on 1994 July 22 by the SpectroCam-10 (henceforth ‘SC-10’) instrument (McGhee, 2000) at the 200-inch (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 m) Hale telescope at the Palomar Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These spectra capture the site of the SL9 G impact approximately 90 hours after the 5  a) SL9 Impact L: 1994-07-20 9 12 15 18 21 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 1 10 Radiance (µW cm-2 sr-1 µm-1) Impact L Off impact b) SL9 Impact L: 1994-07-20 9 12 15 18 21 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 Fractional Residual c) SL9 Impact G: 1994-07-20 9 12 15 18 21 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 1 10 Radiance (µW cm-2 sr-1 µm-1) Impact G Off impact d) SL9 Impact G: 1994-07-20 9 12 15 18 21 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 Fractional Residual e) SL9 Impact K: 1994-07-21 7 8 9 10 11 12 13 14 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 1 10 Radiance (µW cm-2 sr-1 µm-1) Impact K Off impact f) SL9 Impact K: 1994-07-21 7 8 9 10 11 12 13 14 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 Fractional Residual ηιϕκλµνο�π Figure 3: a) IRTF-MIRAC spectrophotometry of the SL9 impact site L (red points) and background atmosphere (blue points) on 1994 July 20, b) the fractional residual between the impact site and background (Equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Similar results are shown for SL9 impact G on the same date (c, d) and SL9 impact K on 1994 July 21 (e, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Horizontal error bars represent the full width at half maximum (FWHM) of the given filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' No Q band images were recorded on 1994 July 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 8 9 10 11 12 13 Wavelength (µm) 0 10 20 30 40 50 60 70 Spatial pixel ηιϕκλµνο�π Figure 4: The spectral-spatial image of the Palomar/SC-10 observation recorded on 1994 July 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Spatial pixels 0 – 50 capture the unperturbed atmosphere west-to-east in the ∼44◦S latitude band , the enhanced emissions of SL9 impact G are evident near the limb at spatial pixels 55-60 and dark sky off planet is captured from spatial pixels 62 – 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' predicted time of impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Table 1 provides further details of the Palomar/SC-10 observations recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We did not examine spectra from the R-impact site, as they were largely dominated by blackbody emission with fewer identifiable spectral features and have already been presented by Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Spectra were obtained using the 1x15” slit with the slit length oriented parallel to Jupiter’s equator and covering the region of peak brightness due to the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The slit position also captured 5 – 6 sky pixels off Jupiter from which the standard deviation was calculated to derive the noise-equivalent radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Each exposure was reduced by subtracting an off-Jupiter exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Bad pixels in the image were removed by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A slight tilt of the spectral lines in the images was corrected such that they were aligned vertically in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The wavelength grid was derived using the known wavelength of telluric ozone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 4 shows the resulting spectral-spatial image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Spatial registration and viewing geometries of each pixel were calculated by identifying the location of Jupiter’s limb in the spectral-spatial image (Figure 4), the known latitude of the G impact (44◦S), the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25” pixel scale of the instrument and the sub-observer latitude and longitude at the time of observation determined by JPL Horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Initially, the absolute calibration was performed using Callisto, just as for the R-impact observations reported by Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, this resulted in radiances (even away from the impact) that were factors of 3 – 4 higher than typical values for Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We suggest differences in atmospheric seeing between the measurements of Jupiter and Callisto could account for these unphysically-large radiances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to calibrate the Palomar/SC-10 data in absolute radiance, we instead scaled the observed radiances to CIRS (Composite Infrared Spectrometer, Kunde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1996) spectra of Jupiter 6  Palomar/SC-10: 1994 July 22 8 9 10 11 12 13 Wavelength (µm) 0 10 20 30 Radiance (µW cm-2 sr-1 µm-1) Impact G µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='292494 Background µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='489952 8 9 10 11 12 13 Wavelength (µm) -2 0 2 4 6 8 10 Fractional residual ηιϕκλµνο�π Figure 5: The top panel shows Palomar/SpectroCam-10 spectra of the SL9 G impact site (red) and an unperturbed region in the same latitude band (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The lower panel shows the fractional residual between the two spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Note the difference in emission angles of the two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' recorded during the 2000-2001 flyby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Using the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 cm−1 resolution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='025 µm resolution at 10 µm) CIRS data, we computed a mean spectrum between 40-45◦S over all sampled longitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The CIRS spectrum was then convolved down to the coarser 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 µm (10 cm−1) resolution of Palomar/SC-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We coadded Palomar/SC-10 spectra over 10 spatial pixels centered over the central meridian, and away from the impact region, which resulted in a mean emission angle similar to the CIRS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A calibration scale factor was derived by computing a ratio of the mean CIRS radiance to the mean Palomar count from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 to 13 µm, and subsequently applied to all Palomar radiances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A coadded spectrum of the G impact site was computed by averaging the spectra obtained in spatial pixels 53 – 56 (approximately longitudes of 23-32◦W), which resulted in a mean emission angle of 73◦ (µ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3, where µ is the cosine of the emission angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The 15” slit of SC-10 samples dark sky off the eastern limb and the 44◦S latitude band east of the central meridian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Unfortunately, no spectra were measured at µ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 west of the central meridian, which would allow a spectrum of the background atmosphere to be computed at a similar viewing geometry as the G impact site for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We coadded spectra over spatial pixels 40 – 50 (or longitudes of 0 – 18◦W) as a compromise between averaging a sufficient number of spectra to increase the effective signal-to-noise ratio and achieving a mean emission angle (µ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5) as similar as possible to the impact spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We anticipate the difference in emission angle between the impact and non-impact region may introduce an offset in computing the fractional residual since different emission angles sound slightly different levels of the atmosphere (at different temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nevertheless, we continued to analyze the spectra such that a spectroscopic analysis of both SL9 and Wesley impacts could be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 5 shows the Palomar spectra of the SL9 G impact site, background atmosphere and the fractional residual (Equation 1) between the two locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As observed by MIRAC at other SL9 sites (Figure 3), the G impact site is enhanced in CH4 emission at ∼8 due to stratospheric heating and between 9 – 11 µm due to NH3 gas lofted in to the stratosphere and non-gaseous compounds produced from the cometary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Wesley impact observations in 2009 Low-resolution (R = 100) N-band (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5-13 µm) and (R = 80) Q-band (17 – 25 µm) spectra were measured using the Thermal-Region Camera Spectrograph, T-ReCS, on the 8-m Gemini South Telescope on 2009 July 24, approximately four (Earth) days or ∼10 Jupiter rotations after the effects of the impact were first observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The slit of the spectrograph was aligned parallel to Jupiter’s equator, centered over the latitude of the Wesley impact (∼60◦S), and measurements were repeated while Jupiter rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 7 N-band and 6 Q-band measurements were recorded over the course of the night, providing multiple emission angle spectra of the impact site (with the core at ∼304◦W) as it rotated across Jupiter’s disk as well spectra of the unperturbed atmosphere in the same latitude band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The measurements are detailed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Spectra were also measured of the Cohen standard HD216032 (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1999), which were used to perform the nominal, absolute calibration of the Jupiter spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These spectra were previously analyzed and presented by Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011) and Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Initially, they found that the N- and Q-band spectra could not be fit simultaneously with a sensible atmospheric model, which was attributed to the wavelength-dependent effects of telluric contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to resolve this, Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011) first calculated the ratio between spectra of the impact and background atmosphere such that telluric contamination is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Second, they computed a series of synthetic spectra from 7 - 25 µm over a range of emission angles using an atmospheric model derived from Cassini-CIRS (Composite Infrared Spectrometer, Kunde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1996) measurements during the 2001 flyby (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The impact-to-background ratios were then applied to synthetic spectra (at the appropriate emission angle) to compute absolute spectra in radiance units without telluric artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Further details of this technique are provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 of Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We likewise adopt and re-analyze this 7  Date Instrument Time Spectrum Airmass CML 2009-Jul-24 Gemini-S/T-ReCS 04:25 N1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='176 278 04:31 N2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='160 281 04:38 N3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='145 285 04:45 N4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='132 290 04:51 N5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='119 293 04:57 N6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='108 297 05:05 N7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='097 302 06:35 Q1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='041 356 06:41 Q2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='042 360 06:48 Q3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='045 4 06:55 Q4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='048 8 07:01 Q5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='052 12 07:08 Q6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='057 16 Table 2: Details of the observations capturing the aftermath of the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Spectra beginning with ‘N’ and ‘Q’ are N-band (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 13 µm) and Q-band (17 - 25 µm) spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' ‘CML’ denotes the central meridian longitude at the time of observation in System III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  (a) T-ReCS N band 8 9 10 11 12 13 Wavelength (µm) 0 2 4 6 8 Radiance (µW cm-2 sr-1 µm-1) Impact µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='592346 Background µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='531442 (b) T-ReCS Q band 18 20 22 24 Wavelength (µm) 0 2 4 6 8 10 12 Radiance (µW cm-2 sr-1 µm-1) Impact µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='292949 Background µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='437701 (c) T-ReCS N band: similar µ 8 9 10 11 12 13 Wavelength (µm) 0 2 4 6 8 Radiance (µW cm-2 sr-1 µm-1) Impact µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='592346 Background µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='576769 (d) T-ReCS Q band: similar µ 18 20 22 24 Wavelength (µm) 0 2 4 6 8 10 12 Radiance (µW cm-2 sr-1 µm-1) Impact µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='391369 Background µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='425395 (e) T-ReCS N band residual 8 9 10 11 12 13 Wavelength (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 Fractional residual (f) T-ReCS Q band residual 18 20 22 24 Wavelength (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 Fractional residual ηιϕκλµνο�π Figure 6: T-ReCS N band (a) and Q band (b) coadded spectra of the Wesley impact (red spectra) and a background region for comparison (blue) as recorded on 2009 Jul 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The cosine of emission angle of the spectra (µ) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Panels (c) and (d) show similar spectra except where individual observations were omitted such that the coadded spectra of the impact and background were similar as possible in emission angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Panels (e) and (f) show the fractional residual (Equation 1) between the spectra shown in panels (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 8  10 15 20 25 Wavelength (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 Normalized fractional residual Wesley: Gemini-S/T-ReCS SL9 G: Palomar/SC-10 SL9 K: IRTF/MIRAC SL9 L: IRTF/MIRAC SL9 G: IRTF/MIRAC ηιϕκλµνο�π 8 9 10 11 12 13 Wavelength (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 Normalized fractional residual ηιϕκλµνο�π Figure 7: Comparison of the fractional residuals of the SL9 and Wesley impact sites from all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The top panels shows the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 25 µm range, the bottom panel focuses on the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 13 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The fractional residuals have been normalized to unity for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' corrected version of the T-ReCS spectra in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In inverting the T-ReCS spectra, Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011) used the multi emission angle observations of the same location to improve the vertical sensitivity of retrieved atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In this work, we instead adopt a single, average spectrum such that the treatment of spectra for the SL9 and Wesley impacts are similar as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nominally, spectra of the impact and background atmosphere were computed by coadding individual spectra recorded at respective longitude ranges of 299 -307◦W and 316-325◦W, as shown in Figure 6a-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While using all available spectra within these longitude ranges results in a higher signal-to-noise ratio, they capture the impact site and background atmosphere at very different mean emission angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  For a direct comparison of the spectra between impact site and background atmosphere, we repeated the coaddition of spectra over the same longitude ranges but omitted a subset of individual spectra such that the impact and background spectra were similar in mean emission angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These are shown in Figure 6c-d and the fractional residual between them are shown in Figure 6e-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The atmosphere at the site of the Wesley impact is enhanced in emission from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 to 12 µm and 17 – 19 µm in comparison to the unperturbed atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Again, the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8-to-12 µm feature was nominally attributed to some mixture of stratospheric NH3 gas (in emission) and non-gaseous material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The 17 – 19 µm feature was also attributed to non-gaseous material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1, we adopt the fractional residual computed from spectra similar in emission angle (Figure 6c-f) as an emissivity spectrum of the impact region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the purpose of comparing relative wavelength dependence between the impacts, the fractional residuals for each dataset were normalized, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This higlights two main contrasts between the SL9 impacts and the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' First, the SL9 impact sites exhibited enhanced 7 – 8 µm stratospheric emissions whereas the Wesley impact site was not enhanced in CH4 emissions outside of uncertainty, which is in agreement with previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011,Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Second, the broad N-band feature extends from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 to 13 µm whereas for SL9, the broad feature extends to shorter wavelengths of ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As we demonstrate in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1, the N-band feature extends to shorter wavelengths in the SL9 impact due to the presence of obsidian (glassy silica) and its absence from the Wesley impact spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Analysis & Results We performed two separate analyses to determine the species responsible for the observed emission features in the SL9 and Wesley impact sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Each analysis has advantages and 9  disadvantages over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' First, the fractional residual between the spectra of the impact region and a region away from the impact was calculated and then normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The normalized fractional residual spectrum was then modelled by performing a least-squares search over a grid of candidate mineral species with varying abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In this part of the analysis, gaseous NH3 in Jupiter’s stratosphere was also treated as a “mineral” and included in the model grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The advantage of this method is that gaseous NH3 and non-gaseous mineral species are constrained simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The disadvantage is that the calculation of the fractional residual assumes the temperature of the impact region and off-impact region are similar and were measured at a similar emission angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In the Wesley impact, negligible stratospheric heating was observed over the impact region (Figure 6, Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011) and both regions were sampled at a range of emission angles, which allowed spectra similar in emission angle to be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, for SL9, strong stratospheric heating was observed over the impact regions and it was impossible to sample the impact and off-impact region at similar emission angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Thus, the assumption of similar temperature and emission angle is valid for the Wesley impact but is a poor assumption for the SL9 impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This analysis is detailed further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  In the second approach, we inverted the spectra using the NEMESIS radiative transfer code (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The vertical profiles of temperature and stratospheric hydrocarbons were allowed to vary and retrievals were performed over a grid of NH3 vertical profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The combination of parameters producing a synthetic spectrum that minimized the goodness-of-fit was interpreted to represent the gaseous component of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Wavelengths where the synthetic spectra did not adequately fit the observed spectra were interpreted to result from non-gaseous emission from impactor material and the wavelength dependence of the non-gaseous emission was compared with the absorptance spectra of several candidate mineral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Unlike Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011), we did not attempt to include mineral species in the radiative transfer inversion to avoid introducing degenerate parameters such as aerosol size distribution and vertical profiles to an already large parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The advantage of this method is that the effects of temperature and emission angle on the planet are fully characterized, a range of vertical profiles of NH3 can be tested, and absolute temperatures and NH3 concentrations are constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The disadvantage is that the inversion could misinterpret non-gaseous emission as gaseous emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This analysis is detailed further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Mineralogical dust modelling Mineralogical spectral modeling was performed by adopting a collection of candidate mineralogical species, performing a grid search to find a combination of species that produced a model emissivity spectrum that minimized the goodness-of-fit (χ2/n, where n is the number of degrees of freedom) to the emergent emissivity spectra of both impacts (Figure 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For an optically thin refractory residue, a composite spectrum can be written as the combination of the contributions from individual species (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009), as in Equation 2 Fv = Bν[Tatm] ∆2 � i � amax amin Qabs,i(a, λ)πa2 dni(r) da da (2) where ∆ is the distance between the observer and the dust, Bν is the Planck function at wavelength λ and atmospheric temperature Tatm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' a is the particle radius, r is the distance from the Sun, Qabs,i is the emission efficiency of the ith species and dn/da is the differential particle size distribution as a function of radius, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The emitted flux depends on the composition (location of spectral features) and the particle size (feature to continuum contrast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The particle size distribution was assumed to be a power law or power law reduced at small particle sizes to model radiation-pressure blowout effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  For the candidate mineralogical species, we nominally considered ∼100 species that have previously been used to fit the mid-infrared spectra of dust produced by comets, asteroids and jovian trojans in our solar system and by exocomets, exoasteroids, and aggregating planetesimals in other nearby star systems (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2006, 2007a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009, 2012, 2017), including silica-rich hyper velocity impact systems like HD172555 (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These species include multiple amorphous silicates with olivine-like and pyroxene-like composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' multiple ferromagnesian silicates (forsterite, fayalite, clino- and ortho-pyroxene, augite, anorthite, bronzite, diopside, and ferrosilite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' silicas, both amorphous and crystalline (obsidian, tektites, quartz, cristobalite, tridymite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' phyllosilicates (such as saponite, serpentine, smectite, montmorillonite, and chlorite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' sulfates (such as gypsum, ferrosulfate, and magnesium sulfate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' oxides (including various aluminas, spinels, hibonite, magnetite, and hematite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Mg/ Fe sulfides (including pyrrohtite, troilite, pyrite, and ningerite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' carbonate minerals (including calcite, aragonite, dolomite, magnesite, and siderite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' water-ice, clean and with carbon dioxide, carbon monoxide, methane, and ammonia clathrates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' carbon dioxide ice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' graphitic and amorphous carbon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' and the neutral and ionized polycyclic aromatic hydrocarbon (PAH) emission models of Draine and Li (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The Supplementary material of Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2006) details the sources of absorption spectra of the aforementioned species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, for both impacts in this study, the wavelengths at which non-gaseous spectral features were and were not observed (Figure 7) ruled out the large majority of these nominal species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Ultimately, we only considered the mineralogical species listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Since gaseous NH3 lofted into the stratosphere has a significant contribution to the observed spectral features, we also included and treated gaseous NH3 as a “mineral” in our spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to compute a emissivity spectra of NH3, we forward modelled spectra of Jupiter with a range of vertical NH3 profiles and computed their fractional difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 provides further details of the radiative transfer model and atmospheric model of Jupiter used to compute forward model spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The nominal, vertical NH3 profile was retrieved from Cassini-CIRS 10  a) NH3 vertical profiles 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 Volume Mixing Ratio 103 102 10 1 10-1 10-2 10-3 Pressure (mbar) f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='35 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='45 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='55 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='65 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='75 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='85 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='95 f=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 b) Forward models 7 8 9 10 11 12 13 Wavelength (µm) 0 2 4 6 8 Radiance (µW cm-2 sr-1 µm-1) c) Emissivity spectra 7 8 9 10 11 12 13 Wavelength (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 Fractional residual v1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 v2: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='55 v3: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 v4: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 v5: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='65 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 v6: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='65 v7: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='55 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 ηιϕκλµνο�π Figure 8: (a) A family of NH3 vertical profiles with varying fractional scale heights, as indicated in the legend, to parameterize stratospheric NH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Panel (b) shows the corresponding forward model spectra at T-ReCS/SC-10 spectral resolution using the same color conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Panel (c) shows seven NH3 emissivity spectra computed by finding the fractional residual between two forward model spectra, as detailed by the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For example, spectrum ‘v1’ was computed by finding the residual between the forward model spectra using fractional scale heights of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' spectra at 45◦S on Jupiter and assumes NH3 is well mixed with a volume mixing ratio of 360 ppmv at pressures higher than 890 mbar, and then decreases in abundance at lower pressures according to a fractional scale height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='21 (see Appendix A for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to parameterize NH3 gas lofted into the stratosphere, we adopted the approach presented by Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  (2011), where the NH3 profile, q, is set to the CIRS-derived profile, qCIRS (p) at pressures higher than a cutoff pressure, p0, and then a decrease in abundance with altitude quantified by a fractional scale height, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This is quantified in Equation 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' q(p ≥ p0) = qCIRS (p) q(p < p0) = q0 � p p0 � 1−f f (3) q0 is the abundance of NH3 at pressure, p0, f is a fractional scale height (between 0 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the purposes of computing emissivity spectra of NH3, we assumed p0 = 330 mbar and varied f between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, we note to readers that the vertical profile of NH3 in the SL9 impact regions may be more complex than the power law parameterization we adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 8a, b show the resulting vertical profiles of NH3 and corresponding forward model spectra at T-ReCS/SC-10 spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We computed a range of NH3 emissivity spectra by calculating the fractional residual between different combinations of forward model spectra, as shown in Figure 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The goal was to produce a range of spectra with varying strengths of the stronger, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3- and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8-µm lines with respect to the weaker lines between 9 - 10 µm and 11 - 12 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In modeling the emission spectra of the SL9 and Wesley impacts, each emissivity spectrum was adopted in turn, and scaled by a factor at all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  We found that emissivity spectrum v2 (the fractional residual between the forward models using fractional scale heights of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='55) yielded the best fits to the Palomar/SC-10 spectra of the SL9 G impact, whereas v6 yielded the best fits to the Gemini/T-ReCS spectra of the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' After testing the spectral mineralogical fitting with several different size distributions, we found that a single power law of dn/da ∼ a−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='20 optimized the fit to the Palomar/SC-10 spectra of the SL9 G impact whereas a power law index of -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='30 best reproduced the Gemini-S/T-ReCS spectra of the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Both are extremely steep, as expected for impact-produced dust populations dominated by small, sub-micron- to micron-sized particles (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Takasawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While T-ReCS spectra of the Wesley impact recorded both the N and Q band, Palomar spectra of the SL9 impact only recorded the N band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order for the analysis to be as consistent as possible for both impacts, we included the IRTF-MIRAC broadband 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2-, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8-µm measurements of the SL9 G impact (Figures 3) in the fitting of the Palomar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The MIRAC measurements were scaled by a factor of 4 such that the difference in fractional residual between the Palomar N-band and MIRAC Q-band measurements was similar as the difference in fractional residual between the MIRAC N- and Q-band measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Finally, for both impacts, we omitted wavelengths 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm from the spectral fitting since these wavelengths are obscured by telluric ozone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For Palomar/SC-10 spectra of SL9, wavelengths shorter than ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm contain strong stratospheric CH4 emission, which we did not want to be interpreted 11  Species SL9 G (1994) Wesley (2009) % ∆χ2 % ∆χ2 Amorphous Olivine 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='65 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 Amorphous Pyroxene 0 0 0 0 Obsidian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 0 0 Fosterite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='01 0 Fayalite 0 0 0 0 Diopside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 0 0 Ferrosite 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='06 Ortho Enstatite 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='01 Australite Tektite 0 0 0 0 Bediasite Tektite 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0 0 MgFeS 0 0 0 0 PAH 0 0 0 0 NH3 (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 Amorphous Carbon 0 0 0 0 H2O (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 0 0 χ2/n 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='91 Table 3: The list of species included in the spectral fitting, the derived proportion of each mineral (%) and the change in the reduced χ2 (∆χ2) when that species is omitted from the spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Results for the SL9 G impact site are shown in the middle column and for the Wesley impact in the right-hand column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Palomar/SC-10, 1994-Jul-22 9 12 15 18 Wavelength  (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Fractional residual -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Normalized absorptance Amorphous Olivine Obsidian Fosterite Diopsene Bedias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Tektite NH3 (g) H2O (s) Total χ2/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='437 Palomar/SC-10, 1994-Jul-22 8 9 10 11 12 13 Wavelength  (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Fractional residual -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Normalized absorptance ηιϕκλµνο�π Figure 9: The observed fractional residual of SL9 Impact G using Palomar/SC-10 spectroscopy and the best-fitting total mineralogical spectrum (solid grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The top panel shows the full wavelength range, the lower panel focuses on the 8 - 13 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The individual species that contribute to the total mineralogical curve (see Table 3) are shown and colored according to the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Areas covered by 45◦-lines denote the wavelengths omitted in performing the mineralogical fitting due to contamination by telluric O3 or jovian, stratospheric (gaseous) CH4 emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 12  Gemini/T-ReCS, 2009-Jul-24 9 12 15 18 21 24 Wavelength  (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Fractional residual -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Normalized absorptance Amorphous Olivine Fosterite Ferrosite Ortho Enstatite NH3 (g) Total χ2/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='870 Gemini/T-ReCS, 2009-Jul-24 8 9 10 11 12 13 Wavelength  (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Fractional residual -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Normalized absorptance ηιϕκλµνο�π Figure 10: The observed fractional residual of the Wesley impact using Gemini-S/T-ReCS spectroscopy and the best-fitting total mineralogical spectrum (solid grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The top panel shows the full wavelength range, the lower panel focuses on the 8 - 13 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The individual species that contribute to the total mineralogical curve (see Table 3) are shown and colored according to the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Areas covered by 45◦-lines denote the wavelengths omitted in performing the mineralogical fitting due to contamination by telluric O3 or jovian, stratospheric (gaseous) CH4 emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' as non-gaseous emission by the spectral fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the Gemini/T-ReCS spectrum of the Wesley impact, we found that including the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm drove unphysical solutions for mineralogical compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to analyze both impacts/observations as consistently as possible, we chose to omit wavelengths shorter than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm in the fitting of both observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Table 3 details the best-fitting compositions of minerals for each impact and Figures 9 and 10 compare the fractional residuals of both impacts with the best-fitting mineralogical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the SL9 G impact, we found that the main components producing significant emissivities were amorphous olivine, silica (in the form of obsidian), and gaseous NH3 in the stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A small amount of the water ice, diopside, and forsterite typically found in comets was present in the D/SL9 case, but their absence from the synthetic spectrum had only a small, statistically-insignificant effect on the reduced χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We found no evidence of amorphous pyroxene in the D/SL9 case, which was initially unexpected since comets typically have a ∼1:1 ratio of pyroxenaceous to olivinaceous species (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2006, 2007b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As we discuss in Section 4, we suggest that all the cometary pyroxene was converted into silicas by the strong heating (> 1000 K) associated with the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A similar finding was seen in the HD172555 exodisk system by Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' By contrast, the Wesley impact emission appears composed of almost pure amorphous olivine, with some ferromagnesian pyroxenes mixed in at an ∼8:1 ratio, together with gaseous NH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' No emission due to silica is apparent in our new reduction and spectral analysis, a marked difference from the findings of Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Without any evidence of silica, it is impossible to understand this mixture of materials being formed at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to explain the conversion of the usually dominant crystalline olivine into amorphous olivine, we suggest that the Wesley residue was formed at low temperatures and high pressures, as discussed further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figures 9 and 10 also demonstrate the a posteriori fit of the best-fitting mineralogical composition to the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm spectral region that was omitted from the least-squares search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For SL9, the fit to the Palomar/SC-10 spectrum between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm is adequate (within uncertainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The fit is expectedly poorer from 8 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 µm, which captures stratospheric CH4 emission that was not included in the grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the Wesley impact, the fit of the best-fitting mineralogical composition does not adequately fit the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This suggests a missing (and currently unknown) source of opacity from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 µm in the impact region, which is in 13  absorption rather than in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Radiative transfer inversions We adopt the NEMESIS (Non-linear optimal Estimator for MultivariatE spectral analysis, Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2008)) radiative transfer code to perform inversions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This code has been used extensively in previous investigations (Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2009b, 2010, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Upper-tropospheric temperatures were sensed using H2-related collision-induced absorption (CIA) continuum that dominates wavelengths in the 17-25 µm range, with coefficients derived by ab initio models for H2-H2 (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2018a), H2-He (Birnbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1996), and H2-CH4 (Borysow and Frommhold, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The spectroscopic line information for CH4 and its isotopologues, C2H2, C2H4, C2H6, NH3 and PH3, which are the relevant and dominant gaseous species at the wavelengths of this study, were adopted from Supplementary Table 1 of Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For computational efficiency, we chose to perform forward models and inversions using the correlated-k method, where absorption coefficients of all relevant gases within a wavenumber are sorted in order of line strength and the distributions of line strengths, or k distribution, is computed (Lacis and Oinas, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Rodgers, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  In order to model both Gemini/ T-ReCS and Palomar/SC-10 spectra, the aforementioned spectroscopic line information was convolved with a triangular line function with a FWHM of ∆ν= 10 cm−1 and ∆ν = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='55 cm−1 to model the N-band (7 – 13 µm) and Q-band (17 – 25 µm), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' K-distributions in each band were calculated and concatenated such that the N- and Q-band spectra could be modelled simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Modeling Palomar/SC-10 spectra Adopting a initial guess or a priori atmosphere derived from Cassini-CIRS (Composite Infrared Spectrometer) measurements of 40◦S in 2001 (see Appendix A), we fit the observed Palomar/SC-10 8–13 µm spectra of the G impact site (Figure 5) by allowing the vertical profiles of temperature, NH3, the abundance of tropospheric aerosol and C2H4 to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The tropospheric aerosol and stratospheric C2H4 abundances were parameterized by retrieving a scale factor applied to all altitudes of their respective CIRS-derived profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Temperature was allowed to vary continuously at all altitudes with wavelengths 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm providing sensitivity to the upper troposphere (1– 5 mbar) and CH4 (8 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 µm) and C2H6 emission (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm) providing sensitivity to the lower stratosphere (50 mbar < p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Outside of the range of sensitivity, retrieved temperatures tend back to a priori (CIRS-derived) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1, we parameterized the vertical profile of NH3 using the approach presented in Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  A 2-dimensional model grid was computed with 13 p0 values ranging from 50 mbar to 7 mbar in increments of 50 mbar and fractional scale heights from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 to 1 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The vertical profiles of NH3 of the model grid were adopted in turn, fixed and the SC-10 spectrum of the SL9 G impact was inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In previous work analyzing observations of the SL9 impacts, a range of parameterizations of the stratospheric NH3 vertical profile were adopted, including a step profile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2002) and a double-peaked profile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, the observations analyzed in those studies were of high spectral resolving powers (∼104 to 107) and therefore better resolve the vertical structure in the impact regions compared to the ∼102 resolving powers analyzed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We therefore chose to parameterize the vertical profile of NH3 using a simple power law index (two parameters) to avoid a highly degenerate parameter space and to be consistent with our parameterization of the NH3 vertical profile in analyzing T-ReCS spectra of the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' At mid-infrared wavelengths, the magnitude of stratospheric hydrocarbon emissions is modulated both by the vertical temperature profile and the abundance of the emitting species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Sinclair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  2020), stratospheric temperatures are normally derived from fitting CH4 emission at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm since the vertical profile of CH4 is generally assumed to be horizontally homogeneous in the lower stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, in this work, there are only a few spectral points capturing CH4 emission in this wavelength range and with a poorer signal-to-noise ratio (SNR) due to telluric obscuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As a result, the vertical temperature profile is largely constrained from fitting C2H6 emission, since it is sampled by a greater number of spectral points and at a higher SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Our analysis assumes the CIRS-derived C2H6 profile at 45◦S derived from Cassini-CIRS measurements recorded during the 2001 Jupiter flyby, however, stratospheric abundances of C2H6 do vary temporally due radiative forcing and dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Melin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2018, Hue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Thus, absolute stratospheric temperatures and abundances should be interpreted with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 11a-d shows the reduced χ2 fits to the Palomar spectrum at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm (capturing continuum and stratospheric CH4 emission), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm (predominantly NH3 gaseous emission and non-gaseous impact residue ), 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm (continuum and stratospheric C2H6 emission) and over all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We find there is no model atmosphere within the described model grid that fits all wavelengths adequately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' with χ2/n ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm spectral range is relatively better fit (χ2/n ∼ 20) with models assuming higher values of p0 and f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  higher NH3 abundances over a larger vertical range of atmosphere), the same models produce too much NH3 emission and therefore a poorer fit in the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This is also demonstrated in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We attribute the inability to fit all wavelengths with the same temperature, aerosol and ammonia profile due to the presence of significant non-gaseous emission predominantly in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Given the inability to fit all wavelengths of the spectrum due to non-gaseous emission, we instead modelled the spectrum in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The retrievals of temperature and aerosol abundance were repeated across the same model grid of NH3 14  a) Fit from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='399 χ2/n b) Fit from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Fractional scale height 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Fractional scale height 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='0 Fractional scale height 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='71 χ2/n h) All wavelengths 100 200 300 400 500 600 700 Deviation pressure (mbar) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Fractional scale height 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 χ2/n ηιϕκλµνο�π Figure 11: Reduced χ2 values in fitting the Palomar/SC-10 spectra as a function of deviation pressure, p0 and fractional scale height of the NH3 (gas) profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The left column shows results derived from inverting the full 8 – 13 µm spectral range, the right column shows results derived from inverting the 8 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm spectral range (and radiances in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm range were forward modelled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' are shown in the left column, Results are shown at wavelengths of 8 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm (R branch of CH4 emission, continuum), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm (NH3 and non-gaseous emission), 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm (C2H6 emission) and all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' White lines denote χ2/n increments of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' White crosses denote model results that fit the spectra within absolute χ2 min + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 (the 1-σ confidence level) and reproduce the observed strength of NH3 emission at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='36 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='75 µm within uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' profiles (see above) but using only the 8 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm spectral ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These ranges were chosen because they capture CH4 and C2H6 emission features, and also correspond to wavelength ranges where the candidate mineralogical species (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1) have significantly weaker absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The retrieved atmosphere was then forward-modelled across the entire 8 – 13 µm spectral range and the reduced χ2/n values were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The goodness-of-fit values for these results are shown in 11e-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In omitting the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm spectral range, the retrieval is able to adequately fit (χ2/n ∼ 1) the 8 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 and 15  Palomar/SC-10, 1994-Jul-22 8 9 10 11 12 13 Wavelength (µm) 0 10 20 30 40 Radiance (µW cm-2 sr-1 µm-1) Best-fitting 8 - 13 µm retrieval Best-fitting 8 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 13 µm retrieval 100 120 140 160 180 Retrieved Temperature (K) 103 102 10 1 10-1 10-2 10-3 Pressure (mbar) 10-12 10-10 10-8 10-6 10-4 10-2 NH3 VMR ηιϕκλµνο�π Figure 12: The top panel shows the observed Palomar/SC-10 spectrum (black points with error bars) and model spectra are shown as solid, colored lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The red spectrum corresponds to the “best-fitting” result when inverting the full 8 – 13 µm range, which produces too much NH3 emission at wavelengths longer than 11 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The blue spectrum is the best-fitting result when inverting only the 8 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm spectral range and that produces NH3 emission features of a similar strength as observed within uncertainty - see the text for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The blue synthetic spectrum is considered to represent the component of the observed spectrum formed by gaseous emission, as detailed further in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The bottom panel shows the corresponding temperature and NH3 profiles (solid lines) associated with the synthetic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Dotted lines represent 1-σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The black, solid lines denote the a priori (CIRS-derived) temperature and NH3 profiles adopted in inverting the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Hashed regions at pressures lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar indicate where the observations have little/no sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm spectral range with the same atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  In order to calculate the component of the spectrum produced by gaseous emission, we searched the model grid results for results that satisfied the following criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' First, the synthetic spectrum must fit the observed spectrum in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm ranges with an absolute χ2 value of less than χ2 min + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3, which denotes the 1-σ confidence level (Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1992) when varying two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Second, the synthetic spectrum must reproduce the line-to-continuum radiance difference of the strong NH3 emission features at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='36 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='75 µm within uncertainty on the radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The models that satisfy these criteria are shown as white crosses in Figure 11e-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A model atmosphere with the NH3 profile deviating from the CIRS-derived profile at 650 mbar and with a fractional scale height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='75 yields the best-fitting spectrum that satisfy these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The spectrum associated with this model was adopted as the gaseous emission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The uncertainty of the gaseous component of the spectrum was determined by calculating the standard deviation of the synthetic spectra that satisfied the aforementioned criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Similarly, at each atmospheric level, the standard deviation in NH3 values from the range of the model atmosphere that satisfied the aforementioned criteria were calculated and adopted as their respective uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  We performed a similar calculation to determine the uncertainty on temperature but found that the standard deviation was smaller than the uncertainty on temperature from the best-fitting retrieval and so the latter was adopted as the uncertainty on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The gaseous emission spectrum and the associated vertical profiles of temperature and NH3 and their uncertainties are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Relative to the CIRS-derived a priori temperature profile, heating of the atmosphere is evident from ∼100 mbar to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar (with no sensitivity to temperature at lower pressures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The warmest temperatures of 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 K were retrieved at the 2-mbar level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The observations have limited sensitivity to temperature at pressures lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar and so we cannot rule out heating of the atmosphere at lower pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' At 30 mbar and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar, we derive NH3 concentrations of 38+7 −19 ppmv and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 ppmv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The non-gaseous emission spectrum was derived by calculating the fractional residual (Equation 1) between the observed spectrum and the best-fitting gaseous emission spectrum (Figure 12 in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 13 compares the non-gaseous fractional residual with the features of candidate mineralogical species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The derived non-gaseous emission exhibits a broad feature from wavelengths of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1, the dominant opacity sources responsible for the broad feature are a mixture of amorphous olivine, obsidian, and tekt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Superimposed on this broad feature are narrower features at wavelengths of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm with (normalized) fractional residuals of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='12, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We do not consider the feature at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 µm real due to its proximity to telluric O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Although the observed feature at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm is consistent in wavelength with forsterite, the mineral exhibits a much broader feature (FWHM ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 µm) compared to what is observed (FWHM ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In addition, forsterite has a feature at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='25 µm, which is twice as strong as its feature at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm, where no significant residual is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In taking into account the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 µm (R = λ/∆λ = 100) accuracy of the wavelength grid, the features at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 µm are near-coincident with those of fayalite at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, in acknowledging the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1-µm accuracy of the wavelength grid, we also note that the features at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 µm are within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 µm of NH3 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Thus, it remains inconclusive whether the identified narrow features 16  Impact G, 1994-Jul-22, Palomar/SC-10 8 9 10 11 12 13 Wavelength  (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Fractional Residual -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Normalized Absorptance Amorphous Olivine Amorphous Pyroxene Obsidian Fosterite Fayalite Diopsene Ferrosite Ortho Enstatite Austr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Tektite Bedias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Tektite NH3 (g) Amorphous Carbon H2O (s) ηιϕκλµνο�π Figure 13: Comparison of the fractional residual between observed and modelled SC-10 spectra of the SL9 G impact site, which denotes the non-gaseous component of the spectrum, and the absorptance spectra of candidate mineralogical species according to the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The fractional residual was normalized to 1 and all mineral absorptance spectra have been normalized to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='65 – 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm range is hatched due to telluric ozone and so we do not consider any apparent features in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' result from non-gaseous species or poor fitting of the NH3 emission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Modeling Gemini/T-ReCS spectra The atmosphere derived from Cassini-CIRS measurements of 60◦S in 2001 (see Appendix A) was adopted as the background atmosphere and a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The T-ReCS spectra were then inverted over the same model grid of NH3 profiles adopted in inverting the SC-10 spectra (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to optimize the signal-to-noise ratio, we adopted the coadded spectrum in Figure 6, where all individual spectra sampling the impact region were coadded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Initially, we inverted the entire T-ReCS spectral range from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 to 13 µm and 17 to 25 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 14a-d shows the reduced χ2 distributions of the model grid in fitting the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Overall, the fit to the observations is optimized when the vertical profile of NH3 deviates from the CIRS profile at 100 – 200 mbar and decreases in abundance with a high fractional scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While we can adequately fit the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm spectrum with variations in temperature, NH3 and tropospheric aerosol, we find that the 17 – 19 µm feature cannot be adequately fit with a physically-sensible atmospheric model and cannot be adequately fit using an atmospheric model that simultaneously fits the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This is further suggestive the 17 – 19 µm enhancement of the Wesley impact site, with respect to the background atmosphere, results from a non-gaseous species in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As shown in Figure 14b-c, when the 17 – 19 µm was included in the spectral inversion, we found that the overall fit to 17 – 25 µm spectral range was poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We performed a second set of retrievals where the 17 – 19 µm spectral range was omitted from the spectral inversion and found that the fit to the 19 – 25 µm range significantly improved (comparing Figures 14c and g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In the remainder of this section, we will therefore discuss results from retrievals omitting the 17 – 19 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The best-fitting (χ2/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='629) NH3 profile deviates from the CIRS-derived profile at p0 = 200 mbar, and decreases with a fractional scale height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, we note that a range of solutions with p0 ranging from 150 to 350 mbar and fractional scale heights from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='75 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 also fit the observed spectrum with (absolute) χ2 values less than χ2 min + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3, which corresponds to the 1-σ confidence level when varying two parameters (Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These models are marked in Figure 14e – h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A fractional scale height of 1 imposes a constant abundance of NH3 with altitude from the 200-mbar level to the top of the atmosphere, which is physically unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Instead, the vertical profile of NH3 is likely to be approximately constant with altitude from p0 = 2 mbar to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar, over which the observations have sensitivity, but then decreases significantly at higher altitudes outside the vertical range of sensitivity of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We adopted the best-fitting spectrum (see above) as the gaseous component of the impact region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The uncertainty of the gaseous spectrum was determined by calculating the the standard deviation of the synthetic spectra that had (absolute) χ2 less than χ2 min + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' At each atmospheric level, the standard deviation in NH3 values from the range of the model atmosphere that satisfied the aforementioned criteria were calculated and adopted as their respective uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  We performed a similar calculation to determine the temperature uncertainty but found that the 17  a) Fit from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 13 µm 100 200 300 400 500 600 700 Deviation pressure (mbar) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Fractional scale height 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='30 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='01 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='72 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='44 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='15 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='87 χ2/n b) Fit from 17 - 19 µm 100 200 300 400 500 600 700 Deviation pressure (mbar) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 Fractional scale height 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='01 χ2/n c) Fit from 19 - 25 µm 100 200 300 400 500 600 700 Deviation pressure (mbar) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content='34 χ2/n ηιϕκλµνο�π Figure 14: Reduced χ2 distributions calculated from the observed and modelled T-ReCS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Results in the left column are for retrievals where the entire spectral range available (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 13, 17 - 25 µm) was included in the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  χ2/n are shown at wavelengths from a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 13 µm, b) 17 – 19 µm, c) 19 – 25 µm and d) all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Similar results are shown in panels e) – h) but where the 17 – 19 µm range was excluded from the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' White contours denote increments of χ/n of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' White crosses denote model combinations that allow the 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 13 µm and 19 - 25 µm spectral ranges to be fit within (absolute) χ2 min + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3, which corresponds to the 1-σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' standard deviation was smaller than the temperature uncertainty from the best-fitting retrieval and so the latter was adopted as the real temperature uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Figure 15 compares the observed T-ReCS spectra with the gaseous spectrum noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This result was forward modelled at 17 – 19 µm (omitted from the inversion as noted above) so this spectral range could be included in the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' At 30 mbar, we derive NH3 abundances of 150+338 −121 ppbv, which is in agreement (within uncertainty) of the results derived by Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As also noted in previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011, Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2011), there is no evidence of stratospheric heating associated with the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In fact, stratospheric temperatures are ∼10 K cooler than those derived from Cassini measurements in 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In the 18  2009-Jul-24, Wesley Impact, N band 8 9 10 11 12 13 Wavelength (µm) 0 1 2 3 4 5 6 7 Radiance (µW cm-2 sr-1 µm-1) 2009-Jul-24, Wesley Impact, Q band 18 20 22 24 Wavelength (µm) 6 7 8 9 10 11 Radiance (µW cm-2 sr-1 µm-1) 8 9 10 11 12 13 Wavelength (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 Fractional Residual 18 20 22 24 Wavelength (µm) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 Fractional Residual 100 120 140 160 180 Retrieved Temperature (K) 103 102 10 1 10-1 10-2 10-3 Pressure (mbar) 10-14 10-12 10-10 10-8 10-6 10-4 10-2 NH3 VMR ηιϕκλµνο�π Figure 15: The top row compares observed T-ReCS spectra (black points with error bars) and the best-fitting synthetic spectra (solid, red lines) in the N (left) and Q (right) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The middle row shows the fractional residual between the observed and modelled spectra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The bottom row shows the a priori (solid, back) and retrieved temperature profile (solid red, with 1-σ uncertainty shown as dotted lines) according the lower axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The best-fitting NH3 profile (solid black) is shown according the upper axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Hashed regions at pressures lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar indicate where the observations have little/no sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Q band, the model spectra cannot adequately fit the observed spectra from 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 – 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For example, at 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm, the fractional residual is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3% and therefore greater than 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, we find we can adequately fit the majority of the spectrum in the N band by varying only the vertical profiles of temperature, NH3 and scaling the abundance of C2H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Discussion Using methods as consistently as possible, our analysis of mid-infrared data capturing the Jovian atmosphere in the aftermath of the SL9 and Wesley impacts have revealed similarities and differences between the two events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In both impacts, NH3, normally concentrated below Jupiter’s tropopause, was transported to stratospheric altitudes by ballistic splashback, which produced strong NH3 emission features in the N band with the strongest lines at ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 and ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 µm (see Figure 7, as seen in previous work (Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In inverting the spectra of both impacts, we parameterized the NH3 vertical profile by using the CIRS-derived profile (see Appendix A) at pressures higher than a deviation pressure, p0 and a decrease in abundance with altitude according to a fractional scale height, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The fit to the continua and the strength of the NH3 emission lines in the Palomar/SC-10 spectra of SL9’s G impact was optimized using p0 = 650 mbar and a fractional scale height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This results in NH3 concentrations of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 ppmv and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='66+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='54 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='82 ppmv at 30 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' These are broadly-consistent with the findings of Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (1997), who derived ∼1 ppmv NH3 concentrations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar using data recorded ∼6 days after K impact, and Fast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2002), who derived ∼3 ppmv NH3 concentrations at pressures lower than 1 mbar using data recorded ∼4 days after the G impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The fit to Gemini-South/T-ReCS spectra of the Wesley impact was optimized using p0 = 200 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' At 30 mbar, we derive NH3 abundances of 150+338 −121 ppbv, which is in agreement within uncertainty of the results derived by Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, we note to the reader the our inversion of the spectra did not include the opacities of non-gaseous features and so 19  these concentrations represent an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nevertheless, the higher values of p0 and higher concentrations of NH3 required to fit the spectra of the SL9 impact, compared to the Wesley impact, is consistent with the SL9 fragments entering the atmosphere at angles closer to the local zenith, and reaching deeper, NH3-richer altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In both events, material introduced by the impactors into the atmosphere was evident as non-gaseous emission features in the spectra of the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to determine the mineral species responsible for these non-gaseous emission features and their relative abundances, we performed a least-squares search of the mineral species listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In the impact sites of SL9, we would have expected a preponderance of amorphous olivine and pyroxene, plus water ice and amorphous carbon as well as NH3 gas lofted to stratospheric altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The results of our mineralogical least-squares fitting indicate that amorphous olivine and NH3 gas were significant opacity sources in the impact site of SL9’s G fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, our analysis detected silica (in the form of obsidian) and an absence of amorphous pyroxene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This was initially surprising since comets are expected to have similar abundances of olivinaceous to pyroxenacous species (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2006, 2007b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We suggest that high temperatures and pressures produced by the cometary impacts converted all the cometary pyroxene into silicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  While amorphous pyroxene is stable up to temperatures of ∼1000 K, amorphous olivine is stable up to ∼1500 K and so presumably, the material was heated to a temperature between 1000 - 1500 K such that amorphous olivine remained unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A similar finding was seen in the HD172555 exodisk system by Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As far as we are aware, silica was not detected or reported in previous studies of the SL9 impact residue though this is unsurprising given the majority of mid-infrared studies focused on high-resolution spectroscopy of NH3 lines between 10 - 11 µm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1997, Fast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2002) and did not sample the 9 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (1995) presented Palomar/SC-10 spectra (the same dataset used in this paper) of the R impact, also observing a broad N-band feature, but this was interpreted as silicate emission, not silica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' However, their study did not attempt to derive an emissivity spectrum of the impact (by computing a residual between the impact and a non-impact region) or model the features using a least-squares search of mineral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' If the Wesley impactor was a comet, we would also have expected a composition of predominantly amorphous olivine and pyroxene, plus water ice and amorphous carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' If the Wesley impactor was an asteroid, as suggested previously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Hammel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Orton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  2011, we would have expected a preponderance of crystalline olivine and phyllosilicates plus impact-alteration-produced silicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Instead, our analysis found that the Wesley impact emission appears composed of almost pure amorphous olivine, with some ferromagnesian pyroxenes mixed in at an ∼8:1 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' No emission due to silicas (expected from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 µm) is apparent in our new reduction and spectral analysis, a marked difference from the findings of Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011) who used the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We believe our results differ due to differences in how the spectra were coadded and modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011) inverted spectra of the impact obtained over a range of emission angles and found the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1-µm signature was most evident in the higher emission-angle observations, whereas, in this work, we averaged individual T-ReCS spectra over a range of emission angles to compute a single, coadded spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Given the presence of silicas is analysis-dependent, we consider its non-detection in this work a tentative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' If we assume the non-detection of silica is correct, we must therefore assume residue from the Wesley impactor did not reach the required temperatures of ∼1000-1500K in order to convert rocky silicates into silicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Instead, we must conclude from our new study that the Wesley residue was formed at low temperatures and high pressures such that the usually dominant crystalline olivine was readily converted into amorphous olivine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Whether these pressures were achieved due to the shock of impact, or due to the pressures achieved at depth inside a large differentiated body, is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We currently favor the impact induced shock amorphization explanation, as all reports have the Wesley impactor as being relatively small in size (<10 km radius), which would be too small to thoroughly self-pressure alter its olivine throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In addition, such an explanation would require removal of the body’s low pressure, near-surface crystalline olivine without heating the high pressure phase enough to produce silica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This still belabors the question of how a body impinging on Jupiter with ∼65 km/sec (Jupiter’s escape speed) relative velocity and a huge specific energy per kg of material can manage to NOT melt and transform its matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As noted above, SL9 was seen to create large amounts of glassy silica (a material not found in comets) from amorphous pyroxene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' A remote possibility is that the Wesley 2009 impactor entered the atmosphere at an extremely shallow angle, which would allow material to spall off the impactor at relatively low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' If we take the velocity of impact required to convert silicates into silicas as ∼10 km/sec (Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2009) and references therein), an impact angle of less than tan−1(10/65) ∼ 9◦ with respect to the local horizontal would be required in order to impart the impactors kinetic energy slow enough to the body that it could have a chance to retain all its silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  This is outside the 20◦ ± 5◦ estimated by S´anchez-Lavega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2010) and Pond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2012) in order to explain the lack of stratospheric heating (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010) and the observed debris deposition between 3- 10 mbar (de Pater et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Hammel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' An impact on Jupiter by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5- to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5-km objects, similar in size to the fragments of SL9 and the Wesley impactor, is likely in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While Schenk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2004) inferred from the cratering record on the Galilean moons that SL9/Wesley-sized impacts on Jupiter will occur once every ∼100 years, S´anchez-Lavega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2010) instead calculated a rate of once every 10 - 20 years from the observed, recent rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For future impact events, we believe a mid-infrared characterization of the impact site would allow the impactor and its effect on the atmosphere to be characterized in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' First, the relative proportions of amorphous pyroxanaceous and olivinaceous species would be constrained 20  from their features at 9, 10, and 19 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Second, the detection or non-detection of silica would indicate whether temperatures and pressures produced by the impact reached the ∼1500 K or 4 - 6 GPa levels required to temperature alter silicate species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Third, if stratospheric NH3 emission is detected, concentrations of NH3 could be derived, which in turn would constrain the terminal depth of the impactor material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' While broadband imaging and low-resolution spectroscopy readily allowed non-gaseous features from the impactor to be identified, the coarse spectral resolving power (R = 100) presented in this work introduced uncertainty in identifying gaseous vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' non-gaseous spectral features and allowed only simple (two variable) parameterizations for the vertical profile of NH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Characterization of future impacts would therefore benefit from N- and Q-band spectroscopy performed near-simultaneously and at higher spectral resolving powers (1000 < 15000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For Earth-based observations, R > 1000 would also better allow CH4 emissions from the planet to be disentangled relative to telluric CH4 features (assuming the Earth-Jupiter relative velocity exceeds ∼ 15 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This would improve the retrieval of the stratospheric temperature field since the CH4 is generally considered to be horizontally homogeneous in the lower stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  Conclusions We performed a retrospective analysis of mid-infrared Earth-based observations of Jupiter recorded in July 1994 and July 2009 in order to compare the effects on Jupiter’s atmosphere by Comet D/Shoemaker-Levy 9 (SL9) and the Wesley impactor quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Spectrophotometry by IRTF/MIRAC and Palomar/Spectrocam-10 in 1994 and Gemini/T-ReCS in 2009 were reduced, calibrated and analyzed using methods as consistently as possible in order to facilitate robust comparisons between the two impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' As presented in previous work, we find the 8 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 µm spectral range of both impact sites is significantly enhanced due to stratospheric emission from NH3, having been lofted from the upper troposphere to the stratosphere by ballistic “blowout”, as well as non-gaseous impact material (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Also, in agreement with previous work, we find that the SL9 impacts exhibited enhanced stratospheric CH4 emissions whereas no significant enhancement was evident in the site above the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In new findings, we determine that the sites of the SL9 and Wesley impacts both exhibit enhanced emissions at 18 – 19 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For example, at 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='9 µm, the SL9 L impact site is enhanced by 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5% and the Wesley impact site is by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' In order to detect and quantify the species responsible for these observed spectral features, we performed two separate but complementary analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  First, we performed mineralogical spectral modeling by adopting a collection of candidate mineralogical species and performing a grid search to find a combination of species that produced a model emissivity spectrum that minimized the goodness-of-fit to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the impact site of SL9’s G fragment, we find that stratospheric NH3 emission, amorphous olivine and silica (in the form of obsidian) are the dominant opacity sources responsible for the observed spectral features at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 - 11 µm and 18 - 19 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We find no evidence of amorphous pyroxene in the site of SL9’s G fragment, which was initially unexpected since comets typically exhibit a ∼1:1 ratio of pyroxenaceous species to olivinaceous species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We suggest that the high pressures and temperatures produced by the impact readily converted cometary pyroxene into silicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the Wesley impact, we find that stratospheric NH3 and amorphous olivine are the dominant opacity sources responsible for the observed spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We find no evidence of silicas in the aftermath of the Wesley impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This is in contrast to previous work (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2011) who used the same data but processed/analyzed differently, which suggests the detection or non-detection of silica is analyssis-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We therefore consider the non-detection of silica a tentative finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  If the non-detection of silica is correct, it would require that material from the Wesley impactor was modified at relatively lower temperatures such that species retain all their silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' This would require that the Wesley impactor entered the atmosphere at a very shallow angle of 9◦, which is shallower than the 20 ± 5◦ estimated previously (S´anchez-Lavega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Pond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Our second analysis involved inverting the spectra of both impacts using radiative transfer software in order to constrain NH3 abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For SL9’s G impact, we constrain NH3 concentrations of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='4+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='66+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='54 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='82 ppmv at 30 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 mbar, respectively, which is broadly consistent with previous work (Griffith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2002) though our parameterization of NH3’s vertical profile differed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For the Wesley impact, we derive an NH3 concentration of 150 +338 −121 ppbv at 30 mbar, which is consistent (within uncertainty) with Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Higher concentrations of NH3 in the SL9 impacts is consistent with the fragments reaching a deeper terminal depth compared to the Wesley impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Acknowledgements Some of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  During the course of this research, Meera Krishnamorthy was a student at the California Institute of Technology and worked at JPL as an intern in the Caltech Summer Undergraduate Research Fellowship (SURF) program, supported by these funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Fletcher was supported by a European Research Council Consolidator Grant (under the European Union’s Horizon 2020 research and innovation programme, grant agreement No 723890) at the University of Leicester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' We thank Mike Chaffin and Sonal Jain for creating IDL-compatible color tables1 that are perceptually-uniform, colorblind-friendly, and print correctly in black and white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Emma Dahl was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, administered by Oak Ridge Associated Universities under contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='com/planetarymike/IDL-Colorbars 21  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Data Availability Archiving of the IRTF/MIRAC images and the Gemini-South/T-ReCS spectra at the Planetary Data System (PDS) is in progress (at the time of writing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Until they are available on the PDS, the data will be made available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The calibrated spatial-spectral Palomar/SC-10 image of Jupiter is available at the Mendeley data archive2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Raw data files and the software used to process the Palomar data would also be made available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+page_content=' The Collision of Jupiter and Comet Shoemaker-Levy 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Icarus 108, 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1006/icar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Cassini-CIRS retrievals During Cassini’s 2000/2001 flyby of Jupiter, four maps of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 cm−1 ‘MIRMAP’ observations - ‘ATMOS02A’, ‘ATMOS02B’, ‘ATMOS02C’ and ‘ATMOS02D’ - were acquired from January 1st to Jan 11th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Spectral from all four maps were sorted into 2◦ latitude bins, nyquist-sampled by 1◦S and coadded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The effective noise on these spatially-binned observations was assumed to be the largest of either: 1) the standard deviation of the mean or 2) the noise-equivalent spectral radiance (NESR) spectrum of CIRS by a factor of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5/∆˜ν)(1/( √ N) + 1/( √ M)), where ∆˜ν is the spectral resolution of the target spectra, N is the number of averaged target spectra and M is the number of deep-space spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='  The spectra at 40◦S (the latitude of the SL9 impacts) and 60◦S (the latitude of the Wesley impact) were inverted by varying temperature, NH3, tropospheric aerosol, C2H2, C2H4 and C2H6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Inversions were performed using the NEMESIS radiative transfer code (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The sources of line data, the atmospheric model are identical to those described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The vertical temperature profile was allowed to vary continuously at all altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The vertical profile of NH3 was parameterized with three variables: 1) a deep volume mixing ratio (qdeep) at pressures higher than a knee pressure, 2) the knee pressure (pknee) and 3) a fractional scale height ( fNH3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' For tropospheric aerosol, a grey absorber was assumed with an a priori base pressure (pbase) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='7 bar, an optical depth (τ1) of 1 and a fractional scale height ( fcloud) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2, were assumed, and all three parameters were varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The vertical profiles of C2H2, C2H4 and C2H6 were adopted from a photochemical model of Jupiter (Moses and Poppe, 2017) and varied by applying a scale factor to all altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Comparisons of the observed and synthetic spectra and the corresponding retrieval parameters are shown in Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='16 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The retrieved atmospheric parameters at 40◦S and 60◦S are adopted as a priori values for inversions of the Palomar/SC-10 and Gemini-S/T-ReCS spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 24  Cassini-CIRS (January 2001), 40oS 7 8 9 10 11 Wavelength (µm) 0 2 4 6 8 Radiance (µW cm-2 sr-1 µm-1) 11 12 13 14 15 16 17 Wavelength (µm) 0 2 4 6 8 10 Radiance (µW cm-2 sr-1 µm-1) 100 120 140 160 180 Retrieved Temperature (K) 103 102 10 1 10-1 10-2 10-3 Pressure (mbar) 10-12 10-10 10-8 10-6 10-4 10-2 NH3 VMR τ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='3 p1 = 700 mbar f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 qNH3 = 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='8 ± 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 ppmv pknee = 885 mbar fNH3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='01 C2H2  x (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='05) C2H4  x (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='05) C2H6  x (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='03) χ2/n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1726409 ηιϕκλµνο�π Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='16: Inversions of Cassini-CIRS spectra recorded during the 2001 Jupiter flyby at 40◦S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The top and middle panels show observed spectra (black circles with error bars) from 7 - 11 µm and 11 - 17 µm, and synthetic spectra are shown as solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The retrieved atmospheric parameters corresponding to the synthetic spectra are shown in the bottom panel and the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Black, solid lines represent a priori values, solid lines denote retrieved values and dotted lines outline the 1-σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 25  Cassini-CIRS (January 2001), 60oS 7 8 9 10 11 Wavelength (µm) 0 1 2 3 4 5 Radiance (µW cm-2 sr-1 µm-1) 11 12 13 14 15 16 17 Wavelength (µm) 0 1 2 3 4 5 6 7 Radiance (µW cm-2 sr-1 µm-1) 100 120 140 160 180 Retrieved Temperature (K) 103 102 10 1 10-1 10-2 10-3 Pressure (mbar) 10-12 10-10 10-8 10-6 10-4 10-2 NH3 VMR τ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 p1 = 700 mbar f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='2 qNH3 = 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='5 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='1 ppmv pknee = 700 mbar fNH3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='04 C2H2  x (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='03) C2H4  x (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='05) C2H6  x (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='03) χ2/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='76112759 ηιϕκλµνο�π Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content='17: Inversions of Cassini-CIRS spectra recorded during the 2001 Jupiter flyby at 60◦S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The top and middle panels show observed spectra (black circles with error bars) from 7 - 11 µm and 11 - 17 µm, and synthetic spectra are shown as solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' The retrieved atmospheric parameters corresponding to the synthetic spectra are shown in the bottom panel and the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' Black, solid lines represent a priori values, solid lines denote retrieved values and dotted lines outline the 1-σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
+page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfZPw6/content/2301.01347v1.pdf'}
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+Relative sea level and coastal vertical movements in relation with volcano-
+tectonic processes in Mayotte 
+Julien Gargani1, 2 
+1Université Paris-Saclay, CNRS, Geops, Orsay, France. 
+2Université Paris-Saclay, Centre d’Alembert, Orsay, France. 
+ 
+Highlights 
+ An abrupt subsidence of 10 m occurred in Mayotte at 9.3 kyr B.P., followed by an uplift 
+of 10 m from 8.1 kyr to 7 kyr B.P. 
+ Volcano-tectonic activity is responsible for these vertical movements. 
+ Deep reservoir overpressure has been recorded over a period of 1 kyr. 
+ 
+Abstract 
+During the last 10 kyr, significant subsidence and uplift occurred on Mayotte Island in the 
+Comoros archipelago, but the role of volcanic processes in Holocene vertical movements has 
+been neglected. Here, we show that an abrupt subsidence of 10 m occurred 9.3 kyr ago and then 
+an uplift of the same amplitude occurred at a rate of 9 mm/yr from 8.1 kyr to 7 kyr ago, and we 
+compare the relative sea level of Mayotte with a reference sea level curve. Using a modeling 
+approach, this study shows that an increase and a decrease in pressure in a deep magma reservoir 
+was responsible for the ~10 m subsidence and the 10 m uplift, whereas the loading by new 
+volcanic edifices has caused the subsidence that has occurred during the past 6.1 kyr. The 
+surface movement was caused by pulses from the deep mantle that were probably related to 
+superplume activity. 
+ 
+Keywords: uplift, subsidence, volcanic edifice, sea level 
+ 
+1. Introduction 
+Recent 
+volcano-seismic 
+events 
+in 
+unexpected places or with significant 
+impacts challenge our understanding of 
+volcanic activity. First, these events suggest 
+reconsidering the role of volcanic activity or 
+sea-level rise/inundation on local natural 
+hazards [Roger, 2019: Audru et al., 2010] 
+and promoting specific studies on the 
+reduction in risk [Deves et al., 2022; 
+Pasquon et al., 2022] associated with 
+volcanic activity. Second, the potential 
+interrelation between short-term volcanic 
+activity and long-term volcano-tectonic 
+evolution also makes it challenging to 
+understand volcanic processes and Earth 
+surface evolution. 
+The risk associated with volcanic activities 
+is not only due to magmatic explosions and 
+lava flows but also to a consequence of 
+induced landslides [Ward and Day, 2001; 
+Gargani, 2020] and subsidence [Lambeck, 
+1981], 
+thereby 
+increasing 
+marine 
+submersion risk and tsunamis [Ward and 
+Day, 2001]. Subsidence in a volcanic 
+context may be associated with volcanic 
+edifice loading [Lambeck, 1981; Gargani, 
+2022b] or magma reservoir deflation 
+[Gargani, 2006]. The relationship between 
+volcanic activity and significant relative sea 
+
+level variation has been highlighted in 
+previous 
+studies 
+on 
+contemporary 
+observations as well as on past events 
+[Sternai et al., 2017; Duggen et al., 2003]. 
+Mayotte Island (Comoros archipelago, SW 
+Indian 
+Ocean, 
+between 
+Africa 
+and 
+Madagascar) has experienced significant 
+seismic activity since May 2018, with more 
+than 11000 earthquakes of up to magnitude 
+5.9, unusually long-period events (more 
+than 400) and surface deflation up to 200 
+mm/yr [Lemoine et al., 2020; Cesca et al., 
+2021]. This seismic activity indicates deep 
+magma reservoirs at 20-35 km, 50 km and 
+70 km from Mayotte’s Petite Terre Island 
+[Lemoine et al., 2020; Cesca et al., 2021; 
+Feuillet et al., 2021]. This latter reservoir is 
+connected to the 50 km-depth reservoir 
+through rapid magma propagation in the 
+lithosphere [Lavassyere et al., 2022]. The 
+magma reservoir diameters are 10-15 km at 
+depths of 20-35 km and 20-30 km at depths 
+of ~50 km [Lemoine et al., 2020; Cesca et 
+al., 2021; Feuillet et al., 2021; Lavassyere et 
+al., 2022] (Figure 1). During the 2018-2021 
+seismo-volcanic event, a deflation of the 
+magma reservoir located at a depth of ~50 
+km occurred (Lavassyere et al., 2022). This 
+seismo-volcanic activity caused the birth of 
+an 820 m tall, 5 km3 deep sea volcanic 
+edifice documented on the eastern insular 
+slopes of Mayotte at the tip of a 50 km-long 
+volcanic ridge composed of many other 
+recent edifices and lava flows [Feuillet et 
+al., 2021]. 
+ 
+Figure 1: Mayotte volcanic context. (A) Grand Terre (GT) and Petite Terre (PT) islands in 
+Mayotte archipelago, with a volcanic edifice, lava flows and a caldera, (B) schematic 
+representation of the lithosphere below Mayotte. 
+Before the 2018-2021 event, the most recent 
+volcanic 
+activity 
+in 
+the 
+archipelago 
+occurred on the Comoros Islands (Karthala 
+volcano), 200 km away from Mayotte 
+[Bachèlery et al., 2016]. Studies have 
+suggested that the presence of a hotspot 
+explains the E‒W sublinear trend of the 
+Comoros volcanic archipelago [Emerick 
+and Duncan, 1982; Ebinger and Sleep, 
+1998; Class et al., 2005], but this 
+explanation is controversial [Bertil et al., 
+2021; Quidelleur et al., 2022]. Researchers 
+have 
+interpreted 
+the 
+E‒W 
+seismic 
+alignment in the region as highlighting the 
+activity of structures associated with the 
+continuity of the East African Rift and in 
+
+Mayotte
+GT
+PT
+Caldera
+44.6°
+45°
+45.4°
+45.8°
+FO
+2018-2021
+-12.4°
+A
+Eruption
+-10-
+GT
+Mayotte
+-12.6°
+-20-
+Caldera
+Crust
+2018-2021
+Depth (km)
+-12.8°
+-30-
+Eruption
+Magma
+-40-
+reservoir
+-13°
+-50-
+1500
+Mantle
+-13.2°
+-60-
+-70-
+-13.4°
+0
+20 km
+3500
+-80-
+B
+Asthenosphere
+20
+40
+Distance (km)
+60relation to the Madagascar drift on an E‒W-
+trending axis [Bertil et al., 2021]. 
+The main volcanic building phase of 
+Mayotte Island is estimated to have 
+occurred from 10.6 Ma to 1.25 Ma. The lava 
+content demonstrates lithospheric but also 
+mantle sources, suggesting lithospheric 
+melt as well as the existence of an ascending 
+mantle plume [Pelleter et al., 2014]. The 
+mantle depths range from 17 km to 70 km 
+below Mayotte [Feuillet et al., 2021]. On 
+Mayotte Island, volcanic events have been 
+dated to 6.5-7 kyr B.P. (ash), 8 and 2.2 kyr 
+B.P. (phonolite), 4 kyr and 4.12 ± 0.04 kyr 
+B.P. (pumice stone), and 5.98 ± 0.14 kyr 
+B.P. (volcanic mud) [Zinke et al., 2003a, 
+2000; Camoin et al., 2004]. These volcanic 
+activities highlight the existence of volcanic 
+edifice building during the Holocene. 
+Nevertheless, the volume of volcanic rock 
+that erupted during this phase is unknown. 
+Furthermore, the duration of the complete 
+volcanic cycle from magma ascent to 
+volcanic edifice construction in Mayotte 
+during the Holocene has never been 
+estimated. 
+Relative sea level is useful to quantify 
+vertical movements [Gargani, 2022a and 
+2022b]. Previous studies on coral reefs have 
+obtained accurate data on relative sea levels 
+in Mayotte during the Holocene [Zinke et 
+al. 2001; Zinke et al., 2003b; Zinke et al., 
+2003a; Zinke et al., 2005; Camoin et al., 
+1997; Camoin et al., 2004]. These studies 
+have discussed the role of climate variation 
+and postglacial floods, as well as glacio-
+hydro-isostatic adjustment, on relative sea 
+level. However, the potential effect of 
+volcanic activity on relative sea level rise 
+and fall has not been analyzed in Mayotte. 
+ 
+FIGURE 2: Relative sea level in Mayotte (red cross) compared to the sea level curve 
+constructed using data from Tahiti and Barbados (light blue). Data from Mayotte: Zinke et al. 
+(2003); data from Tahiti: Bar et al. (1990, 1996, 2010); Hallman et al. (2018); Deschamps et 
+al. (2012); Pirazzoli et al. (1985); data from Barbados: Peltier et al. (2006); Abdul et al. 
+(2016). Before 6 kyr B.P., the subsidence rate in Tahiti is corrected with an uplift rate of 0.25 
+
+0
+XMAH
+MAYOTTE Zinke et al (2003) 
+H4
+-10
+-20
+SEALEVEL (m)
+-30
+-40
+王
+-50
+-60
+2
+4
+6
+8
+10
+12
+TIME (kyr B.P.)mm/yr, and afterward, it is corrected using a subsidence rate of 0.15 mm/yr. Before 11.18 kyr, 
+the uplift rate is corrected with an uplift rate of 0.34 mm/yr in Barbados, and after 11.18 kyr 
+B.P., an uplift rate of 0.8 mm/yr is used. 14C ages are corrected by a U/Th ratio of 1.14. 
+This study analyzes the vertical movement 
+in Mayotte during the last 10 kyr by 
+modeling magma reservoir pressure as well 
+as volcanic edifice loading and compares 
+these parameters with relative sea level. 
+More precisely, volcanic activities, such as 
+magma reservoir overpressure (amplitude 
+and duration) and the volume of volcanic 
+edifice during the last 10 kyr in Mayotte, are 
+quantified. The implications of these results 
+at the regional scale are discussed. 
+ 
+2. Method 
+2.1 Overpressure and surface uplift 
+The vertical displacement Uz of a magma 
+reservoir of radius rs at a depth –Z with an 
+overpressure P is given by the following 
+equation [Battaglia et al., 2013; McTigue, 
+1987]: 
+Uz = (1-) (P rs3 Z) / [G (R2+Z2)]3/2 × [1 - (rs/Z)3 × 
+[(1-) / [2 (7-5)] - [15 (2-)] / [4 (7-5)] × 
+Z2/(R2+Z2) ]] 
+where is the Poisson's ratio (=0.25) and 
+R is the distance from the expansion source 
+embedded at position  to point j on the free 
+surface and corresponds to the radial 
+distance from the source. G is the shear 
+modulus (15 GPa). 
+The depths of the magma reservoir that are 
+considered in this study are 20 km, 50 km 
+and 70 km, which are in agreement with the 
+estimation by Feuillet et al. (2021). The 
+radii of the magma reservoir tested by 
+modeling range from 5 km to 30 km. The 
+influence of the magma reservoir pressure 
+at different depths was also modeled. The 
+overpressure and depressurization tested in 
+this study range from 20 to 150 MPa. The 
+overpressure was estimated using the 
+observed uplift of 10 ± 1 m. 
+2.2 Volcanic edifice loading and isostatic 
+adjustment 
+Loading of volcanic edifices on the 
+lithosphere causes subsidence [Lambeck, 
+1981]. Modeling isostatic adjustment in 
+response to loading is performed using a 
+classical law [Turcotte and Schubert, 2001] 
+that has been applied in various studies 
+[Watts, 2000; Gargani, 2004 et 2010]. 
+Turcotte and Schubert (2001) used the 
+equation 𝜵2(𝐷∙𝜵2𝑤(𝑥))+(𝜌𝑎+𝜌v)𝑔∙𝑤(𝑥) = 
+𝜌v𝑔[𝑧init(𝑥)+ 𝑧(𝑥)] to model the flexure of 
+the lithosphere, where D is the rigidity of 
+the lithosphere, w(x) represents the uplift, 
+ρa and ρv are the densities of the 
+asthenosphere and volcanic rock, g is the 
+acceleration of gravity, 𝑧init(𝑥) is the initial 
+topography and 𝑧(𝑥) is the topography after 
+the construction of the new volcanic edifice. 
+The rigidity is defined by D = ETe3/[12(1-2)], 
+where E is Young's modulus and Te is the 
+effective elastic thickness. 
+The E‒W sublinear trend of the Comoros 
+volcanic archipelago permits us to analyze 
+the symmetry axis of this 200 km-long 
+structure and to simplify the numerical 
+problem using a 2-D approach. The elastic 
+thickness in Mayotte is 40 ± 5 km [Calmant 
+et al., 1990], which corresponds to a 
+flexural rigidity that ranges from 3.8×1022 
+N.m2 to 8.1×1022 N.m2. In the calculation, E 
+= 1010 Pa.  
+The resulting vertical motion rates are 
+calculated by considering a constant 
+displacement during the 10 kyr after the 
+abrupt mass displacement, which is in 
+agreement with the time necessary to relax 
+the viscous properties of the lithosphere 10 
+± 5 kyr after isostatic adjustment [Mitrovica 
+et al., 2000; Van der Wal et al., 2010]. The 
+average velocity that is obtained is a first-
+order estimation sufficient to describe the 
+main dynamics of the system. 
+Different loading masses corresponding to 
+various volumes of volcanic rock involved 
+in volcanic edifice building were simulated. 
+Loading was simulated by several volcanic 
+
+edifices that were 200 m tall and had a 
+diameter of 3.5 km, corresponding to a 
+conical volume of 0.6 km3 and a mass of 1.8 
+× 1012 kg. This study discusses the loading 
+mass that explains the observed subsidence. 
+Two 
+different 
+locations 
+have 
+been 
+considered (0 km and 20 km from the 
+coast). The results have been compared with 
+observed vertical movements. 
+2.3 Relative sea level and vertical 
+movement 
+Well-preserved coral reefs were collected in 
+Mayotte and dated [Camoin et al., 1997; 
+Zinke et al., 2004a; Camoin et al., 2004]. A 
+relative sea level curve for Mayotte was 
+obtained [Zinke et al., 2003a; Camoin et al., 
+2004]. Comparing the relative sea level 
+curve of Mayotte with a reference sea level 
+curve permits us to estimate the vertical 
+movement in Mayotte from 10 kyr to 1 kyr 
+B.P. 
+The reference sea level curve used in this 
+study is based on Bard et al. [1990; 1996; 
+2010], Deschamps et al. [2012] and 
+Hallman et al. [2018] for Tahiti and Peltier 
+et al. [2006] and Abdul et al. [2016] for 
+Barbados. For accuracy, only U–Th ages 
+are used for the reference sea level curve. 
+For coherency, only the coral of the 
+Acropora palmata species are included in 
+the reference curve. For the construction of 
+the sea level curve using Tahiti data, 
+subsidence rate corrections of 0.15 mm/yr 
+after 6 kyr B.P. [Deschamps et al., 2018] 
+and of 0.25 mm/yr before 6 kyr B.P. [Bard 
+et al., 1990] are used. For Barbados, uplift 
+rates of 0.34 mm/yr before 11.18 kyr B.P. 
+and 0.8 mm/yr before 11.18 kyr B.P. are 
+used [Bard et al., 2016; Gargani, 2022]. 
+ 
+3. Results and interpretation 
+A comparison between the relative sea level 
+in Mayotte and the reference sea level 
+suggests that a subsidence of approximately 
+10 ± 1 m occurred abruptly at 9.3 kyr B.P. 
+in Mayotte in less than 200 yr at a minimum 
+subsidence rate of 50 mm/yr (Figure 3). 
+Then, an uplift of +10 m occurred from 8.1 
+to 7 kyr B.P. during 1 kyr at a rate of 9 
+mm/yr. Afterward, progressive subsidence 
+occurred from 6.1 kyr B.P. to 1 kyr B.P. at 
+a rate of 0.4 mm/yr (i.e., 2 m in 5 kyr). 
+
+ 
+Figure 3: Comparing the relative sea level variation in Mayotte with a reconstructed sea level 
+curve (light blue curve; see the methods for details) using a modeling approach. Abrupt 
+depressurization of a magma reservoir occurred at 9.3 kyr B.P. and caused a subsidence of ~10 
+m. Then, a progressive pressure increase in the magma reservoir occurred from 8.1 to 7 kyr 
+B.P. and caused an uplift rate of 9 mm/yr. At 6.1 kyr B.P., new volcanic edifices loaded the 
+lithosphere.
+An increase in the pressure in the magma 
+reservoir associated with magma ascent can 
+explain the uplift of the surface observed 
+between 8.1 and 7 kyr ago. More precisely, 
+a magma reservoir overpressure of 80 ± 20 
+MPa at a depth of 50 km with a size of 20 
+km could explain an uplift of 10 ± 1 m 
+(Figure 4). Nevertheless, an overpressure of 
+80 ± 20 MPa at a depth of 70 km for a 
+magma reservoir with a radius of 25 km 
+could also explain this uplift (Fig. 5). 
+The cause of the subsidence of 10 m that 
+occurred 9.3 kyr ago in Mayotte is well 
+explained by a decrease in the magma 
+reservoir pressure by the same amount (i.e., 
+a pressure decrease of 80 ± 20 MPa at a 
+depth of 50 km with a magma reservoir size 
+of 20 km, or a pressure decrease of 80 ± 20 
+MPa at a depth of 70 km for a magma 
+reservoir with a radius of 25 km). With a 
+maximum duration of 200 yr for abrupt 
+subsidence, the minimum subsidence rate 
+was 5 cm/yr. 
+ 
+
+MAYOTTEZinke etal (2003)
+0
+HOMPHN
+MODEL
+-5
+Pressure
+-10
+decreases
+SEALEVEL (m)
+-15
+-20
+-25
+-30
+Newvolcanicedificeloading+Pressure decreases
+-35
+Overpressure
+-40
+1
+2
+3
+4
+5
+6
+7
+8
+9
+TIME (kvr B.P. 
+FIGURE 4: Modeling of the vertical displacement Uz caused by magma reservoirs with various 
+overpressures P. The magma reservoir (not to scale) has a radius rs and is located at a depth 
+–Z at a distance R from the expansion source embedded at position  to point j. The shear 
+modulus G=15 GPa, =0.25, rs=50 km, and Z=50 km. The observed uplift is 10 ± 1 m. 
+ 
+ 
+
+18
+16
+14
+zn
+12
+UPLIFT (m)
+10
+Observed
+8
+75MPa
+60.MPa
+100 MPa
+6
+45 MPa
+4
+2
+20 MPa
+0
+0
+10
+20
+30
+40
+50
+DISTANCE (km)
+-z1
+R 
+Figure 5: Modeling the vertical displacement Uz caused by magma reservoir overpressure P 
+at a depth Z of 70 km with a radius rs of 25 km. The shear modulus G=15 GPa, and =0.25. 
+The observed uplift is 10 ± 1 m. 
+The subsidence that occurred from 6.1 kyr 
+B.P. to 1 kyr B.P. is due to a progressive 
+deflation of the magma reservoir and/or a 
+loading of the lithosphere by one or more 
+new volcanic edifices. The loading by a 
+volcanic edifice located at a distance of 0 
+km from Mayotte with a height of 200 m 
+caused a subsidence of 1.4 ± 0.2 m (Figure 
+6). This value corresponds to a subsidence 
+rate of 0.14 mm/yr, assuming a duration of 
+10 kyr for isostatic adjustment. Considering 
+a second edifice construction of a height of 
+200 m at a maximum distance of 40 km 
+from Mayotte (GT), the total subsidence 
+could reach 2.5 m. To obtain the observed 
+subsidence rate of 0.4 mm/yr, it is necessary 
+to load the lithosphere by a mass of 5.4 × 
+1012 kg, corresponding to a volume of 1.8 
+km3 or a deflation of the magma reservoir 
+of 30-40 MPa for a chamber at depths of 50-
+70 km. 
+
+20
+G=15GPa,Z-70km, Rs=30000m, deltaP-100MPa
+G=15GPa, Z=70km, Rs=25000m, deltaP=80MPa
+G=15GPa, Z=70km, Rs=25000m, deltaP=60MPa
+G=15GPa, Z=70km, Rs=25000m, deltaP=40MPa
+G=15GPa, Z=70km, Rs=25000m, deltaP=20MPa
+15
+UPLIFT (m)
+10
+5
+0
+10
+20
+30
+40
+50
+DISTANCE (km) 
+Figure 6: Isostatic adjustment modeling. (A) Isostatic adjustment modeling for different elastic 
+thicknesses after loading by volcanic edifices located in Petite Terre and the offshore Mayotte 
+archipelago are in gray. (B) Details of the modeling results. =0.25, ρa =3179 kg/m3, ρv = 3000 kg/m3, E 
+= 1010 Pa, and 35 km < Te < 45 km.
+4. Discussion 
+The complete vertical movement sequence 
+is characterized by an abrupt subsidence 
+phase of approximately 10 m due to magma 
+chamber deflation at a minimum subsidence 
+rate of 5 cm/yr at 9.3 kyr ago, followed by a 
+progressive uplift of 10 m at a rate of 
+
+500
+A
+WITHOUT iSOSTASY
+Te =45km
+Te=40km
+0
+Te=35km
+-500
+-1000
+(w)
+HEIGHT
+-1500
+-2000
+-2500
+-3000
+-3500
+-60
+-40
+-20
+0
+20
+40
+60
+DISTANCE (km)
+B
+WITHOUT ISOSTASY
+Te=45km
+Te=40km
+Te=35km
+Te =45km, 2 newvolcanoes
+1
+Te = 40 km,2 new volcanoes
+Te =35km,2 new volcanoes
+0
+()
+HEIGHT
+-1
+-2
+-3
+13.4
+13.5
+13.6
+13.7
+13.8
+13.9
+14
+14.1
+DISTANCE (km)approximately 9 mm/yr in relation to a 
+magma reservoir pressure increase from 8.1 
+to 7 kyr ago. Then, slow subsidence caused 
+by magma reservoir deflation and/or 
+loading by new volcanic edifices occurred 
+after 6.1 kyr. Finally, a new abrupt 
+subsidence rate increase of more than 5 
+cm/yr started in 2018. 
+The abrupt subsidence rate that occurred 9.3 
+kyr ago is of the same order as the 
+subsidence that occurred from May 2018 to 
+January 2020 in Mayotte. The estimated 
+downward surface deformation during the 
+apex of the recent volcanic activity was 186 
+mm/yr, and the observed mean subsidence 
+was 12 cm in 1.5 yr using the Global 
+Navigation Satellite System (GNSS) and 
+Interferometric Synthetic Aperture Radar 
+(INSAR) during the 2018-2021 event 
+[Lemoine et al., 2020]. 
+A magma reservoir pressure decrease at 9.3 
+kyr B.P. or an increase from 8.1 kyr to 7.1 
+kyr is coherent with the volcanic context 
+and explains the vertical movements. The 
+observation of altered phonolite at 7.965 ± 
+45 kyr also suggests increased volcanic 
+activity approximately 8 kyr ago [Zinke et 
+al., 2003a]. Seismological findings show a 
+complex geometry of a plume of hot 
+upwelling rock rooted in the deep mantle, 
+which is indicative of hotspot volcanoes 
+beneath the Comoros archipelago [French 
+and Romanowicz, 2015; O’Connor et al., 
+2019]. 
+Geophysical 
+observations 
+are 
+consistent with geochemical data showing a 
+nonvolatile superplume isotopic signature 
+in the Comoros–Mayotte and Madagascar 
+systems [O’Connor et al., 2019]. The 
+geochemical 
+and 
+petrographic 
+characteristics of lavas from Mayotte 
+indicate a deep mantle source [Spath et al., 
+1996] that suggests a complex hotspot 
+origin for Mayotte volcanic activity 
+[Emerick and Duncan, 1982; Debeuf, 
+2004]. The relatively long duration (1 kyr) 
+of the deep overpressure (50-70 km) 
+suggests that the magmatic pulse that has 
+occurred during the past 10 kyr has been 
+caused by deep mantle processes beneath 
+Mayotte along the Comoros archipelago. 
+This 
+mantle/asthenospheric 
+flow 
+is 
+potentially a pulse from a complex hotspot 
+system. 
+The abrupt and relatively short (< 0.2 kyr) 
+subsidence that occurred 9.3 kyr ago cannot 
+be explained by the loading of new volcanic 
+edifices due to the duration of isostatic 
+adjustment (10 ± 5 kyr). Glacio-isostatic 
+adjustment should be correlated with 
+specific climatic change [Austermann et al., 
+2013], which is not the case here. Similarly, 
+the significant uplift that occurred from 8.1 
+kyr to 7.1 kyr cannot be explained by abrupt 
+mass unloading in this case. Landslides or 
+erosion could cause isostatic adjustment, 
+thereby influencing the subsidence or uplift 
+rates and relative sea level curves, but over 
+a longer duration [Gargani, 2022a and b]. 
+These phenomena cannot explain an uplift 
+with a duration of 1 kyr because the isostatic 
+adjustment duration is approximately 10 
+times longer [Mitrovica et al., 2000; Van 
+der Wal et al., 2010]. 
+The estimated subsidence rate of 0.4 ± 0.1 
+mm/yr of Mayotte that occurred from 6.1 
+kyr to 1 kyr B.P. could have been caused by 
+loading by new volcanic edifices and/or 
+magma reservoir deflation. The slow 
+subsidence of the island is considered active 
+[Zincke et al., 2003; Camoin et al., 2004]. 
+The estimated long-term subsidence rate is 
+slightly 
+higher 
+than 
+the 
+long-term 
+subsidence rate that is estimated to range 
+from 0.13 mm/yr to 0.25 mm/yr over a 
+longer duration [Camoin et al., 2004]. 
+Neglecting vertical movement fluctuations 
+of -10 m at 9.3 kyr B.P. followed by +10 m 
+that occurred from 8.1 kyr B.P. to 7 kyr BP 
+permits us to obtain an average subsidence 
+value of approximately 2 m in 10 kyr B.P., 
+which is in agreement with previous studies 
+[Camoin et al., 2004]. 
+This study suggests that new volcanic 
+edifices were built between 6.1 kyr and 1 
+kyr ago. The volcanic activity during this 
+period is indicated by relatively young 
+volcanic morphological features east of 
+Mayotte, along the slope of the main 
+
+structure and by several volcanic rocks with 
+ages ranging from 2.2 kyr to 7 kyr [Zinke et 
+al., 2000, 2003; Camoin et al., 2004]. There 
+is no published age concerning submarine 
+edifices along the Mayotte slope to our 
+knowledge.  
+ 
+5. Conclusion 
+During the last 10 kyr, significant volcanic 
+activity caused uplift and subsidence in 
+Mayotte. The subsidence of -10 m was due 
+to a magma reservoir overpressure decrease 
+at 9.3 kyr B.P., whereas the uplift of +10 m 
+was caused by a magma reservoir pressure 
+increase of 80 ± 20 MPa from 50-70 km 
+deep. The significant duration of the 
+overpressure was caused by a quasi-
+permanent deep magmatic flow during 1 
+kyr and suggests a deep magmatic root. The 
+magma reservoir pressure increase was 
+contemporaneous 
+with 
+the 
+observed 
+volcanic activity. The accumulation of 
+volcanic material at Mayotte contributed to 
+the slow subsidence of 2.5 m after 6.1 kyr 
+B.P. The recent volcanic activity that started 
+in 2018 could represent the last sequence of 
+the deep-sourced cycle. The long volcanic 
+cycle of approximately 10 kyr was recorded 
+by relative sea level evolution and provides 
+new insight for understanding the volcanic 
+activity of the Comoros archipelago. 
+Transient hotspot processes could have 
+caused this long-term cycle. 
+ 
+Acknowledgment: 
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+
diff --git a/etAyT4oBgHgl3EQfjfjE/content/tmp_files/load_file.txt b/etAyT4oBgHgl3EQfjfjE/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf,len=1072
+page_content='Relative sea level and coastal vertical movements in relation with volcano-  tectonic processes in Mayotte  Julien Gargani1, 2  1Université Paris-Saclay, CNRS, Geops, Orsay, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  2Université Paris-Saclay, Centre d’Alembert, Orsay, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Highlights  An abrupt subsidence of 10 m occurred in Mayotte at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', followed by an uplift  of 10 m from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr to 7 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Volcano-tectonic activity is responsible for these vertical movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Deep reservoir overpressure has been recorded over a period of 1 kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Abstract  During the last 10 kyr, significant subsidence and uplift occurred on Mayotte Island in the  Comoros archipelago, but the role of volcanic processes in Holocene vertical movements has  been neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Here, we show that an abrupt subsidence of 10 m occurred 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr ago and then  an uplift of the same amplitude occurred at a rate of 9 mm/yr from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr to 7 kyr ago, and we  compare the relative sea level of Mayotte with a reference sea level curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Using a modeling  approach, this study shows that an increase and a decrease in pressure in a deep magma reservoir  was responsible for the ~10 m subsidence and the 10 m uplift, whereas the loading by new  volcanic edifices has caused the subsidence that has occurred during the past 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  surface movement was caused by pulses from the deep mantle that were probably related to  superplume activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Keywords: uplift, subsidence, volcanic edifice, sea level  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Introduction  Recent  volcano-seismic  events  in  unexpected places or with significant  impacts challenge our understanding of  volcanic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' First, these events suggest  reconsidering the role of volcanic activity or  sea-level rise/inundation on local natural  hazards [Roger, 2019: Audru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2010]  and promoting specific studies on the  reduction in risk [Deves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Pasquon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2022] associated with  volcanic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Second, the potential  interrelation between short-term volcanic  activity and long-term volcano-tectonic  evolution also makes it challenging to  understand volcanic processes and Earth  surface evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The risk associated with volcanic activities  is not only due to magmatic explosions and  lava flows but also to a consequence of  induced landslides [Ward and Day, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Gargani, 2020] and subsidence [Lambeck,  1981],  thereby  increasing  marine  submersion risk and tsunamis [Ward and  Day, 2001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Subsidence in a volcanic  context may be associated with volcanic  edifice loading [Lambeck, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Gargani,  2022b] or magma reservoir deflation  [Gargani, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The relationship between  volcanic activity and significant relative sea  level variation has been highlighted in  previous  studies  on  contemporary  observations as well as on past events  [Sternai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Duggen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Mayotte Island (Comoros archipelago, SW  Indian  Ocean,  between  Africa  and  Madagascar) has experienced significant  seismic activity since May 2018, with more  than 11000 earthquakes of up to magnitude  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='9, unusually long-period events (more  than 400) and surface deflation up to 200  mm/yr [Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Cesca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' This seismic activity indicates deep  magma reservoirs at 20-35 km, 50 km and  70 km from Mayotte’s Petite Terre Island  [Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Cesca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' This latter reservoir is  connected to the 50 km-depth reservoir  through rapid magma propagation in the  lithosphere [Lavassyere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  magma reservoir diameters are 10-15 km at  depths of 20-35 km and 20-30 km at depths  of ~50 km [Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Cesca et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Lavassyere et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2022] (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' During the 2018-2021  seismo-volcanic event, a deflation of the  magma reservoir located at a depth of ~50  km occurred (Lavassyere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' This  seismo-volcanic activity caused the birth of  an 820 m tall, 5 km3 deep sea volcanic  edifice documented on the eastern insular  slopes of Mayotte at the tip of a 50 km-long  volcanic ridge composed of many other  recent edifices and lava flows [Feuillet et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Figure 1: Mayotte volcanic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (A) Grand Terre (GT) and Petite Terre (PT) islands in  Mayotte archipelago, with a volcanic edifice, lava flows and a caldera, (B) schematic  representation of the lithosphere below Mayotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Before the 2018-2021 event, the most recent  volcanic  activity  in  the  archipelago  occurred on the Comoros Islands (Karthala  volcano), 200 km away from Mayotte  [Bachèlery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Studies have  suggested that the presence of a hotspot  explains the E‒W sublinear trend of the  Comoros volcanic archipelago [Emerick  and Duncan, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Ebinger and Sleep,  1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Class et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2005], but this  explanation is controversial [Bertil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Quidelleur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Researchers  have  interpreted  the  E‒W  seismic  alignment in the region as highlighting the  activity of structures associated with the  continuity of the East African Rift and in  Mayotte  GT  PT  Caldera  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='6°  45°  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4°  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8°  FO  2018-2021  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4°  A  Eruption  10-  GT  Mayotte  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='6°  20-  Caldera  Crust  2018-2021  Depth (km)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8°  30-  Eruption  Magma  40-  reservoir  13°  50-  1500  Mantle  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2°  60-  70-  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4°  0  20 km  3500  80-  B  Asthenosphere  20  40  Distance (km)  60relation to the Madagascar drift on an E‒W-  trending axis [Bertil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The main volcanic building phase of  Mayotte Island is estimated to have  occurred from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='6 Ma to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25 Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The lava  content demonstrates lithospheric but also  mantle sources, suggesting lithospheric  melt as well as the existence of an ascending  mantle plume [Pelleter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  mantle depths range from 17 km to 70 km  below Mayotte [Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' On  Mayotte Island, volcanic events have been  dated to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='5-7 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (ash), 8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (phonolite), 4 kyr and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='04 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (pumice stone), and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='14 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (volcanic mud) [Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2003a,  2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' These volcanic  activities highlight the existence of volcanic  edifice building during the Holocene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Nevertheless, the volume of volcanic rock  that erupted during this phase is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Furthermore, the duration of the complete  volcanic cycle from magma ascent to  volcanic edifice construction in Mayotte  during the Holocene has never been  estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Relative sea level is useful to quantify  vertical movements [Gargani, 2022a and  2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Previous studies on coral reefs have  obtained accurate data on relative sea levels  in Mayotte during the Holocene [Zinke et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  2003a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' These studies  have discussed the role of climate variation  and postglacial floods, as well as glacio-  hydro-isostatic adjustment, on relative sea  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' However, the potential effect of  volcanic activity on relative sea level rise  and fall has not been analyzed in Mayotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  FIGURE 2: Relative sea level in Mayotte (red cross) compared to the sea level curve  constructed using data from Tahiti and Barbados (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Data from Mayotte: Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' data from Tahiti: Bar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (1990, 1996, 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Hallman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Deschamps et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Pirazzoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' data from Barbados: Peltier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Abdul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Before 6 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', the subsidence rate in Tahiti is corrected with an uplift rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25  0  XMAH  MAYOTTE Zinke et al (2003)  H4  10  20  SEALEVEL (m)  30  40  王  50  60  2  4  6  8  10  12  TIME (kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' )mm/yr, and afterward, it is corrected using a subsidence rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='15 mm/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Before 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='18 kyr,  the uplift rate is corrected with an uplift rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='34 mm/yr in Barbados, and after 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='18 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', an uplift rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8 mm/yr is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' 14C ages are corrected by a U/Th ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  This study analyzes the vertical movement  in Mayotte during the last 10 kyr by  modeling magma reservoir pressure as well  as volcanic edifice loading and compares  these parameters with relative sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  More precisely, volcanic activities, such as  magma reservoir overpressure (amplitude  and duration) and the volume of volcanic  edifice during the last 10 kyr in Mayotte, are  quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The implications of these results  at the regional scale are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Method  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 Overpressure and surface uplift  The vertical displacement Uz of a magma  reservoir of radius rs at a depth –Z with an  overpressure \uf044P is given by the following  equation [Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=" McTigue,  1987]:  Uz = (1-\uf06e) (\uf044P rs3 Z) / [G (R2+Z2)]3/2 × [1 - (rs/Z)3 ×  [(1-\uf06e) / [2 (7-5\uf06e)] - [15 (2-\uf06e)] / [4 (7-5\uf06e)] ×  Z2/(R2+Z2) ]]  where\uf020\uf06e is the Poisson's ratio (\uf06e=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25) and  R is the distance from the expansion source  embedded at position \uf078 to point j on the free  surface and corresponds to the radial  distance from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' G is the shear  modulus (15 GPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The depths of the magma reservoir that are  considered in this study are 20 km, 50 km  and 70 km, which are in agreement with the  estimation by Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  radii of the magma reservoir tested by  modeling range from 5 km to 30 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  influence of the magma reservoir pressure  at different depths was also modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  overpressure and depressurization tested in  this study range from 20 to 150 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  overpressure was estimated using the  observed uplift of 10 ± 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2 Volcanic edifice loading and isostatic  adjustment  Loading of volcanic edifices on the  lithosphere causes subsidence [Lambeck,  1981].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Modeling isostatic adjustment in  response to loading is performed using a  classical law [Turcotte and Schubert, 2001]  that has been applied in various studies  [Watts, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Gargani, 2004 et 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Turcotte and Schubert (2001) used the  equation 𝜵2(𝐷∙𝜵2𝑤(𝑥))+(𝜌𝑎+𝜌v)𝑔∙𝑤(𝑥) =  𝜌v𝑔[𝑧init(𝑥)+ 𝑧(𝑥)] to model the flexure of  the lithosphere, where D is the rigidity of  the lithosphere, w(x) represents the uplift,  ρa and ρv are the densities of the  asthenosphere and volcanic rock, g is the  acceleration of gravity, 𝑧init(𝑥) is the initial  topography and 𝑧(𝑥) is the topography after  the construction of the new volcanic edifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content="  The rigidity is defined by D = ETe3/[12(1-\uf06e2)],  where E is Young's modulus and Te is the  effective elastic thickness." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The E‒W sublinear trend of the Comoros  volcanic archipelago permits us to analyze  the symmetry axis of this 200 km-long  structure and to simplify the numerical  problem using a 2-D approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The elastic  thickness in Mayotte is 40 ± 5 km [Calmant  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 1990], which corresponds to a  flexural rigidity that ranges from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8×1022  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='m2 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1×1022 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' In the calculation, E  = 1010 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The resulting vertical motion rates are  calculated by considering a constant  displacement during the 10 kyr after the  abrupt mass displacement, which is in  agreement with the time necessary to relax  the viscous properties of the lithosphere 10  ± 5 kyr after isostatic adjustment [Mitrovica  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Van der Wal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  average velocity that is obtained is a first-  order estimation sufficient to describe the  main dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Different loading masses corresponding to  various volumes of volcanic rock involved  in volcanic edifice building were simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Loading was simulated by several volcanic  edifices that were 200 m tall and had a  diameter of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='5 km, corresponding to a  conical volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='6 km3 and a mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8  × 1012 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' This study discusses the loading  mass that explains the observed subsidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Two  different  locations  have  been  considered (0 km and 20 km from the  coast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The results have been compared with  observed vertical movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 Relative sea level and vertical  movement  Well-preserved coral reefs were collected in  Mayotte and dated [Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' A  relative sea level curve for Mayotte was  obtained [Zinke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2003a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Comparing the relative sea level  curve of Mayotte with a reference sea level  curve permits us to estimate the vertical  movement in Mayotte from 10 kyr to 1 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The reference sea level curve used in this  study is based on Bard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  2010], Deschamps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [2012] and  Hallman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [2018] for Tahiti and Peltier  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [2006] and Abdul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [2016] for  Barbados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' For accuracy, only U–Th ages  are used for the reference sea level curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  For coherency, only the coral of the  Acropora palmata species are included in  the reference curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' For the construction of  the sea level curve using Tahiti data,  subsidence rate corrections of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='15 mm/yr  after 6 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [Deschamps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2018]  and of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25 mm/yr before 6 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' [Bard  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 1990] are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' For Barbados, uplift  rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='34 mm/yr before 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='18 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8 mm/yr before 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='18 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' are  used [Bard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Gargani, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Results and interpretation  A comparison between the relative sea level  in Mayotte and the reference sea level  suggests that a subsidence of approximately  10 ± 1 m occurred abruptly at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  in Mayotte in less than 200 yr at a minimum  subsidence rate of 50 mm/yr (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Then, an uplift of +10 m occurred from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  to 7 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' during 1 kyr at a rate of 9  mm/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Afterward, progressive subsidence  occurred from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' to 1 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' at  a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4 mm/yr (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2 m in 5 kyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Figure 3: Comparing the relative sea level variation in Mayotte with a reconstructed sea level  curve (light blue curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' see the methods for details) using a modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Abrupt  depressurization of a magma reservoir occurred at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' and caused a subsidence of ~10  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Then, a progressive pressure increase in the magma reservoir occurred from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 to 7 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' and caused an uplift rate of 9 mm/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' At 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', new volcanic edifices loaded the  lithosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  An increase in the pressure in the magma  reservoir associated with magma ascent can  explain the uplift of the surface observed  between 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 and 7 kyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' More precisely,  a magma reservoir overpressure of 80 ± 20  MPa at a depth of 50 km with a size of 20  km could explain an uplift of 10 ± 1 m  (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Nevertheless, an overpressure of  80 ± 20 MPa at a depth of 70 km for a  magma reservoir with a radius of 25 km  could also explain this uplift (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The cause of the subsidence of 10 m that  occurred 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr ago in Mayotte is well  explained by a decrease in the magma  reservoir pressure by the same amount (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  a pressure decrease of 80 ± 20 MPa at a  depth of 50 km with a magma reservoir size  of 20 km, or a pressure decrease of 80 ± 20  MPa at a depth of 70 km for a magma  reservoir with a radius of 25 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' With a  maximum duration of 200 yr for abrupt  subsidence, the minimum subsidence rate  was 5 cm/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  MAYOTTEZinke etal (2003)  0  HOMPHN  MODEL  5  Pressure  10  decreases  SEALEVEL (m)  15  20  25  30  Newvolcanicedificeloading+Pressure decreases  35  Overpressure  40  1  2  3  4  5  6  7  8  9  TIME (kvr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  FIGURE 4: Modeling of the vertical displacement Uz caused by magma reservoirs with various  overpressures \uf044P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The magma reservoir (not to scale) has a radius rs and is located at a depth  –Z at a distance R from the expansion source embedded at position \uf078 to point j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The shear  modulus G=15 GPa, \uf06e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25, rs=50 km, and Z=50 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The observed uplift is 10 ± 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  18  16  14  zn  12  UPLIFT (m)  10  Observed  8  75MPa  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='MPa  100 MPa  6  45 MPa  4  2  20 MPa  0  0  10  20  30  40  50  DISTANCE (km)  z1  R  Figure 5: Modeling the vertical displacement Uz caused by magma reservoir overpressure \uf044P  at a depth Z of 70 km with a radius rs of 25 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The shear modulus G=15 GPa, and \uf06e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The observed uplift is 10 ± 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The subsidence that occurred from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' to 1 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' is due to a progressive  deflation of the magma reservoir and/or a  loading of the lithosphere by one or more  new volcanic edifices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The loading by a  volcanic edifice located at a distance of 0  km from Mayotte with a height of 200 m  caused a subsidence of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2 m (Figure  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' This value corresponds to a subsidence  rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='14 mm/yr, assuming a duration of  10 kyr for isostatic adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Considering  a second edifice construction of a height of  200 m at a maximum distance of 40 km  from Mayotte (GT), the total subsidence  could reach 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' To obtain the observed  subsidence rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4 mm/yr, it is necessary  to load the lithosphere by a mass of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4 ×  1012 kg, corresponding to a volume of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8  km3 or a deflation of the magma reservoir  of 30-40 MPa for a chamber at depths of 50-  70 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  20  G=15GPa,Z-70km, Rs=30000m, deltaP-100MPa  G=15GPa, Z=70km, Rs=25000m, deltaP=80MPa  G=15GPa, Z=70km, Rs=25000m, deltaP=60MPa  G=15GPa, Z=70km, Rs=25000m, deltaP=40MPa  G=15GPa, Z=70km, Rs=25000m, deltaP=20MPa  15  UPLIFT (m)  10  5  0  10  20  30  40  50  DISTANCE (km)  Figure 6: Isostatic adjustment modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (A) Isostatic adjustment modeling for different elastic  thicknesses after loading by volcanic edifices located in Petite Terre and the offshore Mayotte  archipelago are in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (B) Details of the modeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' \uf06e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25, ρa =3179 kg/m3, ρv = 3000 kg/m3, E  = 1010 Pa, and 35 km < Te < 45 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Discussion  The complete vertical movement sequence  is characterized by an abrupt subsidence  phase of approximately 10 m due to magma  chamber deflation at a minimum subsidence  rate of 5 cm/yr at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr ago, followed by a  progressive uplift of 10 m at a rate of  500  A  WITHOUT iSOSTASY  Te =45km  Te=40km  0  Te=35km  500  1000  (w)  HEIGHT  1500  2000  2500  3000  3500  60  40  20  0  20  40  60  DISTANCE (km)  B  WITHOUT ISOSTASY  Te=45km  Te=40km  Te=35km  Te =45km, 2 newvolcanoes  1  Te = 40 km,2 new volcanoes  Te =35km,2 new volcanoes  0  ()  HEIGHT  1  2  3  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='6  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='7  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='8  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='9  14  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  DISTANCE (km)approximately 9 mm/yr in relation to a  magma reservoir pressure increase from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  to 7 kyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Then, slow subsidence caused  by magma reservoir deflation and/or  loading by new volcanic edifices occurred  after 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Finally, a new abrupt  subsidence rate increase of more than 5  cm/yr started in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The abrupt subsidence rate that occurred 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3  kyr ago is of the same order as the  subsidence that occurred from May 2018 to  January 2020 in Mayotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The estimated  downward surface deformation during the  apex of the recent volcanic activity was 186  mm/yr, and the observed mean subsidence  was 12 cm in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='5 yr using the Global  Navigation Satellite System (GNSS) and  Interferometric Synthetic Aperture Radar  (INSAR) during the 2018-2021 event  [Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  A magma reservoir pressure decrease at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3  kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' or an increase from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  kyr is coherent with the volcanic context  and explains the vertical movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  observation of altered phonolite at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='965 ±  45 kyr also suggests increased volcanic  activity approximately 8 kyr ago [Zinke et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2003a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Seismological findings show a  complex geometry of a plume of hot  upwelling rock rooted in the deep mantle,  which is indicative of hotspot volcanoes  beneath the Comoros archipelago [French  and Romanowicz, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' O’Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Geophysical  observations  are  consistent with geochemical data showing a  nonvolatile superplume isotopic signature  in the Comoros–Mayotte and Madagascar  systems [O’Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  geochemical  and  petrographic  characteristics of lavas from Mayotte  indicate a deep mantle source [Spath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  1996] that suggests a complex hotspot  origin for Mayotte volcanic activity  [Emerick and Duncan, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Debeuf,  2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The relatively long duration (1 kyr)  of the deep overpressure (50-70 km)  suggests that the magmatic pulse that has  occurred during the past 10 kyr has been  caused by deep mantle processes beneath  Mayotte along the Comoros archipelago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  This  mantle/asthenospheric  flow  is  potentially a pulse from a complex hotspot  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The abrupt and relatively short (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2 kyr)  subsidence that occurred 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr ago cannot  be explained by the loading of new volcanic  edifices due to the duration of isostatic  adjustment (10 ± 5 kyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Glacio-isostatic  adjustment should be correlated with  specific climatic change [Austermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  2013], which is not the case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Similarly,  the significant uplift that occurred from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  kyr to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr cannot be explained by abrupt  mass unloading in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Landslides or  erosion could cause isostatic adjustment,  thereby influencing the subsidence or uplift  rates and relative sea level curves, but over  a longer duration [Gargani, 2022a and b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  These phenomena cannot explain an uplift  with a duration of 1 kyr because the isostatic  adjustment duration is approximately 10  times longer [Mitrovica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Van  der Wal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The estimated subsidence rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  mm/yr of Mayotte that occurred from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1  kyr to 1 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' could have been caused by  loading by new volcanic edifices and/or  magma reservoir deflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The slow  subsidence of the island is considered active  [Zincke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  The estimated long-term subsidence rate is  slightly  higher  than  the  long-term  subsidence rate that is estimated to range  from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='13 mm/yr to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='25 mm/yr over a  longer duration [Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Neglecting vertical movement fluctuations  of -10 m at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' followed by +10 m  that occurred from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' to 7 kyr BP  permits us to obtain an average subsidence  value of approximately 2 m in 10 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  which is in agreement with previous studies  [Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  This study suggests that new volcanic  edifices were built between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr and 1  kyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The volcanic activity during this  period is indicated by relatively young  volcanic morphological features east of  Mayotte, along the slope of the main  structure and by several volcanic rocks with  ages ranging from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2 kyr to 7 kyr [Zinke et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2000, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Camoin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' There  is no published age concerning submarine  edifices along the Mayotte slope to our  knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Conclusion  During the last 10 kyr, significant volcanic  activity caused uplift and subsidence in  Mayotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The subsidence of -10 m was due  to a magma reservoir overpressure decrease  at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='3 kyr B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', whereas the uplift of +10 m  was caused by a magma reservoir pressure  increase of 80 ± 20 MPa from 50-70 km  deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The significant duration of the  overpressure was caused by a quasi-  permanent deep magmatic flow during 1  kyr and suggests a deep magmatic root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The  magma reservoir pressure increase was  contemporaneous  with  the  observed  volcanic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The accumulation of  volcanic material at Mayotte contributed to  the slow subsidence of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='5 m after 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1 kyr  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The recent volcanic activity that started  in 2018 could represent the last sequence of  the deep-sourced cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' The long volcanic  cycle of approximately 10 kyr was recorded  by relative sea level evolution and provides  new insight for understanding the volcanic  activity of the Comoros archipelago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Transient hotspot processes could have  caused this long-term cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Acknowledgment:  References  Audru, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Guennoc, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Thinon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' &  Abellard, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=" Bathymay : la structure sous-  marine de Mayotte révélée par l'imagerie  multifaisceaux." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Comptes  Rendus  Geoscience,  338,  1240-1249,  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='crte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='010 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Audru J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Bitri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Desprats J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  Dominique P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Eucher G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Hachim S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Jossot  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Mathon C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Nédellec J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Sabourault P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  Sedan O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Stollsteiner P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Terrier-Sedan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Major natural hazards in a tropical  volcanic island : a review for Mayotte  island, Comoros archipelago, Indian Ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Engineering geology, 114, 264-381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Austermann J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Mittrovica J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Latychev  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' & Milne G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (2013), Barbados based  estimate of ice volume at Last Glacial  Maximum affected by subducted plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Geoscience, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' 553-537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Bache F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Gargani J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', J-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Suc, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Gorini,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='Rabineau, S-M Popescu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Leroux, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Do Couto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='Jouannic, J-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Rubino, J-L  Olivet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Clauzon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='Tadeu Dos Reis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Aslanian, (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Messinian evaporite  deposition during sea level rise in the Gulf  of Lions (Western Mediterranean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Marine  and Petroleum Geology, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='66, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='262-277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Bachèlery, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',Morin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Villeneuve, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  Soulé, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Nassor, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', and Ali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  Structure and eruptive history of Karthala  volcano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  In  Bachelery,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=',  Lenat,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
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+page_content=', Dullo  W-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', Thomassin B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=', 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Facies and  faunal assemblage changes in response to  the Holocene transgression in the Lagoon of  Mayotte (Comoro Archipelago, SW Indian  Ocean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
+page_content=' Facies, 50, 391-408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etAyT4oBgHgl3EQfjfjE/content/2301.00417v1.pdf'}
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+arXiv:2301.01839v1  [math.RT]  4 Jan 2023
+The dimension of an orbitope based on a solution to the
+Legendre pair problem
+Kristopher N. Kilpatricka, Dursun A. Bulutoglua
+aDepartment of Mathematics and Statistics, Air Force Institute of Technology,
+Wright-Patterson Air Force Base, Ohio 45433, USA
+Abstract
+The Legendre pair problem is a particular case of a rank-1 semidefinite description
+problem that seeks to find a pair of vectors pu, vq each of length ℓ such that the
+vector puJ, vJqJ satisfies the rank-1 semidefinite description. The group pZℓ ˆ
+Zℓq ¸ Zˆ
+ℓ acts on the solutions satisfying the rank-1 semidefinite description by
+ppi, jq, kqpu, vq “ ppi, kqu, pj, kqvq for each ppi, jq, kq P pZℓ ˆZℓq¸Zˆ
+ℓ . By applying
+the methods based on representation theory in Bulutoglu [Discrete Optim. 45
+(2022)], and results in Ingleton [Journal of the London Mathematical Society s(1-
+31) (1956), 445-460] and Lam and Leung [Journal of Algebra 224 (2000), 91-109],
+for a given solution puJ, vJqJ satisfying the rank-1 semidefinite description, we
+show that the dimension of the convex hull of the orbit of u under the action of
+Zℓ or Zℓ ¸ Zˆ
+ℓ is ℓ ´ 1 provided that ℓ “ pn or ℓ “ pqi for i “ 1, 2, any positive
+integer n, and any two odd primes p, q. Our results lead to the conjecture that
+this dimension is ℓ ´ 1 in both cases. We also show that the dimension of the
+convex hull of all feasible points of the Legendre pair problem of length ℓ is 2ℓ ´ 2
+provided that it has at least one feasible point.
+Keywords:
+Circulant matrix, Convex hull, Ramanujan sums, Rational
+representations, Vanishing sums of roots of unity
+2000 MSC: 90C22 68R05 20C15 05E20 20G05 11L03
+1. Introduction
+A Hadamard matrix can be constructed by finding a solution to a system of
+constraints for a pair of vectors. To define this system of constraints, let Zℓ “
+t0, . . . , ℓ ´ 1u denote the integers mod ℓ. Two length ℓ vectors pu, vq is a Legendre
+Email addresses: kristopher.n.kilpatrick@gmail.com (Kristopher N. Kilpatrick),
+dursun.bulutoglu@gmail.com (Dursun A. Bulutoglu)
+Preprint submitted to Elsevier
+January 6, 2023
+
+pair (LP) if
+Pupjq ` Pvpjq “ ´2, @j P Zℓ ´ t0u,
+1Ju “ 1Jv, u, v P t´1, 1uℓ,
+(1.1)
+where Pxpjq “ ř
+iPZℓ xpiqxpi´jq is the periodic autocorrelation function of a vector
+x P Cℓ. The problem of finding solutions to the system of constraints (1.1) is known
+as the LP problem. System of constraints (1.1) can be cast as the following rank-1
+constrained semidefinite description (rank-1 CSDD).
+ÿ
+i´j“r pmod ℓq
+1ďi,jďℓ
+Yij `
+ÿ
+i´j“r pmod ℓq
+ℓ`1ďi,jď2ℓ
+Yij
+“
+´2,
+for r “ 1, . . . , ℓ,
+Yjj
+“
+1,
+for j “ 1, . . . , 2ℓ,
+Y
+ľ
+0,
+rankpYq
+“
+1,
+(1.2)
+ℓ´1
+ÿ
+j“0
+Y1j`1
+“
+ℓ´1
+ÿ
+j“0
+Y1pℓ`j`1q,
+where Y ľ 0 means Y is non-negative definite. In this rank-1 CSDD for LPs, an
+LP pu, vq is obtained by taking uj “ Y1pj`1q and vj “ Y1pℓ`j`1q for j “ 0, . . . , ℓ ´ 1.
+For the definition of the feasible set of a rank-1 constrained semidefinite program-
+ming problem, i.e. rank-1 CSDD, see [3].
+Let QZℓ be the vector space of all functions from Zℓ to Q. A circulant shift
+of u P QZℓ by j P Zℓ, denoted by cju, is a transformation such that cjupiq “
+upi ´ jq, i P Zℓ. The circulant matrix of u P QZℓ, denoted by Cu, is a matrix such
+that pj ` 1qth row of Cu is pcjuqJ. If pu, vq is an LP with 1Ju “ 1Jv “ ´1, then
+H “
+»
+——–
+1
+1
+1J
+1J
+1
+´1
+1J
+´1J
+1
+1
+Cv
+Cu
+1
+´1
+CJ
+u
+´CJ
+v
+fi
+ffiffifl
+is a p2ℓ`2qˆp2ℓ`2q Hadamard matrix, where 1 is the vector of all 1s of length ℓ [2].
+A Hadamard matrix can also be constructed by using an LP with 1Ju “ 1Jv “ 1.
+Hence, to construct a p2ℓ ` 2q ˆ p2ℓ ` 2q Hadamard matrix for some odd ℓ, it
+suffices to find an LP of length ℓ. It is conjectured that an LP of length ℓ exists for
+each odd ℓ, where this conjecture implies the Hadamard conjecture. It is shown
+in Arasu et al. [2] that an LP pu, vq must satisfy
+1Ju “ 1Jv “ ˘1.
+2
+
+In this paper, we choose all LPs pu, vq to satisfy
+1Ju “ 1Jv “ ´1.
+(1.3)
+Let N ¸ H be the semidirect product of the two groups N and H as defined in
+Rotman [12]. Then, the group Zℓ ¸Zˆ
+ℓ acts on u P QZℓ by pj, kqupiq “ uppj, kq´1iq
+for each pj, kq P Zℓ ¸ Zˆ
+ℓ , i P Zℓ, where pa, bqi “ bi ` a for pa, bq P Zℓ ¸ Zˆ
+ℓ , i P Zℓ.
+The group pZℓ ˆ Zℓq ¸ Zˆ
+ℓ acts on any pair pu, vq P QZℓ ‘ QZℓ by
+ppi, jq, kqpu, vq “ ppi, kqu, pj, kqvq
+for each ppi, jq, kq P pZℓ ˆ Zℓq ¸ Zˆ
+ℓ . Two pairs pu, vq, pu1, v1q of vectors of length ℓ
+are equivalent if either pu1, v1q or pv1, u1q is in the orbit of pu, vq under the action
+of pZℓ ˆZℓq¸Zˆ
+ℓ . If pu, vq is an LP and pu1, v1q is equivalent to pu, vq, then pu1, v1q
+is also an LP [2]. The symmetry group of a rank-constrained CSDD is the set of
+all permutations of its solution vector that send solutions to solutions. Hence, the
+symmetry group of rank-1 CSDD for LPs contains the group pZℓ ˆ Zℓq ¸ Zˆ
+ℓ .
+Throughout the paper, for each n P Zě1, let rns “ t1, . . . , nu, and
+td : d | ℓu “ td P rℓs : d | ℓu.
+For a set of vectors S in a vector space over the field of scalars F, SpanFpS) is
+the span, AffFpS) is the affine hull, and dimFpS) is the dimension of the affine
+hull of the vectors in S over F. For a matrix M, ColFpMq is the column space of
+M over F. If F is not provided, then F “ R. Also, let Conv(S) be the convex
+hull of the vectors in S. For a finite group G acting linearly on a vector space V ,
+G v is the orbit of v P V under the action of G, and ConvpG vq is the orbitope
+of G through v [13]. Moreover, let Fℓ be the feasible set of all pairs puJ, vJqJ
+satisfying the rank-1 CSDD for the LP problem.
+In this paper, we determine
+dimpConvpFℓqq, and investigate all possible values of dimpConvppZℓ ¸ Zˆ
+ℓ q uqq and
+dimpConvppZℓ¸Zˆ
+ℓ q vqq when pu, vq is an LP. The following theorems and corollary
+are our main results.
+Theorem 1.1. Let ℓ be an odd positive integer and Fℓ be the feasible set of all
+pairs puJ, vJqJ satisfying the rank-1 CSDD for the LP problem. If Fℓ ‰ H, then
+dimpConvpFℓqq “ 2ℓ ´ 2.
+The following corollary follows immediately from Theorem 1.1.
+Corollary 1.2. Let puJ, vJqJ satisfy the rank-1 CSDD for the LP problem. Then
+two linear constraints that are linearly independent from 1Ju “ 1Jv implied by
+having constraint (1.2) in the rank-1 CSDD for the LP problem are 1Ju “ ˘1.
+Furthermore, no linear constraint that is linearly independent from constraints
+1Ju “ ˘1 and 1Ju “ 1Jv is implied.
+3
+
+Proof. First, the rank-1 CSDD for the LP problem is feasible. Then, any other
+linear constraint that is linearly independent from the constraints 1Ju “ ˘1 and
+1Ju “ 1Jv would necessarily imply that dimpConvpFℓqq ă 2ℓ ´ 2, contradicting
+Theorem 1.1.
+Theorem 1.3. Let pu, vq be an LP, and either G “ Zℓ or G “ Zℓ ¸ Zˆ
+ℓ . Then
+there exists U1, U2 Ď td : d | ℓu such that U1 X U2 “ H and
+dimpConvpG uqq “ ℓ ´ 1 ´
+`ř
+dPU1 φ
+` ℓ
+d
+˘˘
+,
+dimpConvpG vqq “ ℓ ´ 1 ´
+`ř
+dPU2 φ
+` ℓ
+d
+˘˘
+.
+Theorem 1.4. Let p, q be distinct odd primes and n P Zě1. Let ℓ “ pn, or ℓ “ pqi
+for i “ 1, 2. Let pu, vq be an LP of length ℓ, and either G “ Zℓ or G “ Zℓ ¸ Zˆ
+ℓ .
+Then
+dimpConvpG uqq “ dimpConvpG vqq “ ℓ ´ 1.
+In Section 2, we present necessary background in representation theory, the
+power spectral density, vanishing sums of roots of unity, and affine geometry. In
+Section 3, we prove our main results and conjecture that the restrictions on ℓ in
+Theorem 1.4 other than ℓ being an odd positive integer can be dropped. Section 4
+is our discussion about future work.
+2. Background theory
+Let G be a finite group and V be a finite-dimensional vector space over a field
+F. Let GLpV q be the F-automorphisms of V . An n-dimensional F-representation
+of G is a pair pρ, V q such that ρ: G Ñ GLpV q is a homomorphism and V is an n-
+dimensional vector space over F. A subspace W of V is an F-subrepresentation of V
+if ρpgqW Ď W @g P G. A representation pρ, V q of G is an irreducible representation
+if the only subrepresentations of V are SpanFp0q and V . Two F-representations
+ρ1 : G Ñ GLpWq and ρ2 : G Ñ GLpW 1q of G are equivalent if there is an invertible
+linear transformation φ : W Ñ W 1 Q φpρ1pgqwq “ ρ2pgqφpwq @w P W and g P G.
+For a decomposition V “ m1V1 k ¨ ¨ ¨ k mbVb into irreducible representations of a
+group G, mi is the number of times the irreducible C-representation Vi appears up
+to equivalence. Such mi ě 1 is called the multiplicity of Vi.
+The only fields F that we consider in this paper are Q, R, and C, the rational,
+real, and complex numbers, respectively. An F-representation ρ : G Ñ GLpV q
+is unitary with respect to an inner product x¨, ¨y in V if xρpgqv, ρpgquy “ xu, vy
+for all u, v P V . Every representation is unitary with respect to a complex inner
+product [14]. We use the convention xαu | vy “ ¯αxu | vy for α P F and u, v P V .
+If a representation pρ, V q of G is such that ρpgq permutes a basis of V for all
+4
+
+g P G, then pρ, V q is called a permutation F-representation. Every permutation
+F-representation is unitary with respect to the complex dot product. Throughout
+the paper, every group G is finite, every vector space V is finite-dimensional, and
+every representation of any specified group is a permutation representation and
+hence is unitary with respect to the complex dot product.
+Theorem 2.1. [Maschke] Every representation of a finite group is a direct sum
+of orthogonal irreducible subrepresentations with respect to some inner product.
+The decomposition in Theorem 2.1 is said to be multiplicity-free if each irre-
+ducible appears only once. The character of an F-representation pρ, V q of G is the
+map χρ : G Ñ F defined by χρpgq “ Trpρpgqq, where Trpρpgqq is the trace of ρpgq.
+We say that the character is an irreducible character if the character belongs to an
+irreducible representation. Two F-representations of a finite group are equivalent
+if and only if they have the same character [5]. We may simply write the character
+as χ if the representation is clear from the context. The following theorem is from
+Serre [14].
+Theorem 2.2. Let pρ, V q be a C-representation of a group G. Let
+V “ m1V1 k . . . k mhVh
+with mi P Zě1 be a decomposition of V into irreducibles pρi, Viq with characters χi
+@i P rhs, where χi “ χρi. Then the orthogonal projection Pi of V onto miVi “
+Àmi
+k“1 Vi is given by
+Pi “ dimCpViq
+|G|
+ÿ
+gPG
+χipgqMρpgq.
+where Mρpgq is the matrix of ρpgq in some orthonormal basis for V .
+The exponent m of a group G is the smallest nonnegative integer such that
+gm “ e @g P G. Let pρ, V q be a C-representation of a group G with exponent m.
+Then ρpgqm “ id @g P G, where id is the identity mapping on V . Therefore, the
+eigenvalues of ρpgq are mth roots of unity. Throughout the paper, let ζm “ e2πi{m.
+Since χpgq is the trace of ρpgq, χpgq P Qpζmq, where Qpζmq is the field extension of
+Q obtained by adjoining ζm.
+Let Zˆ
+m “ tr P Zm | pr, mq “ 1u be the multiplicative group of Zm, where pr, mq
+is the greatest common divisor of r and m. The automorphism group AutpQpζmqq
+of Qpζmq is the Galois group of the field extension Qpζmq over Q denoted by
+GalpQpζmq{Qq. The elements of GalpQpζmq{Qq can be indexed by the elements of
+Zˆ
+m, where for σr P GalpQpζmq{Qq
+σrpζmq “ ζr
+m, σrpqq “ q for each q P Q, and r P Zˆ
+m.
+5
+
+The map σr Ñ r is an isomorphism between GalpQpζmq{Qq and Zˆ
+m. Let AutpCq
+be the group of all automorphisms of C and
+AutpCqQpζmq “ tτ|Qpζmq | τ P AutpCqu,
+where τ|Qpζmq is the restriction of τ P AutpCq to Qpζmq. For each τ P AutpCq,
+pτpζmqqm “ τppζmqmq “ τp1q “ 1, and τpqq “ q for each q P Q. Hence, we have
+AutpCqQpζmq Ď AutpQpζmqq. Since any automorphism of a subfield of C can be
+extended to an automorphism of C [15], each σr P AutpQpζmqq can be extended to
+a σ1
+r P AutpCq. Then AutpQpζmqq Ď AutpCqQpζmq implying
+AutpQpζmqq “ AutpCqQpζmq.
+Since the entries of the matrix of ρpgq are in Qpζmq, AutpQpζmqq (AutpCq) acts on
+the (characters of the) C-representations of G of the same dimension.
+Let V “ FG be the space of all functions from G to F. Then the standard basis
+egphq “
+#
+1
+if g “ h,
+0
+otherwise
+for g, h P G spans V . We further equip V with the usual inner product x¨ | ¨y
+(the complex dot product) so that tegugPG is an orthonormal basis.
+Then the
+regular F-representation of G is the pair pρ, V q such that ρ: G Ñ GLpV q is the
+homomorphism determined by the action of G on the basis of V , where ρphqeg “
+ehg @h, g P G. Clearly, the regular F-representation of G is also a permutation
+representation.
+The complexification of a vector space U over Q is defined as UC “ C bQ U. It
+is clear that if U is a Q-representation, then UC is the C-representation obtained
+by extending the field of scalars Q of U to C. Let x¨ | ¨y be the inner product of
+V over F. It is plain to show that an inner product on VC may be defined as
+xz b v | z1 b v1y “ zz1xv | v1y. Then the following lemma follows immediately.
+Lemma 2.3. Let V be an inner product space over Q and U, W subspaces of V .
+Then U is orthogonal to W if and only if UC is orthogonal to WC.
+The following theorem is well-known.
+Theorem 2.4. Let
+`
+R, QZℓ˘
+be the regular Q-representation of Zℓ. Then
+`
+QZℓ˘
+C “
+ë
+iPZℓ
+Vi
+is the decomposition of
+`
+QZℓ˘
+C into the one-dimensional irreducible C-representations
+of Zℓ, where the C-representations pρk, Vkq, k P Zℓ are given by ρkpjq: Vk Ñ Vk,
+v ÞÑ ζ
+jk
+ℓ v, and χkpjq “ Trpρkpjqq “ ζ
+jk
+ℓ
+for each k, j P Zℓ.
+6
+
+For each k P Zℓ, vk “ ř
+jPZℓ ζjk
+ℓ ej is the basis vector for the one-dimensional
+C-representation Vk of Zℓ. It is well-known that tv0, . . . , vℓ´1u is a basis of CZℓ.
+Throughout the paper, we call the vectors v0, . . . , vℓ´1 the discrete Fourier basis
+vectors.
+The following theorem characterizes the orbits of the one-dimensional repre-
+sentations in Theorem 2.4 under the action of AutpCq.
+Theorem 2.5. For k P Zℓ, let χk be the irreducible character of the C-representation
+of Zℓ. For each divisor d of ℓ, let Od “ tχk | pk, ℓq “ d and k P Zℓu. Then
+(i) |Od| “ φpℓ{dq, where φ is Euler’s totient function.
+(ii) Od “ AutpCq χd “ AutpQpζℓ{dqq χd.
+Proof. Let d be a divisor of ℓ. By the definition of Euler’s totient function, (i)
+follows immediately. To prove (ii),
+σrpχdpjqq “ ζ
+jrd
+ℓ
+“ χrdpjq
+for each σr P AutpQpζℓ{dqq, r P Zˆ
+ℓ{d, and j P Zℓ.
+Hence, σrpχdq “ χrd, and
+prd, ℓq “ d implies χrd P Od, consequently AutpQpζℓ{dqq χd Ď Od.
+Let χk P Od such that k “ rd and pr, ℓ{dq “ 1. Then σr P AutpQpζℓ{dqq and
+χk “ χrd “ σrχd
+Hence, Od Ď AutpQpζℓ{dqq χd.
+The following theorem follows from Isaacs [8, Theorem 9.21], Theorem 2.2, and
+Lemma 2.3.
+Theorem 2.6. Let pρ, V q be a Q-representation of a group G with exponent m.
+Let
+V “ W1 k ¨ ¨ ¨ k Wb
+be a decomposition of V into irreducible Q-subrepresentations. Let VC, WiC be the
+C-representations obtained from V, Wi by extending the field of scalars of V, Wi to
+C. Let
+VC “ Wp1,1q k ¨ ¨ ¨ k Wp1,r1q k ¨ ¨ ¨ k Wpb,1q k ¨ ¨ ¨ k Wpb,rbq
+be a decomposition of VC into
+`
+ρi,j, Wpi,jq
+˘
+irreducible C-subrepresentations with
+characters χρi,j, where
+WiC “ Wpi,1q k ¨ ¨ ¨ k Wpi,riq for i P rbs.
+7
+
+For i P rbs, let Oρi,1 be the AutpQpζmqq-orbit of χρi,1. Let Mρpgq be the matrix of
+ρpgq @g P G, with respect to the standard basis. If VC is multiplicity-free, then
+PWi “ dimCpWpi,1qq
+|G|
+ÿ
+gPG
+¨
+˝ ÿ
+χPOρi,1
+χpgqMρpgq
+˛
+‚
+is the orthogonal projection matrix into the ith irreducible Q-subrepresentation
+subspace ColQpPWiq @i P rbs.
+The following theorem provides the decomposition of QZℓ into irreducible Q-
+subrepresentations under the action of Zℓ.
+Theorem 2.7. Let
+`
+R, QZℓ˘
+be the regular Q-representation of Zℓ. Let
+Pd “ 1
+ℓ
+ÿ
+iPZℓ
+ÿ
+χPOd
+χpiqMRpiq
+for each divisor d of ℓ, where MRpiq is the matrix of Rpiq with respect to the
+standard basis teiuiPZℓ, and Od is defined in Theorem 2.5. Then
+QZℓ “
+ë
+d | ℓ
+ColQpPdq
+is the decomposition of QZℓ into irreducible Q-subrepresentations under the action
+of Zℓ.
+Proof. Let d be a divisor of ℓ.
+By Theorem 2.5, the inner sum of Pd is over
+the orbit of χd under the action of AutpQpζℓ{dqq.
+Further, by Theorem 2.4,
+`
+QZℓ˘
+C is multiplicity-free.
+Then by Theorem 2.6, ColQpPdq is an irreducible
+Q-subrepresentation of QZℓ. Now, by the identity ℓ “ ř
+d | ℓ φpℓ{dq, it suffices to
+show that
+dimQ pColQpPdqq “ φ
+ˆℓ
+d
+˙
+.
+Since R is the regular representation of Zℓ, only Rp0q will contribute to the trace
+8
+
+of Pd. Since Pd is a projection matrix,
+dimQ pColQpPdqq “ rankpPdq “ Tr pPdq “ Tr
+˜
+1
+ℓ
+ÿ
+iPZℓ
+ÿ
+χPOd
+χpiqMRpiq
+¸
+“ 1
+ℓ
+ÿ
+iPZℓ
+ÿ
+χPOd
+χpiq Tr
+`
+MRpiq
+˘
+“ 1
+ℓ
+ÿ
+χPOd
+Tr
+`
+MRp0q
+˘
+“ 1
+ℓ
+ÿ
+χPOd
+ℓ
+“
+ÿ
+χPOd
+1
+“ |Od|
+“ φ
+ˆℓ
+d
+˙
+.
+The group Zℓ ¸ Zˆ
+ℓ acts on Zℓ by pa, bqi “ bi ` a @i P Zℓ and pa, bq P Zℓ ¸ Zˆ
+ℓ .
+Then a representation pT, QZℓq of Zℓ ¸ Zˆ
+ℓ is given by the action on the basis
+Tpa, bqei “ ebi`a. The next theorem shows that QZℓ has the same decomposition
+under the action of Zℓ ¸ Zˆ
+ℓ as in Theorem 2.7, i.e., extending Zℓ to Zℓ ¸ Zˆ
+ℓ does
+not change the decomposition in Theorem 2.7.
+Theorem 2.8. The decomposition QZℓ “ Ë
+d | ℓ ColQpPdq is the decomposition of
+QZℓ into irreducible Q-subrepresentations under the action of Zℓ ¸ Zˆ
+ℓ .
+Proof. Let d be a divisor of ℓ.
+We have ColCpPdq “ Ë
+pk,ℓq“d Vk, where Vk is
+spanned by vk “ ř
+iPZℓ ζki
+ℓ ei. Let pa, bq P Zℓ ¸ Zˆ
+ℓ . Observe
+Tpa, bqvk “
+ÿ
+iPZℓ
+ζki
+ℓ ebi`a “
+ÿ
+jPZℓ
+ζkpj´aqb´1
+ℓ
+ej “ ζ´kab´1
+ℓ
+vb´1k.
+Since pb, ℓq “ 1, pbk, ℓq “ d if and only if pk, ℓq “ d. Therefore, ColCpPdq is a
+C-subrepresentation under the action of Zℓ ¸ Zˆ
+ℓ . Since ColQpPdq is an irreducible
+Q-subrepresentation under the action of Zℓ and Zℓ is a subgroup of Zℓ ¸ Zˆ
+ℓ ,
+ColQpPdq is an irreducible Q-subrepresentation under the action of Zℓ ¸ Zˆ
+ℓ .
+The discrete Fourier transform (DFT) of u P QZℓ is µkpuq “ xu | vky, k P Zℓ.
+Equation µkpuq “ xu | vky, k P Zℓ is the change of basis (variables) formula that
+provides the Fourier coordinates µkpuq in terms of the standard basis coordinates
+9
+
+uj. The power spectral density (PSD) of u P QZℓ is |µkpuq|2, for k P Zℓ. The
+following theorem provides an equivalent condition that characterizes an LP in
+terms of its PSD.
+Theorem 2.9 (Fletcher et al. [6]). Let u, v P t´1, 1uℓ. Then pu, vq is an LP if
+and only if µ0puq “ µ0pvq and
+|µkpuq|2 ` |µkpvq|2 “ 2pℓ ` 1q,
+@k P Zℓ ´ t0u.
+We say that there is a vanishing sum of m ℓth roots of unity, if there exists m
+ℓth roots of unity w1, . . . , wm (not necessarily distinct) satisfying w1`¨ ¨ ¨`wm “ 0.
+Theorem 2.10 (Lam and Leung [11]). Suppose that ℓ “ pn1
+1 . . . pns
+s
+for distinct
+primes p1, . . . , ps and n1, . . . , ns P Zě1. Then there exists a vanishing sum of m
+ℓth roots of unity if and only if m “ a1p1 ` ¨ ¨ ¨ ` asps for some a1, . . . , as P Zě0.
+Let u P t´1, 1uℓ satisfy x1 | uy “ ´1. Let J “ tj P Zℓ | upjq “ ´1u. Then
+µkpuq for k ‰ 0, has two forms
+µkpuq “ ´2
+ÿ
+jPJ
+ζkj
+ℓ
+“ 2
+ÿ
+jRJ
+ζkj
+ℓ ,
+(2.1)
+where |J| “ pℓ ` 1q{2.
+Lemma 2.11. Let ℓ “ pn, p an odd prime, and n P Zě1. Let u P t´1, 1uℓ satisfy
+x1 | uy “ ´1. Then µkpuq ‰ 0 @k P Zℓ.
+Proof. Since µ0puq “ ´1 we need to only verify for k P rℓ ´ 1s. Suppose for a
+contradiction that µkpuq “ 0 for some k P rℓ´1s. By equation (2.1), ř
+jPJ ζkj
+ℓ
+“ 0.
+By Theorem 2.10,
+ℓ ` 1
+2
+“ ap
+for some a P Zě0. This means that p | ppℓ ` 1q{2q, a contradiction. Therefore,
+µkpuq ‰ 0 @k P Zℓ.
+Lemma 2.12. Let ℓ “ pq, p, q distinct odd primes.
+Let u P t´1, 1uℓ satisfy
+x1 | uy “ ´1. Then µkpuq ‰ 0 @k P Zℓ.
+Proof. Since µ0puq “ ´1 we need to only verify for k P rℓ ´ 1s. Suppose for a
+contradiction that µkpuq “ 0 for some k P rℓ ´ 1s. By equation (2.1),
+ÿ
+jPJ
+ζkj
+ℓ
+“
+ÿ
+jRJ
+ζkj
+ℓ
+“ 0.
+(2.2)
+10
+
+By Theorem 2.10,
+ℓ ` 1
+2
+“ ap ` bq and ℓ ´ 1
+2
+“ cp ` dq
+for some a, b, c, d P Zě0. This means
+ℓ “ ℓ ` 1
+2
+` ℓ ´ 1
+2
+“ pa ` cqp ` pb ` dqq.
+If a ` c ‰ 0 and b ` d ‰ 0, then p | pb ` dq and q | pa ` cq. Consequently,
+ℓ “ pa ` cqp ` pb ` dqq ě pq ` pq “ 2ℓ,
+a contradiction.
+Suppose that a ` c “ 0. Then a “ c “ 0 and subsequently
+q | ppℓ`1q{2q, a contradiction. A similar contradiction occurs if b`d “ 0. Therefore,
+µkpuq ‰ 0 @k P Zℓ.
+Lemmas 2.11 and 2.12 do not generalize to arbitrary odd ℓ. There is a u P
+t´1, 1u45 with x1 | uy “ ´1 such that µkpuq “ 0 for some k P Z45 ´ t0u.
+The following lemma is well-known.
+Lemma 2.13. Let V be a vector space over F. Let S Ď V . Then AffFpSq “
+SpanFpSq if and only if 0 P AffFpSq.
+Let pρ, V q be a representation of a group G. Let S Ď V be nonempty. Then the
+barycenter of S is βpSq “ p1{|S|q ř
+vPS v. The fixed space of V under the action
+of G is
+V G “ tv P V | ρpgqv “ v @g P Gu.
+Clearly, V G is a subrepresentation of V . By Theorem 2.1, there exists a subrepre-
+sentation orthogonal to V G.
+Lemma 2.14. Let pρ, V q be a unitary representation of a group G. Let P be the
+orthogonal projection matrix onto V G. Then Px “ βpG xq for x P V .
+Proof. Let u1, . . . , uk be an orthonormal basis for V G. Then
+Py “
+ÿ
+iPrks
+xui | yyui
+for any y P V . Then, for each i P rks and r P G, we have
+xui | ρprqxy “ xρprqui | ρprqxy “ xui | xy.
+11
+
+Then
+xui | βpG xqy “
+1
+|G x|
+ÿ
+ρprqxPG x
+xui | ρprqxy “
+1
+|G x|
+ÿ
+ρprqxPG x
+xui | xy “ xui | xy.
+Since βpG xq P V G,
+βpG xq “ PβpG xq “
+ÿ
+iPrks
+xui | βpG xqyui “
+ÿ
+iPrks
+xui | xyui “ Px.
+Lemma 2.15. Let
+`
+R, QZℓ˘
+be the regular Q-representation of Zℓ. Then the pro-
+jection matrix Pℓ onto the fixed space of QZℓ is p1{ℓqJ, where J is the ℓ ˆ ℓ matrix
+of all ones.
+Proof. Notice that the fixed space of V is a direct sum of copies of the trivial
+representation of Zℓ. By Theorem 2.2,
+Pℓ “ 1
+ℓ
+ÿ
+jPZℓ
+χ0pjqMRpjq “ 1
+ℓ
+ÿ
+jPZℓ
+MRpjq “ 1
+ℓJ.
+3. Main results
+We start with the proof of the first of our main results.
+Proof of Theorem 1.1. Let
+F 1
+ℓ “ tu P t´1, 1uℓ | Dv Q pu, vq is an LPu
+and
+F 2
+ℓ “ tv P t´1, 1uℓ | Du Q pu, vq is an LPu.
+By symmetry, F 1
+ℓ “ F 2
+ℓ . Since Fℓ ‰ H, there exists puJ, vJqJ P Fℓ. Then u P F 1
+ℓ
+and v P F 2
+ℓ . Let p “ βpZℓ uq “ ´p1{ℓq1. Then p P ConvpF 1
+ℓ q Ď AffpF 1
+ℓ q. To
+reach the desired conclusion, we observe the following facts:
+(i) Since F 1
+ℓ “ F 2
+ℓ , SpanCpF 1
+ℓ ´ pq “ SpanCpF 2
+ℓ ´ pq.
+(ii) Since both SpanCpF 1
+ℓ ´ pq and SpanCpF 2
+ℓ ´ pq are C-subrepresentations
+of pQZℓqC under the action of Zℓ orthogonal to 1, by Theorem 2.1, both
+SpanCpF 1
+ℓ ´ pq and SpanCpF 2
+ℓ ´ pq must be an orthogonal direct sum of the
+irreducible C-subrepresentations V1, . . . , Vℓ´1 of pQZℓqC.
+12
+
+(iii) Both SpanCpF 1
+ℓ ´ pq and SpanCpF 2
+ℓ ´ pq cannot be orthogonal to an irre-
+ducible Vk for some k P rℓ ´ 1s. For this would imply that the DFT of u
+must satisfy
+µkpuq “ µkpu ´ pq “ 0,
+similarly, µkpvq “ 0, contradicting Theorem 2.9.
+By (i), (ii), and (iii),
+SpanCpF 1
+ℓ ´ pq “ V1 k ¨ ¨ ¨ k Vℓ´1 and SpanCpF 2
+ℓ ´ pq “ V1 k ¨ ¨ ¨ k Vℓ´1.
+Let f “ β
+´
+pZℓ ˆ Zℓq
+`
+uJ, vJ˘J¯
+“ ´p1{ℓq
+`
+1J, 1J˘J.
+Then f P ConvpFℓq Ď
+AffpFℓq. It is evident that
+SpanC pFℓ ´ fq “ SpanC
+`
+F 1
+ℓ ´ p
+˘
+‘ SpanC
+`
+F 2
+ℓ ´ p
+˘
+.
+Then by Theorem 2.7, SpanQpF i
+ℓ ´ pq “
+ë
+d | ℓ
+d‰ℓ
+ColQpPdq for i “ 1, 2, and by fact (i),
+SpanQ pFℓ ´ fq “
+¨
+˚
+˝
+ë
+d | ℓ
+d‰ℓ
+ColQpPdq
+˛
+‹‚‘
+¨
+˚
+˝
+ë
+d | ℓ
+d‰ℓ
+ColQpPdq
+˛
+‹‚.
+Hence,
+dimpConvpFℓqq “ dimpSpanpFℓ´fqq “ dimQpSpanQpFℓ´fqq “ 2pℓ´1q “ 2ℓ´2.
+The following theorem is needed for our next main result.
+Theorem 3.1. Let u P Qℓ satisfy SpanQpuq ‰ SpanQp1q and let y “ u´p1Ju{ℓq1.
+Then there exists T Ď td P rℓs : d | ℓ and d ‰ ℓu such that
+SpanQpZℓ yq “
+ë
+dPT
+ColQpPdq,
+(3.1)
+where Pd is defined in Theorem 2.7. Moreover,
+dimpConvpZℓ uqq “ dimpAffpZℓ uqq “ dimQpSpanQpZℓ yqq “ rankpCyq “
+ÿ
+dPT
+φ
+ˆℓ
+d
+˙
+,
+where Cy is the ℓ ˆ ℓ circulant matrix whose first row is y.
+13
+
+Proof. Since SpanQpZℓ yq is a Q-subrepresentation of QZℓ under the action of Zℓ,
+SpanQpZℓ yq is an orthogonal direct sum of irreducible Q-subrepresentations of QZℓ
+by Theorem 2.1. By Theorem 2.7, the irreducible Q-subrepresentations of QZℓ are
+ColQpPdq for each divisor d of ℓ. Therefore, there exists a set T of divisors of ℓ
+for which equation (3.1) holds. We first show that SpanQpZℓ yq is orthogonal only
+to ColQpPℓq, where Pℓ is the projection onto the fixed space of QZℓ under the
+action of Zℓ. Since ColQpPℓq “ SpanQp1q, xy | 1y “ 0, and the representation QZℓ
+is unitary, we have SpanQpZℓ yq is orthogonal to ColQpPℓq.
+Let z “ u ´ Pℓu. Then, by Lemma 2.15, z “ y “ u ´ p1Ju{ℓq1. Since Pℓy “
+Pℓu ´ P2
+ℓu “ 0 and βpZℓ yq “ Pℓy by Lemma 2.14, 0 “ βpZℓ yq P ConvQpZℓ yq as
+βpZℓ yq is a convex combination of points of Zℓ y. Then by Lemma 2.13, we have
+SpanQpZℓ yq “ AffQpZℓ yq. Observe that
+dimQpAffQpZℓ uqq “ dimQ
+ˆ
+AffQpZℓ uq ´ 1Ju
+ℓ 1
+˙
+“ dimQpAffQpZℓ yqq “ dimQpSpanQpZℓ yqq.
+Then,
+dimpAffpZℓ uqq “ dimQpAffQpZℓ uqq “ dimQpSpanQpZℓ yqq “
+ÿ
+dPT
+φ
+ˆℓ
+d
+˙
+.
+We now prove Theorem 1.3.
+Proof of Theorem 1.3.
+Since Zℓ ă Zℓ ¸ Zˆ
+ℓ , it suffices to prove the result for G “ Zℓ. Let X1 “ Zℓ u,
+X2 “ Zℓ v. Since
+dimpConvpX1qq “ dimpAffpX1qq
+and
+dimpConvpX2qq “ dimpAffpX2qq,
+by Theorem 3.1, it suffices to prove that U1XU2 “ H. For the sake of contradiction,
+let d1 P U1XU2 be such that d1 ‰ ℓ. This implies that for each i “ 1, 2, SpanCpXi`
+p1{ℓq1q is orthogonal to ColCpPd1q. Also, ColCpPd1q Ă pQZℓqC and ColCpPd1q is
+invariant under the action of Zℓ.
+Then by Theorems 2.1 and 2.4, there exists
+U1 Ă Zℓ ´ t0u such that ColCpPd1q “ Ë
+iPU1 Vi. This implies that for each i “ 1, 2,
+SpanCpXi ` p1{ℓq1q is orthogonal to an irreducible Vk for some k P rℓ ´ 1s. Then
+µkpuq “ µk
+ˆ
+u ` 1
+ℓ1
+˙
+“ 0,
+similarly, µkpvq “ 0, contradicting Theorem 2.9.
+Lemma 3.2. Let m, n P Z. If m ´ 2 ě 2, n ´ 2 ě 1 or m ´ 2 ě 1, n ´ 2 ě 2, then
+n ´ 2 ě 2pn ´ 1q{m.
+14
+
+Proof. If m ´ 2 ě 2, n ´ 2 ě 1 or m ´ 2 ě 1, n ´ 2 ě 2, then pm ´ 2qpn ´ 2q ě 2.
+Adding 2pn ´ 2q to both sides of pm ´ 2qpn ´ 2q ě 2 yields mpn ´ 2q ě 2pn ´ 1q.
+Hence n ´ 2 ě 2pn ´ 1q{m.
+Lemma 3.3. Let p, q be distinct odd primes. Then the quotient of 2pq ´ 1qpp ´ 1q
+by division of p is at least q.
+Moreover, for 2pq ´ 1qpp ´ 1q “ sp ` r, where
+s “ 2pq ´ 1q ´ k, r “ kp ´ 2pq ´ 1q, and k is the smallest integer such that
+kp ´ 2pq ´ 1q ě 0, then s and r are the quotient and remainder of 2pq ´ 1qpp ´ 1q
+upon division by p.
+Proof. Since for any integer k, 2pq ´1qpp´1q “ p2pq ´1q´kqp`pkp´2pq ´1qq we
+may choose the smallest such k such that kp´2pq´1q ě 0. Then k “
+P
+2pq´1q{p
+T
+,
+where rns is the ceiling of n. Let 2pq´1qpp´1q “ sp`r, where s “ 2pq´1q´k and
+r “ kp´2pq´1q. Now, as p, q are distinct odd primes, WLOG suppose that p ě 3,
+then q ě 5 ě 4. Since p ´ 2 ě 1 and q ´ 2 ě 2, by Lemma 3.2, q ´ 2 ě 2pq ´ 1q{p.
+Therefore, q ´ 2 ě k. Then s ´ q “ 2pq ´ 1q ´ k ´ q “ q ´ 2 ´ k ě k ´ k “ 0.
+Finally, since k ´ 1 ă 2pq ´ 1q{p, r “ pk ´ 2pq ´ 1q ă p.
+Lemma 3.4. Let ℓ “ pαqβ, where p, q are distinct odd primes, and α, β P Zě1.
+Then φpℓq ą pℓ ´ 1q{2.
+Proof. By Lemma 3.3, 2pq ´ 1qpp ´ 1q “ sp ` r where s ě q. Then
+2φpℓq “ 2pq ´ 1qpp ´ 1qpα´1qβ´1
+“ psp ` rqpα´1qβ´1
+ě pαqβ ` rpα´1qβ´1
+ą pαqβ ´ 1
+“ ℓ ´ 1,
+and the result follows.
+Lemma 3.5. Let u P Qℓ, then
+||µpuq||2 “ ℓ||u||2.
+Proof. Let U be the matrix whose columns are the discrete Fourier basis vectors
+v0, . . . , vℓ´1. Then, since µ “ UJu,
+||µpuq||2 “
+ÿ
+kPZℓ
+|µkpuq|2 “ µ˚µ “ uJU˚Uu “ uJℓIQℓu “ ℓ||u||2,
+where we used the fact that U˚U “ ℓIQℓ.
+15
+
+Corollary 3.6. Let u P QZℓ satisfying ||u||2 “ ℓ and x1 | uy “ ´1. Then
+ℓ´1
+ÿ
+k“1
+|µkpuq|2 “ pℓ ` 1qpℓ ´ 1q.
+Proof. By Lemma 3.5,
+ℓ´1
+ÿ
+k“1
+|µkpuq|2 “ ||µpuq||2 ´ |µ0puq|2 “ ℓ2 ´ 1 “ pℓ ` 1qpℓ ´ 1q.
+For each ℓ, n P Zě1, the Ramanujan’s sum [1] is defined by cℓpnq “ ř
+0ăkďℓ
+pk,ℓq“1 e2πikn{ℓ.
+It is well-known that @ℓ, n P Zě1
+cℓpnq “ µ
+ˆ
+ℓ
+pℓ, nq
+˙
+φpℓq
+φ
+´
+ℓ
+pℓ,nq
+¯ P Z,
+where
+µpnq “
+$
+&
+%
+1
+if n “ 1,
+p´1qr
+if n “ p1 . . . pr for distinct primes p1, . . . , pr,
+0
+if n is divisible by some prime square
+is the M¨obius function.
+Lemma 3.7. Let ℓ ą 1 and u P Qℓ. If d is a divisor of ℓ, then
+ÿ
+0ăkďℓ
+pk,ℓq“d
+|µkpuq|2 “
+ÿ
+sPZℓ
+c ℓ
+dpsqPupsq.
+Proof. First,
+ÿ
+0ăkďℓ
+pk,ℓq“d
+e
+2πikph´jq
+ℓ
+“
+ÿ
+0ărď ℓ
+d
+pr, ℓ
+d q“1
+e
+2πirph´jq
+ℓ
+d
+“ c ℓ
+dph ´ jq.
+Let v0, v1, . . . , vℓ´1 be the discrete Fourier basis vectors. Note that vkv˚
+k “ rak
+hjs,
+where ak
+hj “ e2πikph´jq{ℓ. Then
+ÿ
+0ăkďℓ
+pk,ℓq“d
+vkv˚
+k “ rbd
+hjs,
+where bd
+hj “ cℓ{dph ´ jq. Since µkpuq “ uJvk,
+|µkpuq|2 “ µkpuqµkpuq “ puJvkqpv˚
+kuq “ uJpvkv˚
+kqu.
+16
+
+Then
+ÿ
+0ăkďℓ
+pk,ℓq“d
+|µkpuq|2 “ uJ
+¨
+˚
+˝
+ÿ
+0ăkďℓ
+pk,ℓq“d
+vkv˚
+k
+˛
+‹‚u
+“
+ÿ
+h,jPZℓ
+bd
+hjuhuj
+“
+ÿ
+h,jPZℓ
+c ℓ
+d ph ´ jquhuj
+“
+ÿ
+s,jPZℓ
+c ℓ
+d psqujuj`s
+“
+ÿ
+sPZℓ
+c ℓ
+d psq
+ÿ
+jPZℓ
+ujuj`s
+“
+ÿ
+sPZℓ
+c ℓ
+d psqPupsq.
+For a vector u P Qℓ and a divisor d of ℓ, let
+u
+ℓ
+d pjq “
+ÿ
+iPZd
+u
+ˆℓ
+di ` j
+˙
+,
+@j P Z ℓ
+d.
+Then uℓ{d P Qℓ{d is called the d-compression of u. The following lemma is used to
+prove one of our main results.
+Lemma 3.8. Let ℓ ą 1 and u P Qℓ. If d is a divisor of ℓ, then
+ÿ
+0ăkďℓ
+pk,ℓq“d
+|µkpuq|2 “
+ÿ
+0ăkď ℓ
+d
+pk, ℓ
+d q“1
+|µkpu
+ℓ
+dq|2.
+17
+
+Proof. By Lemma 3.7,
+ÿ
+0ăkďℓ
+pk,ℓq“d
+|µkpuq|2 “
+ÿ
+sPZℓ
+c ℓ
+dpsqPupsq
+“
+ÿ
+jPZ ℓ
+d
+ÿ
+iPZd
+c ℓ
+d
+ˆℓ
+di ` j
+˙
+Pu
+ˆℓ
+di ` j
+˙
+“
+ÿ
+jPZ ℓ
+d
+c ℓ
+d
+ˆℓ
+d ` j
+˙ ÿ
+iPZd
+Pu
+ˆℓ
+di ` j
+˙
+“
+ÿ
+sPZ ℓ
+d
+c ℓ
+dpsqPu
+ℓ
+d psq
+“
+ÿ
+0ăkď ℓ
+d
+pk, ℓ
+d q“1
+|µkpu
+ℓ
+dq|2.
+The following theorem follows from Theorem 3 in [4].
+Theorem 3.9. Let pu, vq be an LP of length ℓ “ dm. Then the corresponding
+m “ ℓ{d-compressed sequences pud, vdq satisfy
+Pudp0q ` Pvdp0q “ 2ℓ ´ 2
+ˆℓ
+d ´ 1
+˙
+“ 2ℓ ` 2 ´ 2ℓ
+d ,
+Pudpjq ` Pvdpjq “ ´2ℓ
+d ,
+@j P Zd ´ t0u.
+We now prove another main result.
+Theorem 3.10. Let pu, vq be an LP of length ℓ such that ||ud||2 “ ||vd||2 for
+some divisor d of ℓ such that 2φpdq ě d ´ 1. Then, |µkpvq|2 ą 0 and |µkpuq|2 ą 0
+@k Q pk, ℓq “ ℓ{d.
+Proof. By symmetry, it suffices to prove the result for v. For the sake of contra-
+diction, let |µk1pvq|2 “ 0 for some k1 Q pk1, ℓq “ ℓ{d for some divisor d of ℓ. Then
+|µkpvq|2 “ 0 @k Q pk, ℓq “ ℓ{d. By Theorem 2.9, |µkpuq|2 “ 2ℓ`2 @k Q pk, ℓq “ ℓ{d.
+Consequently,
+ÿ
+0ăkďℓ
+pk,ℓq“ ℓ
+d
+|µkpuq|2 “ p2ℓ ` 2qφpdq.
+(3.2)
+Then, by Lemma 3.8,
+ÿ
+0ăkďℓ
+pk,ℓq“ ℓ
+d
+|µkpuq|2 “
+ÿ
+0ăkďd
+pk,dq“1
+|µkpudq|2 “ p2ℓ ` 2qφpdq.
+(3.3)
+18
+
+By Theorem 3.9,
+||ud||2 “ ||vd||2 “ ℓ ` 1 ´ ℓ
+d.
+Then, by Lemma 3.5,
+dÿ
+k“1
+|µkpudq|2 “ d||ud||2 ´ |µ0pudq|2 “ d||ud||2 ´ 1.
+(3.4)
+Hence, by equations (3.2), (3.3), and (3.4)
+ÿ
+0ăkďd
+pk,dq“1
+|µkpudq|2 “ p2ℓ ` 2qφpdq ă
+dÿ
+k“1
+|µkpudq|2 “ pℓ ` 1qpd ´ 1q
+which implies
+2φpdq ă pd ´ 1q,
+a contradiction.
+The following corollary follows from Lemma 3.4 and Theorem 3.10.
+Corollary 3.11. Let pu, vq be an LP of length ℓ “ pαqβ, where p, q are distinct
+odd primes, and α, β P Zě1. Let
+T “
+␣
+d P rℓs : d ‰ 1, d | ℓ, and ||ud||2 ‰ ||vd||2 (
+.
+Then
+dimpConvpZℓ uqq “ ℓ ´ 1 ´
+ÿ
+dPT1
+φ pdq ,
+dimpConvpZℓ vqq “ ℓ ´ 1 ´
+ÿ
+dPT2
+φ pdq
+for some T1 Ď T and T2 Ď T. Equivalently
+dimpConvpZℓ uqq ě
+ÿ
+dPT 1
+φ pdq ,
+dimpConvpZℓ vqq ě
+ÿ
+dPT 1
+φ pdq ,
+where
+T 1 “
+␣
+d P rℓs : d ‰ 1, d | ℓ, and ||ud||2 “ ||vd||2(
+.
+(3.5)
+19
+
+Proof. By symmetry, it suffices to prove the result for u. Let y “ u ` p1{ℓq1.
+First, we prove that
+ColQpP ℓ
+d1 q Ď SpanQpZℓ yq @d1 P T 1.
+The subspace SpanQpZℓ yq is a Q-subrepresentation of Qℓ, and thus either
+SpanQpZℓ yq K ColQ
+´
+P ℓ
+d1
+¯
+or
+ColQ
+´
+P ℓ
+d1
+¯
+Ď SpanQpZℓ yq.
+If SpanQpZℓ yq K ColQ
+`
+Pℓ{d1˘
+, then SpanCpZℓ yq K ColC
+`
+Pℓ{d1˘
+, and consequently
+µkpuq “ µkpyq “ 0 for some k Q pk, ℓq “ ℓ{d1. By Lemma 3.4, this contradicts
+Theorem 3.10. Hence, ColQ
+`
+Pℓ{d1˘
+Ď SpanQ pZℓ yq and the result now follows from
+Theorem 3.1.
+Since ||uℓ||2 “ ||vℓ||2 “ ℓ for u, v P t´1, 1uℓ, the following corollary follows
+from the proof of Corollary 3.11.
+Corollary 3.12. Let pu, vq be an LP of length ℓ “ pαqβ, where p, q are distinct
+odd primes, and α, β P Zě1. Then
+(i) ColQ pP1q Ď SpanQpZℓ yq for y “ u ` p1{ℓq1 and y “ v ` p1{ℓq1,
+(ii) dimpConvpZℓ uqq ě φ pℓq and dimpConvpZℓ vqq ě φ pℓq.
+Larger |T 1| in equation (3.5) in Corollary 3.11 yields larger lower bounds for
+dimpConvpZℓ uqq and dimpConvpZℓ vqq. There is evidence that in general T 1 ‰ H.
+In fact, based on known LPs, it was recently conjectured in [10] that for each
+positive ℓ divisible by 5
+5 P
+␣
+d P rℓs : d ‰ 1, d | ℓ, ||ud||2 “ ||vd||2, and pu, vq is an LP of length ℓ
+(
+.
+On the other hand, there are LPs pu, vq such that ||ud||2 ‰ ||vd||2 for some divisor
+d of ℓ, see [9, 10].
+The following theorem gives us a general lower bound on
+dimpConvpZℓ uqq for u P t´1, 1uℓ such that x1 | uy “ ´1.
+Theorem 3.13. Let ℓ “ pn1
+1 . . . pns
+s
+where p1, . . . , ps are distinct odd primes and
+n1, . . . , ns P Zě1. Let u P t´1, 1uℓ satisfy x1 | uy “ ´1, then
+dimpConvpZℓ uqq ě
+ÿ
+jPrss
+ÿ
+iPrnjs
+φppi
+jq.
+20
+
+Proof. Let y “ u ` p1{ℓq1. We will first prove that ColQpPdj,iq Ď SpanQpZℓ yq
+for each dj,i “ pn1
+1 . . . pi
+j . . . pns
+s , j “ 1, . . . , s, i “ 0, . . . , nj ´ 1. Assume otherwise,
+and let µkpyq “ 0 for some k Q pk, ℓq “ dj,i.
+Then µkpuq “ µkpyq “ 0, and
+by equation (2.1) and Theorem 2.10, pj | pℓ ` 1q{2, a contradiction. Therefore,
+ColQpPdj,iq Ď SpanQpZℓ yq, and by Theorem 3.1,
+dimpConvpZℓ uqq “ dimpAffpZℓ uqq “ dimpSpanpZℓ yqq
+“ dimQpSpanQpZℓ yqq ě
+ÿ
+jPrss
+ÿ
+iPrnjs
+φppi
+jq.
+The following corollary follows by combining Corollary 3.12 and the proof of
+Theorem 3.13.
+Corollary 3.14. Let pu, vq be an LP of length ℓ “ pαqβ, where p, q are distinct
+odd primes, and α, β P Zě1. Then
+dimpConvpZℓ uqq ě
+ÿ
+jPrαs
+φppjq `
+ÿ
+iPrβs
+φpqiq ` φpℓq,
+dimpConvpZℓ vqq ě
+ÿ
+jPrαs
+φppjq `
+ÿ
+iPrβs
+φpqiq ` φpℓq.
+A different set of lower bounds on dimpConvpZℓuqq can be obtained by using
+the results in Ingleton [7]. Next, we derive such lower bounds when ℓ “ pqβ for
+some distinct odd primes p, q and positive integer β. However, first we need the
+concept of non-recurrent matrices.
+Let Cu be the circulant matrix of u P Rℓ. Then Cu is non-recurrent if ℓ is
+the only divisor d of ℓ such that ui “ uj whenever i ” j pmod dq. The following
+lemma shows that Cu non-recurrent if there is a v such that pu, vq is an LP.
+Lemma 3.15. Let u P t´1, 1uℓ and x1 | uy “ ¯1. Then Cu is non-recurrent.
+Proof. First note that the constraint x1 | uy “ ¯1 implies that there are pℓ ˘ 1q{2
+´1’s and pℓ ¯ 1q{2 1’s. Suppose for contradiction that Cu is s-recurrent. Then
+s | ℓ and ui “ uj whenever i ” j pmod sq. Since u0, . . . , us´1 each appear d “ ℓ{s
+times and d divides neither pℓ ˘ 1q{2 nor pℓ ¯ 1q{2, we get a contradiction.
+The following lemma and the subsequent corollary are needed to derive lower
+bounds on the rank of circulant matrices Cu for u P t´1, 1uℓ based on rank lower
+bounds in [7] for circulant Cu with u P t0, 1uℓ.
+Lemma 3.16. Let Cu be the circulant matrix of u P Rℓ, J be the ℓ ˆ ℓ all 1s
+matrix, and λ P R. Then
+21
+
+(i) the discrete Fourier basis tv0, v1, . . . , vℓ´1u is an eigenbasis for both Cu and
+Cu ` λJ,
+(ii) if the eigenvalue of Cu corresponding to eigenvector vi is µi for i “ 0, . . . , ℓ´
+1, then the eigenvalue of Cu `λJ corresponding to eigenvector v0 and vi are
+µ0 ` ℓλ and µi for i “ 1, . . . , ℓ ´ 1, respectively, where µ0 “ řℓ´1
+i“0 ui.
+Proof. It is straightforward to check the statements of the lemma.
+The following corollary follows from Lemma 3.16.
+Corollary 3.17. Let u, Cu, J, λ, and µ0 be as in Lemma 3.16. Then
+(i)
+rankpCu ` λJq “
+$
+’
+&
+’
+%
+rankpCuq
+if
+pµ0 ` ℓλqµ0 ‰ 0,
+rankpCuq ` 1
+if
+µ0 “ 0 and λ ‰ 0,
+rankpCuq ´ 1
+if
+µ0 ` ℓλ “ 0 and λ ‰ 0,
+(ii) for odd ℓ and u P t0, 1uℓ,
+rankpC2u´1q “ rankpCuq.
+Proof. By Lemma 3.16, (i) follows immediately. Since C2u´1 “ 2Cu ´ J, and for
+odd ℓ,
+rankp2Cu ´ Jq “ rankp2Cuq “ rankpCuq,
+(ii) follows.
+For ℓ “ pαpα1
+1 . . . pαm´1
+m´1 where p, p1, . . . , pm´1 are distinct primes and α, α1, . . . ,
+αm´1 P Zě1, let
+τpℓ, pq “ 1 ` ǫpmqǫpαqφppq ` φppαq `
+ÿ
+iPrm´1s
+φpppαi
+i q,
+where ǫp1q “ 0 and ǫpkq “ 1 for k P Zě1. Now, the following lemma follows from
+the corresponding result for circulant ℓ ˆ ℓ non-recurrent circulant matrix with
+entries from t0, 1u in Ingleton [7] and Corollary 3.17.
+Lemma 3.18. Let ℓ “ pqβ, where p, q are distinct odd primes, and β P Zě1. Let
+u P t´1, 1uℓ satisfy x1 | uy “ ´1. Then rankpCuq ě mintτpℓ, pq, τpℓ, qqu.
+If β ě 2, then the following lemma implies that the rank is at least τpℓ, qq.
+22
+
+Lemma 3.19. Let ℓ “ pqβ for distinct odd primes p, q, and β P Zě2.
+Then
+τpℓ, qq ă τpℓ, pq.
+Proof. Now
+τpℓ, qq “ 1 ` ǫp2qǫpβqφpqq ` φpqβq ` φppqq “ 1 ` φpqq ` φpqβq ` φppqq,
+and
+τpℓ, pq “ 1 ` φppq ` φppqβq.
+Then
+τpℓ, pq ´ τpℓ, qq “ φpqqpqβ´1pφppq ´ 1q ´ pφppq ` 1qq ` φppq
+“ φpqqpqβ´1pp ´ 2q ´ pq ` φppq.
+We consider two cases: p “ 3 and p ě 5. If p “ 3, then
+τpℓ, pq ´ τpℓ, qq “ φpqqpqβ´1 ´ 3q ` φp3q ě φpqqp5β´1 ´ 3q ` φp3q ą 0.
+Suppose that p ě 5. Then since qβ´1 ´ 1 ě 1 and p ´ 3 ě 1, pqβ´1 ´ 1qpp ´ 3q ě 1.
+Then qβ´1pp ´ 2q ´ p ě qβ´1 ´ 2 ě 3β´1 ´ 2 ą 0. Implying
+τpℓ, pq ´ τpℓ, qq “ φpqqpqβ´1pp ´ 2q ´ pqq ` φppq ą 0.
+Now, the following corollary follows from Theorem 3.13, Corollary 3.17, and
+Lemmas 3.18 and 3.19.
+Corollary 3.20. Let ℓ “ pqβ for distinct odd primes p, q, and β P Zě2. Let Cu
+with u P t´1, 1uℓ be the circulant matrix of u such that Cu is non-recurrent. Then
+rankpCuq ě Lppqβq “ max
+$
+&
+%φppqq ` φpqq ` φpqβq ` 1, φppq `
+ÿ
+iPrβs
+φpqiq ` 1
+,
+.
+- .
+If φppqq ` φpqq ` φpqβq ą φppq ` ř
+iPrβs
+φpqiq, we may improve the lower bound in
+Corollary 3.20. We have the following theorem.
+Theorem 3.21. Let ℓ “ pqβ for distinct odd primes p, q, and β P Zě2. Let Cu
+with u P t´1, 1uℓ be the circulant matrix of u such that Cu is non-recurrent. Then
+(i) rankpCuq is at least
+φppq ` φppqq `
+ÿ
+iPrβs
+φpqiq ` 1
+if φppqq ` φpqq ` φpqβq ą φppq `
+ÿ
+iPrβs
+φpqiq,
+φppq `
+ÿ
+iPrβs
+φpqiq ` 1
+otherwise,
+23
+
+(ii) dimpConvpZℓuqq is at least
+φppq ` φppqq `
+ÿ
+iPrβs
+φpqiq
+if φppqq ` φpqq ` φpqβq ą φppq `
+ÿ
+iPrβs
+φpqiq,
+φppq `
+ÿ
+iPrβs
+φpqiq
+otherwise.
+Proof. We first show that (ii) follows from (i). Let y “ u ´ p1Ju{ℓq1. Then by
+Theorem 3.1,
+dimpConvpZℓ uqq “ dimpAffpZℓ uqq “ dimpSpanpZℓ yqq “ rankpCyq.
+Since Cy “ Cu ´ pp1Juq{ℓqJ, by part (i) of Corollary 3.17,
+dimpConvpZℓ uqq “ rankpCyq “ rankpCuq ´ 1.
+To prove (i), suppose that φppqq`φpqq`φpqβq ą φppq` ř
+iPrβs
+φpqiq. By Corollary 3.17
+and Theorem 3.1,
+rankpCuq “ rankpCyq ` 1 “ 1 ` φppq `
+ÿ
+iPrβs
+φpqiq `
+ÿ
+iPrβs
+aiφppqiq,
+where ai P t0, 1u, i P rβs. If ai “ 0 @i P rβs, then
+rankpCuq “ 1 ` φppq `
+ÿ
+iPrβs
+φpqiq ă 1 ` φppqq ` φpqq ` φpqβq,
+contradicting Corollary 3.20. Hence, at least one of the ais must be equal to 1.
+Since φppqi1q ą φppqi2q if and only if i1 ą i2, a1 “ 1, ai “ 0 for i “ 2, . . . , β results
+in the smallest value of rankpCuq that we can not rule out.
+The following corollary follows immediately from Theorems 3.1 and 3.21.
+Corollary 3.22. Let ℓ “ pq2, where p, q are distinct odd primes. Let Cu with
+u P t´1, 1uℓ be the circulant matrix of u such that Cu is non-recurrent. Then
+rankpCuq is at least
+φppq ` φppqq ` φpqq ` φpq2q ` 1 “ ℓ ´ φpℓq.
+Consequently, either
+dimpConvpZℓ uqq “ ℓ ´ 1 ´ φpℓq,
+or
+dimpConvpZℓ uqq “ ℓ ´ 1.
+24
+
+The following corollary follows from Lemmas 2.11, 2.12, and 2.3, and Theo-
+rem 3.1.
+Corollary 3.23. Let p, q be distinct odd primes and n P Zě1. Suppose that ℓ “ pn
+or ℓ “ pq. Let u P t´1, 1uℓ satisfy x1 | uy “ ´1 and let y “ u ` p1{ℓq1. Then
+SpanQpZℓ yq “
+ë
+d | ℓ
+d‰ℓ
+ColQpPdq,
+where Pd is defined in Theorem 2.7. Moreover,
+dimpConvpZℓ uqq “ dimpAffpZℓ uqq “
+ÿ
+d | ℓ
+d‰ℓ
+dimQpColQpPdqq “ ℓ ´ 1.
+(3.6)
+Proof. Let d ‰ ℓ.
+Since pColQpPdqqC “ ColCpPdq is spanned by the discrete
+Fourier basis vectors vk for k P Zℓ Q pk, ℓq “ d, and µkpyq “ µkpuq. By Lem-
+mas 2.11 and 2.12, µkpyq ‰ 0. Since this holds @k Q pk, ℓq “ d, by Lemma 2.3,
+SpanQpZℓyq is not orthogonal to ColQpPdq. As this holds for each divisor d ‰ ℓ of
+ℓ, by Theorem 3.1, we must have
+SpanQpZℓ yq “
+ë
+d | ℓ
+d‰ℓ
+ColQpPdq.
+Equation (3.6) now follows from Theorem 3.1.
+The following lemma follows from Corollaries 3.14 and 3.22.
+Lemma 3.24. Let ℓ “ pq2, and pu, vq be an LP of length ℓ, then
+dimpConvpZℓ uqq “ dimpConvpZℓ vqq “ ℓ ´ 1.
+We now prove Theorem 1.4.
+Proof of Theorem 1.4. Since Zℓ ă pZℓ ¸ Zˆ
+ℓ q, it suffices to prove the result for
+G “ Zℓ. By Corollary 3.23 and Lemma 3.24,
+ℓ ´ 1 “ dimQ pAffQ pZℓ uqq “ dimpAffpZℓ uqq
+“ dimpConvpZℓ uqq.
+The result now follows from Theorem 1.3 as U “ H is the only possibility.
+Based on our partial results, we end the paper with the following conjecture.
+25
+
+Conjecture 3.25. Let pu, vq be a LP of length ℓ. Then
+dimpConvpG uqq “ dimpConvpG vqq “ ℓ ´ 1
+for G “ Zℓ and G “ Zℓ ¸ Zˆ
+ℓ .
+It suffices to prove Conjecture 3.25 for only one of G “ Zℓ or G “ Zℓ ¸ Zˆ
+ℓ .
+This is because by Theorem 2.8 the decomposition of QZℓ into irreducible Q-
+subrepresentations is the same under G “ Zℓ or G “ Zℓ ¸ Zˆ
+ℓ .
+4. Discussion
+In this paper, we made progress towards proving the conjecture that for an LP
+pu, vq of length ℓ,
+dimpConvpG uqq “ dimpConvpG vqq “ ℓ ´ 1
+for G “ Zℓ and G “ Zℓ ¸Zˆ
+ℓ . The immediate implication of this conjecture is that
+the search space for LPs cannot be decreased by imposing additional linearly inde-
+pendent linear equality constraints on u or v without rendering the rank-1 CSDD
+for the LP problem infeasible or causing its symmetry group to exclude a non-
+empty subset of Zℓ, provided that this rank-1 CSDD is feasible in the first place.
+Future research will involve proving Conjecture 3.25 with or without additional
+assumptions.
+Acknowledgments
+The views expressed in this article are those of the authors, and do not reflect
+the official policy or position of the United States Air Force, Department of De-
+fense, or the U.S. Government.
+References
+[1] T.M. Apostol. Introduction to Analytic Number Theory. Springer Science &
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+
diff --git a/f9AzT4oBgHgl3EQf3_7V/content/tmp_files/load_file.txt b/f9AzT4oBgHgl3EQf3_7V/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..853623374ad83d8c8571f9fa4682dc999e552328
--- /dev/null
+++ b/f9AzT4oBgHgl3EQf3_7V/content/tmp_files/load_file.txt
@@ -0,0 +1,881 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf,len=880
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='01839v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='RT]  4 Jan 2023  The dimension of an orbitope based on a solution to the  Legendre pair problem  Kristopher N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Kilpatricka, Dursun A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Bulutoglua  aDepartment of Mathematics and Statistics, Air Force Institute of Technology,  Wright-Patterson Air Force Base, Ohio 45433, USA  Abstract  The Legendre pair problem is a particular case of a rank-1 semidefinite description  problem that seeks to find a pair of vectors pu, vq each of length ℓ such that the  vector puJ, vJqJ satisfies the rank-1 semidefinite description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The group pZℓ ˆ  Zℓq ¸ Zˆ  ℓ acts on the solutions satisfying the rank-1 semidefinite description by  ppi, jq, kqpu, vq “ ppi, kqu, pj, kqvq for each ppi, jq, kq P pZℓ ˆZℓq¸Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By applying  the methods based on representation theory in Bulutoglu [Discrete Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' 45  (2022)], and results in Ingleton [Journal of the London Mathematical Society s(1-  31) (1956), 445-460] and Lam and Leung [Journal of Algebra 224 (2000), 91-109],  for a given solution puJ, vJqJ satisfying the rank-1 semidefinite description, we  show that the dimension of the convex hull of the orbit of u under the action of  Zℓ or Zℓ ¸ Zˆ  ℓ is ℓ ´ 1 provided that ℓ “ pn or ℓ “ pqi for i “ 1, 2, any positive  integer n, and any two odd primes p, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Our results lead to the conjecture that  this dimension is ℓ ´ 1 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We also show that the dimension of the  convex hull of all feasible points of the Legendre pair problem of length ℓ is 2ℓ ´ 2  provided that it has at least one feasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Keywords:  Circulant matrix, Convex hull, Ramanujan sums, Rational  representations, Vanishing sums of roots of unity  2000 MSC: 90C22 68R05 20C15 05E20 20G05 11L03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Introduction  A Hadamard matrix can be constructed by finding a solution to a system of  constraints for a pair of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' To define this system of constraints, let Zℓ “  t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ℓ ´ 1u denote the integers mod ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Two length ℓ vectors pu, vq is a Legendre  Email addresses: kristopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='kilpatrick@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='com (Kristopher N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Kilpatrick),  dursun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='bulutoglu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='com (Dursun A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Bulutoglu)  Preprint submitted to Elsevier  January 6, 2023  pair (LP) if  Pupjq ` Pvpjq “ ´2, @j P Zℓ ´ t0u,  1Ju “ 1Jv, u, v P t´1, 1uℓ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1)  where Pxpjq “ ř  iPZℓ xpiqxpi´jq is the periodic autocorrelation function of a vector  x P Cℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The problem of finding solutions to the system of constraints (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1) is known  as the LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' System of constraints (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1) can be cast as the following rank-1  constrained semidefinite description (rank-1 CSDD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  ÿ  i´j“r pmod ℓq  1ďi,jďℓ  Yij `  ÿ  i´j“r pmod ℓq  ℓ`1ďi,jď2ℓ  Yij  “  ´2,  for r “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ℓ,  Yjj  “  1,  for j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , 2ℓ,  Y  ľ  0,  rankpYq  “  1,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2)  ℓ´1  ÿ  j“0  Y1j`1  “  ℓ´1  ÿ  j“0  Y1pℓ`j`1q,  where Y ľ 0 means Y is non-negative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' In this rank-1 CSDD for LPs, an  LP pu, vq is obtained by taking uj “ Y1pj`1q and vj “ Y1pℓ`j`1q for j “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ℓ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  For the definition of the feasible set of a rank-1 constrained semidefinite program-  ming problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' rank-1 CSDD, see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let QZℓ be the vector space of all functions from Zℓ to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' A circulant shift  of u P QZℓ by j P Zℓ, denoted by cju, is a transformation such that cjupiq “  upi ´ jq, i P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The circulant matrix of u P QZℓ, denoted by Cu, is a matrix such  that pj ` 1qth row of Cu is pcjuqJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If pu, vq is an LP with 1Ju “ 1Jv “ ´1, then  H “  »  ——–  1  1  1J  1J  1  ´1  1J  ´1J  1  1  Cv  Cu  1  ´1  CJ  u  ´CJ  v  fi  ffiffifl  is a p2ℓ`2qˆp2ℓ`2q Hadamard matrix, where 1 is the vector of all 1s of length ℓ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  A Hadamard matrix can also be constructed by using an LP with 1Ju “ 1Jv “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Hence, to construct a p2ℓ ` 2q ˆ p2ℓ ` 2q Hadamard matrix for some odd ℓ, it  suffices to find an LP of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It is conjectured that an LP of length ℓ exists for  each odd ℓ, where this conjecture implies the Hadamard conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It is shown  in Arasu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' [2] that an LP pu, vq must satisfy  1Ju “ 1Jv “ ˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  2  In this paper, we choose all LPs pu, vq to satisfy  1Ju “ 1Jv “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3)  Let N ¸ H be the semidirect product of the two groups N and H as defined in  Rotman [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then, the group Zℓ ¸Zˆ  ℓ acts on u P QZℓ by pj, kqupiq “ uppj, kq´1iq  for each pj, kq P Zℓ ¸ Zˆ  ℓ , i P Zℓ, where pa, bqi “ bi ` a for pa, bq P Zℓ ¸ Zˆ  ℓ , i P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The group pZℓ ˆ Zℓq ¸ Zˆ  ℓ acts on any pair pu, vq P QZℓ ‘ QZℓ by  ppi, jq, kqpu, vq “ ppi, kqu, pj, kqvq  for each ppi, jq, kq P pZℓ ˆ Zℓq ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Two pairs pu, vq, pu1, v1q of vectors of length ℓ  are equivalent if either pu1, v1q or pv1, u1q is in the orbit of pu, vq under the action  of pZℓ ˆZℓq¸Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If pu, vq is an LP and pu1, v1q is equivalent to pu, vq, then pu1, v1q  is also an LP [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The symmetry group of a rank-constrained CSDD is the set of  all permutations of its solution vector that send solutions to solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Hence, the  symmetry group of rank-1 CSDD for LPs contains the group pZℓ ˆ Zℓq ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Throughout the paper, for each n P Zě1, let rns “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , nu, and  td : d | ℓu “ td P rℓs : d | ℓu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  For a set of vectors S in a vector space over the field of scalars F, SpanFpS) is  the span, AffFpS) is the affine hull, and dimFpS) is the dimension of the affine  hull of the vectors in S over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For a matrix M, ColFpMq is the column space of  M over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If F is not provided, then F “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Also, let Conv(S) be the convex  hull of the vectors in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For a finite group G acting linearly on a vector space V ,  G v is the orbit of v P V under the action of G, and ConvpG vq is the orbitope  of G through v [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Moreover, let Fℓ be the feasible set of all pairs puJ, vJqJ  satisfying the rank-1 CSDD for the LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  In this paper, we determine  dimpConvpFℓqq, and investigate all possible values of dimpConvppZℓ ¸ Zˆ  ℓ q uqq and  dimpConvppZℓ¸Zˆ  ℓ q vqq when pu, vq is an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The following theorems and corollary  are our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ be an odd positive integer and Fℓ be the feasible set of all  pairs puJ, vJqJ satisfying the rank-1 CSDD for the LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If Fℓ ‰ H, then  dimpConvpFℓqq “ 2ℓ ´ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following corollary follows immediately from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let puJ, vJqJ satisfy the rank-1 CSDD for the LP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  two linear constraints that are linearly independent from 1Ju “ 1Jv implied by  having constraint (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2) in the rank-1 CSDD for the LP problem are 1Ju “ ˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Furthermore, no linear constraint that is linearly independent from constraints  1Ju “ ˘1 and 1Ju “ 1Jv is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' First, the rank-1 CSDD for the LP problem is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then, any other  linear constraint that is linearly independent from the constraints 1Ju “ ˘1 and  1Ju “ 1Jv would necessarily imply that dimpConvpFℓqq ă 2ℓ ´ 2, contradicting  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP, and either G “ Zℓ or G “ Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  there exists U1, U2 Ď td : d | ℓu such that U1 X U2 “ H and  dimpConvpG uqq “ ℓ ´ 1 ´  `ř  dPU1 φ  ` ℓ  d  ˘˘  ,  dimpConvpG vqq “ ℓ ´ 1 ´  `ř  dPU2 φ  ` ℓ  d  ˘˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let p, q be distinct odd primes and n P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pn, or ℓ “ pqi  for i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP of length ℓ, and either G “ Zℓ or G “ Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then  dimpConvpG uqq “ dimpConvpG vqq “ ℓ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  In Section 2, we present necessary background in representation theory, the  power spectral density, vanishing sums of roots of unity, and affine geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' In  Section 3, we prove our main results and conjecture that the restrictions on ℓ in  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4 other than ℓ being an odd positive integer can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Section 4  is our discussion about future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Background theory  Let G be a finite group and V be a finite-dimensional vector space over a field  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let GLpV q be the F-automorphisms of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' An n-dimensional F-representation  of G is a pair pρ, V q such that ρ: G Ñ GLpV q is a homomorphism and V is an n-  dimensional vector space over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' A subspace W of V is an F-subrepresentation of V  if ρpgqW Ď W @g P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' A representation pρ, V q of G is an irreducible representation  if the only subrepresentations of V are SpanFp0q and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Two F-representations  ρ1 : G Ñ GLpWq and ρ2 : G Ñ GLpW 1q of G are equivalent if there is an invertible  linear transformation φ : W Ñ W 1 Q φpρ1pgqwq “ ρ2pgqφpwq @w P W and g P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  For a decomposition V “ m1V1 k ¨ ¨ ¨ k mbVb into irreducible representations of a  group G, mi is the number of times the irreducible C-representation Vi appears up  to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Such mi ě 1 is called the multiplicity of Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The only fields F that we consider in this paper are Q, R, and C, the rational,  real, and complex numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' An F-representation ρ : G Ñ GLpV q  is unitary with respect to an inner product x¨, ¨y in V if xρpgqv, ρpgquy “ xu, vy  for all u, v P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Every representation is unitary with respect to a complex inner  product [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We use the convention xαu | vy “ ¯αxu | vy for α P F and u, v P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  If a representation pρ, V q of G is such that ρpgq permutes a basis of V for all  4  g P G, then pρ, V q is called a permutation F-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Every permutation  F-representation is unitary with respect to the complex dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Throughout  the paper, every group G is finite, every vector space V is finite-dimensional, and  every representation of any specified group is a permutation representation and  hence is unitary with respect to the complex dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' [Maschke] Every representation of a finite group is a direct sum  of orthogonal irreducible subrepresentations with respect to some inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The decomposition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1 is said to be multiplicity-free if each irre-  ducible appears only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The character of an F-representation pρ, V q of G is the  map χρ : G Ñ F defined by χρpgq “ Trpρpgqq, where Trpρpgqq is the trace of ρpgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We say that the character is an irreducible character if the character belongs to an  irreducible representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Two F-representations of a finite group are equivalent  if and only if they have the same character [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We may simply write the character  as χ if the representation is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The following theorem is from  Serre [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pρ, V q be a C-representation of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  V “ m1V1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' k mhVh  with mi P Zě1 be a decomposition of V into irreducibles pρi, Viq with characters χi  @i P rhs, where χi “ χρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the orthogonal projection Pi of V onto miVi “  Àmi  k“1 Vi is given by  Pi “ dimCpViq  |G|  ÿ  gPG  χipgqMρpgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  where Mρpgq is the matrix of ρpgq in some orthonormal basis for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The exponent m of a group G is the smallest nonnegative integer such that  gm “ e @g P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pρ, V q be a C-representation of a group G with exponent m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then ρpgqm “ id @g P G, where id is the identity mapping on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Therefore, the  eigenvalues of ρpgq are mth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Throughout the paper, let ζm “ e2πi{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since χpgq is the trace of ρpgq, χpgq P Qpζmq, where Qpζmq is the field extension of  Q obtained by adjoining ζm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let Zˆ  m “ tr P Zm | pr, mq “ 1u be the multiplicative group of Zm, where pr, mq  is the greatest common divisor of r and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The automorphism group AutpQpζmqq  of Qpζmq is the Galois group of the field extension Qpζmq over Q denoted by  GalpQpζmq{Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The elements of GalpQpζmq{Qq can be indexed by the elements of  Zˆ  m, where for σr P GalpQpζmq{Qq  σrpζmq “ ζr  m, σrpqq “ q for each q P Q, and r P Zˆ  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  5  The map σr Ñ r is an isomorphism between GalpQpζmq{Qq and Zˆ  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let AutpCq  be the group of all automorphisms of C and  AutpCqQpζmq “ tτ|Qpζmq | τ P AutpCqu,  where τ|Qpζmq is the restriction of τ P AutpCq to Qpζmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For each τ P AutpCq,  pτpζmqqm “ τppζmqmq “ τp1q “ 1, and τpqq “ q for each q P Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Hence, we have  AutpCqQpζmq Ď AutpQpζmqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since any automorphism of a subfield of C can be  extended to an automorphism of C [15], each σr P AutpQpζmqq can be extended to  a σ1  r P AutpCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then AutpQpζmqq Ď AutpCqQpζmq implying  AutpQpζmqq “ AutpCqQpζmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since the entries of the matrix of ρpgq are in Qpζmq, AutpQpζmqq (AutpCq) acts on  the (characters of the) C-representations of G of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let V “ FG be the space of all functions from G to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the standard basis  egphq “  #  1  if g “ h,  0  otherwise  for g, h P G spans V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We further equip V with the usual inner product x¨ | ¨y  (the complex dot product) so that tegugPG is an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then the  regular F-representation of G is the pair pρ, V q such that ρ: G Ñ GLpV q is the  homomorphism determined by the action of G on the basis of V , where ρphqeg “  ehg @h, g P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Clearly, the regular F-representation of G is also a permutation  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The complexification of a vector space U over Q is defined as UC “ C bQ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It  is clear that if U is a Q-representation, then UC is the C-representation obtained  by extending the field of scalars Q of U to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let x¨ | ¨y be the inner product of  V over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It is plain to show that an inner product on VC may be defined as  xz b v | z1 b v1y “ zz1xv | v1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the following lemma follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let V be an inner product space over Q and U, W subspaces of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then U is orthogonal to W if and only if UC is orthogonal to WC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  `  R, QZℓ˘  be the regular Q-representation of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  `  QZℓ˘  C “  ë  iPZℓ  Vi  is the decomposition of  `  QZℓ˘  C into the one-dimensional irreducible C-representations  of Zℓ, where the C-representations pρk, Vkq, k P Zℓ are given by ρkpjq: Vk Ñ Vk,  v ÞÑ ζ  jk  ℓ v, and χkpjq “ Trpρkpjqq “ ζ  jk  ℓ  for each k, j P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  6  For each k P Zℓ, vk “ ř  jPZℓ ζjk  ℓ ej is the basis vector for the one-dimensional  C-representation Vk of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It is well-known that tv0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , vℓ´1u is a basis of CZℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Throughout the paper, we call the vectors v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , vℓ´1 the discrete Fourier basis  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem characterizes the orbits of the one-dimensional repre-  sentations in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4 under the action of AutpCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For k P Zℓ, let χk be the irreducible character of the C-representation  of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For each divisor d of ℓ, let Od “ tχk | pk, ℓq “ d and k P Zℓu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  (i) |Od| “ φpℓ{dq, where φ is Euler’s totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (ii) Od “ AutpCq χd “ AutpQpζℓ{dqq χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let d be a divisor of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By the definition of Euler’s totient function, (i)  follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' To prove (ii),  σrpχdpjqq “ ζ  jrd  ℓ  “ χrdpjq  for each σr P AutpQpζℓ{dqq, r P Zˆ  ℓ{d, and j P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Hence, σrpχdq “ χrd, and  prd, ℓq “ d implies χrd P Od, consequently AutpQpζℓ{dqq χd Ď Od.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let χk P Od such that k “ rd and pr, ℓ{dq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then σr P AutpQpζℓ{dqq and  χk “ χrd “ σrχd  Hence, Od Ď AutpQpζℓ{dqq χd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem follows from Isaacs [8, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='21], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2, and  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pρ, V q be a Q-representation of a group G with exponent m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let  V “ W1 k ¨ ¨ ¨ k Wb  be a decomposition of V into irreducible Q-subrepresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let VC, WiC be the  C-representations obtained from V, Wi by extending the field of scalars of V, Wi to  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  VC “ Wp1,1q k ¨ ¨ ¨ k Wp1,r1q k ¨ ¨ ¨ k Wpb,1q k ¨ ¨ ¨ k Wpb,rbq  be a decomposition of VC into  `  ρi,j, Wpi,jq  ˘  irreducible C-subrepresentations with  characters χρi,j, where  WiC “ Wpi,1q k ¨ ¨ ¨ k Wpi,riq for i P rbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  7  For i P rbs, let Oρi,1 be the AutpQpζmqq-orbit of χρi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let Mρpgq be the matrix of  ρpgq @g P G, with respect to the standard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If VC is multiplicity-free, then  PWi “ dimCpWpi,1qq  |G|  ÿ  gPG  ¨  ˝ ÿ  χPOρi,1  χpgqMρpgq  ˛  ‚  is the orthogonal projection matrix into the ith irreducible Q-subrepresentation  subspace ColQpPWiq @i P rbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem provides the decomposition of QZℓ into irreducible Q-  subrepresentations under the action of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  `  R, QZℓ˘  be the regular Q-representation of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  Pd “ 1  ℓ  ÿ  iPZℓ  ÿ  χPOd  χpiqMRpiq  for each divisor d of ℓ, where MRpiq is the matrix of Rpiq with respect to the  standard basis teiuiPZℓ, and Od is defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  QZℓ “  ë  d | ℓ  ColQpPdq  is the decomposition of QZℓ into irreducible Q-subrepresentations under the action  of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let d be a divisor of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5, the inner sum of Pd is over  the orbit of χd under the action of AutpQpζℓ{dqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Further, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4,  `  QZℓ˘  C is multiplicity-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='6, ColQpPdq is an irreducible  Q-subrepresentation of QZℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Now, by the identity ℓ “ ř  d | ℓ φpℓ{dq, it suffices to  show that  dimQ pColQpPdqq “ φ  ˆℓ  d  ˙  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since R is the regular representation of Zℓ, only Rp0q will contribute to the trace  8  of Pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since Pd is a projection matrix,  dimQ pColQpPdqq “ rankpPdq “ Tr pPdq “ Tr  ˜  1  ℓ  ÿ  iPZℓ  ÿ  χPOd  χpiqMRpiq  ¸  “ 1  ℓ  ÿ  iPZℓ  ÿ  χPOd  χpiq Tr  `  MRpiq  ˘  “ 1  ℓ  ÿ  χPOd  Tr  `  MRp0q  ˘  “ 1  ℓ  ÿ  χPOd  ℓ  “  ÿ  χPOd  1  “ |Od|  “ φ  ˆℓ  d  ˙  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The group Zℓ ¸ Zˆ  ℓ acts on Zℓ by pa, bqi “ bi ` a @i P Zℓ and pa, bq P Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then a representation pT, QZℓq of Zℓ ¸ Zˆ  ℓ is given by the action on the basis  Tpa, bqei “ ebi`a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The next theorem shows that QZℓ has the same decomposition  under the action of Zℓ ¸ Zˆ  ℓ as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=', extending Zℓ to Zℓ ¸ Zˆ  ℓ does  not change the decomposition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The decomposition QZℓ “ Ë  d | ℓ ColQpPdq is the decomposition of  QZℓ into irreducible Q-subrepresentations under the action of Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let d be a divisor of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We have ColCpPdq “ Ë  pk,ℓq“d Vk, where Vk is  spanned by vk “ ř  iPZℓ ζki  ℓ ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pa, bq P Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Observe  Tpa, bqvk “  ÿ  iPZℓ  ζki  ℓ ebi`a “  ÿ  jPZℓ  ζkpj´aqb´1  ℓ  ej “ ζ´kab´1  ℓ  vb´1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since pb, ℓq “ 1, pbk, ℓq “ d if and only if pk, ℓq “ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Therefore, ColCpPdq is a  C-subrepresentation under the action of Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since ColQpPdq is an irreducible  Q-subrepresentation under the action of Zℓ and Zℓ is a subgroup of Zℓ ¸ Zˆ  ℓ ,  ColQpPdq is an irreducible Q-subrepresentation under the action of Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The discrete Fourier transform (DFT) of u P QZℓ is µkpuq “ xu | vky, k P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Equation µkpuq “ xu | vky, k P Zℓ is the change of basis (variables) formula that  provides the Fourier coordinates µkpuq in terms of the standard basis coordinates  9  uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The power spectral density (PSD) of u P QZℓ is |µkpuq|2, for k P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The  following theorem provides an equivalent condition that characterizes an LP in  terms of its PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='9 (Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u, v P t´1, 1uℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then pu, vq is an LP if  and only if µ0puq “ µ0pvq and  |µkpuq|2 ` |µkpvq|2 “ 2pℓ ` 1q,  @k P Zℓ ´ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We say that there is a vanishing sum of m ℓth roots of unity, if there exists m  ℓth roots of unity w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , wm (not necessarily distinct) satisfying w1`¨ ¨ ¨`wm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10 (Lam and Leung [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Suppose that ℓ “ pn1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' pns  s  for distinct  primes p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ps and n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ns P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then there exists a vanishing sum of m  ℓth roots of unity if and only if m “ a1p1 ` ¨ ¨ ¨ ` asps for some a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , as P Zě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let u P t´1, 1uℓ satisfy x1 | uy “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let J “ tj P Zℓ | upjq “ ´1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  µkpuq for k ‰ 0, has two forms  µkpuq “ ´2  ÿ  jPJ  ζkj  ℓ  “ 2  ÿ  jRJ  ζkj  ℓ ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1)  where |J| “ pℓ ` 1q{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pn, p an odd prime, and n P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P t´1, 1uℓ satisfy  x1 | uy “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then µkpuq ‰ 0 @k P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since µ0puq “ ´1 we need to only verify for k P rℓ ´ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Suppose for a  contradiction that µkpuq “ 0 for some k P rℓ´1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1), ř  jPJ ζkj  ℓ  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10,  ℓ ` 1  2  “ ap  for some a P Zě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' This means that p | ppℓ ` 1q{2q, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Therefore,  µkpuq ‰ 0 @k P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pq, p, q distinct odd primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let u P t´1, 1uℓ satisfy  x1 | uy “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then µkpuq ‰ 0 @k P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since µ0puq “ ´1 we need to only verify for k P rℓ ´ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Suppose for a  contradiction that µkpuq “ 0 for some k P rℓ ´ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1),  ÿ  jPJ  ζkj  ℓ  “  ÿ  jRJ  ζkj  ℓ  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2)  10  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10,  ℓ ` 1  2  “ ap ` bq and ℓ ´ 1  2  “ cp ` dq  for some a, b, c, d P Zě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' This means  ℓ “ ℓ ` 1  2  ` ℓ ´ 1  2  “ pa ` cqp ` pb ` dqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  If a ` c ‰ 0 and b ` d ‰ 0, then p | pb ` dq and q | pa ` cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Consequently,  ℓ “ pa ` cqp ` pb ` dqq ě pq ` pq “ 2ℓ,  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Suppose that a ` c “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then a “ c “ 0 and subsequently  q | ppℓ`1q{2q, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' A similar contradiction occurs if b`d “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Therefore,  µkpuq ‰ 0 @k P Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='12 do not generalize to arbitrary odd ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' There is a u P  t´1, 1u45 with x1 | uy “ ´1 such that µkpuq “ 0 for some k P Z45 ´ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following lemma is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let V be a vector space over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let S Ď V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then AffFpSq “  SpanFpSq if and only if 0 P AffFpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let pρ, V q be a representation of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let S Ď V be nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the  barycenter of S is βpSq “ p1{|S|q ř  vPS v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The fixed space of V under the action  of G is  V G “ tv P V | ρpgqv “ v @g P Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Clearly, V G is a subrepresentation of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1, there exists a subrepre-  sentation orthogonal to V G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pρ, V q be a unitary representation of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let P be the  orthogonal projection matrix onto V G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then Px “ βpG xq for x P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , uk be an orthonormal basis for V G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  Py “  ÿ  iPrks  xui | yyui  for any y P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then, for each i P rks and r P G, we have  xui | ρprqxy “ xρprqui | ρprqxy “ xui | xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  11  Then  xui | βpG xqy “  1  |G x|  ÿ  ρprqxPG x  xui | ρprqxy “  1  |G x|  ÿ  ρprqxPG x  xui | xy “ xui | xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since βpG xq P V G,  βpG xq “ PβpG xq “  ÿ  iPrks  xui | βpG xqyui “  ÿ  iPrks  xui | xyui “ Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  `  R, QZℓ˘  be the regular Q-representation of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the pro-  jection matrix Pℓ onto the fixed space of QZℓ is p1{ℓqJ, where J is the ℓ ˆ ℓ matrix  of all ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Notice that the fixed space of V is a direct sum of copies of the trivial  representation of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2,  Pℓ “ 1  ℓ  ÿ  jPZℓ  χ0pjqMRpjq “ 1  ℓ  ÿ  jPZℓ  MRpjq “ 1  ℓJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Main results  We start with the proof of the first of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  F 1  ℓ “ tu P t´1, 1uℓ | Dv Q pu, vq is an LPu  and  F 2  ℓ “ tv P t´1, 1uℓ | Du Q pu, vq is an LPu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  By symmetry, F 1  ℓ “ F 2  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since Fℓ ‰ H, there exists puJ, vJqJ P Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then u P F 1  ℓ  and v P F 2  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let p “ βpZℓ uq “ ´p1{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then p P ConvpF 1  ℓ q Ď AffpF 1  ℓ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' To  reach the desired conclusion, we observe the following facts:  (i) Since F 1  ℓ “ F 2  ℓ , SpanCpF 1  ℓ ´ pq “ SpanCpF 2  ℓ ´ pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (ii) Since both SpanCpF 1  ℓ ´ pq and SpanCpF 2  ℓ ´ pq are C-subrepresentations  of pQZℓqC under the action of Zℓ orthogonal to 1, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1, both  SpanCpF 1  ℓ ´ pq and SpanCpF 2  ℓ ´ pq must be an orthogonal direct sum of the  irreducible C-subrepresentations V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , Vℓ´1 of pQZℓqC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  12  (iii) Both SpanCpF 1  ℓ ´ pq and SpanCpF 2  ℓ ´ pq cannot be orthogonal to an irre-  ducible Vk for some k P rℓ ´ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For this would imply that the DFT of u  must satisfy  µkpuq “ µkpu ´ pq “ 0,  similarly, µkpvq “ 0, contradicting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  By (i), (ii), and (iii),  SpanCpF 1  ℓ ´ pq “ V1 k ¨ ¨ ¨ k Vℓ´1 and SpanCpF 2  ℓ ´ pq “ V1 k ¨ ¨ ¨ k Vℓ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let f “ β  ´  pZℓ ˆ Zℓq  `  uJ, vJ˘J¯  “ ´p1{ℓq  `  1J, 1J˘J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then f P ConvpFℓq Ď  AffpFℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It is evident that  SpanC pFℓ ´ fq “ SpanC  `  F 1  ℓ ´ p  ˘  ‘ SpanC  `  F 2  ℓ ´ p  ˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7, SpanQpF i  ℓ ´ pq “  ë  d | ℓ  d‰ℓ  ColQpPdq for i “ 1, 2, and by fact (i),  SpanQ pFℓ ´ fq “  ¨  ˚  ˝  ë  d | ℓ  d‰ℓ  ColQpPdq  ˛  ‹‚‘  ¨  ˚  ˝  ë  d | ℓ  d‰ℓ  ColQpPdq  ˛  ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Hence,  dimpConvpFℓqq “ dimpSpanpFℓ´fqq “ dimQpSpanQpFℓ´fqq “ 2pℓ´1q “ 2ℓ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem is needed for our next main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P Qℓ satisfy SpanQpuq ‰ SpanQp1q and let y “ u´p1Ju{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then there exists T Ď td P rℓs : d | ℓ and d ‰ ℓu such that  SpanQpZℓ yq “  ë  dPT  ColQpPdq,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1)  where Pd is defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Moreover,  dimpConvpZℓ uqq “ dimpAffpZℓ uqq “ dimQpSpanQpZℓ yqq “ rankpCyq “  ÿ  dPT  φ  ˆℓ  d  ˙  ,  where Cy is the ℓ ˆ ℓ circulant matrix whose first row is y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since SpanQpZℓ yq is a Q-subrepresentation of QZℓ under the action of Zℓ,  SpanQpZℓ yq is an orthogonal direct sum of irreducible Q-subrepresentations of QZℓ  by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7, the irreducible Q-subrepresentations of QZℓ are  ColQpPdq for each divisor d of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Therefore, there exists a set T of divisors of ℓ  for which equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We first show that SpanQpZℓ yq is orthogonal only  to ColQpPℓq, where Pℓ is the projection onto the fixed space of QZℓ under the  action of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since ColQpPℓq “ SpanQp1q, xy | 1y “ 0, and the representation QZℓ  is unitary, we have SpanQpZℓ yq is orthogonal to ColQpPℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let z “ u ´ Pℓu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='15, z “ y “ u ´ p1Ju{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since Pℓy “  Pℓu ´ P2  ℓu “ 0 and βpZℓ yq “ Pℓy by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='14, 0 “ βpZℓ yq P ConvQpZℓ yq as  βpZℓ yq is a convex combination of points of Zℓ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='13, we have  SpanQpZℓ yq “ AffQpZℓ yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Observe that  dimQpAffQpZℓ uqq “ dimQ  ˆ  AffQpZℓ uq ´ 1Ju  ℓ 1  ˙  “ dimQpAffQpZℓ yqq “ dimQpSpanQpZℓ yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then,  dimpAffpZℓ uqq “ dimQpAffQpZℓ uqq “ dimQpSpanQpZℓ yqq “  ÿ  dPT  φ  ˆℓ  d  ˙  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since Zℓ ă Zℓ ¸ Zˆ  ℓ , it suffices to prove the result for G “ Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let X1 “ Zℓ u,  X2 “ Zℓ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since  dimpConvpX1qq “ dimpAffpX1qq  and  dimpConvpX2qq “ dimpAffpX2qq,  by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1, it suffices to prove that U1XU2 “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For the sake of contradiction,  let d1 P U1XU2 be such that d1 ‰ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' This implies that for each i “ 1, 2, SpanCpXi`  p1{ℓq1q is orthogonal to ColCpPd1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Also, ColCpPd1q Ă pQZℓqC and ColCpPd1q is  invariant under the action of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then by Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4, there exists  U1 Ă Zℓ ´ t0u such that ColCpPd1q “ Ë  iPU1 Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' This implies that for each i “ 1, 2,  SpanCpXi ` p1{ℓq1q is orthogonal to an irreducible Vk for some k P rℓ ´ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  µkpuq “ µk  ˆ  u ` 1  ℓ1  ˙  “ 0,  similarly, µkpvq “ 0, contradicting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let m, n P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If m ´ 2 ě 2, n ´ 2 ě 1 or m ´ 2 ě 1, n ´ 2 ě 2, then  n ´ 2 ě 2pn ´ 1q{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  14  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If m ´ 2 ě 2, n ´ 2 ě 1 or m ´ 2 ě 1, n ´ 2 ě 2, then pm ´ 2qpn ´ 2q ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Adding 2pn ´ 2q to both sides of pm ´ 2qpn ´ 2q ě 2 yields mpn ´ 2q ě 2pn ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Hence n ´ 2 ě 2pn ´ 1q{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let p, q be distinct odd primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the quotient of 2pq ´ 1qpp ´ 1q  by division of p is at least q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Moreover, for 2pq ´ 1qpp ´ 1q “ sp ` r, where  s “ 2pq ´ 1q ´ k, r “ kp ´ 2pq ´ 1q, and k is the smallest integer such that  kp ´ 2pq ´ 1q ě 0, then s and r are the quotient and remainder of 2pq ´ 1qpp ´ 1q  upon division by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since for any integer k, 2pq ´1qpp´1q “ p2pq ´1q´kqp`pkp´2pq ´1qq we  may choose the smallest such k such that kp´2pq´1q ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then k “  P  2pq´1q{p  T  ,  where rns is the ceiling of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let 2pq´1qpp´1q “ sp`r, where s “ 2pq´1q´k and  r “ kp´2pq´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Now, as p, q are distinct odd primes, WLOG suppose that p ě 3,  then q ě 5 ě 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since p ´ 2 ě 1 and q ´ 2 ě 2, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2, q ´ 2 ě 2pq ´ 1q{p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Therefore, q ´ 2 ě k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then s ´ q “ 2pq ´ 1q ´ k ´ q “ q ´ 2 ´ k ě k ´ k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Finally, since k ´ 1 ă 2pq ´ 1q{p, r “ pk ´ 2pq ´ 1q ă p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pαqβ, where p, q are distinct odd primes, and α, β P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then φpℓq ą pℓ ´ 1q{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3, 2pq ´ 1qpp ´ 1q “ sp ` r where s ě q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  2φpℓq “ 2pq ´ 1qpp ´ 1qpα´1qβ´1  “ psp ` rqpα´1qβ´1  ě pαqβ ` rpα´1qβ´1  ą pαqβ ´ 1  “ ℓ ´ 1,  and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P Qℓ, then  ||µpuq||2 “ ℓ||u||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let U be the matrix whose columns are the discrete Fourier basis vectors  v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , vℓ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then, since µ “ UJu,  ||µpuq||2 “  ÿ  kPZℓ  |µkpuq|2 “ µ˚µ “ uJU˚Uu “ uJℓIQℓu “ ℓ||u||2,  where we used the fact that U˚U “ ℓIQℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  15  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P QZℓ satisfying ||u||2 “ ℓ and x1 | uy “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  ℓ´1  ÿ  k“1  |µkpuq|2 “ pℓ ` 1qpℓ ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5,  ℓ´1  ÿ  k“1  |µkpuq|2 “ ||µpuq||2 ´ |µ0puq|2 “ ℓ2 ´ 1 “ pℓ ` 1qpℓ ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  For each ℓ, n P Zě1, the Ramanujan’s sum [1] is defined by cℓpnq “ ř  0ăkďℓ  pk,ℓq“1 e2πikn{ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  It is well-known that @ℓ, n P Zě1  cℓpnq “ µ  ˆ  ℓ  pℓ, nq  ˙  φpℓq  φ  ´  ℓ  pℓ,nq  ¯ P Z,  where  µpnq “  $  &  %  1  if n “ 1,  p´1qr  if n “ p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' pr for distinct primes p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , pr,  0  if n is divisible by some prime square  is the M¨obius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ ą 1 and u P Qℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If d is a divisor of ℓ, then  ÿ  0ăkďℓ  pk,ℓq“d  |µkpuq|2 “  ÿ  sPZℓ  c ℓ  dpsqPupsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' First,  ÿ  0ăkďℓ  pk,ℓq“d  e  2πikph´jq  ℓ  “  ÿ  0ărď ℓ  d  pr, ℓ  d q“1  e  2πirph´jq  ℓ  d  “ c ℓ  dph ´ jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , vℓ´1 be the discrete Fourier basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Note that vkv˚  k “ rak  hjs,  where ak  hj “ e2πikph´jq{ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  ÿ  0ăkďℓ  pk,ℓq“d  vkv˚  k “ rbd  hjs,  where bd  hj “ cℓ{dph ´ jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since µkpuq “ uJvk,  |µkpuq|2 “ µkpuqµkpuq “ puJvkqpv˚  kuq “ uJpvkv˚  kqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  16  Then  ÿ  0ăkďℓ  pk,ℓq“d  |µkpuq|2 “ uJ  ¨  ˚  ˝  ÿ  0ăkďℓ  pk,ℓq“d  vkv˚  k  ˛  ‹‚u  “  ÿ  h,jPZℓ  bd  hjuhuj  “  ÿ  h,jPZℓ  c ℓ  d ph ´ jquhuj  “  ÿ  s,jPZℓ  c ℓ  d psqujuj`s  “  ÿ  sPZℓ  c ℓ  d psq  ÿ  jPZℓ  ujuj`s  “  ÿ  sPZℓ  c ℓ  d psqPupsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  For a vector u P Qℓ and a divisor d of ℓ, let  u  ℓ  d pjq “  ÿ  iPZd  u  ˆℓ  di ` j  ˙  ,  @j P Z ℓ  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then uℓ{d P Qℓ{d is called the d-compression of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The following lemma is used to  prove one of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ ą 1 and u P Qℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If d is a divisor of ℓ, then  ÿ  0ăkďℓ  pk,ℓq“d  |µkpuq|2 “  ÿ  0ăkď ℓ  d  pk, ℓ  d q“1  |µkpu  ℓ  dq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7,  ÿ  0ăkďℓ  pk,ℓq“d  |µkpuq|2 “  ÿ  sPZℓ  c ℓ  dpsqPupsq  “  ÿ  jPZ ℓ  d  ÿ  iPZd  c ℓ  d  ˆℓ  di ` j  ˙  Pu  ˆℓ  di ` j  ˙  “  ÿ  jPZ ℓ  d  c ℓ  d  ˆℓ  d ` j  ˙ ÿ  iPZd  Pu  ˆℓ  di ` j  ˙  “  ÿ  sPZ ℓ  d  c ℓ  dpsqPu  ℓ  d psq  “  ÿ  0ăkď ℓ  d  pk, ℓ  d q“1  |µkpu  ℓ  dq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem follows from Theorem 3 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP of length ℓ “ dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then the corresponding  m “ ℓ{d-compressed sequences pud, vdq satisfy  Pudp0q ` Pvdp0q “ 2ℓ ´ 2  ˆℓ  d ´ 1  ˙  “ 2ℓ ` 2 ´ 2ℓ  d ,  Pudpjq ` Pvdpjq “ ´2ℓ  d ,  @j P Zd ´ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We now prove another main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP of length ℓ such that ||ud||2 “ ||vd||2 for  some divisor d of ℓ such that 2φpdq ě d ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then, |µkpvq|2 ą 0 and |µkpuq|2 ą 0  @k Q pk, ℓq “ ℓ{d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By symmetry, it suffices to prove the result for v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' For the sake of contra-  diction, let |µk1pvq|2 “ 0 for some k1 Q pk1, ℓq “ ℓ{d for some divisor d of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  |µkpvq|2 “ 0 @k Q pk, ℓq “ ℓ{d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='9, |µkpuq|2 “ 2ℓ`2 @k Q pk, ℓq “ ℓ{d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Consequently,  ÿ  0ăkďℓ  pk,ℓq“ ℓ  d  |µkpuq|2 “ p2ℓ ` 2qφpdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2)  Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='8,  ÿ  0ăkďℓ  pk,ℓq“ ℓ  d  |µkpuq|2 “  ÿ  0ăkďd  pk,dq“1  |µkpudq|2 “ p2ℓ ` 2qφpdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3)  18  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='9,  ||ud||2 “ ||vd||2 “ ℓ ` 1 ´ ℓ  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5,  dÿ  k“1  |µkpudq|2 “ d||ud||2 ´ |µ0pudq|2 “ d||ud||2 ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4)  Hence, by equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4)  ÿ  0ăkďd  pk,dq“1  |µkpudq|2 “ p2ℓ ` 2qφpdq ă  dÿ  k“1  |µkpudq|2 “ pℓ ` 1qpd ´ 1q  which implies  2φpdq ă pd ´ 1q,  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following corollary follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP of length ℓ “ pαqβ, where p, q are distinct  odd primes, and α, β P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  T “  ␣  d P rℓs : d ‰ 1, d | ℓ, and ||ud||2 ‰ ||vd||2 (  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then  dimpConvpZℓ uqq “ ℓ ´ 1 ´  ÿ  dPT1  φ pdq ,  dimpConvpZℓ vqq “ ℓ ´ 1 ´  ÿ  dPT2  φ pdq  for some T1 Ď T and T2 Ď T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Equivalently  dimpConvpZℓ uqq ě  ÿ  dPT 1  φ pdq ,  dimpConvpZℓ vqq ě  ÿ  dPT 1  φ pdq ,  where  T 1 “  ␣  d P rℓs : d ‰ 1, d | ℓ, and ||ud||2 “ ||vd||2(  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5)  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By symmetry, it suffices to prove the result for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let y “ u ` p1{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  First, we prove that  ColQpP ℓ  d1 q Ď SpanQpZℓ yq @d1 P T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The subspace SpanQpZℓ yq is a Q-subrepresentation of Qℓ, and thus either  SpanQpZℓ yq K ColQ  ´  P ℓ  d1  ¯  or  ColQ  ´  P ℓ  d1  ¯  Ď SpanQpZℓ yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  If SpanQpZℓ yq K ColQ  `  Pℓ{d1˘  , then SpanCpZℓ yq K ColC  `  Pℓ{d1˘  , and consequently  µkpuq “ µkpyq “ 0 for some k Q pk, ℓq “ ℓ{d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4, this contradicts  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Hence, ColQ  `  Pℓ{d1˘  Ď SpanQ pZℓ yq and the result now follows from  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since ||uℓ||2 “ ||vℓ||2 “ ℓ for u, v P t´1, 1uℓ, the following corollary follows  from the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP of length ℓ “ pαqβ, where p, q are distinct  odd primes, and α, β P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  (i) ColQ pP1q Ď SpanQpZℓ yq for y “ u ` p1{ℓq1 and y “ v ` p1{ℓq1,  (ii) dimpConvpZℓ uqq ě φ pℓq and dimpConvpZℓ vqq ě φ pℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Larger |T 1| in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='5) in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11 yields larger lower bounds for  dimpConvpZℓ uqq and dimpConvpZℓ vqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' There is evidence that in general T 1 ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  In fact, based on known LPs, it was recently conjectured in [10] that for each  positive ℓ divisible by 5  5 P  ␣  d P rℓs : d ‰ 1, d | ℓ, ||ud||2 “ ||vd||2, and pu, vq is an LP of length ℓ  (  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  On the other hand, there are LPs pu, vq such that ||ud||2 ‰ ||vd||2 for some divisor  d of ℓ, see [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following theorem gives us a general lower bound on  dimpConvpZℓ uqq for u P t´1, 1uℓ such that x1 | uy “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pn1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' pns  s  where p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ps are distinct odd primes and  n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ns P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P t´1, 1uℓ satisfy x1 | uy “ ´1, then  dimpConvpZℓ uqq ě  ÿ  jPrss  ÿ  iPrnjs  φppi  jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  20  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let y “ u ` p1{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We will first prove that ColQpPdj,iq Ď SpanQpZℓ yq  for each dj,i “ pn1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' pi  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' pns  s , j “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , s, i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , nj ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Assume otherwise,  and let µkpyq “ 0 for some k Q pk, ℓq “ dj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then µkpuq “ µkpyq “ 0, and  by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='10, pj | pℓ ` 1q{2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Therefore,  ColQpPdj,iq Ď SpanQpZℓ yq, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1,  dimpConvpZℓ uqq “ dimpAffpZℓ uqq “ dimpSpanpZℓ yqq  “ dimQpSpanQpZℓ yqq ě  ÿ  jPrss  ÿ  iPrnjs  φppi  jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following corollary follows by combining Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='12 and the proof of  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be an LP of length ℓ “ pαqβ, where p, q are distinct  odd primes, and α, β P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  dimpConvpZℓ uqq ě  ÿ  jPrαs  φppjq `  ÿ  iPrβs  φpqiq ` φpℓq,  dimpConvpZℓ vqq ě  ÿ  jPrαs  φppjq `  ÿ  iPrβs  φpqiq ` φpℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  A different set of lower bounds on dimpConvpZℓuqq can be obtained by using  the results in Ingleton [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Next, we derive such lower bounds when ℓ “ pqβ for  some distinct odd primes p, q and positive integer β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' However, first we need the  concept of non-recurrent matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Let Cu be the circulant matrix of u P Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then Cu is non-recurrent if ℓ is  the only divisor d of ℓ such that ui “ uj whenever i ” j pmod dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The following  lemma shows that Cu non-recurrent if there is a v such that pu, vq is an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P t´1, 1uℓ and x1 | uy “ ¯1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then Cu is non-recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' First note that the constraint x1 | uy “ ¯1 implies that there are pℓ ˘ 1q{2  ´1’s and pℓ ¯ 1q{2 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Suppose for contradiction that Cu is s-recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  s | ℓ and ui “ uj whenever i ” j pmod sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , us´1 each appear d “ ℓ{s  times and d divides neither pℓ ˘ 1q{2 nor pℓ ¯ 1q{2, we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following lemma and the subsequent corollary are needed to derive lower  bounds on the rank of circulant matrices Cu for u P t´1, 1uℓ based on rank lower  bounds in [7] for circulant Cu with u P t0, 1uℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let Cu be the circulant matrix of u P Rℓ, J be the ℓ ˆ ℓ all 1s  matrix, and λ P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  21  (i) the discrete Fourier basis tv0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , vℓ´1u is an eigenbasis for both Cu and  Cu ` λJ,  (ii) if the eigenvalue of Cu corresponding to eigenvector vi is µi for i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ℓ´  1, then the eigenvalue of Cu `λJ corresponding to eigenvector v0 and vi are  µ0 ` ℓλ and µi for i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , ℓ ´ 1, respectively, where µ0 “ řℓ´1  i“0 ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' It is straightforward to check the statements of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following corollary follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u, Cu, J, λ, and µ0 be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  (i)  rankpCu ` λJq “  $  ’  &  ’  %  rankpCuq  if  pµ0 ` ℓλqµ0 ‰ 0,  rankpCuq ` 1  if  µ0 “ 0 and λ ‰ 0,  rankpCuq ´ 1  if  µ0 ` ℓλ “ 0 and λ ‰ 0,  (ii) for odd ℓ and u P t0, 1uℓ,  rankpC2u´1q “ rankpCuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='16, (i) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since C2u´1 “ 2Cu ´ J, and for  odd ℓ,  rankp2Cu ´ Jq “ rankp2Cuq “ rankpCuq,  (ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  For ℓ “ pαpα1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' pαm´1  m´1 where p, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , pm´1 are distinct primes and α, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' ,  αm´1 P Zě1, let  τpℓ, pq “ 1 ` ǫpmqǫpαqφppq ` φppαq `  ÿ  iPrm´1s  φpppαi  i q,  where ǫp1q “ 0 and ǫpkq “ 1 for k P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Now, the following lemma follows from  the corresponding result for circulant ℓ ˆ ℓ non-recurrent circulant matrix with  entries from t0, 1u in Ingleton [7] and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pqβ, where p, q are distinct odd primes, and β P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let  u P t´1, 1uℓ satisfy x1 | uy “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then rankpCuq ě mintτpℓ, pq, τpℓ, qqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  If β ě 2, then the following lemma implies that the rank is at least τpℓ, qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  22  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pqβ for distinct odd primes p, q, and β P Zě2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then  τpℓ, qq ă τpℓ, pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Now  τpℓ, qq “ 1 ` ǫp2qǫpβqφpqq ` φpqβq ` φppqq “ 1 ` φpqq ` φpqβq ` φppqq,  and  τpℓ, pq “ 1 ` φppq ` φppqβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then  τpℓ, pq ´ τpℓ, qq “ φpqqpqβ´1pφppq ´ 1q ´ pφppq ` 1qq ` φppq  “ φpqqpqβ´1pp ´ 2q ´ pq ` φppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We consider two cases: p “ 3 and p ě 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If p “ 3, then  τpℓ, pq ´ τpℓ, qq “ φpqqpqβ´1 ´ 3q ` φp3q ě φpqqp5β´1 ´ 3q ` φp3q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Suppose that p ě 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then since qβ´1 ´ 1 ě 1 and p ´ 3 ě 1, pqβ´1 ´ 1qpp ´ 3q ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Then qβ´1pp ´ 2q ´ p ě qβ´1 ´ 2 ě 3β´1 ´ 2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Implying  τpℓ, pq ´ τpℓ, qq “ φpqqpqβ´1pp ´ 2q ´ pqq ` φppq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Now, the following corollary follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='13, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='17, and  Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='18 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pqβ for distinct odd primes p, q, and β P Zě2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let Cu  with u P t´1, 1uℓ be the circulant matrix of u such that Cu is non-recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  rankpCuq ě Lppqβq “ max  $  &  %φppqq ` φpqq ` φpqβq ` 1, φppq `  ÿ  iPrβs  φpqiq ` 1  ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  If φppqq ` φpqq ` φpqβq ą φppq ` ř  iPrβs  φpqiq, we may improve the lower bound in  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pqβ for distinct odd primes p, q, and β P Zě2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let Cu  with u P t´1, 1uℓ be the circulant matrix of u such that Cu is non-recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  (i) rankpCuq is at least  φppq ` φppqq `  ÿ  iPrβs  φpqiq ` 1  if φppqq ` φpqq ` φpqβq ą φppq `  ÿ  iPrβs  φpqiq,  φppq `  ÿ  iPrβs  φpqiq ` 1  otherwise,  23  (ii) dimpConvpZℓuqq is at least  φppq ` φppqq `  ÿ  iPrβs  φpqiq  if φppqq ` φpqq ` φpqβq ą φppq `  ÿ  iPrβs  φpqiq,  φppq `  ÿ  iPrβs  φpqiq  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' We first show that (ii) follows from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let y “ u ´ p1Ju{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then by  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1,  dimpConvpZℓ uqq “ dimpAffpZℓ uqq “ dimpSpanpZℓ yqq “ rankpCyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since Cy “ Cu ´ pp1Juq{ℓqJ, by part (i) of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='17,  dimpConvpZℓ uqq “ rankpCyq “ rankpCuq ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  To prove (i), suppose that φppqq`φpqq`φpqβq ą φppq` ř  iPrβs  φpqiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='17  and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1,  rankpCuq “ rankpCyq ` 1 “ 1 ` φppq `  ÿ  iPrβs  φpqiq `  ÿ  iPrβs  aiφppqiq,  where ai P t0, 1u, i P rβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' If ai “ 0 @i P rβs, then  rankpCuq “ 1 ` φppq `  ÿ  iPrβs  φpqiq ă 1 ` φppqq ` φpqq ` φpqβq,  contradicting Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Hence, at least one of the ais must be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since φppqi1q ą φppqi2q if and only if i1 ą i2, a1 “ 1, ai “ 0 for i “ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' , β results  in the smallest value of rankpCuq that we can not rule out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following corollary follows immediately from Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pq2, where p, q are distinct odd primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let Cu with  u P t´1, 1uℓ be the circulant matrix of u such that Cu is non-recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  rankpCuq is at least  φppq ` φppqq ` φpqq ` φpq2q ` 1 “ ℓ ´ φpℓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Consequently, either  dimpConvpZℓ uqq “ ℓ ´ 1 ´ φpℓq,  or  dimpConvpZℓ uqq “ ℓ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  24  The following corollary follows from Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='12, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3, and Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let p, q be distinct odd primes and n P Zě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Suppose that ℓ “ pn  or ℓ “ pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let u P t´1, 1uℓ satisfy x1 | uy “ ´1 and let y “ u ` p1{ℓq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  SpanQpZℓ yq “  ë  d | ℓ  d‰ℓ  ColQpPdq,  where Pd is defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Moreover,  dimpConvpZℓ uqq “ dimpAffpZℓ uqq “  ÿ  d | ℓ  d‰ℓ  dimQpColQpPdqq “ ℓ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let d ‰ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Since pColQpPdqqC “ ColCpPdq is spanned by the discrete  Fourier basis vectors vk for k P Zℓ Q pk, ℓq “ d, and µkpyq “ µkpuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Lem-  mas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='11 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='12, µkpyq ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since this holds @k Q pk, ℓq “ d, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3,  SpanQpZℓyq is not orthogonal to ColQpPdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' As this holds for each divisor d ‰ ℓ of  ℓ, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1, we must have  SpanQpZℓ yq “  ë  d | ℓ  d‰ℓ  ColQpPdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='6) now follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The following lemma follows from Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let ℓ “ pq2, and pu, vq be an LP of length ℓ, then  dimpConvpZℓ uqq “ dimpConvpZℓ vqq “ ℓ ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  We now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Since Zℓ ă pZℓ ¸ Zˆ  ℓ q, it suffices to prove the result for  G “ Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='23 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='24,  ℓ ´ 1 “ dimQ pAffQ pZℓ uqq “ dimpAffpZℓ uqq  “ dimpConvpZℓ uqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  The result now follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='3 as U “ H is the only possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Based on our partial results, we end the paper with the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  25  Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Let pu, vq be a LP of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Then  dimpConvpG uqq “ dimpConvpG vqq “ ℓ ´ 1  for G “ Zℓ and G “ Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  It suffices to prove Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='25 for only one of G “ Zℓ or G “ Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  This is because by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='8 the decomposition of QZℓ into irreducible Q-  subrepresentations is the same under G “ Zℓ or G “ Zℓ ¸ Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' Discussion  In this paper, we made progress towards proving the conjecture that for an LP  pu, vq of length ℓ,  dimpConvpG uqq “ dimpConvpG vqq “ ℓ ´ 1  for G “ Zℓ and G “ Zℓ ¸Zˆ  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content=' The immediate implication of this conjecture is that  the search space for LPs cannot be decreased by imposing additional linearly inde-  pendent linear equality constraints on u or v without rendering the rank-1 CSDD  for the LP problem infeasible or causing its symmetry group to exclude a non-  empty subset of Zℓ, provided that this rank-1 CSDD is feasible in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Future research will involve proving Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='25 with or without additional  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
+page_content='  Acknowledgments  The views expressed in this article are those of the authors, and do not reflect  the official policy or position of the United States Air Force, Department of De-  fense, or the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AzT4oBgHgl3EQf3_7V/content/2301.01839v1.pdf'}
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diff --git a/gdA0T4oBgHgl3EQfH_8L/content/tmp_files/2301.02067v1.pdf.txt b/gdA0T4oBgHgl3EQfH_8L/content/tmp_files/2301.02067v1.pdf.txt
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@@ -0,0 +1,1721 @@
+arXiv:2301.02067v1  [math.DG]  5 Jan 2023
+DIFFUSIVE STABILITY AND SELF-SIMILAR DECAY
+FOR THE HARMONIC MAP HEAT FLOW
+TOBIAS LAMM AND GUIDO SCHNEIDER
+Abstract. In this paper we study the harmonic map heat flow on the eu-
+clidean space Rd and we show an unconditional uniqueness result for maps with
+small initial data in the homogeneous Besov space ˙B
+d
+p
+p,∞(Rd) where d < p < ∞.
+As a consequence we obtain decay rates for solutions of the harmonic map flow
+of the form ∥∇u(t)∥L∞(Rd) ≤ Ct− 1
+2 .
+Additionally, under the assumption of a stronger spatial localization of the
+initial conditions, we show that the temporal decay happens in a self-similar
+way. We also explain that similar results hold for the biharmonic map heat
+flow and the semilinear heat equation with a power-type nonlinearity.
+1. Introduction
+The diffusive stability method is a tool which allows us to prove stability results
+for parabolic partial differential equations in the case of a linearization possessing
+essential spectrum up to the imaginary axis. It is based on algebraic decay rates
+which hold for the heat semigroup (et∆)t≥0 on Rd, defined by
+(et∆u0)(x) =
+1
+(4πt)d/2
+�
+Rd e− |x−y|2
+4t
+u0(y)dy.
+We have the decay
+∥et∆∥Lp→L∞ ≤ Ct− d
+2p ,
+for p ∈ [1, ∞), due to the diffusion which occurs for spatially localized initial con-
+ditions and the decay
+∥et∆∇∥L∞→L∞ ≤ Ct− 1
+2
+due to smoothing. These decays can be used for instance to prove stability results
+in (Lp ∩ C1
+b )(Rd) of the trivial solution u = 0 in semilinear heat equations
+∂tu = ∆u + f(u, ∇u),
+with x ∈ Rd, t ≥ 0, u(x, t) ∈ R, and a smooth nonlinear function f containing terms
+of the form um(∂1u)m1 . . . (∂du)md and satisfying f(0, 0) = 0. Since ∂tu ∼ t− d
+2p −1
+and ∆u ∼ t− d
+2p −1 for t → ∞ we have that the nonlinear terms behave like
+um(∂1u)m1 . . . (∂du)md ∼ t−
+(m+m1+...+md)d
+2p
+−
+m1+...+md
+2
+Date: January 6, 2023.
+We thank Peer Kunstmann for many helpful discussions on Besov spaces.
+Funded by the
+Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 258734477
+- SFB 1173.
+1
+
+2
+T. LAMM AND G. SCHNEIDER
+for t → ∞. Hence, they are asymptotically irrelevant with respect to the linear
+diffusion if
+d
+2p + 1 < (m + m1 + . . . + md)d
+2p
++ m1 + . . . + md
+2
+.
+As an example for d = 1 and p = 1 all such terms are irrelevant except for u2, u3,
+and u∂xu. With this idea the stability of u = 0 in (Lp ∩ C1
+b )(Rd) follows for
+∂tu = ∆u + um(∂x1u)m1 . . . (∂xdu)md,
+if the above condition is satisfied. An abstract existence result in this direction can
+be found in [19].
+After the introduction of this method in the papers of Fujita and Weisler [6, 22],
+the last decades saw a successful application of this tool to stability questions arising
+for pattern forming systems, see e.g. [3, 15, 14, 8]. A summary and overview of
+various aspects of this diffusive stability method can also be found in the monograph
+[16].
+It is the goal of this paper to use this tool for gaining new stability and unique-
+ness results for dissipative semilinear geometric flow problems on Rd, in particular
+for the harmonic map heat flow.
+In the following we let N be a smooth Riemannian manifold which we assume
+to be isometrically embedded into some euclidean space Rm.
+For δ > 0 suf-
+ficiently small we have a well defined nearest point projection π : Nδ → N,
+where Nδ := {x ∈ Rm : dist(x, N) < δ} and we define the second fundamen-
+tal form A(y) : TyN × TyN → T ⊥
+y N for every y ∈ N and v1, v2 ∈ TyN by
+A(y)(v1, v2) = −D2π(y)(v1, v2). A map u : Rd × [0, T ) → N is then a solution of
+the harmonic map heat flow with initial data u(·, 0) = u0 : Rd → N if it solves the
+parabolic partial differential equation
+∂tu − ∆u = A(u)(∇u, ∇u).
+(1)
+For instance, in case of N = Sm−1 = {x ∈ Rm : ∥x∥Rm = 1}, we compute
+A(u)(∇u, ∇u)
+=
+−(∆u, u)u = −(∂2
+kuα)uαu
+=
+−∂k((∂kuα)uα)u + (∂kuα)(∂kuα)u = u|∇u|2,
+where here and in the following we use Einstein’s summation convention and the
+fact that (∂kuα)uα = 0 which follows from the fact that uαuα = 1.
+Rather general existence results for the harmonic map heat flow on euclidean
+space have been obtained by Koch and the first author [9] for initial maps with
+a small oscillation and by Wang [20] for initial maps with a small BMO-norm.
+These solutions were constructed by applying a fixed point argument on suitably
+constructed Banach spaces. As a byproduct of this method, the authors obtained a
+conditional uniqueness result for the harmonic map heat flow under the smallness
+assumptions on the initial data mentioned above.
+One of our main goals of this paper is to use the diffusive stability method in
+order to improve this conditional uniqueness result to an unconditional uniqueness
+result. The price we have to pay is that our initial data have to be small in a strict
+subspace of the above mentioned space of functions of bounded mean oscillation.
+More precisely, we require our initial data to be small in a homogeneous Besov
+space, see Theorem 2.7 for further details.
+
+DIFFUSIVE STABILITY
+3
+The deviation v of a trivial spatially constant equilibrium u∗ satisfies a semilinear
+diffusion equation of the form
+(2)
+∂tv − ∆v = A(u∗ + v)(∇v, ∇v).
+As above we have A(u∗ +v)(∇v, ∇v) ∼ t− d
+p −1 for t → ∞, and so the nonlinearity is
+irrelevant w.r.t. to linear diffusion if p ∈ [1, ∞). For p = ∞ the linear and nonlinear
+terms are of the same order and a stability result on exponentially long time scales
+can be established, see Theorem 2.9.
+For slightly stronger localized initial conditions the associated solutions of the
+linear heat equation decay in a self-similar way towards zero. The same is true for
+semilinear heat equations with nonlinearities which are irrelevant in L1 ∩ C1
+b (Rd)
+with the above counting. Using the discrete renormalization approach, cf. [2, 3],
+gives that the renormalized solution td/2v(x
+√
+t, t) of (2) converges towards a mul-
+tiple of a Gaussian e−|x|2/4 for t → ∞. In the discrete renormalization approach
+instead of solving the PDE (2) directly we consider an equivalent sequence of prob-
+lems which converges formally towards the linear diffusion equation. Let
+vn(x, τ) = Ldnv(Lnx, L2nτ),
+with L > 1 fixed and n ∈ N. Then vn satisfies
+∂τvn − ∆vn
+=
+Ln(d+2−2−2d)D2π(u∗ + L−dnv)(∇vn, ∇vn)
+(3)
+=
+L−ndD2π(u∗ + L−dnv)(∇vn, ∇vn)
+for τ ∈ [L−2, 1], with
+(4)
+vn(x, L−2) = Ldvn−1(Lx, 1).
+The influence of the nonlinear terms vanishes for n → ∞ with a geometric rate. In
+the limit we obtain a linear diffusion equation. With a simple fixed point argument
+the convergence of vn|τ=1 towards the limit of the renormalized linear diffusion
+problem, namely a multiple of the Gaussian e−|x|2/4 can be established, cf. Theorem
+4.1.
+A different approach for the harmonic map heat flow is the consideration of
+w = ∇v which satisfies
+(5)
+∂tw − ∆w = ∇(A(u∗ + v)(w, w)).
+We show that the above mentioned discrete renormalization approach can also be
+applied to this system of equations.
+Finally, we show that our technique can be extended to various other semilinear
+heat equations on euclidean space, such as equations with a power-type nonlinearity
+or higher order equations, such as the biharmonic map heat flow.
+A similar handling of dissipative quasilinear geometric flow problems, such as the
+Ricci-DeTurck flow, the mean curvature flow, the Yamabe flow, and the Willmore
+flow of graphs, will be the topic of future research.
+Notation. In the following many possibly different constants are denoted with
+the same symbol C if they can be chosen independent of time t.
+
+4
+T. LAMM AND G. SCHNEIDER
+2. Stability and uniqueness results for the harmonic map flow
+Our first main result about the harmonic map heat flow is given by an asymp-
+totic stability and uniqueness result assuming smallness conditions on a certain
+homogeneous Besov norm of the inital map u0.
+We note that due to the unboundedness of the domain the linearization around
+u = const. for (1) possesses essential spectrum up to the imaginary axis and so the
+principle of linearized stability does not apply directly in this situation. Therefore,
+we use the diffusive stability method for establishing the following results. We start
+by extending a stability result for the harmonic map flow which was originally due
+to Soyeur [17]. Using the notation ˙W 1,p(Rd, N) for the homogeneous Sobolev space,
+his result is as follows
+Theorem 2.1. There exists a constant ε > 0 depending only on d and N such that
+if u0 ∈ ˙W 1,p(Rd, N) with d < p < ∞ and ∥∇u0∥Ld < ε, then we have the estimate
+t
+1
+2 − d
+2p ∥∇u(t)∥Lp ≤ C∥∇u0∥Ld,
+(6)
+where C = C(d, m, p), for every solution u ∈ C0([0, ∞), ˙W 1,p(Rd, N)) of (1).
+Remark 2.2. In the same paper Soyeur also showed the existence of a solution
+u ∈ C0([0, ∞), ˙W 1,p(Rd, N)) for small initial data in the above sense. Additionally,
+the smallness assumption on the Ld-norm of ∇u0 implies that the BMO-norm of
+u0 is small and hence, by a paper of C. Wang [20], this implies the existence of a
+global smooth solution of the harmonic map flow.
+Remark 2.3. This result can be extended to maps from complete Riemannian
+manifolds for which one has good heat kernel estimates as it was done in Li and
+Wang [11] for (6).
+Examples of such manifolds include manifolds with positive
+Ricci curvature.
+The original result of Soyeur already extended a previous result of Struwe [18],
+who was the first one to show a convergence result for the harmonic map heat flow
+on Rd to a constant map with additional decay properties for the first derivative.
+More precisely, he showed in Theorem 7.1 of the paper mentioned above, that for
+smooth initial maps u0 so that ∥∇u0∥L∞ is bounded and ∥∇u0∥L2 is sufficiently
+small, the unique smooth solution of the harmonic map heat flow satisfies for t large
+enough ∥∇u(·, t)∥L∞ ≤ Ct− 1
+2 and hence converges to a constant map.
+As a first result we improve Theorem 2.1 for initial data satisfying a slightly
+less restrictive smallness condition. More precisely, for a map v : Rd → Rm we
+denote by G(t)v := Gt ⋆ v its heat extension, i.e. G(t)v is a solution of the heat
+equation with initial data v. Here G(x, t) =
+1
+(4πt)d/2 e
+−|x|2
+4t , (x, t) ∈ Rd × (0, ∞), is
+the standard heat kernel. For d < p ≤ ∞ we define the homogeneous Besov space
+˙B
+d
+pp,∞(Rd) as the set of functions for which the norm
+∥v∥
+˙B
+d
+p
+p,∞
+:= sup
+t>0
+t
+1
+2 − d
+2p ∥∇(G(t)v)∥Lp
+is finite. It follows that
+˙W 1,d(Rd) ⊂ ˙B
+d
+p
+p,∞(Rd) ⊂ BMO(Rd) ⊂ ˙B0
+∞,∞(Rd)
+for every d < p < ∞, where each of these inclusions is strict (see e.g. [4]).
+Now we are in a position to state our first main result.
+
+DIFFUSIVE STABILITY
+5
+Theorem 2.4. There exists a constant ε > 0 depending only on d and N such that
+if u0 ∈ ˙B
+d
+pp,∞(Rd, N) with d < p < ∞, 2 ≤ p and ∥u0∥
+˙B
+d
+p
+p,∞
+< ε, then we have the
+estimate
+t
+1
+2 − d
+2p ∥∇u(t)∥Lp ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+,
+where C = C(d, m, p), for every solution u : (0, ∞) × Rd → N of (1) with
+t
+1
+2 − d
+2p ∇u(t, x) ∈ C0([0, ∞), Lp(Rd))
+and
+lim
+tց0 t
+1
+2 − d
+2p ∥∇u(t)∥Lp = 0.
+Remark 2.5. Note that the existence of such solutions follows for example from
+the work of Soyeur under the assumption that u0 ∈ ˙W 1,p(Rd, N) with a sufficiently
+small norm.
+In the proof of this result we need a standard estimate on convolutions of the
+heat kernel. Note that it relies on Young’s convolution inequality.
+Lemma 2.6. Let G(x, t) =
+1
+(4πt)d/2 e
+−|x|2
+4t , (x, t) ∈ Rd × (0, ∞), be the heat kernel.
+Using the notation G(t)f = G(t) ⋆ f we have the estimate
+∥∇σG(t)∥L(Lr,Lq) ≤ Ct− d
+2 ( 1
+r − 1
+q )− σ
+2 ,
+(7)
+where 1 ≤ r ≤ q ≤ ∞, σ ∈ N0 and C < ∞ is a constant. Additionally, we have
+that for some constant C < ∞
+� t
+0
+(t − s)−αs−β ds = Ct1−α−β
+(8)
+if both 0 < α, β < 1.
+Now we are in a position to prove Theorem 2.4.
+Proof of Theorem 2.4: We start by noting that
+∇(u(x, t) − G(t) ⋆ u0(x)) =
+� t
+0
+�
+Rd ∇G(x − y, t − s)(A(u)(∇u, ∇u)(y, s)) dy ds.
+Using the estimate (7) with σ = 1, r = p
+2 and q = p we get for every t > 0
+∥∇(u(t) − G(t)u0)∥Lp ≤C
+� t
+0
+∥|∇u(s)|2∥L
+p
+2 (t − s)−( 1
+2 + d
+2p ) ds
+≤C
+� t
+0
+∥∇u(s)∥2
+Lp(t − s)−( 1
+2 + d
+2p ) ds.
+Setting
+m(T ) = sup
+0≤t≤T
+t
+1
+2 − d
+2p ∥∇u(t)∥Lp,
+the last inequality implies for all t > 0
+t
+1
+2 − d
+2p ∥∇u(t)∥Lp ≤C∥u0∥
+˙B
+d
+p
+p,∞
++ Ct
+1
+2 − d
+2p m(T )2
+� t
+0
+(t − s)−( 1
+2 + d
+2p )s−1+ d
+p ds
+≤C∥u0∥
+˙B
+d
+p
+p,∞
++ Cm(T )2,
+
+6
+T. LAMM AND G. SCHNEIDER
+where we used the identity (8) with 0 < α = 1
+2 + d
+2p < 1 and 0 < β = 1 − d
+p < 1.
+Note that it is exactly here that we have to use the assumption d < p < ∞.
+Taking the supremum over all 0 ≤ t ≤ T we thus obtain for every 0 < T < ∞
+Cm(T )2 − m(T ) + C∥u0∥
+˙B
+d
+p
+p,∞
+≥ 0
+and for ε > 0 small enough this finally implies for all T < ∞
+m(T ) ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+≤ Cε.
+Here we used that m(0) = 0. By definition this estimate implies that
+t
+1
+2 − d
+2p ∥∇u(t)∥Lp ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+for every 0 < t < ∞, as claimed.
+□
+In the next theorem we improve a uniqueness result for solutions of the harmonic
+map heat flow with initial data with small BMO-norm. For this we define the
+Banach space X as the space of functions such that the norm
+∥u∥X := sup
+0<t<∞
+�
+∥u(t)∥L∞(Rd) + t
+1
+2 ∥∇u(t)∥L∞(Rd)
+�
++ sup
+x∈Rd
+sup
+0<R<∞
+R− d
+2 ∥∇u∥L2(BR(x)×(0,R2))
+is finite. It was shown by the first author and Koch [9] that there exist constants
+ε > 0, C > 0 such that for every u0 : Rd → N satisfying ∥u0 − P∥L∞ < ε, where
+P ∈ N is some arbitrary point, there exists a global solution u ∈ P + X of (1).
+Moreover, the solution is unique in the ball
+BX(P, Cε) = {u : ∥u − P∥X ≤ Cε}.
+This result was later on extended to initial data u0 ∈ BMO(Rd) with small
+BMO-norm by Wang [20]. He obtained the uniqueness of the solution in the set
+BX(G(t)u0, Cε).
+As our second main result for the harmonic map flow we improve this conditional
+uniqueness result to an unconditional uniqueness result (i.e. we no longer have to
+restrict to the class of solutions in X with small X-norm) but the price we have to
+pay is to go from BMO-initial data to homogeneous Besov initial data.
+Theorem 2.7. Let N be a smooth and closed manifold. There exists ε > 0, C > 0
+such that for every u0 ∈ ˙B
+d
+p
+p,∞(Rd, N) with d < p < ∞, ∥u0∥
+˙B
+d
+p
+p,∞
+< ε and every
+solution u : (0, ∞) × Rd → N of (1) with
+t
+1
+2 − d
+2p ∇u(t, x) ∈ C0([0, ∞), Lp(Rd))
+and
+lim
+tց0 t
+1
+2 − d
+2p ∥∇u(t)∥Lp = 0,
+we have the estimate
+∥u − G(t)u0∥X ≤ Cε,
+i.e. there exists only one solution under these assumptions.
+
+DIFFUSIVE STABILITY
+7
+Remark 2.8. Since
+sup
+t>0
+t
+1
+2 ∥∇(G(t)u0)∥L∞(Rd) ≃ ∥u0∥ ˙B0
+∞,∞,
+respectively
+sup
+x∈Rd
+sup
+0<R<∞
+R− d
+2 ∥∇(G(t)u0)∥L2(BR(x)×(0,R2)) ≃ ∥u0∥BMO
+and since we have the inclusions
+˙B
+d
+p
+p,∞ ⊂ BMO ⊂ ˙B0
+∞,∞
+for every d < p < ∞, it follows from the above estimates that for every solution u ∈
+C0([0, ∞), ˙W 1,p(Rd, N)) of (1) as above with initial data u0 satisfying ∥u0∥
+˙B
+d
+p
+p,∞
+< ε
+we have
+sup
+t>0
+t
+1
+2 ∥∇u∥L∞(Rd) + sup
+x∈Rd
+sup
+0<R<∞
+R− d
+2 ∥∇u∥L2(BR(x)×(0,R2)) ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+.
+Proof of Theorem 2.7: We fix p > d and we note that we get from Theorem 2.4 the
+estimate
+sup
+t>0
+t
+1
+2 − d
+2p ∥∇u(t)∥Lp ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+.
+Next we use again the estimate (7) with σ = 1, r = p
+2 and q = ∞ to conclude for
+every t > 0
+∥∇(u(t) − G(t)u0)∥L∞ ≤C
+� t
+0
+∥|∇u(s)|2∥L
+p
+2 (t − s)−( 1
+2 + d
+p ) ds
+≤C
+� t
+0
+∥∇u(s)∥2
+Lp(t − s)−( 1
+2 + d
+p ) ds
+≤C∥u0∥2
+˙B
+d
+p
+p,∞
+� t
+0
+s
+d
+p −1(t − s)−( 1
+2 + d
+p ) ds
+≤Ct− 1
+2 ∥u0∥2
+˙B
+d
+p
+p,∞
+,
+where we used (8) in the last estimate.
+Similarly, we obtain
+∥u(t) − G(t)u0∥L∞ ≤C
+� t
+0
+∥|∇u(s)|2∥L
+p
+2 (t − s)− d
+p ds
+≤C
+� t
+0
+∥∇u(s)∥2
+Lp(t − s)− d
+p ds
+≤C∥u0∥2
+˙B
+d
+p
+p,∞
+� t
+0
+s
+d
+p −1(t − s)− d
+p ds
+≤C∥u0∥2
+˙B
+d
+p
+p,∞
+.
+It remains to establish the bound
+sup
+x∈Rd
+sup
+0<R<∞
+R− d
+2 ∥∇(u − G(·)u0)∥L2(BR(x)×(0,R2)) ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+.
+
+8
+T. LAMM AND G. SCHNEIDER
+Using H¨older’s inequality, we estimate for every x ∈ Rd and some p > d
+�
+BR(x)
+|∇(u(t) − G(t)u0)|2 ≤C(
+�
+BR(x)
+|∇(u(t) − G(t)u0)|p)
+2
+p Rd(1− 2
+p )
+≤Ct
+d
+p −1Rd(1− 2
+p )∥u0∥2
+˙B
+d
+p
+p,∞
+.
+From this it is easy to see that
+sup
+x∈Rd
+sup
+0<R<∞
+R− d
+2 ∥∇(u − G(·)u0)∥L2(BR(x)×(0,R2)) ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+.
+□
+Finally, we show a new stability result on exponentially long time scales for the
+harmonic map flow starting near a constant map. Note that in this result we do
+not assume that the initial values decay in space.
+Theorem 2.9. For every C > 0 there exist b > 0 and ε0 > 0 such that for every
+ε ∈ (0, ε0) and every solution u : Rd × [0, ∞) → N of (1) with ∥u0 − P∥W 1,∞ ≤ ε,
+where P ∈ N is an arbitrary point, we have
+sup
+t∈[0,exp( b
+ε)−1]
+∥u(t) − P∥C0 +
+sup
+t∈[0,exp( b
+ε)−1]
+(1+t)
+1
+2 ∥∇u(t)∥C0 ≤ C.
+Proof. We define the functions
+a(t)
+=
+sup
+0≤s≤t
+∥u(s) − P∥L∞
+resp.
+b(t)
+=
+sup
+0≤s≤t
+∥(1+s)
+1
+2 ∇u(s)∥L∞,
+and we estimate (using the fact that ∥G(t)(u0 − P)∥L∞ ≤ ∥u0 − P∥L∞)
+∥u(t) − P∥L∞
+≤
+C∥u0 − P∥L∞ + C
+����
+� t
+0
+G(·, t − s) ⋆ A(u)(∇u, ∇u)(s)ds
+����
+L∞
+≤
+Ca(0) + C
+� t
+0
+∥|∇u(s)|2∥L∞ds
+≤
+Ca(0) + Cb2(t)
+� t
+0
+(1+s)−1 ds
+≤
+Ca(0) + Cb2(t) log(1 + t)
+and
+(1 + t)
+1
+2 ∥∇u(t)∥L∞ ≤C(a(0) + b(0)) + Cb2(t)(1 + t)
+1
+2
+� t
+0
+(t − s)− 1
+2 (1 + s)−1 ds
+≤C(a(0) + b(0)) + Cb2(t)(1 + log(1 + t)),
+where we used that
+∥∇(G(·, t) ⋆ (u0 − P))∥L∞ ≤
+�
+C∥∇u0∥L∞,
+for t ≤ 1,
+Ct− 1
+2 ∥u0 − P∥L∞,
+for t > 1.
+Next we define the rescaled functions ˜a(t) = ε−1a(t) resp. ˜b(t) = ε−1b(t) and we
+conclude
+˜a(t) + ˜b(t) ≤ C1(˜a(0) + ˜b(0)) + C2ε(1 + log(1 + t))˜b2(t)
+
+DIFFUSIVE STABILITY
+9
+for some constants C1, C2 > 0. Since ˜a(0) + ˜b(0) ≤ 1 we conclude
+˜a(t) + ˜b(t) ≤ 2C1
+as long as
+1 + t ≤ exp
+�
+1
+4C1C2ε − 1
+�
+.
+Hence
+a(t) + b(t) ≤ 2C1ε
+for all 0 ≤ t ≤ exp
+�
+1
+4C1C2ε − 1
+�
+− 1 where we note that for ε > 0 sufficiently
+small, the last number satisfies the required properties in the statement of the
+Theorem.
+□
+3. Related equations
+In this section we want to show that the results obtained for the harmonic map
+heat flow can be extended to two other parabolic equations to either simplify proofs
+of existing results or yield new results. We note that in the case of the Navier-Stokes
+equation similar results have been obtained earlier by Cannone [5], Planchon [13]
+and Koch-Tataru [10].
+3.1. Equations with power-type nonlinearities. Here we study the very clas-
+sical semilinear equation
+∂tu − ∆u = f(u)
+(9)
+where u : Rd × [0, ∞) → R with u(·, 0) = u0 and f : R → R is a smooth function
+satisfying |f(s)| ≤ |s|q for some q > 1 and all s ∈ R. It was already shown in [6, 22]
+that for non-negative initial data u0 so that the norm ∥u0∥
+L
+d(q−1)
+2
+is sufficiently
+small (here we also assume d(q−1)
+2
+> 1), there exists a non-negative global solution
+of (9) satisfying
+sup
+t>0
+t
+1
+q−1 − d
+2p ∥u(t)∥Lp ≤ C∥u0∥
+L
+d(q−1)
+2
+for every d(q−1)
+2
+< p < dq(q−1)
+2
+. This result should be directly compared (and was in
+fact a motivation) to the one of Soyeur [17] (see Theorem 2.1) for the harmonic map
+heat flow. In fact, with the same technique that we used in the previous section,
+we are able to extend these results and we obtain
+Theorem 3.1. There exists a constant ε > 0 depending only on d and q such that
+if u0 ∈ B
+d
+p −
+2
+q−1
+p,∞
+(Rd) with 1 < q < ∞, 1 < d(q−1)
+2
+and if ∥u0∥
+B
+d
+p −
+2
+q−1
+p,∞
+< ε, then we
+have the estimate
+sup
+t>0
+t
+1
+q−1 − d
+2p ∥u(t)∥Lp ≤ C∥u0∥
+B
+d
+p −
+2
+q−1
+p,∞
+,
+where C = C(d, p), for every d(q−1)
+2
+< p < dq(q−1)
+2
+and every solution u : (0, ∞) ×
+Rd → R of (9) with
+t
+1
+q−1 − d
+2p u(t, x) ∈ C0((0, ∞), Lp(Rd))
+and
+lim
+tց0 t
+1
+q−1 − d
+2p ∥u(t)∥Lp = 0.
+
+10
+T. LAMM AND G. SCHNEIDER
+If we know additionally that q > max{2, 1 + 2
+d} then we also get the estimate
+sup
+t>0
+t
+1
+q−1 ∥u(t)∥L∞ ≤ C∥u0∥
+B
+d
+p −
+2
+q−1
+p,∞
+.
+Proof. Arguing as in the proof of Theorem 2.4, we obtain
+∥u(t) − G(t)u0∥Lp ≤C
+� t
+0
+∥|u(s)|q∥
+L
+p
+q (t − s)− d(q−1)
+2p
+ds
+≤C
+� t
+0
+∥u(s)∥q
+Lp(t − s)− d(q−1)
+2p
+ds
+≤Cm(T )q
+� t
+0
+s
+dq
+2p −
+q
+q−1 (t − s)− d(q−1)
+2p
+ds
+≤Cm(T )q,
+since 0 > dq
+2p −
+q
+q−1 > −1 and 0 > − d(q−1)
+2p
+> −1. Note that here we let
+m(T ) = sup
+0≤t≤T
+t
+1
+q−1 − d
+2p ∥u(t)∥Lp.
+As above, this estimate implies the desired claim since
+sup
+t>0
+t
+1
+q−1 − d
+2p ∥G(t)u0∥Lp ∼= ∥u0∥
+B
+d
+p −
+2
+q−1
+p,∞
+.
+Next, if q > 2 there exists a p satisfying dq
+2 < p < dq(q−1)
+2
+and for one such value
+of p we note that for every 0 < t < ∞
+∥u(t) − G(t)u0∥L∞ ≤C
+� t
+0
+∥|u(s)|q∥
+L
+p
+q (t − s)− dq
+2p ds
+≤C
+� t
+0
+∥u(s)∥q
+Lp(t − s)− dq
+2p ds
+≤Cm(T )q
+� t
+0
+s
+dq
+2p −
+q
+q−1 (t − s)− dq
+2p ds
+≤Ct−
+1
+q−1 m(T )q,
+and hence we also get the estimate
+sup
+t>0
+t
+1
+q−1 ∥u(t)∥L∞ ≤ C∥u0∥
+B
+d
+p −
+2
+q−1
+p,∞
+.
+□
+Note that the Lp-estimate in the previous theorem has been obtained earlier by
+Miao, Yuan and Zhang [12]. Related estimates have also been obtained by Blatt
+and Struwe [1] for small initial data in Morrey spaces.
+3.2. Biharmonic map flow. As another example we consider solutions u : Rd ×
+(0, T ) → Sm−1 ⊂ Rm of the (extrinsic) biharmonic map heat flow governed by
+(∂t + ∆2)u = u(|∆u|2 − ∆|∇u|2 − 2div⟨∆u, ∇u⟩)
+= u|∆u|2 + ∇u · (∇|∇u|2 + 2⟨∆u, ∇u⟩) − div(u∇|∇u|2 + 2u⟨∆u, ∇u⟩)
+(10)
+=: f1[u] + divf2[u],
+
+DIFFUSIVE STABILITY
+11
+where we can estimate
+|f1[u]| ≤ C(|∇2u|2 + |∇u|4)
+and
+|f2[u]| ≤ C|∇u||∇2u|.
+For the sake of simplicity we restrict ourselves to sphere-valued maps. The ar-
+guments below can easily be extended to general closed manifolds N and maps
+u : Rd × (0, T ) → N. Additionally, there exists another version of the biharmonic
+map flow, the so called intrinsic biharmonic map. All of the results shown below
+carry over directly to this fourth order parabolic PDE.
+Again we are interested in an unconditional uniqueness resp. stability result
+for small initial data in a suitable Besov space. As before, the principle of linear
+stability cannot be applied directly due to the fact that the linearization again
+possesses an essential spectrum up to the imaginary axis.
+But as before some diffusive behavior can be used to control the nonlinear terms
+which are irrelevant w.r.t.
+this diffusive behavior.
+In the following we let b :
+Rd × (0, ∞) → R denote the biharmonic heat kernel. It follows from the results of
+[9] that Lemma 2.6 can be extended to this case, namely we have
+Lemma 3.2. Let b : Rd × (0, ∞) → R be the biharmonic heat kernel. Using the
+notation b(t)f = bt ⋆ f we have the estimate
+∥∇σb(t)∥L(Lr,Lq) ≤ Ct− d
+4 ( 1
+r − 1
+q )− σ
+4 ,
+(11)
+where 1 ≤ r ≤ q ≤ ∞, σ ∈ N0 and C < ∞ is a constant.
+Before we now state our main results for the biharmonic map flow, we note that
+we have another equivalent characterization of the homogeneous Besov space via
+the biharmonic heat kernel given by
+∥u0∥
+˙B
+d
+p
+p,∞
+:= sup
+t>0
+�
+t
+1
+4 − d
+4p ∥∇(b(t)u0)∥Lp + t
+1
+2 − d
+2p ∥∇2(b(t)u0)∥L
+p
+2
+�
+.
+Here we note that the first term on the right hand-side indeed defines an equivalent
+norm on the homogeneous Besov space ˙B
+d
+pp,∞, whereas the second term defines an
+equivalent norm on the homogeneous space ˙B
+2d
+p
+p
+2 ,∞. Since the second homogeneous
+Besov space is embedded in the first one we choose our norm as a definition of the
+larger space.
+Theorem 3.3. There exists a constant ε > 0 depending only on d and n such that
+if u0 ∈
+˙B
+d
+pp,∞(Rd, Sm−1) with d < p < ∞ and ∥u0∥
+˙B
+d
+p
+p,∞
+< ε, then we have the
+estimate
+t
+1
+4 − d
+4p ∥∇u(t)∥Lp + t
+1
+2 − d
+2p ∥∇2u(t)∥L
+p
+2 ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+,
+where C = C(d, m, p), for every solution u : (0, ∞) × Rd → Sm−1 of (1) with
+t
+1
+4 − d
+4p ∇u ∈ C0([0, ∞), Lp(Rd)), t
+1
+2 − d
+2p ∇2u ∈ C0([0, ∞), L
+p
+2 (Rd))
+and
+lim
+tց0(t
+1
+4 − d
+4p ∥∇u(t)∥Lp + t
+1
+2 − d
+2p ∥∇2u(t)∥L
+p
+2 ) = 0.
+
+12
+T. LAMM AND G. SCHNEIDER
+Proof. We follow closely the proof of Theorem 2.4 by observing that
+∇i(u(x, t) − b(t) ⋆ u0(x)) =
+� t
+0
+�
+Rd ∇ib(x − y, t − s)f1[u] dy ds
+−
+� t
+0
+�
+Rd ∇i+1b(x − y, t − s)f2[u](s) dy ds =: Ii + IIi.
+Using the estimate (11) with σ = 1, r = p
+4 and q = p we get for every t > 0
+∥I1∥Lp ≤C
+� t
+0
+�
+∥|∇u(s)|4∥L
+p
+4 + ∥|∇2u(s)|2∥L
+p
+4
+�
+(t − s)−( 1
+4 + 3d
+4p ) ds
+≤C
+� t
+0
+(∥∇u(s)∥4
+Lp + ∥∇2u(s)∥2
+L
+p
+2 )(t − s)−( 1
+4 + 3d
+4p ) ds.
+Setting
+m(T ) = sup
+0≤t≤T
+�
+t
+1
+4 − d
+4p ∥∇u(t)∥Lp + t
+1
+2 − d
+2p ∥∇2u(t)∥L
+p
+2
+�
+,
+the last inequality implies for all t > 0
+t
+1
+4 − d
+4p ∥I∥Lp ≤Ct
+1
+4 − d
+4p m(T )2(1 + m(T )2)
+� t
+0
+(t − s)−( 1
+4 + 3d
+4p )s−1+ d
+p ds
+≤Cm(T )2(1 + m(T )2),
+where we used again the identity (8).
+Similarly, using (11) with σ = 2, r = p
+3 and q = p we get for every t > 0
+∥II1∥Lp ≤C
+� t
+0
+∥|∇u(s)||∇2u(s)|∥L
+p
+3 (t − s)−( 1
+2 + d
+2p ) ds
+≤C
+� t
+0
+(∥∇u(s)∥Lp∥∇2u(s)∥L
+p
+2 )(t − s)−( 1
+2 + d
+2p ) ds.
+Thus, we obtain
+t
+1
+4 − d
+4p ∥II∥Lp ≤Ct
+1
+4 − d
+4p m(T )2
+� t
+0
+(t − s)−( 1
+2 + d
+2p )s− 3
+4 + 3d
+4p ds
+≤Cm(T )2,
+which implies
+t
+1
+4 − d
+4p ∥∇u(t)∥Lp ≤ C∥u0∥
+˙B
+d
+p
+p,∞
++ Cm(T )2(1 + m(T )2).
+For i = 2 we argue similarly, only this time we choose σ = 2, r = p
+4 and q = p
+2 for
+I2 resp. σ = 3, r = p
+3 and q = p
+2 for II2 to obtain
+t
+1
+2 − d
+2p ∥∇2u(t)∥L
+p
+2 ≤ C∥u0∥
+˙B
+d
+p
+p,∞
++ Cm(T )2(1 + m(T )2).
+Taking the supremum over all 0 ≤ t ≤ T we thus obtain for every 0 < T < ∞
+Cm(T )2(1 + m(T )2) − m(T ) + C∥u0∥
+˙B
+d
+p
+p,∞
+≥ 0
+and for ε > 0 small enough this finally implies for all T < ∞ that
+m(T ) ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+≤ Cε.
+
+DIFFUSIVE STABILITY
+13
+Here we used that m(0) = 0. By definition this estimate implies that
+t
+1
+4 − d
+4p ∥∇u(t)∥Lp + t
+1
+2 − d
+2p ∥∇2u(t)∥L
+p
+2 ≤ C∥u0∥
+˙B
+d
+p
+p,∞
+for every 0 < t < ∞, as claimed.
+□
+As in the case of the harmonic map heat flow this result can now be used to
+improve the uniqueness result for solutions of the biharmonic map heat flow with
+small initial data in BMO due to Wang [21]. For this we have to define again a
+suitable function space. This time it is given by
+∥u∥Xb := sup
+0<t<∞
+�
+∥u(t)∥L∞(Rd) +
+2
+�
+i=1
+t
+i
+4 ∥∇iu(t)∥L∞(Rd)
+�
++ sup
+x∈Rd
+sup
+0<R<∞
+2
+�
+i=1
+R− di
+4 ∥∇iu∥L
+4
+i (BR(x)×(0,R4))
+is finite.
+Similarly to Theorem 2.7 we then obtain
+Theorem 3.4. There exists ε > 0, C > 0 such that for every u0 ∈ ˙B
+d
+p
+p,∞(Rd, Sm−1)
+with d < p < ∞, ∥u0∥
+˙B
+d
+p
+p,∞
+< ε and every solution u : (0, ∞) × Rd → Sm−1 of (10)
+with
+t
+1
+4 − d
+4p ∇u(t, x) ∈ C0([0, ∞), Lp(Rd)), t
+1
+2 − d
+2p ∇2u(t, x) ∈ C0([0, ∞), L
+p
+2 (Rd))
+and
+lim
+tց0
+�
+t
+1
+4 − d
+4p ∥∇u(t)∥Lp + t
+1
+2 − d
+2p ∥∇2u(t)∥L
+p
+2
+�
+= 0,
+we have the estimate
+∥u − b(t)u0∥Xb ≤ Cε,
+i.e., there exists only one solution of (10) under these assumptions.
+Additionally, one obtains the bounds
+sup
+t>0
+2
+�
+i=1
+t
+i
+4 ∥∇iu(t)∥L∞(Rd) + sup
+x∈Rd
+sup
+0<R<∞
+2
+�
+i=1
+R− di
+4 ∥∇iu∥L
+4
+i (BR(x)×(0,R4))
+≤ C∥u0∥
+˙B
+d
+p
+p,∞
+.
+4. Self-similar decay
+In this section we collect a number of results which show an asymptotic self-
+similar decay of the deviations v for the harmonic and bi-harmonic map heat flow.
+We start with the v-equation (2) for the harmonic map heat flow. The transfer of
+the presented method to the biharmonic map heat flow can be found below. We
+close this section with some remarks about similar statements for the w-equation
+(5).
+
+14
+T. LAMM AND G. SCHNEIDER
+4.1. Harmonic map heat flow. It is well known that for spatially localized initial
+conditions the solutions of the linear diffusion equation
+∂tv = ∆v,
+v|t=0 = v0
+decay asymptotically in a self similar way, i.e., for x ∈ Rd the renormalized solution
+td/2v(x
+√
+t, t) converges towards a multiple of a Gaussian, Vlime−|x|2/4 with Vlim ∈ R.
+The same is true for the deviation v of a trivial spatially constant equilibrium u∗ for
+the harmonic map heat flow (1). As we have seen in the introduction, the deviation
+v satisfies a semilinear diffusion equation of the form
+(12)
+∂tv − ∆v = A(u∗ + v)(∇v, ∇v),
+where we recall that A(u∗ + v)(·, ·) is a smooth bounded bilinear mapping acting
+on the d-dimensional tangent space. Our first result reads as follows
+Theorem 4.1. There exist δ, C > 0, such that for all solutions v of (12) with
+∥v|t=0∥Hm
+r ≤ δ where m > d/2 + 1 and r > d/2 + 1 we have a Vlim ∈ Rd such that
+∥td/2 v(·
+√
+t, t) − Vlime−|·|2/4∥Hm
+r ≤ C(1+t)−d/2
+for all t ≥ 0, where
+∥v∥Hm
+r = ∥vσr∥Hm,
+with
+σ(x) = (1 + x2)1/2.
+Proof. We use the discrete renormalization approach which was introduced for such
+problems at the beginning of the 1990s, cf. [2, 3]. Although the proof is documented
+in the literature for similar problems, we recall its main steps since to our knowledge
+such results have not been formulated for geometric flow problems so far.
+Instead of solving (12) directly we consider an equivalent sequence of problems
+which converges formally towards the linear diffusion equation. For this we let
+(13)
+vn(ξ, τ) = Lndv(Lnξ, L2nτ),
+with L > 1 fixed and n ∈ N. Then vn satisfies
+(∂τvn − ∆ξvn) = Ln(d+2)(∂tv − ∆xv)
+=
+Ln(d+2)A(u∗ + v)(∇xv, ∇xv)
+=
+Ln(d+2)A(u∗ + L−ndvn)(L−n(d+1)∇ξvn, L−n(d+1)∇ξvn)
+=
+L−ndA(u∗ + L−ndvn)(∇ξvn, ∇ξvn)
+for τ ∈ [L−2, 1], with initial condition
+(14)
+vn(ξ, L−2) = Ldvn−1(Lξ, 1).
+The influence of the nonlinear terms vanishes with a geometric rate for n → ∞. In
+the limit n → ∞ we obtain a linear diffusion equation. With a simple fixed point
+argument the convergence of vn|τ=1 towards the limit of the renormalized linear
+diffusion problem, namely multiples of the Gaussian ψ(ξ) = (2π)−d/2e−|ξ|2/4, can
+be established.
+Notation. In the rest of the proof we use the symbol C for constants which can
+be chosen independent of L and n.
+We need the projection
+Πvn =
+�
+vn(ξ)dξ,
+
+DIFFUSIVE STABILITY
+15
+respectively �Π�vn = �vn|k=0 in Fourier space. The bound
+|Πvn| ≤ C|�vn|k=0| ≤ C∥�vn∥C0
+b ≤ C∥�vn∥Hr
+m ≤ C∥vn∥Hm
+r
+holds due to Sobolev’s embedding theorem Hr
+m ⊂ C0
+b for r > d/2. We set
+Vn+1 = Πvn|τ=1
+and introduce the deviation of vn(ξ, 1) from Vn+1ψ(ξ) by
+ρn+1(ξ) = vn(ξ, 1) − Vn+1ψ(ξ).
+By construction we have
+�
+ρn(ξ)dξ = 0, resp. �ρn|k=0 = 0. For proving the con-
+vergence of (Vn)n∈N towards a Vlim ∈ R and the decay of (ρn)n∈N towards zero we
+establish a number of inequalities.
+Lemma 4.2. We have
+|Vn+1 − Vn|
+≤
+CL−ndR2
+n,
+∥ρn+1∥Hm
+r
+≤
+(C/L)∥ρn∥Hm
+r + CL−ndR2
+n + C|Vn+1 − Vn|
+where
+Rn =
+sup
+τ∈[1/L2,1]
+∥vn(τ)∥Hm
+r .
+Proof. We consider the variation of constants formula
+(15)
+vn(ξ, τ) = e(τ−1/L2)∆ξLdvn−1(Lξ, 1) + gn(τ)
+for τ ∈ [1/L2, 1], where
+gn(τ) = L−nd
+� τ
+1/L2 e(τ−s)∆ξA(u∗ + L−ndvn)(∇ξvn, ∇ξvn)(s)ds.
+Applying the projection Π in case τ = 1 yields
+(16)
+Vn+1 = Vn + Πgn(1),
+where we used
+�
+e(τ−1/L2)∆ξf(ξ)dξ = e−(τ−1/L2)k2|k=0 �f(0) = �f(0) =
+�
+f(ξ)dξ
+and
+�
+Ldvn−1(Lξ, 1)dξ =
+�
+vn−1(ξ, 1)dξ = Vn.
+Rearranging the terms in the variation of constant formula additionally yields
+ρn+1(ξ)
+=
+e(1−1/L2)∆ξLdρn(Lξ)
+(17)
++gn(1) + e(1−1/L2)∆ξLdVnψ(Lξ) − Vn+1ψ(ξ),
+We estimate the terms on the right hand side of (16) and (17).
+i) Using
+e−(1−1/L2)|k|2e−|k/L|2 = e−|k|2
+yields
+∥e(1−1/L2)∆ξLdVnψ(L·) − Vn+1ψ(·)∥Hm
+r =∥(Vn − Vn+1)ψ(·)∥Hm
+r
+≤C|Vn+1 − Vn|.
+
+16
+T. LAMM AND G. SCHNEIDER
+ii) Since ρ has mean value zero and since �ρ ∈ Hr
+m ⊂ C1
+b for r > d/2 + 1 we have
+∥e(1−1/L2)∆ξLdρn(L·)∥Hm
+r
+≤
+C∥e−(1−1/L2)|k|2 �ρn(·/L)∥Hrm
+≤
+(C/L)∥�ρn∥Hrm ≤ (C/L)∥ρn∥Hm
+r .
+iii) For the integral term we obtain
+∥ gn(τ)∥Hm
+r
+≤
+L−nd
+� τ
+1/L2 ∥e(τ−s)∆∥Hm−1
+2
+→Hm
+r
+×∥ A(u∗ + L−ndvn)(∇ξvn, ∇ξvn)(s)∥Hm−1
+2
+ds
+≤
+CL−ndR2
+n
+� τ
+1/L2(τ − s)−1/2ds ≤ CL−ndR2
+n.
+□
+Using the first inequality of Lemma 4.2 to eliminate |An+1 − An| in the second
+inequality of Lemma 4.2 gives
+Corollary 4.3.
+|Vn+1 − Vn|
+≤
+CL−ndR2
+n,
+∥ρn+1∥Hm
+r
+≤
+(C/L)∥ρn∥Hm
+r + CL−ndR2
+n.
+For obtaining a closed system of inequalities in Vn and ρn, we have to estimate
+Rn in terms of Vn and ρn.
+Lemma 4.4. There exists C1 > 0 and C2 > 0, such that for all δ > 0 and L > 1
+with L5/2δ < C1 the following holds. For vn−1|τ=1 ∈ Hm
+r
+with
+∥vn−1|τ=1∥Hm
+r < δ
+we have
+Rn ≤ C2L5/2(|Vn| + rn)
+where rn = ∥ρn∥Hm
+r .
+Proof. We consider the variation of constant formula (15), use the estimate iii) from
+Lemma 4.2, replace ii) from Lemma 4.2 by
+∥e(τ−1/L2)∆ξLdvn−1(L·)∥Hm
+r ≤ C∥Ldvn−1(L·)∥Hm
+r ≤ CL5/2∥vn−1∥Hm
+r ,
+to obtain
+Rn ≤ CL5/2∥vn−1∥Hm
+r + CL−σnR2
+n.
+Since there exist two intersection points R < R of the line Rn �→ Rn and the curve
+Rn �→ CL5/2∥vn−1∥Hm
+r + CL−ndR2
+n for L5/2δ > 0 sufficiently small, we have
+Rn ≤ R ≤ 2CL5/2∥vn−1∥Hm
+r ≤ 4CL5/2(|Vn| + rn).
+Obviously there is local existence and uniqueness of solutions in the space Hm
+r .
+Since the above estimate gives an a priori bound in Hm
+r , the solution can be ex-
+tended to the whole interval [1/L2, 1].
+□
+
+DIFFUSIVE STABILITY
+17
+Corollary 4.3 and Lemma 4.4 yield
+|Vn+1 − Vn|
+≤
+CL−nd(L5/2)2(|Vn| + rn)2,
+rn
+≤
+(C/L)rn + CL−nd(L5/2)2(|Vn| + rn)2,
+as long as
+(18)
+C(|Vn| + rn) ≤ δ
+and
+L5/2δ ≤ C1.
+We choose L0 > 1 so large that (C/L0) ≤ 1/10 and then n0 > 0 so large that
+CL−n0d
+0
+(L5/2
+0
+)2 ≤ 1. Then, for all n > n0 und L > L0 we have
+|Vn+1 − Vn|
+≤
+L−d(n−n0)(Vn + rn)2,
+rn+1
+≤
+rn/10 + L−(n−n0)d(Vn + rn)2.
+Now we choose an L > L0 and a δ > 0 so small, that
+L5/2δ ≤ C1.
+Since the quantities |Vn| and rn grow only for n0 steps we can choose the initial
+conditions so small that
+max
+n=1,...,n0 C(|Vn| + rn) ≤ δ/4.
+Then (Vn)n∈N converges with a geometric rate towards a Vlim ∈ Rd and we have
+limn→∞ rn = 0. The estimates further guarantee that
+sup
+n≥n0
+rn ≤ δ/4
+und
+sup
+n≥n0
+|Vn| ≤ δ/2.
+Therefore, all necessary conditions for the validity of the previous estimates are
+satisfied. Since convergence holds for all L ∈ [L0, L2
+0], Theorem 4.1 follows.
+□
+With a slight modification of this proof similar results can be established for the
+w-equation of the harmonic map heat flow (see (5)) and for the biharmonic map
+heat flow (see (10)). In the following we sketch the ideas for the equation (5) which
+we recall to be
+∂tw − ∆w
+=
+∇(A(u∗ + v)(w, w))
+(19)
+=
+2A(u∗ + v)(w, ∇w) + A′(u∗ + v)(w, w, w),
+with A′(u∗+v)(·, ·, ·) a smooth and bounded trilinear mapping acting on the tangent
+space and with w = ∇v. For the scaling
+(20)
+wn(ξ, τ) = Lndw(Lnξ, L2nτ)
+with L > 1 fixed and n ∈ N the previous analysis applies almost line by line. By
+this choice we have
+vn(ξ, τ) = Ln(d−1)v(Lnξ, L2nτ),
+
+18
+T. LAMM AND G. SCHNEIDER
+and so wn satisfies
+(∂τwn − ∆ξwn) = Ln(d+2)(∂tw − ∆xw)
+=
+2Ln(d+2)A(u∗ + v)(w, ∇xw) + Ln(d+2)A′(u∗ + v)(w, w, w)
+=
+2Ln(d+2)A(u∗ + L−n(d−1)vn)(L−ndwn, L−n(d+1)∇ξwn)
++Ln(d+2)A′(u∗ + L−n(d−1)vn)(L−ndwn, L−ndwn, L−ndwn)
+=
+2L−n(d−1)A(u∗ + L−n(d−1)vn)(wn, ∇ξwn)
++L−2n(d−1)A′(u∗ + L−n(d−1)vn)(wn, wn, wn)
+for τ ∈ [L−2, 1], with
+(21)
+wn(ξ, L−2) = Ldwn−1(Lξ, 1).
+Hence, the nonlinear terms are asymptotically irrelevant if d > 1 and so the rest of
+the previous proof allows us to establish the result
+Theorem 4.5. There exist δ, C > 0, such that for all solutions w of (19) with
+∥w|t=0∥Hm
+r ≤ δ where m > d/2 + 1 and r > d/2 + 1, we have a Wlim ∈ Rd×d such
+that
+∥td/2 w(·
+√
+t, t) − Wlime−|·|2/4∥Hm
+r ≤ C(1+t)−d/2 for all t ≥ 0.
+However, since w = ∇v the choice w ∈ Hs
+r excludes a number of interesting
+solutions. A typical example of an initial condition for v : R2 → R2 considered in
+Theorem 4.5 which can not have been handled by Theorem 4.1 or Theorem 4.5 is
+v1(x1, x2) = v0erf(x1)e−x2
+2.
+We find
+(w1j)j=1,2(x1, x2) = ∇v(x1, x2) = (v0e−x2
+1/4e−x2
+2, −2x2v0erf(x1)e−x2
+2)
+which is not an element of Hs
+r. We remark without any proof that initial conditions
+of the form v = c1erf(x1) + �v with c1 ∈ R and �v ∈ Hs
+r can be included easily into
+the formulation and proof of Theorem 4.1.
+4.2. Biharmonic map heat flow. Next we come to the biharmonic map heat
+flow in case u : Rd → Sm−1 ⊂ Rm. Similar to the linear diffusion equation we have
+for the linearized problem
+∂tv = −∆2v,
+v|t=0 = v0
+that for spatially localized initial conditions the solutions decay asymptotically in
+a self similar way, i.e., for x ∈ Rd the renormalized solution td/4v(x
+√
+t, t) converges
+towards a multiple of a universal limit function, Vlimϕ(x) with Vlim ∈ R and ϕ =
+F−1 �ϕ where �ϕ(k) = e−|k|4.
+The same is true for the deviation v of a trivial
+spatially constant equilibrium u∗ for the bi-harmonic map heat flow (1). Note that
+the deviation v satisfies a semilinear parabolic equation of the form
+∂tv
+=
+−∆2v − (∆(∇v, ∇v) + ∇ · (∆v, ∇v) + (∇∆v, ∇v))e1
+(22)
+−(∆(∇v, ∇v) + ∇ · (∆v, ∇v) + (∇∆v, ∇v))v.
+Then we have:
+
+DIFFUSIVE STABILITY
+19
+Theorem 4.6. There exist δ, C > 0, such that for all solutions v of (22) with
+∥v|t=0∥Hm
+r ≤ δ where m > d/2 + 3 and r > d/2 + 1 we have a Vlim ∈ Rd such that
+∥td/4 v(·
+√
+t, t) − Vlimϕ(·)∥Hm
+r ≤ C(1+t)−d/4
+for all t ≥ 0.
+Proof. As above instead of solving (12) directly we consider an equivalent sequence
+of problems which converges formally towards the linearized equation. For this we
+let
+(23)
+vn(ξ, τ) = Lndv(L2nξ, L4nτ),
+with L > 1 fixed and n ∈ N. Then vn satisfies
+(∂τvn − ∆ξvn) = Ln(d+8)(∂tv − ∆xv)
+=
+Ln(d+8)(−(∆x(∇xv, ∇xv) + ∇x · (∆xv, ∇xv) + (∇x∆xv, ∇xv))e1
+−(∆x(∇xv, ∇xv) + ∇x · (∆xv, ∇xv) + (∇x∆xv, ∇xv))v)
+=
+Ln(d+8)(−(L−4n∆ξ(L−n(d+2)∇ξv, L−n(d+2)∇ξv)
++L−2n∇ξ · (L−n(d+4)∆ξv, L−n(d+2)∇ξv)
++(L−n(d+6)∇ξ∆ξv, L−n(d+2)∇ξv))e1
+−(L−4n∆ξ(L−n(d+2)∇ξv, L−n(d+2)∇ξv)
++L−2n∇ξ · (L−n(d+4)∆ξv, L−n(d+2)∇ξv)
++(L−n(d+6)∇ξ∆ξv, L−n(d+2)∇ξv))L−ndv)
+=
+L−nd(−(∆ξ(∇ξv, ∇ξv) + ∇ξ · (∆ξv, ∇ξv) + (∇ξ∆ξv, ∇ξv)))e1
++L−2nd(−(∆ξ(∇ξv, ∇ξv) + ∇ξ · (∆ξv, ∇ξv) + (∇ξ∆ξv, ∇ξv))v)
+for τ ∈ [L−4, 1], with initial condition
+(24)
+vn(ξ, L−4) = Ldvn−1(L2ξ, 1).
+As for the harmonic map heat flow the influence of the nonlinear terms vanishes
+with a geometric rate for n → ∞. In the limit n → ∞ we obtain the linearized
+problem. The rest of the proof follows the associated parts of the proof of Theorem
+4.1 almost line for line.
+□
+References
+[1] S. Blatt and M. Struwe. Well-posedness of the supercritical Lane-Emden heat flow in Morrey
+spaces. ESAIM Control Optim. Calc. Var., 22(4):1370—1381, 2016.
+[2] J. Bricmont and A. Kupiainen. Renormalization group and the Ginzburg-Landau equation.
+Comm. Math. Phys., 150(1):193–208, 1992.
+[3] J. Bricmont, A. Kupiainen, and G. Lin. Renormalization group and asymptotics of solutions
+of nonlinear parabolic equations. Comm. Pure Appl. Math., 6:893–922, 1994.
+[4] M. Cannone Ondelettes, paraproduits et Navier-Stokes. Diderot Editeur, Paris, 1995.
+[5] M. Cannone A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat.
+Iberoamericana, 13:515—541, 1997.
+[6] H. Fujita. On the blowing up of solutions of the Cauchy problem for ut = ∆u+u1+α. J. Fac.
+Sci. Univ. Tokyo Sect. I, 13:109–124, 1966.
+[7] T. Gallay. Local stability of critical fronts in nonlinear parabolic partial differential equations.
+Nonlinearity, 7(3):741–764, 1994.
+[8] M. Johnson, P. Noble, L. Rodrigues, and K. Zumbrun. Behavior of periodic solutions of
+viscous conservation laws under localized and nonlocalized perturbations. Invent. Math.,
+197(1):115–213, 2014.
+
+20
+T. LAMM AND G. SCHNEIDER
+[9] H. Koch and T. Lamm. Geometric flows with rough initial data. Asian J. Math., 16(2):209–
+235, 2012.
+[10] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math., 157:
+22-–35, 2001.
+[11] J. Li and S. Wang. The heat flow of harmonic maps from noncompact manifolds. Chinese
+Ann. Math. Ser. B, 21(1):121–130, 2000.
+[12] C. Miao, B. Yuan, and B. Zhang. Strong solutions to the nonlinear heat equation in homo-
+geneous Besov spaces. Nonlinear Anal., 67:1329–1343, 2007.
+[13] F. Planchon Asymptotic behavior of global solutions to the Navier-Stokes equations in R3.
+Rev. Mat. Iberoamericana, 14:71—93, 1998.
+[14] B. Sandstede, A. Scheel, G. Schneider, and H. Uecker. Diffusive mixing of periodic wave trains
+in reaction-diffusion systems. J. Differ. Equations, 252(5):3541–3574, 2012.
+[15] G. Schneider. Nonlinear stability of Taylor-vortices in infinite cylinders. Arch. Rat. Mech.
+Anal., 144(2):121–200, 1998.
+[16] G. Schneider and H. Uecker. Nonlinear PDEs: A dynamical systems approach, volume 182
+of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2017.
+[17] A. Soyeur. A global existence result for the heat flow of harmonic maps. Comm. Partial
+Differential Equations, 15(2):237–244, 1990.
+[18] M. Struwe. On the evolution of harmonic maps in higher dimensions. J. Differential Geom.,
+28(3):485–502, 1988.
+[19] H. Uecker. Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg
+equation. Comm. Partial Differential Equations, 24:2109–2146, 1999.
+[20] C. Wang. Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with
+rough initial data. Arch. Ration. Mech. Anal., 200(1):1–19, 2011.
+[21] C. Wang. Well-posedness for the heat flow of biharmonic maps with rough initial data. J.
+Geom. Anal., 22(1):223–243, 2012.
+[22] F.B. Weissler. Existence and nonexistence of global solutions for a semilinear heat equation.
+Israel Journal of Mathematics, 38:29–40, 1981.
+(T. Lamm) Institute for Analysis, Karlsruhe Institute of Technology, Englerstr.
+2, 76131 Karlsruhe, Germany
+Email address: tobias.lamm@kit.edu
+(G. Schneider) Institut f¨ur Analysis, Dynamik und Modellierung, Universit¨at Stuttgart,
+Pfaffenwaldring 57, 70569 Stuttgart, Germany
+Email address: guido.schneider@mathematik.uni-stuttgart.de
+
diff --git a/gdA0T4oBgHgl3EQfH_8L/content/tmp_files/load_file.txt b/gdA0T4oBgHgl3EQfH_8L/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8195d09a5f716b9f5d67575dc86e8e8d48092812
--- /dev/null
+++ b/gdA0T4oBgHgl3EQfH_8L/content/tmp_files/load_file.txt
@@ -0,0 +1,588 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf,len=587
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='02067v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='DG]  5 Jan 2023  DIFFUSIVE STABILITY AND SELF-SIMILAR DECAY  FOR THE HARMONIC MAP HEAT FLOW  TOBIAS LAMM AND GUIDO SCHNEIDER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In this paper we study the harmonic map heat flow on the eu-  clidean space Rd and we show an unconditional uniqueness result for maps with  small initial data in the homogeneous Besov space ˙B  d  p  p,∞(Rd) where d < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As a consequence we obtain decay rates for solutions of the harmonic map flow  of the form ∥∇u(t)∥L∞(Rd) ≤ Ct− 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Additionally, under the assumption of a stronger spatial localization of the  initial conditions, we show that the temporal decay happens in a self-similar  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We also explain that similar results hold for the biharmonic map heat  flow and the semilinear heat equation with a power-type nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Introduction  The diffusive stability method is a tool which allows us to prove stability results  for parabolic partial differential equations in the case of a linearization possessing  essential spectrum up to the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' It is based on algebraic decay rates  which hold for the heat semigroup (et∆)t≥0 on Rd, defined by  (et∆u0)(x) =  1  (4πt)d/2  �  Rd e− |x−y|2  4t  u0(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We have the decay  ∥et∆∥Lp→L∞ ≤ Ct− d  2p ,  for p ∈ [1, ∞), due to the diffusion which occurs for spatially localized initial con-  ditions and the decay  ∥et∆∇∥L∞→L∞ ≤ Ct− 1  2  due to smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' These decays can be used for instance to prove stability results  in (Lp ∩ C1  b )(Rd) of the trivial solution u = 0 in semilinear heat equations  ∂tu = ∆u + f(u, ∇u),  with x ∈ Rd, t ≥ 0, u(x, t) ∈ R, and a smooth nonlinear function f containing terms  of the form um(∂1u)m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' (∂du)md and satisfying f(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Since ∂tu ∼ t− d  2p −1  and ∆u ∼ t− d  2p −1 for t → ∞ we have that the nonlinear terms behave like  um(∂1u)m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' (∂du)md ∼ t−  (m+m1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='+md)d  2p  −  m1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='+md  2  Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We thank Peer Kunstmann for many helpful discussions on Besov spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Funded by the  Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 258734477  SFB 1173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  1  2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  for t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Hence, they are asymptotically irrelevant with respect to the linear  diffusion if  d  2p + 1 < (m + m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' + md)d  2p  + m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' + md  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As an example for d = 1 and p = 1 all such terms are irrelevant except for u2, u3,  and u∂xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' With this idea the stability of u = 0 in (Lp ∩ C1  b )(Rd) follows for  ∂tu = ∆u + um(∂x1u)m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' (∂xdu)md,  if the above condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' An abstract existence result in this direction can  be found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  After the introduction of this method in the papers of Fujita and Weisler [6, 22],  the last decades saw a successful application of this tool to stability questions arising  for pattern forming systems, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' [3, 15, 14, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' A summary and overview of  various aspects of this diffusive stability method can also be found in the monograph  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  It is the goal of this paper to use this tool for gaining new stability and unique-  ness results for dissipative semilinear geometric flow problems on Rd, in particular  for the harmonic map heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  In the following we let N be a smooth Riemannian manifold which we assume  to be isometrically embedded into some euclidean space Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  For δ > 0 suf-  ficiently small we have a well defined nearest point projection π : Nδ → N,  where Nδ := {x ∈ Rm : dist(x, N) < δ} and we define the second fundamen-  tal form A(y) : TyN × TyN → T ⊥  y N for every y ∈ N and v1, v2 ∈ TyN by  A(y)(v1, v2) = −D2π(y)(v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' A map u : Rd × [0, T ) → N is then a solution of  the harmonic map heat flow with initial data u(·, 0) = u0 : Rd → N if it solves the  parabolic partial differential equation  ∂tu − ∆u = A(u)(∇u, ∇u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  (1)  For instance, in case of N = Sm−1 = {x ∈ Rm : ∥x∥Rm = 1}, we compute  A(u)(∇u, ∇u)  =  −(∆u, u)u = −(∂2  kuα)uαu  =  −∂k((∂kuα)uα)u + (∂kuα)(∂kuα)u = u|∇u|2,  where here and in the following we use Einstein’s summation convention and the  fact that (∂kuα)uα = 0 which follows from the fact that uαuα = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Rather general existence results for the harmonic map heat flow on euclidean  space have been obtained by Koch and the first author [9] for initial maps with  a small oscillation and by Wang [20] for initial maps with a small BMO-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  These solutions were constructed by applying a fixed point argument on suitably  constructed Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' As a byproduct of this method, the authors obtained a  conditional uniqueness result for the harmonic map heat flow under the smallness  assumptions on the initial data mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  One of our main goals of this paper is to use the diffusive stability method in  order to improve this conditional uniqueness result to an unconditional uniqueness  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The price we have to pay is that our initial data have to be small in a strict  subspace of the above mentioned space of functions of bounded mean oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  More precisely, we require our initial data to be small in a homogeneous Besov  space, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='7 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  DIFFUSIVE STABILITY  3  The deviation v of a trivial spatially constant equilibrium u∗ satisfies a semilinear  diffusion equation of the form  (2)  ∂tv − ∆v = A(u∗ + v)(∇v, ∇v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As above we have A(u∗ +v)(∇v, ∇v) ∼ t− d  p −1 for t → ∞, and so the nonlinearity is  irrelevant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' to linear diffusion if p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For p = ∞ the linear and nonlinear  terms are of the same order and a stability result on exponentially long time scales  can be established, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  For slightly stronger localized initial conditions the associated solutions of the  linear heat equation decay in a self-similar way towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The same is true for  semilinear heat equations with nonlinearities which are irrelevant in L1 ∩ C1  b (Rd)  with the above counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Using the discrete renormalization approach, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' [2, 3],  gives that the renormalized solution td/2v(x  √  t, t) of (2) converges towards a mul-  tiple of a Gaussian e−|x|2/4 for t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In the discrete renormalization approach  instead of solving the PDE (2) directly we consider an equivalent sequence of prob-  lems which converges formally towards the linear diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Let  vn(x, τ) = Ldnv(Lnx, L2nτ),  with L > 1 fixed and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Then vn satisfies  ∂τvn − ∆vn  =  Ln(d+2−2−2d)D2π(u∗ + L−dnv)(∇vn, ∇vn)  (3)  =  L−ndD2π(u∗ + L−dnv)(∇vn, ∇vn)  for τ ∈ [L−2, 1], with  (4)  vn(x, L−2) = Ldvn−1(Lx, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  The influence of the nonlinear terms vanishes for n → ∞ with a geometric rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In  the limit we obtain a linear diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' With a simple fixed point argument  the convergence of vn|τ=1 towards the limit of the renormalized linear diffusion  problem, namely a multiple of the Gaussian e−|x|2/4 can be established, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  A different approach for the harmonic map heat flow is the consideration of  w = ∇v which satisfies  (5)  ∂tw − ∆w = ∇(A(u∗ + v)(w, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We show that the above mentioned discrete renormalization approach can also be  applied to this system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Finally, we show that our technique can be extended to various other semilinear  heat equations on euclidean space, such as equations with a power-type nonlinearity  or higher order equations, such as the biharmonic map heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  A similar handling of dissipative quasilinear geometric flow problems, such as the  Ricci-DeTurck flow, the mean curvature flow, the Yamabe flow, and the Willmore  flow of graphs, will be the topic of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In the following many possibly different constants are denoted with  the same symbol C if they can be chosen independent of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  4  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Stability and uniqueness results for the harmonic map flow  Our first main result about the harmonic map heat flow is given by an asymp-  totic stability and uniqueness result assuming smallness conditions on a certain  homogeneous Besov norm of the inital map u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We note that due to the unboundedness of the domain the linearization around  u = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' for (1) possesses essential spectrum up to the imaginary axis and so the  principle of linearized stability does not apply directly in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Therefore,  we use the diffusive stability method for establishing the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We start  by extending a stability result for the harmonic map flow which was originally due  to Soyeur [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Using the notation ˙W 1,p(Rd, N) for the homogeneous Sobolev space,  his result is as follows  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists a constant ε > 0 depending only on d and N such that  if u0 ∈ ˙W 1,p(Rd, N) with d < p < ∞ and ∥∇u0∥Ld < ε, then we have the estimate  t  1  2 − d  2p ∥∇u(t)∥Lp ≤ C∥∇u0∥Ld,  (6)  where C = C(d, m, p), for every solution u ∈ C0([0, ∞), ˙W 1,p(Rd, N)) of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In the same paper Soyeur also showed the existence of a solution  u ∈ C0([0, ∞), ˙W 1,p(Rd, N)) for small initial data in the above sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Additionally,  the smallness assumption on the Ld-norm of ∇u0 implies that the BMO-norm of  u0 is small and hence, by a paper of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Wang [20], this implies the existence of a  global smooth solution of the harmonic map flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' This result can be extended to maps from complete Riemannian  manifolds for which one has good heat kernel estimates as it was done in Li and  Wang [11] for (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Examples of such manifolds include manifolds with positive  Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  The original result of Soyeur already extended a previous result of Struwe [18],  who was the first one to show a convergence result for the harmonic map heat flow  on Rd to a constant map with additional decay properties for the first derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  More precisely, he showed in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1 of the paper mentioned above, that for  smooth initial maps u0 so that ∥∇u0∥L∞ is bounded and ∥∇u0∥L2 is sufficiently  small, the unique smooth solution of the harmonic map heat flow satisfies for t large  enough ∥∇u(·, t)∥L∞ ≤ Ct− 1  2 and hence converges to a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As a first result we improve Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1 for initial data satisfying a slightly  less restrictive smallness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' More precisely, for a map v : Rd → Rm we  denote by G(t)v := Gt ⋆ v its heat extension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' G(t)v is a solution of the heat  equation with initial data v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Here G(x, t) =  1  (4πt)d/2 e  −|x|2  4t , (x, t) ∈ Rd × (0, ∞), is  the standard heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For d < p ≤ ∞ we define the homogeneous Besov space  ˙B  d  pp,∞(Rd) as the set of functions for which the norm  ∥v∥  ˙B  d  p  p,∞  := sup  t>0  t  1  2 − d  2p ∥∇(G(t)v)∥Lp  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' It follows that  ˙W 1,d(Rd) ⊂ ˙B  d  p  p,∞(Rd) ⊂ BMO(Rd) ⊂ ˙B0  ∞,∞(Rd)  for every d < p < ∞, where each of these inclusions is strict (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Now we are in a position to state our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  DIFFUSIVE STABILITY  5  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists a constant ε > 0 depending only on d and N such that  if u0 ∈ ˙B  d  pp,∞(Rd, N) with d < p < ∞, 2 ≤ p and ∥u0∥  ˙B  d  p  p,∞  < ε, then we have the  estimate  t  1  2 − d  2p ∥∇u(t)∥Lp ≤ C∥u0∥  ˙B  d  p  p,∞  ,  where C = C(d, m, p), for every solution u : (0, ∞) × Rd → N of (1) with  t  1  2 − d  2p ∇u(t, x) ∈ C0([0, ∞), Lp(Rd))  and  lim  tց0 t  1  2 − d  2p ∥∇u(t)∥Lp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Note that the existence of such solutions follows for example from  the work of Soyeur under the assumption that u0 ∈ ˙W 1,p(Rd, N) with a sufficiently  small norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  In the proof of this result we need a standard estimate on convolutions of the  heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Note that it relies on Young’s convolution inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Let G(x, t) =  1  (4πt)d/2 e  −|x|2  4t , (x, t) ∈ Rd × (0, ∞), be the heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Using the notation G(t)f = G(t) ⋆ f we have the estimate  ∥∇σG(t)∥L(Lr,Lq) ≤ Ct− d  2 ( 1  r − 1  q )− σ  2 ,  (7)  where 1 ≤ r ≤ q ≤ ∞, σ ∈ N0 and C < ∞ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Additionally, we have  that for some constant C < ∞  � t  0  (t − s)−αs−β ds = Ct1−α−β  (8)  if both 0 < α, β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Now we are in a position to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4: We start by noting that  ∇(u(x, t) − G(t) ⋆ u0(x)) =  � t  0  �  Rd ∇G(x − y, t − s)(A(u)(∇u, ∇u)(y, s)) dy ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Using the estimate (7) with σ = 1, r = p  2 and q = p we get for every t > 0  ∥∇(u(t) − G(t)u0)∥Lp ≤C  � t  0  ∥|∇u(s)|2∥L  p  2 (t − s)−( 1  2 + d  2p ) ds  ≤C  � t  0  ∥∇u(s)∥2  Lp(t − s)−( 1  2 + d  2p ) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Setting  m(T ) = sup  0≤t≤T  t  1  2 − d  2p ∥∇u(t)∥Lp,  the last inequality implies for all t > 0  t  1  2 − d  2p ∥∇u(t)∥Lp ≤C∥u0∥  ˙B  d  p  p,∞  + Ct  1  2 − d  2p m(T )2  � t  0  (t − s)−( 1  2 + d  2p )s−1+ d  p ds  ≤C∥u0∥  ˙B  d  p  p,∞  + Cm(T )2,  6  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  where we used the identity (8) with 0 < α = 1  2 + d  2p < 1 and 0 < β = 1 − d  p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Note that it is exactly here that we have to use the assumption d < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Taking the supremum over all 0 ≤ t ≤ T we thus obtain for every 0 < T < ∞  Cm(T )2 − m(T ) + C∥u0∥  ˙B  d  p  p,∞  ≥ 0  and for ε > 0 small enough this finally implies for all T < ∞  m(T ) ≤ C∥u0∥  ˙B  d  p  p,∞  ≤ Cε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Here we used that m(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' By definition this estimate implies that  t  1  2 − d  2p ∥∇u(t)∥Lp ≤ C∥u0∥  ˙B  d  p  p,∞  for every 0 < t < ∞, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  In the next theorem we improve a uniqueness result for solutions of the harmonic  map heat flow with initial data with small BMO-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For this we define the  Banach space X as the space of functions such that the norm  ∥u∥X := sup  0<t<∞  �  ∥u(t)∥L∞(Rd) + t  1  2 ∥∇u(t)∥L∞(Rd)  �  + sup  x∈Rd  sup  0<R<∞  R− d  2 ∥∇u∥L2(BR(x)×(0,R2))  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' It was shown by the first author and Koch [9] that there exist constants  ε > 0, C > 0 such that for every u0 : Rd → N satisfying ∥u0 − P∥L∞ < ε, where  P ∈ N is some arbitrary point, there exists a global solution u ∈ P + X of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Moreover, the solution is unique in the ball  BX(P, Cε) = {u : ∥u − P∥X ≤ Cε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  This result was later on extended to initial data u0 ∈ BMO(Rd) with small  BMO-norm by Wang [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' He obtained the uniqueness of the solution in the set  BX(G(t)u0, Cε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As our second main result for the harmonic map flow we improve this conditional  uniqueness result to an unconditional uniqueness result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' we no longer have to  restrict to the class of solutions in X with small X-norm) but the price we have to  pay is to go from BMO-initial data to homogeneous Besov initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Let N be a smooth and closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists ε > 0, C > 0  such that for every u0 ∈ ˙B  d  p  p,∞(Rd, N) with d < p < ∞, ∥u0∥  ˙B  d  p  p,∞  < ε and every  solution u : (0, ∞) × Rd → N of (1) with  t  1  2 − d  2p ∇u(t, x) ∈ C0([0, ∞), Lp(Rd))  and  lim  tց0 t  1  2 − d  2p ∥∇u(t)∥Lp = 0,  we have the estimate  ∥u − G(t)u0∥X ≤ Cε,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' there exists only one solution under these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  DIFFUSIVE STABILITY  7  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Since  sup  t>0  t  1  2 ∥∇(G(t)u0)∥L∞(Rd) ≃ ∥u0∥ ˙B0  ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  respectively  sup  x∈Rd  sup  0<R<∞  R− d  2 ∥∇(G(t)u0)∥L2(BR(x)×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='R2)) ≃ ∥u0∥BMO  and since we have the inclusions  ˙B  d  p  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞ ⊂ BMO ⊂ ˙B0  ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞  for every d < p < ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' it follows from the above estimates that for every solution u ∈  C0([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ˙W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='p(Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' N)) of (1) as above with initial data u0 satisfying ∥u0∥  ˙B  d  p  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞  < ε  we have  sup  t>0  t  1  2 ∥∇u∥L∞(Rd) + sup  x∈Rd  sup  0<R<∞  R− d  2 ∥∇u∥L2(BR(x)×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='R2)) ≤ C∥u0∥  ˙B  d  p  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='7: We fix p > d and we note that we get from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4 the  estimate  sup  t>0  t  1  2 − d  2p ∥∇u(t)∥Lp ≤ C∥u0∥  ˙B  d  p  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Next we use again the estimate (7) with σ = 1, r = p  2 and q = ∞ to conclude for  every t > 0  ∥∇(u(t) − G(t)u0)∥L∞ ≤C  � t  0  ∥|∇u(s)|2∥L  p  2 (t − s)−( 1  2 + d  p ) ds  ≤C  � t  0  ∥∇u(s)∥2  Lp(t − s)−( 1  2 + d  p ) ds  ≤C∥u0∥2  ˙B  d  p  p,∞  � t  0  s  d  p −1(t − s)−( 1  2 + d  p ) ds  ≤Ct− 1  2 ∥u0∥2  ˙B  d  p  p,∞  ,  where we used (8) in the last estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Similarly, we obtain  ∥u(t) − G(t)u0∥L∞ ≤C  � t  0  ∥|∇u(s)|2∥L  p  2 (t − s)− d  p ds  ≤C  � t  0  ∥∇u(s)∥2  Lp(t − s)− d  p ds  ≤C∥u0∥2  ˙B  d  p  p,∞  � t  0  s  d  p −1(t − s)− d  p ds  ≤C∥u0∥2  ˙B  d  p  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  It remains to establish the bound  sup  x∈Rd  sup  0<R<∞  R− d  2 ∥∇(u − G(·)u0)∥L2(BR(x)×(0,R2)) ≤ C∥u0∥  ˙B  d  p  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  8  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  Using H¨older’s inequality, we estimate for every x ∈ Rd and some p > d  �  BR(x)  |∇(u(t) − G(t)u0)|2 ≤C(  �  BR(x)  |∇(u(t) − G(t)u0)|p)  2  p Rd(1− 2  p )  ≤Ct  d  p −1Rd(1− 2  p )∥u0∥2  ˙B  d  p  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  From this it is easy to see that  sup  x∈Rd  sup  0<R<∞  R− d  2 ∥∇(u − G(·)u0)∥L2(BR(x)×(0,R2)) ≤ C∥u0∥  ˙B  d  p  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  Finally, we show a new stability result on exponentially long time scales for the  harmonic map flow starting near a constant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Note that in this result we do  not assume that the initial values decay in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For every C > 0 there exist b > 0 and ε0 > 0 such that for every  ε ∈ (0, ε0) and every solution u : Rd × [0, ∞) → N of (1) with ∥u0 − P∥W 1,∞ ≤ ε,  where P ∈ N is an arbitrary point, we have  sup  t∈[0,exp( b  ε)−1]  ∥u(t) − P∥C0 +  sup  t∈[0,exp( b  ε)−1]  (1+t)  1  2 ∥∇u(t)∥C0 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We define the functions  a(t)  =  sup  0≤s≤t  ∥u(s) − P∥L∞  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  b(t)  =  sup  0≤s≤t  ∥(1+s)  1  2 ∇u(s)∥L∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  and we estimate (using the fact that ∥G(t)(u0 − P)∥L∞ ≤ ∥u0 − P∥L∞)  ∥u(t) − P∥L∞  ≤  C∥u0 − P∥L∞ + C  ����  � t  0  G(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' t − s) ⋆ A(u)(∇u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇u)(s)ds  ����  L∞  ≤  Ca(0) + C  � t  0  ∥|∇u(s)|2∥L∞ds  ≤  Ca(0) + Cb2(t)  � t  0  (1+s)−1 ds  ≤  Ca(0) + Cb2(t) log(1 + t)  and  (1 + t)  1  2 ∥∇u(t)∥L∞ ≤C(a(0) + b(0)) + Cb2(t)(1 + t)  1  2  � t  0  (t − s)− 1  2 (1 + s)−1 ds  ≤C(a(0) + b(0)) + Cb2(t)(1 + log(1 + t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  where we used that  ∥∇(G(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' t) ⋆ (u0 − P))∥L∞ ≤  �  C∥∇u0∥L∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  for t ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Ct− 1  2 ∥u0 − P∥L∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  for t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Next we define the rescaled functions ˜a(t) = ε−1a(t) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ˜b(t) = ε−1b(t) and we  conclude  ˜a(t) + ˜b(t) ≤ C1(˜a(0) + ˜b(0)) + C2ε(1 + log(1 + t))˜b2(t)  DIFFUSIVE STABILITY  9  for some constants C1, C2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Since ˜a(0) + ˜b(0) ≤ 1 we conclude  ˜a(t) + ˜b(t) ≤ 2C1  as long as  1 + t ≤ exp  �  1  4C1C2ε − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Hence  a(t) + b(t) ≤ 2C1ε  for all 0 ≤ t ≤ exp  �  1  4C1C2ε − 1  �  − 1 where we note that for ε > 0 sufficiently  small, the last number satisfies the required properties in the statement of the  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Related equations  In this section we want to show that the results obtained for the harmonic map  heat flow can be extended to two other parabolic equations to either simplify proofs  of existing results or yield new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We note that in the case of the Navier-Stokes  equation similar results have been obtained earlier by Cannone [5], Planchon [13]  and Koch-Tataru [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Equations with power-type nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Here we study the very clas-  sical semilinear equation  ∂tu − ∆u = f(u)  (9)  where u : Rd × [0, ∞) → R with u(·, 0) = u0 and f : R → R is a smooth function  satisfying |f(s)| ≤ |s|q for some q > 1 and all s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' It was already shown in [6, 22]  that for non-negative initial data u0 so that the norm ∥u0∥  L  d(q−1)  2  is sufficiently  small (here we also assume d(q−1)  2  > 1), there exists a non-negative global solution  of (9) satisfying  sup  t>0  t  1  q−1 − d  2p ∥u(t)∥Lp ≤ C∥u0∥  L  d(q−1)  2  for every d(q−1)  2  < p < dq(q−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' This result should be directly compared (and was in  fact a motivation) to the one of Soyeur [17] (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1) for the harmonic map  heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In fact, with the same technique that we used in the previous section,  we are able to extend these results and we obtain  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists a constant ε > 0 depending only on d and q such that  if u0 ∈ B  d  p −  2  q−1  p,∞  (Rd) with 1 < q < ∞, 1 < d(q−1)  2  and if ∥u0∥  B  d  p −  2  q−1  p,∞  < ε, then we  have the estimate  sup  t>0  t  1  q−1 − d  2p ∥u(t)∥Lp ≤ C∥u0∥  B  d  p −  2  q−1  p,∞  ,  where C = C(d, p), for every d(q−1)  2  < p < dq(q−1)  2  and every solution u : (0, ∞) ×  Rd → R of (9) with  t  1  q−1 − d  2p u(t, x) ∈ C0((0, ∞), Lp(Rd))  and  lim  tց0 t  1  q−1 − d  2p ∥u(t)∥Lp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  10  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  If we know additionally that q > max{2, 1 + 2  d} then we also get the estimate  sup  t>0  t  1  q−1 ∥u(t)∥L∞ ≤ C∥u0∥  B  d  p −  2  q−1  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Arguing as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4, we obtain  ∥u(t) − G(t)u0∥Lp ≤C  � t  0  ∥|u(s)|q∥  L  p  q (t − s)− d(q−1)  2p  ds  ≤C  � t  0  ∥u(s)∥q  Lp(t − s)− d(q−1)  2p  ds  ≤Cm(T )q  � t  0  s  dq  2p −  q  q−1 (t − s)− d(q−1)  2p  ds  ≤Cm(T )q,  since 0 > dq  2p −  q  q−1 > −1 and 0 > − d(q−1)  2p  > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Note that here we let  m(T ) = sup  0≤t≤T  t  1  q−1 − d  2p ∥u(t)∥Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As above, this estimate implies the desired claim since  sup  t>0  t  1  q−1 − d  2p ∥G(t)u0∥Lp ∼= ∥u0∥  B  d  p −  2  q−1  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Next, if q > 2 there exists a p satisfying dq  2 < p < dq(q−1)  2  and for one such value  of p we note that for every 0 < t < ∞  ∥u(t) − G(t)u0∥L∞ ≤C  � t  0  ∥|u(s)|q∥  L  p  q (t − s)− dq  2p ds  ≤C  � t  0  ∥u(s)∥q  Lp(t − s)− dq  2p ds  ≤Cm(T )q  � t  0  s  dq  2p −  q  q−1 (t − s)− dq  2p ds  ≤Ct−  1  q−1 m(T )q,  and hence we also get the estimate  sup  t>0  t  1  q−1 ∥u(t)∥L∞ ≤ C∥u0∥  B  d  p −  2  q−1  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  Note that the Lp-estimate in the previous theorem has been obtained earlier by  Miao, Yuan and Zhang [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Related estimates have also been obtained by Blatt  and Struwe [1] for small initial data in Morrey spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Biharmonic map flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' As another example we consider solutions u : Rd ×  (0, T ) → Sm−1 ⊂ Rm of the (extrinsic) biharmonic map heat flow governed by  (∂t + ∆2)u = u(|∆u|2 − ∆|∇u|2 − 2div⟨∆u, ∇u⟩)  = u|∆u|2 + ∇u · (∇|∇u|2 + 2⟨∆u, ∇u⟩) − div(u∇|∇u|2 + 2u⟨∆u, ∇u⟩)  (10)  =: f1[u] + divf2[u],  DIFFUSIVE STABILITY  11  where we can estimate  |f1[u]| ≤ C(|∇2u|2 + |∇u|4)  and  |f2[u]| ≤ C|∇u||∇2u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  For the sake of simplicity we restrict ourselves to sphere-valued maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The ar-  guments below can easily be extended to general closed manifolds N and maps  u : Rd × (0, T ) → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Additionally, there exists another version of the biharmonic  map flow, the so called intrinsic biharmonic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' All of the results shown below  carry over directly to this fourth order parabolic PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Again we are interested in an unconditional uniqueness resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' stability result  for small initial data in a suitable Besov space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' As before, the principle of linear  stability cannot be applied directly due to the fact that the linearization again  possesses an essential spectrum up to the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  But as before some diffusive behavior can be used to control the nonlinear terms  which are irrelevant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  this diffusive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  In the following we let b :  Rd × (0, ∞) → R denote the biharmonic heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' It follows from the results of  [9] that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='6 can be extended to this case, namely we have  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Let b : Rd × (0, ∞) → R be the biharmonic heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Using the  notation b(t)f = bt ⋆ f we have the estimate  ∥∇σb(t)∥L(Lr,Lq) ≤ Ct− d  4 ( 1  r − 1  q )− σ  4 ,  (11)  where 1 ≤ r ≤ q ≤ ∞, σ ∈ N0 and C < ∞ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Before we now state our main results for the biharmonic map flow, we note that  we have another equivalent characterization of the homogeneous Besov space via  the biharmonic heat kernel given by  ∥u0∥  ˙B  d  p  p,∞  := sup  t>0  �  t  1  4 − d  4p ∥∇(b(t)u0)∥Lp + t  1  2 − d  2p ∥∇2(b(t)u0)∥L  p  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Here we note that the first term on the right hand-side indeed defines an equivalent  norm on the homogeneous Besov space ˙B  d  pp,∞, whereas the second term defines an  equivalent norm on the homogeneous space ˙B  2d  p  p  2 ,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Since the second homogeneous  Besov space is embedded in the first one we choose our norm as a definition of the  larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists a constant ε > 0 depending only on d and n such that  if u0 ∈  ˙B  d  pp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞(Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Sm−1) with d < p < ∞ and ∥u0∥  ˙B  d  p  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞  < ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' then we have the  estimate  t  1  4 − d  4p ∥∇u(t)∥Lp + t  1  2 − d  2p ∥∇2u(t)∥L  p  2 ≤ C∥u0∥  ˙B  d  p  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='∞  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  where C = C(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' for every solution u : (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∞) × Rd → Sm−1 of (1) with  t  1  4 − d  4p ∇u ∈ C0([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Lp(Rd)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' t  1  2 − d  2p ∇2u ∈ C0([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L  p  2 (Rd))  and  lim  tց0(t  1  4 − d  4p ∥∇u(t)∥Lp + t  1  2 − d  2p ∥∇2u(t)∥L  p  2 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  12  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We follow closely the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4 by observing that  ∇i(u(x, t) − b(t) ⋆ u0(x)) =  � t  0  �  Rd ∇ib(x − y, t − s)f1[u] dy ds  −  � t  0  �  Rd ∇i+1b(x − y, t − s)f2[u](s) dy ds =: Ii + IIi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Using the estimate (11) with σ = 1, r = p  4 and q = p we get for every t > 0  ∥I1∥Lp ≤C  � t  0  �  ∥|∇u(s)|4∥L  p  4 + ∥|∇2u(s)|2∥L  p  4  �  (t − s)−( 1  4 + 3d  4p ) ds  ≤C  � t  0  (∥∇u(s)∥4  Lp + ∥∇2u(s)∥2  L  p  2 )(t − s)−( 1  4 + 3d  4p ) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Setting  m(T ) = sup  0≤t≤T  �  t  1  4 − d  4p ∥∇u(t)∥Lp + t  1  2 − d  2p ∥∇2u(t)∥L  p  2  �  ,  the last inequality implies for all t > 0  t  1  4 − d  4p ∥I∥Lp ≤Ct  1  4 − d  4p m(T )2(1 + m(T )2)  � t  0  (t − s)−( 1  4 + 3d  4p )s−1+ d  p ds  ≤Cm(T )2(1 + m(T )2),  where we used again the identity (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Similarly, using (11) with σ = 2, r = p  3 and q = p we get for every t > 0  ∥II1∥Lp ≤C  � t  0  ∥|∇u(s)||∇2u(s)|∥L  p  3 (t − s)−( 1  2 + d  2p ) ds  ≤C  � t  0  (∥∇u(s)∥Lp∥∇2u(s)∥L  p  2 )(t − s)−( 1  2 + d  2p ) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Thus, we obtain  t  1  4 − d  4p ∥II∥Lp ≤Ct  1  4 − d  4p m(T )2  � t  0  (t − s)−( 1  2 + d  2p )s− 3  4 + 3d  4p ds  ≤Cm(T )2,  which implies  t  1  4 − d  4p ∥∇u(t)∥Lp ≤ C∥u0∥  ˙B  d  p  p,∞  + Cm(T )2(1 + m(T )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  For i = 2 we argue similarly, only this time we choose σ = 2, r = p  4 and q = p  2 for  I2 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' σ = 3, r = p  3 and q = p  2 for II2 to obtain  t  1  2 − d  2p ∥∇2u(t)∥L  p  2 ≤ C∥u0∥  ˙B  d  p  p,∞  + Cm(T )2(1 + m(T )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Taking the supremum over all 0 ≤ t ≤ T we thus obtain for every 0 < T < ∞  Cm(T )2(1 + m(T )2) − m(T ) + C∥u0∥  ˙B  d  p  p,∞  ≥ 0  and for ε > 0 small enough this finally implies for all T < ∞ that  m(T ) ≤ C∥u0∥  ˙B  d  p  p,∞  ≤ Cε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  DIFFUSIVE STABILITY  13  Here we used that m(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' By definition this estimate implies that  t  1  4 − d  4p ∥∇u(t)∥Lp + t  1  2 − d  2p ∥∇2u(t)∥L  p  2 ≤ C∥u0∥  ˙B  d  p  p,∞  for every 0 < t < ∞, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  As in the case of the harmonic map heat flow this result can now be used to  improve the uniqueness result for solutions of the biharmonic map heat flow with  small initial data in BMO due to Wang [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For this we have to define again a  suitable function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' This time it is given by  ∥u∥Xb := sup  0<t<∞  �  ∥u(t)∥L∞(Rd) +  2  �  i=1  t  i  4 ∥∇iu(t)∥L∞(Rd)  �  + sup  x∈Rd  sup  0<R<∞  2  �  i=1  R− di  4 ∥∇iu∥L  4  i (BR(x)×(0,R4))  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Similarly to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='7 we then obtain  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists ε > 0, C > 0 such that for every u0 ∈ ˙B  d  p  p,∞(Rd, Sm−1)  with d < p < ∞, ∥u0∥  ˙B  d  p  p,∞  < ε and every solution u : (0, ∞) × Rd → Sm−1 of (10)  with  t  1  4 − d  4p ∇u(t, x) ∈ C0([0, ∞), Lp(Rd)), t  1  2 − d  2p ∇2u(t, x) ∈ C0([0, ∞), L  p  2 (Rd))  and  lim  tց0  �  t  1  4 − d  4p ∥∇u(t)∥Lp + t  1  2 − d  2p ∥∇2u(t)∥L  p  2  �  = 0,  we have the estimate  ∥u − b(t)u0∥Xb ≤ Cε,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=', there exists only one solution of (10) under these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Additionally, one obtains the bounds  sup  t>0  2  �  i=1  t  i  4 ∥∇iu(t)∥L∞(Rd) + sup  x∈Rd  sup  0<R<∞  2  �  i=1  R− di  4 ∥∇iu∥L  4  i (BR(x)×(0,R4))  ≤ C∥u0∥  ˙B  d  p  p,∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Self-similar decay  In this section we collect a number of results which show an asymptotic self-  similar decay of the deviations v for the harmonic and bi-harmonic map heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We start with the v-equation (2) for the harmonic map heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The transfer of  the presented method to the biharmonic map heat flow can be found below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We  close this section with some remarks about similar statements for the w-equation  (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  14  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Harmonic map heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' It is well known that for spatially localized initial  conditions the solutions of the linear diffusion equation  ∂tv = ∆v,  v|t=0 = v0  decay asymptotically in a self similar way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=', for x ∈ Rd the renormalized solution  td/2v(x  √  t, t) converges towards a multiple of a Gaussian, Vlime−|x|2/4 with Vlim ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  The same is true for the deviation v of a trivial spatially constant equilibrium u∗ for  the harmonic map heat flow (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' As we have seen in the introduction, the deviation  v satisfies a semilinear diffusion equation of the form  (12)  ∂tv − ∆v = A(u∗ + v)(∇v, ∇v),  where we recall that A(u∗ + v)(·, ·) is a smooth bounded bilinear mapping acting  on the d-dimensional tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Our first result reads as follows  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exist δ, C > 0, such that for all solutions v of (12) with  ∥v|t=0∥Hm  r ≤ δ where m > d/2 + 1 and r > d/2 + 1 we have a Vlim ∈ Rd such that  ∥td/2 v(·  √  t, t) − Vlime−|·|2/4∥Hm  r ≤ C(1+t)−d/2  for all t ≥ 0, where  ∥v∥Hm  r = ∥vσr∥Hm,  with  σ(x) = (1 + x2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We use the discrete renormalization approach which was introduced for such  problems at the beginning of the 1990s, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Although the proof is documented  in the literature for similar problems, we recall its main steps since to our knowledge  such results have not been formulated for geometric flow problems so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Instead of solving (12) directly we consider an equivalent sequence of problems  which converges formally towards the linear diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For this we let  (13)  vn(ξ, τ) = Lndv(Lnξ, L2nτ),  with L > 1 fixed and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Then vn satisfies  (∂τvn − ∆ξvn) = Ln(d+2)(∂tv − ∆xv)  =  Ln(d+2)A(u∗ + v)(∇xv, ∇xv)  =  Ln(d+2)A(u∗ + L−ndvn)(L−n(d+1)∇ξvn, L−n(d+1)∇ξvn)  =  L−ndA(u∗ + L−ndvn)(∇ξvn, ∇ξvn)  for τ ∈ [L−2, 1], with initial condition  (14)  vn(ξ, L−2) = Ldvn−1(Lξ, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  The influence of the nonlinear terms vanishes with a geometric rate for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In  the limit n → ∞ we obtain a linear diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' With a simple fixed point  argument the convergence of vn|τ=1 towards the limit of the renormalized linear  diffusion problem, namely multiples of the Gaussian ψ(ξ) = (2π)−d/2e−|ξ|2/4, can  be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In the rest of the proof we use the symbol C for constants which can  be chosen independent of L and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We need the projection  Πvn =  �  vn(ξ)dξ,  DIFFUSIVE STABILITY  15  respectively �Π�vn = �vn|k=0 in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The bound  |Πvn| ≤ C|�vn|k=0| ≤ C∥�vn∥C0  b ≤ C∥�vn∥Hr  m ≤ C∥vn∥Hm  r  holds due to Sobolev’s embedding theorem Hr  m ⊂ C0  b for r > d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We set  Vn+1 = Πvn|τ=1  and introduce the deviation of vn(ξ, 1) from Vn+1ψ(ξ) by  ρn+1(ξ) = vn(ξ, 1) − Vn+1ψ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  By construction we have  �  ρn(ξ)dξ = 0, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' �ρn|k=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For proving the con-  vergence of (Vn)n∈N towards a Vlim ∈ R and the decay of (ρn)n∈N towards zero we  establish a number of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We have  |Vn+1 − Vn|  ≤  CL−ndR2  n,  ∥ρn+1∥Hm  r  ≤  (C/L)∥ρn∥Hm  r + CL−ndR2  n + C|Vn+1 − Vn|  where  Rn =  sup  τ∈[1/L2,1]  ∥vn(τ)∥Hm  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We consider the variation of constants formula  (15)  vn(ξ, τ) = e(τ−1/L2)∆ξLdvn−1(Lξ, 1) + gn(τ)  for τ ∈ [1/L2, 1], where  gn(τ) = L−nd  � τ  1/L2 e(τ−s)∆ξA(u∗ + L−ndvn)(∇ξvn, ∇ξvn)(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Applying the projection Π in case τ = 1 yields  (16)  Vn+1 = Vn + Πgn(1),  where we used  �  e(τ−1/L2)∆ξf(ξ)dξ = e−(τ−1/L2)k2|k=0 �f(0) = �f(0) =  �  f(ξ)dξ  and  �  Ldvn−1(Lξ, 1)dξ =  �  vn−1(ξ, 1)dξ = Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Rearranging the terms in the variation of constant formula additionally yields  ρn+1(ξ)  =  e(1−1/L2)∆ξLdρn(Lξ)  (17)  +gn(1) + e(1−1/L2)∆ξLdVnψ(Lξ) − Vn+1ψ(ξ),  We estimate the terms on the right hand side of (16) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  i) Using  e−(1−1/L2)|k|2e−|k/L|2 = e−|k|2  yields  ∥e(1−1/L2)∆ξLdVnψ(L·) − Vn+1ψ(·)∥Hm  r =∥(Vn − Vn+1)ψ(·)∥Hm  r  ≤C|Vn+1 − Vn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  16  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  ii) Since ρ has mean value zero and since �ρ ∈ Hr  m ⊂ C1  b for r > d/2 + 1 we have  ∥e(1−1/L2)∆ξLdρn(L·)∥Hm  r  ≤  C∥e−(1−1/L2)|k|2 �ρn(·/L)∥Hrm  ≤  (C/L)∥�ρn∥Hrm ≤ (C/L)∥ρn∥Hm  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  iii) For the integral term we obtain  ∥ gn(τ)∥Hm  r  ≤  L−nd  � τ  1/L2 ∥e(τ−s)∆∥Hm−1  2  →Hm  r  ×∥ A(u∗ + L−ndvn)(∇ξvn, ∇ξvn)(s)∥Hm−1  2  ds  ≤  CL−ndR2  n  � τ  1/L2(τ − s)−1/2ds ≤ CL−ndR2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  Using the first inequality of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2 to eliminate |An+1 − An| in the second  inequality of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2 gives  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  |Vn+1 − Vn|  ≤  CL−ndR2  n,  ∥ρn+1∥Hm  r  ≤  (C/L)∥ρn∥Hm  r + CL−ndR2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  For obtaining a closed system of inequalities in Vn and ρn, we have to estimate  Rn in terms of Vn and ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exists C1 > 0 and C2 > 0, such that for all δ > 0 and L > 1  with L5/2δ < C1 the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For vn−1|τ=1 ∈ Hm  r  with  ∥vn−1|τ=1∥Hm  r < δ  we have  Rn ≤ C2L5/2(|Vn| + rn)  where rn = ∥ρn∥Hm  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We consider the variation of constant formula (15), use the estimate iii) from  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2, replace ii) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2 by  ∥e(τ−1/L2)∆ξLdvn−1(L·)∥Hm  r ≤ C∥Ldvn−1(L·)∥Hm  r ≤ CL5/2∥vn−1∥Hm  r ,  to obtain  Rn ≤ CL5/2∥vn−1∥Hm  r + CL−σnR2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Since there exist two intersection points R < R of the line Rn �→ Rn and the curve  Rn �→ CL5/2∥vn−1∥Hm  r + CL−ndR2  n for L5/2δ > 0 sufficiently small, we have  Rn ≤ R ≤ 2CL5/2∥vn−1∥Hm  r ≤ 4CL5/2(|Vn| + rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Obviously there is local existence and uniqueness of solutions in the space Hm  r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Since the above estimate gives an a priori bound in Hm  r , the solution can be ex-  tended to the whole interval [1/L2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  DIFFUSIVE STABILITY  17  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='3 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='4 yield  |Vn+1 − Vn|  ≤  CL−nd(L5/2)2(|Vn| + rn)2,  rn  ≤  (C/L)rn + CL−nd(L5/2)2(|Vn| + rn)2,  as long as  (18)  C(|Vn| + rn) ≤ δ  and  L5/2δ ≤ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We choose L0 > 1 so large that (C/L0) ≤ 1/10 and then n0 > 0 so large that  CL−n0d  0  (L5/2  0  )2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Then, for all n > n0 und L > L0 we have  |Vn+1 − Vn|  ≤  L−d(n−n0)(Vn + rn)2,  rn+1  ≤  rn/10 + L−(n−n0)d(Vn + rn)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Now we choose an L > L0 and a δ > 0 so small, that  L5/2δ ≤ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Since the quantities |Vn| and rn grow only for n0 steps we can choose the initial  conditions so small that  max  n=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=',n0 C(|Vn| + rn) ≤ δ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Then (Vn)n∈N converges with a geometric rate towards a Vlim ∈ Rd and we have  limn→∞ rn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The estimates further guarantee that  sup  n≥n0  rn ≤ δ/4  und  sup  n≥n0  |Vn| ≤ δ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Therefore, all necessary conditions for the validity of the previous estimates are  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Since convergence holds for all L ∈ [L0, L2  0], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  With a slight modification of this proof similar results can be established for the  w-equation of the harmonic map heat flow (see (5)) and for the biharmonic map  heat flow (see (10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In the following we sketch the ideas for the equation (5) which  we recall to be  ∂tw − ∆w  =  ∇(A(u∗ + v)(w, w))  (19)  =  2A(u∗ + v)(w, ∇w) + A′(u∗ + v)(w, w, w),  with A′(u∗+v)(·, ·, ·) a smooth and bounded trilinear mapping acting on the tangent  space and with w = ∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For the scaling  (20)  wn(ξ, τ) = Lndw(Lnξ, L2nτ)  with L > 1 fixed and n ∈ N the previous analysis applies almost line by line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' By  this choice we have  vn(ξ, τ) = Ln(d−1)v(Lnξ, L2nτ),  18  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' LAMM AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' SCHNEIDER  and so wn satisfies  (∂τwn − ∆ξwn) = Ln(d+2)(∂tw − ∆xw)  =  2Ln(d+2)A(u∗ + v)(w, ∇xw) + Ln(d+2)A′(u∗ + v)(w, w, w)  =  2Ln(d+2)A(u∗ + L−n(d−1)vn)(L−ndwn, L−n(d+1)∇ξwn)  +Ln(d+2)A′(u∗ + L−n(d−1)vn)(L−ndwn, L−ndwn, L−ndwn)  =  2L−n(d−1)A(u∗ + L−n(d−1)vn)(wn, ∇ξwn)  +L−2n(d−1)A′(u∗ + L−n(d−1)vn)(wn, wn, wn)  for τ ∈ [L−2, 1], with  (21)  wn(ξ, L−2) = Ldwn−1(Lξ, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Hence, the nonlinear terms are asymptotically irrelevant if d > 1 and so the rest of  the previous proof allows us to establish the result  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exist δ, C > 0, such that for all solutions w of (19) with  ∥w|t=0∥Hm  r ≤ δ where m > d/2 + 1 and r > d/2 + 1, we have a Wlim ∈ Rd×d such  that  ∥td/2 w(·  √  t, t) − Wlime−|·|2/4∥Hm  r ≤ C(1+t)−d/2 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  However, since w = ∇v the choice w ∈ Hs  r excludes a number of interesting  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' A typical example of an initial condition for v : R2 → R2 considered in  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='5 which can not have been handled by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1 or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='5 is  v1(x1, x2) = v0erf(x1)e−x2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  We find  (w1j)j=1,2(x1, x2) = ∇v(x1, x2) = (v0e−x2  1/4e−x2  2, −2x2v0erf(x1)e−x2  2)  which is not an element of Hs  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' We remark without any proof that initial conditions  of the form v = c1erf(x1) + �v with c1 ∈ R and �v ∈ Hs  r can be included easily into  the formulation and proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Biharmonic map heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Next we come to the biharmonic map heat  flow in case u : Rd → Sm−1 ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Similar to the linear diffusion equation we have  for the linearized problem  ∂tv = −∆2v,  v|t=0 = v0  that for spatially localized initial conditions the solutions decay asymptotically in  a self similar way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=', for x ∈ Rd the renormalized solution td/4v(x  √  t, t) converges  towards a multiple of a universal limit function, Vlimϕ(x) with Vlim ∈ R and ϕ =  F−1 �ϕ where �ϕ(k) = e−|k|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  The same is true for the deviation v of a trivial  spatially constant equilibrium u∗ for the bi-harmonic map heat flow (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Note that  the deviation v satisfies a semilinear parabolic equation of the form  ∂tv  =  −∆2v − (∆(∇v, ∇v) + ∇ · (∆v, ∇v) + (∇∆v, ∇v))e1  (22)  −(∆(∇v, ∇v) + ∇ · (∆v, ∇v) + (∇∆v, ∇v))v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Then we have:  DIFFUSIVE STABILITY  19  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' There exist δ, C > 0, such that for all solutions v of (22) with  ∥v|t=0∥Hm  r ≤ δ where m > d/2 + 3 and r > d/2 + 1 we have a Vlim ∈ Rd such that  ∥td/4 v(·  √  t, t) − Vlimϕ(·)∥Hm  r ≤ C(1+t)−d/4  for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' As above instead of solving (12) directly we consider an equivalent sequence  of problems which converges formally towards the linearized equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' For this we  let  (23)  vn(ξ, τ) = Lndv(L2nξ, L4nτ),  with L > 1 fixed and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Then vn satisfies  (∂τvn − ∆ξvn) = Ln(d+8)(∂tv − ∆xv)  =  Ln(d+8)(−(∆x(∇xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇xv) + ∇x · (∆xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇xv) + (∇x∆xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇xv))e1  −(∆x(∇xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇xv) + ∇x · (∆xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇xv) + (∇x∆xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇xv))v)  =  Ln(d+8)(−(L−4n∆ξ(L−n(d+2)∇ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−n(d+2)∇ξv)  +L−2n∇ξ · (L−n(d+4)∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−n(d+2)∇ξv)  +(L−n(d+6)∇ξ∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−n(d+2)∇ξv))e1  −(L−4n∆ξ(L−n(d+2)∇ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−n(d+2)∇ξv)  +L−2n∇ξ · (L−n(d+4)∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−n(d+2)∇ξv)  +(L−n(d+6)∇ξ∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−n(d+2)∇ξv))L−ndv)  =  L−nd(−(∆ξ(∇ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇ξv) + ∇ξ · (∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇ξv) + (∇ξ∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇ξv)))e1  +L−2nd(−(∆ξ(∇ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇ξv) + ∇ξ · (∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇ξv) + (∇ξ∆ξv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ∇ξv))v)  for τ ∈ [L−4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' with initial condition  (24)  vn(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' L−4) = Ldvn−1(L2ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  As for the harmonic map heat flow the influence of the nonlinear terms vanishes  with a geometric rate for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' In the limit n → ∞ we obtain the linearized  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' The rest of the proof follows the associated parts of the proof of Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='1 almost line for line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  □  References  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Blatt and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Well-posedness of the supercritical Lane-Emden heat flow in Morrey  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' ESAIM Control Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=', 22(4):1370—1381, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
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+page_content='  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with  rough initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=', 200(1):1–19, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  [21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Well-posedness for the heat flow of biharmonic maps with rough initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
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+page_content='  [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Weissler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Existence and nonexistence of global solutions for a semilinear heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  Israel Journal of Mathematics, 38:29–40, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Lamm) Institute for Analysis, Karlsruhe Institute of Technology, Englerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='  2, 76131 Karlsruhe, Germany  Email address: tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='lamm@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='edu  (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content=' Schneider) Institut f¨ur Analysis, Dynamik und Modellierung, Universit¨at Stuttgart,  Pfaffenwaldring 57, 70569 Stuttgart, Germany  Email address: guido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='schneider@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
+page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdA0T4oBgHgl3EQfH_8L/content/2301.02067v1.pdf'}
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+The generalized Clapeyron equation and its application to confined ice growth
+Robert W. Style,1, ∗ Dominic Gerber,1 Alan W. Rempel,2 and Eric R. Dufresne1
+1Department of Materials, ETH Z¨urich, 8093 Z¨urich, Switzerland.
+2 Department of Earth Sciences, University of Oregon, Eugene, Oregon, USA.
+(Dated: January 11, 2023)
+Most theoretical descriptions of stresses induced by freezing are rooted in the (generalized) Clapey-
+ron equation, which predicts the pressure that a solid can exert as it cools below its melting tem-
+perature.
+This equation is central for topics ranging beyond glaciology to geomorphology, civil
+engineering, food storage, and cryopreservation.
+However, it has inherent limitations, requiring
+isotropic solid stresses and conditions near bulk equilibrium. Here, we examine when the Clapeyron
+equation is applicable by providing a rigorous derivation that details all assumptions. We demon-
+strate the natural extension for anisotropic stress states, and we show how the temperature and
+pressure ranges for validity depend on well-defined material properties. Finally, we demonstrate
+how the range of applicability of the (linear) Clapeyron equation can be extended by adding higher-
+order terms, yielding results that are in good agreement with experimental data for the pressure
+melting of ice.
+When water freezes in confined spaces, it can generate large stresses, often resulting in material damage. This
+is important across fields ranging from glaciology to geomorphology, food science, civil engineering and cryopreser-
+vation [1–7]. Broadly speaking, ice can generate stresses via two different mechanisms [8, 9]. The first is due to
+the expansion of water as it freezes: in a closed cavity, freezing will generate pressure (Figure 1a). The second is
+unrelated to the expansion of water and often dominates in porous materials [9]. Here, ice forms in open pores of a
+wet material (Figure 1b), but no pressure builds up during the initial ice-formation process (any pressure is relieved
+by water flow away from the growing ice). However, after the initial ice formation, unfrozen water is sucked back
+towards the ice crystals. When this water freezes onto the existing ice, it causes the ice to wedge open its confining
+pore. This cryosuction process is aided by the presence of thin, mobile layers of water at the surface of ice (known as
+premelted films) [10]. These allow growth of the ice in all directions. In both cases ice will continue to grow, building
+up pressure, until the pressure reaches a maximum value given by a temperature-dependent stall pressure, Pst [9, 11].
+Pst is very similar to the concept of crystallization pressure, found when confined crystals grow from supersaturated
+solutions [12–14] and to the concept of condensation pressure, when phase separation occurs in confinement [15, 16].
+Case (i): Ice growth in a
+confining cavity
+Case (ii): Ice growth in an
+open pore
+Water
+Ice
+(Pl, T)
+(Pl, T)
+Pl
+Water (Pl, T)
+Ice (σ,T)
+-σnn
+-σnn
+A
+B
+FIG. 1: (i) Ice generates pressure as it grows in a closed cavity, due to the expansion of water upon freezing. (ii) Ice growing
+in an open pore is fed by nearby water, and this growth wedges open the cavity, generating stresses.
+Theoretical descriptions of these stress-generation mechanisms are rooted in the (generalized) Clapeyron equation,
+a fundamental equation that describes equilibrium between a solid (ice) at pressure Ps, and a reservoir of liquid
+∗Electronic address: robert.style@mat.ethz.ch
+arXiv:2301.03895v1  [cond-mat.soft]  10 Jan 2023
+
+2
+TABLE I: Ice/water parameter values at atmospheric pressure and 273.15K [19].
+Density of ice
+ρs
+917 kg m−3
+Density of water
+ρl
+997 kg m−3
+Latent heat of fusion
+qm
+334 kJ kg−1
+Melting temperature
+Tm
+273.15 K
+Heat capacity of ice
+cp
+s
+2093 J (kg K)−1
+Heat capacity of water
+cp
+l
+4184 J (kg K)−1
+Bulk modulus of ice
+Ks
+11.33 GPa
+Bulk modulus of water
+Kl
+1.96 GPa
+Coeff. thermal expansion, ice
+αs
+51 × 10−6 K−1.
+Coeff. thermal expansion, water
+αl
+−50 × 10−6 K−1.
+(water) at a different pressure Pl [8, 17, 18]:
+(Ps − P0)
+ρs
+− (Pl − P0)
+ρl
+= qm(Tm − T)
+Tm
+.
+(1)
+Here, ρl and ρs are the densities of water and ice respectively, qm is the specific latent heat of freezing of ice, and
+Tm is the melting temperature at a reference pressure, P0 (often taken as atmospheric pressure). For the freezing
+mechanisms described above, this equation can be used to predict Pst as a function of the temperature, T, as at this
+point, the ice is in equilibrium with the nearby water. For case (i) with ice growing in a closed cavity, the ice and
+water are both at the same pressure (Ps = Pl = Pst), so
+(Pst − P0)
+� 1
+ρs
+− 1
+ρl
+�
+= qm(Tm − T)
+Tm
+.
+(2)
+Using values from Table I, we find that ice can exert pressures of ∼ 11 MPa per degree of undercooling (Tm − T).
+For case (ii), the ice and water need no longer have the same pressure. If the water reservoir is held at the reference
+pressure Pl = P0, then Pst = Ps and
+(Pst − P0)
+ρs
+= qm(Tm − T)
+Tm
+.
+(3)
+In this case, ice can exert pressures of ∼ 1 MPa per degree of undercooling.
+Even when ice is not in equilibrium (e.g. it is growing), the Clapeyron equation gives us useful information. During
+growth, there is no macroscopic equilibrium, but water immediately adjacent to the ice/water interface can often be
+considered to be in equilibrium with the ice [8]. Then, the Clapeyron equation relates the local hydrodynamic pressure
+in the water, Pl, to the local pressure that has been built up in the ice (Pl is just the pressure of a hypothetical,
+reservoir of water in thermodynamic equilibrium with the ice). Water flows along nonhydrostatic gradients in Pl, so
+the Clapeyron equation allows us to predict how water is transported towards (or away from) ice, and thus gives ice
+growth/melting rates [8, 20–23].
+The various applications of the Clapeyron equation make it a key tool for understanding freezing processes [e.g.
+1, 6, 8, 11]. However, it makes a number of assumptions. For example, it assumes that ice can be described by
+an isotropic pressure, whereas ice is often characterized by an anisotropic stress state, σij [24]. It also uses linear
+approximations that are valid only near the bulk melting point of ice (see later). Thus, several key questions arise. In
+particular: What is the appropriate extension of the Clapeyron equation for anisotropically stressed ice? Over what
+range of conditions should the Clapeyron equation be applicable?
+Surprisingly, we are not aware of a systematic derivation of the Clapeyron equation that would allow us to address
+these questions. However, there are several related works. For example, several authors have established the ther-
+modynamic relations that govern the dissolution of anisotropically stressed solids into adjacent fluids [25–27], with
+notable applications to recrystallization and pressure solution processes [e.g. 28]. Although melting was not a focus of
+these works, some of the consequences for ice melting were recognized by Nye [29]. He argued that the phenomenon
+of wire regelation requires a generalization of equation (2) where Ps replaced by the normal stress −σnn, and not
+by the mean of the principal stresses −Tr(σ)/3, as had been argued by others. A further, unflinching critique of
+alternative incorrect theories is given by Kamb [27]. Finally, Sekerka and Cahn [30] examined the special case of a
+solid with σnn = −Pl, to show that anisotropically stressed solids in equilibrium with their melt will recrystallize to
+form a isotropically-stressed state.
+
+3
+Here, we provide a first-principles derivation of the generalized Clapeyron equation, along similar lines to Pater-
+son [28]. We clearly lay out all the underlying assumptions, and present the appropriate extension for the melting
+behavior of anisotropically stressed ice.
+A.
+Deriving the Generalized Clapeyron Equation
+We consider thermodynamic equilibrium for the two scenarios shown in Figure 1, in both of which the temperature
+is held fixed at T < Tm. In case (i), water freezes in a closed cavity, so that the ice and and water both have the same
+pressure, Ps = Pl. In case (ii), ice has frozen in an open cavity, and is in equilibrium with neighbouring bulk water,
+which has pressure Pl. At the same time, the ice exerts a normal stress −σnn on the walls of the cavity, but negligible
+shear forces, due to the presence of premelted films which lubricate the ice/cavity interface [11]. The ice cannot grow
+through the small, connecting pore throat into the neighboring water due to capillarity (i.e. the Gibbs-Thomson effect
+[31, 32]).
+For each scenario, we establish equilibrium behavior by minimizing the relevant free energy of the ice/water system.
+The relevant free energy, Gsys satisfies ∆Gsys = ∆Usys−T∆Ssys+W, where Usys is the internal energy of the ice/water
+system, Ssys is its entropy, and W is the work done by the system on its surroundings. For case (i),
+∆Gsys = 0 = ∆Usys − T∆Ssys + Pl(∆Vs + ∆Vl),
+(4)
+for case (ii),
+∆Gsys = 0 = ∆Usys − T∆Ssys − σnn∆Vs + Pl∆Vl,
+(5)
+where Vs and Vl are the volumes of ice and water, respectively. The first case is just a specialized version of the
+second, where −σnn = Pl. Thus, without loss of generality, we can proceed with equation (5), and the result will
+describe both cases.
+We consider a small perturbation in the system in Figure 1b, where a small mass of ice, ∆m, melts and flows into
+the reservoir. Thus, the volumes of ice and water change as ∆Vs = −vs∆m, and ∆Vl = vl∆m, where vs(σij, T) and
+vl(Pl, T) are the specific volumes of the ice and water, respectively. Then equation (5) becomes
+ul∆m − us∆m − T(sl − ss)∆m + σnnvs∆m + Plvl∆m = 0.
+(6)
+Here, us(σij, T) and ul(Pl, T) are the specific internal energies of the ice and water, respectively, and ss(σij, T) and
+sl(Pl, T) are the respective specific entropies. Dividing through by ∆m, we obtain
+− (σnnvs + Plvl) = (ul − us) − T(sl − ss).
+(7)
+In principle, equation (7) completely describes equilibrium between ice and water. i.e. one could use tabulated
+values of u, v, and s to find −σnn(Pl, T). However, a more convenient form is found by expressing the equation
+relative to the pressure and temperature under bulk melting reference conditions, (P0, Tm). Here, −σnn = Pl = P0 so
+equation (7) becomes
+P0(vo
+s − vo
+l ) = (uo
+l − uo
+s) − Tm(so
+l − so
+s).
+(8)
+The superscript o indicates reference conditions. Subtracting equations 7 and 8, we find
+gl − go
+l = gs − go
+s,
+(9)
+where the specific free energies gl(T, Pl) = ul − Tsl + Plvl, and gs(T, σij) = us − Tss − σnnvs.
+These can be
+Taylor-expanded to obtain the Clapeyron equation (e.g. [1, 33]):
+gl(T, v) = go
+l (Tm, P0) +
+�∂gl
+∂T
+�
+Pl
+(T − Tm) +
+� ∂gl
+∂Pl
+�
+T
+(Pl − P0)
+(10)
+and
+gs(T, σij) = go
+s(Tm, P0) +
+�∂gs
+∂T
+�
+σij
+(T − Tm) +
+� ∂gs
+∂σij
+�
+T
+(σij + P0δij),
+(11)
+
+4
+where δij is the identity matrix. To evaluate the derivatives, we note that ∆gl = −sl∆T + vl∆Pl. Thus, at reference
+conditions,
+�∂gl
+∂T
+�
+Pl
+= −so
+l ,
+� ∂gl
+∂Pl
+�
+T
+= vo
+l ,
+(12)
+and similarly in the solid at reference conditions (σij = −P0δij)
+�∂gs
+∂T
+�
+σij
+= −so
+s.
+(13)
+To calculate the final derivative, we notice that gs = (fs + P0vs) − ¯σnnvs, where fs is the specific Helmholtz free
+energy of the solid, and ¯σij = σij + P0δij. Here, (fs + P0vs)/vo
+s is the free-energy per unit volume of deformations in
+an atmosphere at constant pressure, P0, and thus is the elastic energy per volume of ice in the reference state. As
+such, we can use linear elasticity to write
+gs = 1
+2 ¯σijϵijvo
+s − ¯σnnvo
+s
+�
+1 + Tr(¯σ)
+3Ks
+�
+,
+(14)
+where Ks is now the bulk modulus of the solid. ϵij is the strain of the ice relative to its shape in the reference state
+(Tm, P0), and satsifies the linear-elastic constitutive relationship:
+ϵij = 1
+Es
+[(1 + νs)¯σij − νsδijTr(¯σ)] ,
+(15)
+where Es = 3Ks(1 − 2νs) is the Young’s modulus of the ice and νs is its Poisson ratio. For small strains, vs =
+vo
+s(1 + Tr(ϵ)), and we use this in the second term of equation (14).
+With the two equations above, we can evalulate the remaining derivative at (P0, Tm):
+� ∂gs
+∂σij
+�
+T
+(¯σij = 0) = −ninjvo
+s.
+(16)
+Here, ni is the normal vector to the surface of the ice, so that ¯σnn = ni¯σijnj.
+Finally, we can insert these first-derivative expressions into equations (9–11) to obtain the Clapeyron equation for
+anisotropically stressed solids:
+− (σnn + P0)
+ρos
+− (Pl − P0)
+ρo
+l
+= qm(Tm − T)
+Tm
+.
+(17)
+Here, ρo
+l = 1/vo
+l , and ρo
+s = 1/vo
+s are the densities of water and ice respectively at the bulk melting point, and
+qm ≡ (so
+l − so
+s)Tm. Consistent with the regelation analysis of Nye [29], this version of the Clapeyron equation is
+identical to equation (1), but with Ps replaced by −σnn, and not −Tr(σ)/3, as one might naively assume.
+B.
+Field data supporting the anisotropic Clapeyron equation
+Glaciological field data supporting the form of equation (17) comes from simultaneous measurements of temperatures
+and liquid pressures in glacier boreholes. These measurements show that temperatures increase when changes in the
+hydrologic system cause borehole pressures, Pl, to decrease [e.g. 34, 35].
+The anisotropic Clapeyron equation indeed recovers this correlation. Along borehole walls, σnn = −Pl. Inserting
+this into equation (17), we find that changes in temperature are correlated with changes in borehole pressure by:
+∆T = −Tm
+qm
+� 1
+ρ0s
+− 1
+ρ0
+l
+�
+∆Pl ≈
+�
+−7.16 × 10−8 K Pa−1�
+∆Pl ,
+(18)
+in agreement with the field data.
+By contrast, naively extending the isotropic Clapeyron equation (1), by replacing −Ps = Tr(σ)/3, does not match
+the experimental data. The classic analysis of Nye [36] gives the complete stress tensor at the surface of an idealized
+cylindrical borehole containing liquid at pressure Pl. Far from the borehole, the ice has a far-field isotropic ice pressure
+P∞, and creeps according to Glen’s flow law with exponent n = 3 [37, 38]. In this case, −Tr(σ)/3 = Pl+(P∞ − Pl) /n.
+
+5
+Substituting Ps = −Tr(σ)/3 into the isotropic Clapeyron equation (1) and treating the far-field ice pressure as constant
+leads to
+∆T = −Tm
+qm
+� 1
+ρ0s
+− 1
+ρ0
+l
+−
+1
+nρ0s
+�
+∆Pl ≈
+�
+2.26 × 10−7 K Pa−1�
+∆Pl .
+(19)
+This predicts the opposite of the correlation seen in the field data.
+C.
+Errors in the Clapeyron equation
+In deriving this version of the Clapeyron equation, we have had to make two main assumptions. Firstly, strains in
+the ice are small, so we can use linear elasticity [30]. This is reasonable as the stresses in the ice (which are O(MPa)
+– see introduction) are much less than the ice’s elastic moduli Es, Ks = O(GPa), so strains will be small.
+Secondly, we assume that higher-order terms in the expansions of gl and gs are negligible. We can test this by
+reverting to the case of isotropically-stressed ice (σij = −Psδij). Then, we Taylor-expand equation (9) in T, Pl and
+Ps to obtain the second-order version of the Clapeyron equation:
+(Ps − P0)
+ρos
+− (Pl − P0)
+ρo
+l
+=
+qm(Tm − T)
+Tm
+(20)
+−cp
+l − cp
+s
+2Tm
+(Tm − T)2 −
+1
+2ρo
+l Kl
+(Pl − P0)2 +
+1
+2ρosKs
+(Ps − P0)2
+−αl
+ρo
+l
+(Tm − T)(Pl − P0) + αs
+ρos
+(Tm − T)(Ps − P0).
+Here, we use the following identities [39]:
+∂2g
+∂T 2 = − cp
+Tm
+,
+∂2g
+∂P 2 = −
+1
+Kρo ,
+∂2g
+∂P∂T = α
+ρo .
+(21)
+cp is the heat capacity at constant pressure, K is again the isothermal bulk modulus, and α is the coefficient of
+thermal expansion.
+We can now predict the pressure-melting curve for different freezing scenarios. For bulk ice/water equilibrium
+(Figure 1a), Ps = Pl, and we take atmospheric pressure, Pa, as the reference pressure.
+Figure 2a compares the
+isotropic Clapeyron equation (1) (red, dashed) with experimental data (black, dotted) [40]. There is a significant
+error between the two results for an undercooling of more than ∼ 3 ◦C. However, when we use the full, second-order
+Clapeyron equation (20) (blue), we find good agreement down to an undercooling of at least 15 ◦C. In this situation,
+the terms that are quadratic in pressure dominate the error, and to excellent approximation (Figure 1a, orange
+dash-dotted):
+� 1
+ρos
+− 1
+ρo
+l
+�
+(Ps − P0) +
+�
+1
+2ρo
+l Kl
+−
+1
+2ρosKs
+�
+(Ps − P0)2 = qm(Tm − T)
+Tm
+.
+(22)
+Comparing the first two terms in this equation, we see that the linear equation (17) is only appropriate when:
+|Ps − Pa| ≪ ∆P ∗ =
+�����
+� 1
+ρos
+− 1
+ρo
+l
+� �
+1
+2ρosKs
+−
+1
+2ρo
+l Kl
+�−1����� ≈ 420MPa.
+(23)
+We can perform a similar analysis for freezing in an open system (Figure 1b). We let Pl = P0 = Pa, and assume
+that the ice exerts an isotropic pressure, Ps on its surroundings. Figure 2b compares the prediction of the Clapeyron
+equation (1) (red, dashed) with that obtained when we keep the extra quadratic terms (20) (blue). We are not aware
+of any experimental data precise enough to validate the theory [11]. However, here, the higher-order theory agrees
+well with the linear Clapeyron equation down to large undercoolings. The difference is dominated by the term in
+equation (20) that is quadratic in undercooling. Thus, to excellent approximation (Figure 1a, orange dash-dotted):
+Ps − Pa = ρo
+sqm(Tm − T)
+Tm
+− ρo
+s(cp
+l − cp
+s)
+2Tm
+(Tm − T)2.
+(24)
+
+6
+260
+265
+270
+T [K]
+0
+5
+10
+15
+Ps-P0 [MPa]
+Full 2nd
+order theory
+Case (ii): Ice growth in an open pore
+Pl=Pa
+Linear
+Clapeyron Eq.
+Simplified 2nd
+order theory (Eq. 25)
+0
+50
+100
+150
+200
+250
+Ps-P0 [MPa]
+Simplified 2nd
+order theory (Eq. 23)
+Full 2nd
+order theory
+Linear
+Clapeyron Eq.
+Case (i): ice growth in a closed cavity
+Pl=Ps
+Experimental
+data
+FIG. 2: Evaluating the accuracy of the Clapeyron Equation. A) The pressure of ice in bulk ice/water equilibrium in a closed
+cavity (Pl = Ps = −σnn), as a function of undercooling. The black, dotted curve shows experimental data [40]. B) The stress
+exerted by ice in an open pore, as a function of undercooling. Both figures show the linear Clapeyron equation (dashed red),
+full 2nd-order theory (Eq. 20, blue), and simplified 2nd-order theory (Eqs. 22,24, orange dash-dotted).
+Comparing terms on the right-hand side, shows that we only recover the linear Clapeyron equation (1) if
+|Tm − T| ≪ ∆T ∗ =
+����
+2qm
+cp
+l − cp
+s
+���� ≈ 320K.
+(25)
+This requirement is certainly satisfied for most terrestrial temperatures. Thus, there is some justification for use of
+the linear Clapeyron equation down to relatively large undercoolings to model this type of freezing scenario.
+To summarize, our results suggest that the linearized Clapeyron equation will be valid, provided that |Pl − P0| and
+|Ps − P0| are both small relative to ∆P ∗, while |Tm − T| ≪ ∆T∗. At larger pressures/undercoolings, the quadratic
+terms in equation (20) should be included.
+D.
+Conclusions
+In conclusion, we have derived the linear, Clapeyron equation describing equilibrium between water and ice, clearly
+laying out all the assumptions involved. In particular, this equation is derived using a Taylor expansion around a
+reference temperature and pressure, and ignoring higher-order terms. Thus, it is only valid for a range of pressures
+and and temperatures around the reference conditions. Fortunately, for most naturally-occurring terrestrial freezing
+scenarios, the linear form of the Clapeyron equation should be adequate. For example, at the base of a glacier,
+pressures are typically close to hydrostatic, and thus O(MPa) [41] – this is small enough to lie within the range of
+applicability of the Clapeyron equation. However, more extreme conditions are expected in extraterrestrial settings
+[e.g. 40, 42]. There, the linearized Clapeyron equation will not accurately predict melting temperatures, which could
+
+7
+lead to significant errors in models of ice dynamics (as predicted flow rates are typically based on the departure from
+bulk melting conditions [24]). In this case, the accuracy of the Clapeyron equation can be improved by retaining
+higher-order terms in the Taylor expansion.
+We have also demonstrated the correct form of the Clapeyron equation for the case where ice is anisotropically
+stressed. This is identical to the isotropic form of the Clapeyron equation, but with ice pressure, Ps replaced by the
+normal stress exerted by ice on its surroundings, −σnn. One consequence of this, is that differently stressed faces of
+ice (for example in a polycrystal) will have different melting temperatures.
+While our analysis has focused on ice and water, the results should apply to any processes involving solid/liquid
+equilibrium, for example in the melting and deformation of rocks in geological processes (e.g. [43]). Note however,
+that there are two, key further effects that will likely be important to include in real-world applications. Firstly,
+we have neglected the presence of solutes, which are known to strongly affect the solid/liquid equilibrium [44–47].
+Secondly, we we have ignored the surface energy of the ice [8, 48]. However, we anticipate that both of these effects
+can be incorporated into the results presented here, by including colligative and capillary effects in the analysis above.
+Acknowledgments
+RWS and DG acknowledge support from an ETH Research Grant (Grant No. ETH-38 18-2), and from the Swiss
+National Science Foundation (Grant No. 200021-212066); AWR received funding from NSF-2012468 and a UO Faculty
+Research Award.
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+
diff --git a/hdE2T4oBgHgl3EQfcgcA/content/tmp_files/load_file.txt b/hdE2T4oBgHgl3EQfcgcA/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..340c5e85116fdc2bc5875d9dea96ef375d1b7f6a
--- /dev/null
+++ b/hdE2T4oBgHgl3EQfcgcA/content/tmp_files/load_file.txt
@@ -0,0 +1,434 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf,len=433
+page_content='The generalized Clapeyron equation and its application to confined ice growth  Robert W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Style,1, ∗ Dominic Gerber,1 Alan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Rempel,2 and Eric R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Dufresne1  1Department of Materials, ETH Z¨urich, 8093 Z¨urich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  2 Department of Earth Sciences, University of Oregon, Eugene, Oregon, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (Dated: January 11, 2023)  Most theoretical descriptions of stresses induced by freezing are rooted in the (generalized) Clapey-  ron equation, which predicts the pressure that a solid can exert as it cools below its melting tem-  perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  This equation is central for topics ranging beyond glaciology to geomorphology, civil  engineering, food storage, and cryopreservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  However, it has inherent limitations, requiring  isotropic solid stresses and conditions near bulk equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Here, we examine when the Clapeyron  equation is applicable by providing a rigorous derivation that details all assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' We demon-  strate the natural extension for anisotropic stress states, and we show how the temperature and  pressure ranges for validity depend on well-defined material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Finally, we demonstrate  how the range of applicability of the (linear) Clapeyron equation can be extended by adding higher-  order terms, yielding results that are in good agreement with experimental data for the pressure  melting of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  When water freezes in confined spaces, it can generate large stresses, often resulting in material damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' This  is important across fields ranging from glaciology to geomorphology, food science, civil engineering and cryopreser-  vation [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Broadly speaking, ice can generate stresses via two different mechanisms [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The first is due to  the expansion of water as it freezes: in a closed cavity, freezing will generate pressure (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The second is  unrelated to the expansion of water and often dominates in porous materials [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Here, ice forms in open pores of a  wet material (Figure 1b), but no pressure builds up during the initial ice-formation process (any pressure is relieved  by water flow away from the growing ice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, after the initial ice formation, unfrozen water is sucked back  towards the ice crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' When this water freezes onto the existing ice, it causes the ice to wedge open its confining  pore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' This cryosuction process is aided by the presence of thin, mobile layers of water at the surface of ice (known as  premelted films) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' These allow growth of the ice in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In both cases ice will continue to grow, building  up pressure, until the pressure reaches a maximum value given by a temperature-dependent stall pressure, Pst [9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Pst is very similar to the concept of crystallization pressure, found when confined crystals grow from supersaturated  solutions [12–14] and to the concept of condensation pressure, when phase separation occurs in confinement [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Case (i): Ice growth in a  confining cavity  Case (ii): Ice growth in an  open pore  Water  Ice  (Pl, T)  (Pl, T)  Pl  Water (Pl, T)  Ice (σ,T)  σnn  σnn  A  B  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 1: (i) Ice generates pressure as it grows in a closed cavity, due to the expansion of water upon freezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' (ii) Ice growing  in an open pore is fed by nearby water, and this growth wedges open the cavity, generating stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Theoretical descriptions of these stress-generation mechanisms are rooted in the (generalized) Clapeyron equation,  a fundamental equation that describes equilibrium between a solid (ice) at pressure Ps, and a reservoir of liquid  ∗Electronic address: robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='style@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='ch  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='03895v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='soft]  10 Jan 2023  2  TABLE I: Ice/water parameter values at atmospheric pressure and 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='15K [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Density of ice  ρs  917 kg m−3  Density of water  ρl  997 kg m−3  Latent heat of fusion  qm  334 kJ kg−1  Melting temperature  Tm  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='15 K  Heat capacity of ice  cp  s  2093 J (kg K)−1  Heat capacity of water  cp  l  4184 J (kg K)−1  Bulk modulus of ice  Ks  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='33 GPa  Bulk modulus of water  Kl  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='96 GPa  Coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' thermal expansion, ice  αs  51 × 10−6 K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' thermal expansion, water  αl  −50 × 10−6 K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (water) at a different pressure Pl [8, 17, 18]:  (Ps − P0)  ρs  − (Pl − P0)  ρl  = qm(Tm − T)  Tm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (1)  Here, ρl and ρs are the densities of water and ice respectively, qm is the specific latent heat of freezing of ice, and  Tm is the melting temperature at a reference pressure, P0 (often taken as atmospheric pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For the freezing  mechanisms described above, this equation can be used to predict Pst as a function of the temperature, T, as at this  point, the ice is in equilibrium with the nearby water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For case (i) with ice growing in a closed cavity, the ice and  water are both at the same pressure (Ps = Pl = Pst), so  (Pst − P0)  � 1  ρs  − 1  ρl  �  = qm(Tm − T)  Tm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (2)  Using values from Table I, we find that ice can exert pressures of ∼ 11 MPa per degree of undercooling (Tm − T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  For case (ii), the ice and water need no longer have the same pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' If the water reservoir is held at the reference  pressure Pl = P0, then Pst = Ps and  (Pst − P0)  ρs  = qm(Tm − T)  Tm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (3)  In this case, ice can exert pressures of ∼ 1 MPa per degree of undercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Even when ice is not in equilibrium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' it is growing), the Clapeyron equation gives us useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' During  growth, there is no macroscopic equilibrium, but water immediately adjacent to the ice/water interface can often be  considered to be in equilibrium with the ice [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Then, the Clapeyron equation relates the local hydrodynamic pressure  in the water, Pl, to the local pressure that has been built up in the ice (Pl is just the pressure of a hypothetical,  reservoir of water in thermodynamic equilibrium with the ice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Water flows along nonhydrostatic gradients in Pl, so  the Clapeyron equation allows us to predict how water is transported towards (or away from) ice, and thus gives ice  growth/melting rates [8, 20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  The various applications of the Clapeyron equation make it a key tool for understanding freezing processes [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  1, 6, 8, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, it makes a number of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For example, it assumes that ice can be described by  an isotropic pressure, whereas ice is often characterized by an anisotropic stress state, σij [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' It also uses linear  approximations that are valid only near the bulk melting point of ice (see later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, several key questions arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In  particular: What is the appropriate extension of the Clapeyron equation for anisotropically stressed ice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Over what  range of conditions should the Clapeyron equation be applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Surprisingly, we are not aware of a systematic derivation of the Clapeyron equation that would allow us to address  these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, there are several related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For example, several authors have established the ther-  modynamic relations that govern the dissolution of anisotropically stressed solids into adjacent fluids [25–27], with  notable applications to recrystallization and pressure solution processes [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Although melting was not a focus of  these works, some of the consequences for ice melting were recognized by Nye [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' He argued that the phenomenon  of wire regelation requires a generalization of equation (2) where Ps replaced by the normal stress −σnn, and not  by the mean of the principal stresses −Tr(σ)/3, as had been argued by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' A further, unflinching critique of  alternative incorrect theories is given by Kamb [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Finally, Sekerka and Cahn [30] examined the special case of a  solid with σnn = −Pl, to show that anisotropically stressed solids in equilibrium with their melt will recrystallize to  form a isotropically-stressed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  3  Here, we provide a first-principles derivation of the generalized Clapeyron equation, along similar lines to Pater-  son [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' We clearly lay out all the underlying assumptions, and present the appropriate extension for the melting  behavior of anisotropically stressed ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Deriving the Generalized Clapeyron Equation  We consider thermodynamic equilibrium for the two scenarios shown in Figure 1, in both of which the temperature  is held fixed at T < Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In case (i), water freezes in a closed cavity, so that the ice and and water both have the same  pressure, Ps = Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In case (ii), ice has frozen in an open cavity, and is in equilibrium with neighbouring bulk water,  which has pressure Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' At the same time, the ice exerts a normal stress −σnn on the walls of the cavity, but negligible  shear forces, due to the presence of premelted films which lubricate the ice/cavity interface [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The ice cannot grow  through the small, connecting pore throat into the neighboring water due to capillarity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' the Gibbs-Thomson effect  [31, 32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  For each scenario, we establish equilibrium behavior by minimizing the relevant free energy of the ice/water system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  The relevant free energy, Gsys satisfies ∆Gsys = ∆Usys−T∆Ssys+W, where Usys is the internal energy of the ice/water  system, Ssys is its entropy, and W is the work done by the system on its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For case (i),  ∆Gsys = 0 = ∆Usys − T∆Ssys + Pl(∆Vs + ∆Vl),  (4)  for case (ii),  ∆Gsys = 0 = ∆Usys − T∆Ssys − σnn∆Vs + Pl∆Vl,  (5)  where Vs and Vl are the volumes of ice and water, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The first case is just a specialized version of the  second, where −σnn = Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, without loss of generality, we can proceed with equation (5), and the result will  describe both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  We consider a small perturbation in the system in Figure 1b, where a small mass of ice, ∆m, melts and flows into  the reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, the volumes of ice and water change as ∆Vs = −vs∆m, and ∆Vl = vl∆m, where vs(σij, T) and  vl(Pl, T) are the specific volumes of the ice and water, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Then equation (5) becomes  ul∆m − us∆m − T(sl − ss)∆m + σnnvs∆m + Plvl∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (6)  Here, us(σij, T) and ul(Pl, T) are the specific internal energies of the ice and water, respectively, and ss(σij, T) and  sl(Pl, T) are the respective specific entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Dividing through by ∆m, we obtain  − (σnnvs + Plvl) = (ul − us) − T(sl − ss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (7)  In principle, equation (7) completely describes equilibrium between ice and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' one could use tabulated  values of u, v, and s to find −σnn(Pl, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, a more convenient form is found by expressing the equation  relative to the pressure and temperature under bulk melting reference conditions, (P0, Tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Here, −σnn = Pl = P0 so  equation (7) becomes  P0(vo  s − vo  l ) = (uo  l − uo  s) − Tm(so  l − so  s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (8)  The superscript o indicates reference conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Subtracting equations 7 and 8, we find  gl − go  l = gs − go  s,  (9)  where the specific free energies gl(T, Pl) = ul − Tsl + Plvl, and gs(T, σij) = us − Tss − σnnvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  These can be  Taylor-expanded to obtain the Clapeyron equation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' [1, 33]):  gl(T, v) = go  l (Tm, P0) +  �∂gl  ∂T  �  Pl  (T − Tm) +  � ∂gl  ∂Pl  �  T  (Pl − P0)  (10)  and  gs(T, σij) = go  s(Tm, P0) +  �∂gs  ∂T  �  σij  (T − Tm) +  � ∂gs  ∂σij  �  T  (σij + P0δij),  (11)  4  where δij is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' To evaluate the derivatives, we note that ∆gl = −sl∆T + vl∆Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, at reference  conditions,  �∂gl  ∂T  �  Pl  = −so  l ,  � ∂gl  ∂Pl  �  T  = vo  l ,  (12)  and similarly in the solid at reference conditions (σij = −P0δij)  �∂gs  ∂T  �  σij  = −so  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (13)  To calculate the final derivative, we notice that gs = (fs + P0vs) − ¯σnnvs, where fs is the specific Helmholtz free  energy of the solid, and ¯σij = σij + P0δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Here, (fs + P0vs)/vo  s is the free-energy per unit volume of deformations in  an atmosphere at constant pressure, P0, and thus is the elastic energy per volume of ice in the reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' As  such, we can use linear elasticity to write  gs = 1  2 ¯σijϵijvo  s − ¯σnnvo  s  �  1 + Tr(¯σ)  3Ks  �  ,  (14)  where Ks is now the bulk modulus of the solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' ϵij is the strain of the ice relative to its shape in the reference state  (Tm, P0), and satsifies the linear-elastic constitutive relationship:  ϵij = 1  Es  [(1 + νs)¯σij − νsδijTr(¯σ)] ,  (15)  where Es = 3Ks(1 − 2νs) is the Young’s modulus of the ice and νs is its Poisson ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For small strains, vs =  vo  s(1 + Tr(ϵ)), and we use this in the second term of equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  With the two equations above, we can evalulate the remaining derivative at (P0, Tm):  � ∂gs  ∂σij  �  T  (¯σij = 0) = −ninjvo  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (16)  Here, ni is the normal vector to the surface of the ice, so that ¯σnn = ni¯σijnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Finally, we can insert these first-derivative expressions into equations (9–11) to obtain the Clapeyron equation for  anisotropically stressed solids:  − (σnn + P0)  ρos  − (Pl − P0)  ρo  l  = qm(Tm − T)  Tm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (17)  Here, ρo  l = 1/vo  l , and ρo  s = 1/vo  s are the densities of water and ice respectively at the bulk melting point, and  qm ≡ (so  l − so  s)Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Consistent with the regelation analysis of Nye [29], this version of the Clapeyron equation is  identical to equation (1), but with Ps replaced by −σnn, and not −Tr(σ)/3, as one might naively assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Field data supporting the anisotropic Clapeyron equation  Glaciological field data supporting the form of equation (17) comes from simultaneous measurements of temperatures  and liquid pressures in glacier boreholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' These measurements show that temperatures increase when changes in the  hydrologic system cause borehole pressures, Pl, to decrease [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  The anisotropic Clapeyron equation indeed recovers this correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Along borehole walls, σnn = −Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Inserting  this into equation (17), we find that changes in temperature are correlated with changes in borehole pressure by:  ∆T = −Tm  qm  � 1  ρ0s  − 1  ρ0  l  �  ∆Pl ≈  �  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='16 × 10−8 K Pa−1�  ∆Pl ,  (18)  in agreement with the field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  By contrast, naively extending the isotropic Clapeyron equation (1), by replacing −Ps = Tr(σ)/3, does not match  the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The classic analysis of Nye [36] gives the complete stress tensor at the surface of an idealized  cylindrical borehole containing liquid at pressure Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Far from the borehole, the ice has a far-field isotropic ice pressure  P∞, and creeps according to Glen’s flow law with exponent n = 3 [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In this case, −Tr(σ)/3 = Pl+(P∞ − Pl) /n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  5  Substituting Ps = −Tr(σ)/3 into the isotropic Clapeyron equation (1) and treating the far-field ice pressure as constant  leads to  ∆T = −Tm  qm  � 1  ρ0s  − 1  ρ0  l  −  1  nρ0s  �  ∆Pl ≈  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='26 × 10−7 K Pa−1�  ∆Pl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (19)  This predicts the opposite of the correlation seen in the field data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Errors in the Clapeyron equation  In deriving this version of the Clapeyron equation, we have had to make two main assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Firstly, strains in  the ice are small, so we can use linear elasticity [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' This is reasonable as the stresses in the ice (which are O(MPa)  – see introduction) are much less than the ice’s elastic moduli Es, Ks = O(GPa), so strains will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Secondly, we assume that higher-order terms in the expansions of gl and gs are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' We can test this by  reverting to the case of isotropically-stressed ice (σij = −Psδij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Then, we Taylor-expand equation (9) in T, Pl and  Ps to obtain the second-order version of the Clapeyron equation:  (Ps − P0)  ρos  − (Pl − P0)  ρo  l  =  qm(Tm − T)  Tm  (20)  −cp  l − cp  s  2Tm  (Tm − T)2 −  1  2ρo  l Kl  (Pl − P0)2 +  1  2ρosKs  (Ps − P0)2  −αl  ρo  l  (Tm − T)(Pl − P0) + αs  ρos  (Tm − T)(Ps − P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Here, we use the following identities [39]:  ∂2g  ∂T 2 = − cp  Tm  ,  ∂2g  ∂P 2 = −  1  Kρo ,  ∂2g  ∂P∂T = α  ρo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (21)  cp is the heat capacity at constant pressure, K is again the isothermal bulk modulus, and α is the coefficient of  thermal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  We can now predict the pressure-melting curve for different freezing scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For bulk ice/water equilibrium  (Figure 1a), Ps = Pl, and we take atmospheric pressure, Pa, as the reference pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Figure 2a compares the  isotropic Clapeyron equation (1) (red, dashed) with experimental data (black, dotted) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' There is a significant  error between the two results for an undercooling of more than ∼ 3 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, when we use the full, second-order  Clapeyron equation (20) (blue), we find good agreement down to an undercooling of at least 15 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In this situation,  the terms that are quadratic in pressure dominate the error, and to excellent approximation (Figure 1a, orange  dash-dotted):  � 1  ρos  − 1  ρo  l  �  (Ps − P0) +  �  1  2ρo  l Kl  −  1  2ρosKs  �  (Ps − P0)2 = qm(Tm − T)  Tm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (22)  Comparing the first two terms in this equation, we see that the linear equation (17) is only appropriate when:  |Ps − Pa| ≪ ∆P ∗ =  �����  � 1  ρos  − 1  ρo  l  � �  1  2ρosKs  −  1  2ρo  l Kl  �−1����� ≈ 420MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (23)  We can perform a similar analysis for freezing in an open system (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' We let Pl = P0 = Pa, and assume  that the ice exerts an isotropic pressure, Ps on its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Figure 2b compares the prediction of the Clapeyron  equation (1) (red, dashed) with that obtained when we keep the extra quadratic terms (20) (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' We are not aware  of any experimental data precise enough to validate the theory [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, here, the higher-order theory agrees  well with the linear Clapeyron equation down to large undercoolings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The difference is dominated by the term in  equation (20) that is quadratic in undercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, to excellent approximation (Figure 1a, orange dash-dotted):  Ps − Pa = ρo  sqm(Tm − T)  Tm  − ρo  s(cp  l − cp  s)  2Tm  (Tm − T)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (24)  6  260  265  270  T [K]  0  5  10  15  Ps-P0 [MPa]  Full 2nd  order theory  Case (ii): Ice growth in an open pore  Pl=Pa  Linear  Clapeyron Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Simplified 2nd  order theory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 25)  0  50  100  150  200  250  Ps-P0 [MPa]  Simplified 2nd  order theory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 23)  Full 2nd  order theory  Linear  Clapeyron Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Case (i): ice growth in a closed cavity  Pl=Ps  Experimental  data  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 2: Evaluating the accuracy of the Clapeyron Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' A) The pressure of ice in bulk ice/water equilibrium in a closed  cavity (Pl = Ps = −σnn), as a function of undercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' The black, dotted curve shows experimental data [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' B) The stress  exerted by ice in an open pore, as a function of undercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Both figures show the linear Clapeyron equation (dashed red),  full 2nd-order theory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 20, blue), and simplified 2nd-order theory (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 22,24, orange dash-dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Comparing terms on the right-hand side, shows that we only recover the linear Clapeyron equation (1) if  |Tm − T| ≪ ∆T ∗ =  ����  2qm  cp  l − cp  s  ���� ≈ 320K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  (25)  This requirement is certainly satisfied for most terrestrial temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, there is some justification for use of  the linear Clapeyron equation down to relatively large undercoolings to model this type of freezing scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  To summarize, our results suggest that the linearized Clapeyron equation will be valid, provided that |Pl − P0| and  |Ps − P0| are both small relative to ∆P ∗, while |Tm − T| ≪ ∆T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' At larger pressures/undercoolings, the quadratic  terms in equation (20) should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Conclusions  In conclusion, we have derived the linear, Clapeyron equation describing equilibrium between water and ice, clearly  laying out all the assumptions involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In particular, this equation is derived using a Taylor expansion around a  reference temperature and pressure, and ignoring higher-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Thus, it is only valid for a range of pressures  and and temperatures around the reference conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Fortunately, for most naturally-occurring terrestrial freezing  scenarios, the linear form of the Clapeyron equation should be adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' For example, at the base of a glacier,  pressures are typically close to hydrostatic, and thus O(MPa) [41] – this is small enough to lie within the range of  applicability of the Clapeyron equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, more extreme conditions are expected in extraterrestrial settings  [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' There, the linearized Clapeyron equation will not accurately predict melting temperatures, which could  7  lead to significant errors in models of ice dynamics (as predicted flow rates are typically based on the departure from  bulk melting conditions [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' In this case, the accuracy of the Clapeyron equation can be improved by retaining  higher-order terms in the Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  We have also demonstrated the correct form of the Clapeyron equation for the case where ice is anisotropically  stressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' This is identical to the isotropic form of the Clapeyron equation, but with ice pressure, Ps replaced by the  normal stress exerted by ice on its surroundings, −σnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' One consequence of this, is that differently stressed faces of  ice (for example in a polycrystal) will have different melting temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  While our analysis has focused on ice and water, the results should apply to any processes involving solid/liquid  equilibrium, for example in the melting and deformation of rocks in geological processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Note however,  that there are two, key further effects that will likely be important to include in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' Firstly,  we have neglected the presence of solutes, which are known to strongly affect the solid/liquid equilibrium [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Secondly, we we have ignored the surface energy of the ice [8, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' However, we anticipate that both of these effects  can be incorporated into the results presented here, by including colligative and capillary effects in the analysis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content='  Acknowledgments  RWS and DG acknowledge support from an ETH Research Grant (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' ETH-38 18-2), and from the Swiss  National Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' 200021-212066);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
+page_content=' AWR received funding from NSF-2012468 and a UO Faculty  Research Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hdE2T4oBgHgl3EQfcgcA/content/2301.03895v1.pdf'}
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+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Chin-Yi Cheng 1 Forrest Huang 1 Gang Li 1 Yang Li 1
+Abstract
+Layout design is an important task in various de-
+sign fields, including user interfaces, document,
+and graphic design. As this task requires tedious
+manual effort by designers, prior works have at-
+tempted to automate this process using generative
+models, but commonly fell short of providing intu-
+itive user controls and achieving design objectives.
+In this paper, we build a conditional latent diffu-
+sion model, PLay, that generates parametrically
+conditioned layouts in vector graphic space from
+user-specified guidelines, which are commonly
+used by designers for representing their design
+intents in current practices. Our method outper-
+forms prior works across three datasets on metrics
+including FID and FD-VG, and in user test. More-
+over, it brings a novel and interactive experience
+to professional layout design processes.
+1. Introduction
+Layouts are important artifacts that represent the design and
+arrangements of their encapsulated elements. They are used
+extensively in creative fields ranging from design to engi-
+neering, supporting the authoring processes of numerous
+downstream products, such as user interfaces (UIs), docu-
+ments, posters, architectural floorplans, and even printed
+circuit boards (PCBs). Designing a good layout requires
+thorough consideration of different aspects, such as the func-
+tion, aesthetics, and domain-specific conditions and rules.
+Moreover, many of these aspects and objectives cannot be
+easily and explicitly evaluated and computed. Therefore,
+layout design has been a time-consuming, iterative, and
+manual process.
+Several recent works have tried to improve the layout design
+process by automating it using generative models. However,
+these models are typically unconditional, making them dif-
+ficult for users to adopt in realistic applications. A large
+number of unconditionally, randomly generated layouts are
+1Google Research, Mountain View, United States.
+Corre-
+spondence to: Chin-Yi Cheng <cchinyi@google.com>, Yang
+Li <liyang@google.com>.
+Copyright 2023 by the author(s).
+Figure 1. Sample results. The red lines on the left of each example
+represent the guideline conditions, and the generated layout is
+on the right. The generated layouts are also placed under the
+guidelines to visualize their alignments to the guidelines.
+not useful in practice since it requires users to manually
+check each of them on whether the original objectives and
+constraints have been met. More importantly, for users to
+enjoy co-designing with generative models and trust the gen-
+erated layouts, they need to be able to express their design
+ideas and drive the generation process (Louie et al., 2020).
+Therefore, a model conditioned on user inputs would help
+their adoption in users’ workflows, and the type of supported
+inputs should represent design intention and constraints.
+The key towards achieving our goal of conditional genera-
+tion that can be widely adopted is to choose an adequate type
+of condition for generating design layouts. Different types
+of conditions have been explored in image-based generative
+models, such as text (Saharia et al., 2022b; Rombach et al.,
+2022; Ramesh et al., 2022) and colored semantic segmenta-
+tion (Park et al., 2019), but these might not be ideal choices
+for layout generation. Text conditions have shown to be
+powerful for artistic purposes, but they cannot provide exact
+and detailed control critical for design. On the other hand,
+semantic segmentation allows for precise control, but is te-
+dious and time-consuming for users to manually author all
+segments in the scene, which is equivalent to hand-drawing
+the entire layout in our case of layout design.
+In this work, we investigate layout design workflows in
+several domains, including UI, document, and architectural
+design, and identify a widely used representation: guide-
+arXiv:2301.11529v1  [cs.LG]  27 Jan 2023
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 2. Layout and guidelines. A layout (b) of a UI design
+(a) consists of vector graphic elements with types and geometric
+properties. Guidelines (c) can represent the ideas and constraints.
+lines (e.g., the red lines in Figure 2c), as conditions. Guide-
+lines, also referred to as guides, grids, or partitions, are
+a set of lines that serves multiple purposes, for instance:
+partitioning the design space for preferred proportions and
+alignments; expressing design ideas such as the two-column
+style (top-left example, Figure 1); and showing rules such as
+paddings, margins, and gaps between elements. However, a
+common pain point for designers is that changing guidelines
+(which express design thoughts) would require manually
+redrawing and adjusting the associated design to conform
+to the new guidelines. Therefore, designers currently use
+guidelines only as visual references or a way to document
+their thoughts and yet cannot benefit from the rich design
+information encapsulated by them.
+Hence, using guidelines as the input conditions to our model
+can not only provide a high-level yet precise control to users,
+but also solve a practical pain point in layout design work-
+flows across different domains. We consider the guideline
+condition as a type of parametric conditions, borrowing
+the definition from parametric design in the fields of com-
+putational design and computer-aided design (Monedero,
+2000). In parametric design, the design artifacts are defined
+using algorithms or procedures with parameters as inputs,
+and therefore users can easily make design changes by ma-
+nipulating the parameters. Our work enables an intuitive
+way to instantiate parametric models for layouts by drawing
+guidelines, as opposed to manually setting the relationships
+and heuristics in traditional parametric design tools.
+We introduce PLay, Parametrically Conditioned Layout
+Generation using Guidelines, a two-stage model follow-
+ing the idea of latent diffusion models (LDMs) proposed
+by (Rombach et al., 2022). Compared to (Rombach et al.,
+2022), the main purpose of our first-stage model is to con-
+vert the layout from the discrete vector graphics space to
+the continuous latent space instead of compressing informa-
+tion in original data samples. With this continuous latent
+representation, the second-stage diffusion model can iter-
+atively refine the results similar to image-based diffusion
+models. PLay can generate layout designs conditioned on
+guidelines across three different datasets with significantly
+improved quality. We measure the quality by computing the
+Fr´echet distances using layouts rendered in both image and
+vector graphics domains. We further evaluate the results by
+conducting user testing on professional designers.
+Using guidelines as a way to express both the high and
+low-level ideas, PLay also enables several interactive and
+controllable ways for users to create desired layouts: 1)
+generating and editing layouts by dragging, adding, and
+removing guidelines, similar to drawing a sketch, 2) gener-
+ating variations from an existing layout with different levels
+of similarity controlled by guidelines, and 3) layout inpaint-
+ing. We envision PLay to improve layout design workflows
+in various domains, enable different types of conditions
+and interactions, and potentially solve more general vector
+graphics problems.
+In summary, the main contributions of this paper are:
+• We develop a latent diffusion model in the vector graph-
+ics domain for layout generation, achieving a better
+performance than prior work in multiple metrics with
+large margins.
+• We introduce guidelines as input conditions for the
+latent diffusion model, making the generation process
+parametrically controllable and interactive.
+• We provide a variety of new ways to generate and edit
+layouts, including guideline editing, layout inpainting,
+and generating variations from an existing layout.
+• We propose FID and FD-VG as metrics and conduct
+user testing to examine the quality of generated layouts.
+2. Related Work
+2.1. Layout Generation
+Several prior works have studied generative models for lay-
+outs in the vector graphics domain, and LayoutGAN (Li
+et al., 2019) is among the earliest ones. It uses self-attention
+layers as the generator and argues that the discriminator
+in the image domain can better evaluate the spatial quality,
+such as alignments of the elements. LayoutVAE (Jyothi
+et al., 2019), on the other hand, works purely in the vector
+graphics domain. It aggregates the information across the
+elements using Long Short-Term Memory (LSTM) (Hochre-
+iter & Schmidhuber, 1997) and trains variational autoen-
+coders (VAEs) (Kingma & Welling, 2013) for generation.
+A recent work, VTN (Arroyo et al., 2021), maintains the
+VAE architecture in LayoutVAE but replaces the encoder
+and decoder by Transformers (Vaswani et al., 2017). The
+results from VTN shows that with Transformers, the model
+can learn the proper layout arrangements without mapping
+results to the image domain. Our latent diffusion model in
+
+(a) UI Design
+(b) UI Layout
+(c) Guidelines
+(d) Legend
+##07:27
+Invite school
+Guideline
+Text
+Button
+Container
+Image
+Pictogram
+Tell us a bit more so that we can invite
+your school or daycare.
+List_item
+Toolbar
+Nuddke iddle
+Label
+Text_input
+iddle@gmail..com
+Drawer
+Card_view
+School Phone
+Send anonymously
+Spinner
+Check_box
+Invite!
+Navigation_bar
+Switch
+Page_indicator
+Progress_bar
+Number_stepper
+Date_picker
+口PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+PLay can also be seen as a way to learn a better latent prior
+than VTN.
+Various types of conditions for conditional layout genera-
+tion have also been studied. For example, (Li et al., 2020)
+adds element attributes such as area and aspect ratio as con-
+ditions, and (Lee et al., 2020) uses graph neural networks to
+constrain inter-element relationships. In the image domain,
+House-GAN (Nauata et al., 2020) and House-GAN++ (Nau-
+ata et al., 2021) also explore graph conditions for layout
+generation. However, the shared issues of these approaches
+are 1) users have to tediously assign the properties of rela-
+tionships between each of the elements, and 2) the allowed
+number of elements and relationships for conditions are of-
+ten too small for a complex layout with more than 20 objects.
+As discussed in 1, the guideline conditions in PLay can over-
+come these issues, since one guideline can align multiple
+elements, and multiple guidelines can define complex rules.
+Guidelines can also be seen as space partitioning in 3D. For
+example, (Xu et al., 2021) partitions the 3D space to search
+for potential CAD modeling sequences, and (Chang et al.,
+2021) uses the partitioned space as one of the input condi-
+tions for 3D volume generation. Both works use partitions
+directly as the design or search space, meaning that if the
+given partitions are not good enough, the quality of the re-
+sults will be affected. In contrast, PLay can flexibly attend
+to various guidelines through cross-attention and generate
+elements that do not follow some guidelines if needed.
+In addition, some recent works incorporate other modalities
+such as images into layout generation: CanvasVAE (Yam-
+aguchi, 2021) uses the image information for the elements in
+the layout generation process, and CGL-GAN (Zhou et al.,
+2022) generates layouts that fit the given images. Canvas-
+VAE has a two-stage encoder-decoder architecture inspired
+by DeepSVG (Carlier et al., 2020), with the first-stage for
+the individual elements and the second-stage for the layout.
+In PLay, the element stage is not needed since each element
+in our datasets is composed by its class and bounding box
+coordinates only. However, it is worth experimenting with
+the element stage for PLay in the future, such as incorporat-
+ing hierarchical decoding introduced by (Jiang et al., 2022)
+which can yield better results. This can also augment PLay
+to solve more complex tasks, such as CAD layout genera-
+tion, where each element can be any shapes or curves.
+2.2. Diffusion Models
+Recent advances in Diffusion Models (DMs) (Sohl-
+Dickstein et al., 2015) have shown promising results in
+image generation (Song et al., 2020; Ho et al., 2020; Dhari-
+wal & Nichol, 2021), where classifier guidance (Dhariwal
+& Nichol, 2021) and classifier-free guidance (CFG) (Ho
+& Salimans, 2022) methods not only improve the genera-
+tion quality but also enable the possibility of developing
+conditional diffusion models. Most recent works for text-to-
+image generation use CFG, including GLIDE (Nichol et al.,
+2021), DALL·E2 (Ramesh et al., 2022), Imagen (Saharia
+et al., 2022b), and LDMs (Rombach et al., 2022), and in
+addition to text, LDMs explore other conditions such as
+bounding boxes and semantic maps. We also adopt CFG for
+the guideline conditions in PLay, as it does not require an
+extra classifier and lead to the generation of better results.
+Applying diffusion models outside of the image domain
+has also drawn attention from researchers, such as 3D (Lin
+et al., 2022), video (Ho et al., 2022), and music (Mittal et al.,
+2021) generation. (Mittal et al., 2021) converts the discrete
+melody tokens into a continuous latent space, and trains the
+diffusion model in the latent space. This work inspires us
+to convert the discrete layout elements, composed by con-
+catenating different types of tokens including their class and
+coordinates, to a continuous domain for the diffusion pro-
+cess. A concurrent work, (Strudel et al., 2022), applies the
+same idea for language generation. We also follow LDMs
+(Rombach et al., 2022) to add a small KL-penalty to ensure
+high layout reconstruction quality and avoid arbitrarily large
+variance in the latent space.
+Several recent works explore methods to further control the
+diffusion process. For instance, Prompt-to-Prompt (Hertz
+et al., 2022) fixes the cross-attention maps to preserve scene
+compositions for a new text prompt; SDEdit (Meng et al.,
+2021) injects the stroke-based conditions by adding noise
+to it and denoising it back to a real image. The level of
+added noise becomes a parameter to control the balance
+between image realism and faithfulness to the input drawing.
+Compared to these works, PLay can provide more explicit
+control to generate variations of previously generated layout,
+by extracting and editing the guidelines (5.4.1).
+3. Layout Design
+3.1. Datasets
+We experiment PLay with three publicly available datasets
+for two different domains: UI and document layouts.
+• CLAY (Li et al., 2022) contains about 50K UI layouts
+with 24 classes.
+• RICO-Semantic (Liu et al., 2018) contains about 43K
+UI layouts with 13 classes previously used in VTN.
+• PublayNet (Zhong et al., 2019) contains about 330K
+document layouts with 5 classes.
+The layouts in CLAY are more complex and representative
+of real UI designs compared to RICO-Semantic. Although
+both of them are extracted and processed from RICO (Deka
+et al., 2017), CLAY tries to fix some annotation errors and
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+mismatches between the screenshots and view hierarchies
+and introduces new label systems, whereas RICO-Semantic
+adds semantic annotations for RICO. We suspect that ei-
+ther RICO-Semantics filtered some complex patterns during
+post-processing, or the chosen 13 classes from the 25 origi-
+nal classes largely reduced the complexity.
+To verify that PLay can be applied to other layout domains,
+we also train it on PublayNet to generate document layouts.
+Compared to CLAY, both RICO-Semantic and PublayNet
+have fewer design variations, and their layouts are overall
+simpler and have more repetitive patterns. For example,
+the complexity of each dataset is reflected by its average
+number of elements in each layout: RICO-Semantic=8.79;
+PublayNet=7.90; CLAY=19.62. Therefore, while we evalu-
+ate PLay over all the three datasets, we particularly conduct
+in-depth evaluation and analysis of our model on CLAY
+with qualitative and quantitative results. See the appendix
+for more statistics and examples of the datasets.
+3.2. Layout and Guidelines
+The shared data format across the three datasets for a layout
+is a sequence of elements: E = {e1, e2, ..., eN}, where
+en = {cn, xn
+min, yn
+min, xn
+max, yn
+max} and 1 ≤ n ≤ N. The
+class of an element, cn, and each coordinate (an integer) of
+its bounding box, xn
+min, yn
+min, xn
+max, and yn
+max, is repre-
+sented as a one-hot vector, which are then concatenated to-
+gether as a single vector, and therefore E ∈ {0, 1}N×D. We
+fix the maximum number of elements per layout: N = 128,
+and the layout with fewer elements are padded to the same
+size. We set the dimension of each layout with width = 36
+and height = 64, which result in a fixed D for all layouts.
+The guidelines of a layout is represented as:
+G
+=
+{g1, g2, ..., gM}, where gm = {am, pm} and 1 ≤ m ≤ M.
+Each guideline is composed by its axis am, i.e., horizontal
+versus vertical, and the coordinate position pm. We fix the
+maximum number of guidelines for each layout: M = 128,
+and thus the maximum number of guidelines in each axis is
+M/2. The representation of layouts involving fewer guide-
+lines is padded. To create the layout-guidelines pairs for
+training, for each layout E in the datasets, we can intuitively
+obtain the full guidelines Gfull by extending all the bound-
+ing box edges of each element and removing the duplicated
+values. For the model to learn how to synthesize valid details
+beyond the given conditions, we also have three different
+sampling methods to create random subsets of Gfull during
+training. More details will be discussed in 5.2.
+3.3. Metrics
+There are no universally established metrics to evaluate lay-
+out generation. Prior works such as (Arroyo et al., 2021;
+Li et al., 2019) compute several features in both real and
+generated layouts such as IoU, overlap, and alignment; and
+(Yamaguchi, 2021) computes the feature-wise distribution
+differences. Instead of using feature-based methods, we
+follow the common practice of evaluating Generative Ad-
+versarial Networks (GANs), including several prior layout
+generation works (Nauata et al., 2020; 2021; Lee et al.,
+2020), to measure the Fr´echet Distance (FD) between two
+distributions from latent space. To capture various aspects
+of layout generation, we compute FD in two ways:
+• FID (Heusel et al., 2017): we render the layouts into
+images with the same aspect ratio and add paddings
+if the elements are not fully using the screen space.
+We then feed the images into the pre-trained Incep-
+tion (Szegedy et al., 2017) model to get the activation
+vectors, and compute the FD between the real and gen-
+erated groups.
+• FD-VG: we train a Transformer-based auto-encoder in
+the vector graphics domain, use its encoder to encode
+the layouts, and compute the FD.
+G-Usage: We also evaluate if the generated results satisfy
+the guideline conditions by computing the G (guideline)-
+Usage. We first extract the guidelines G∗ from a generated
+layout, then we calculate the intersection, Ginter, for G∗
+and the given guidelines, G. Finally, we obtain the G-Usage
+as |Ginter|/|G| and take the average across the generated
+samples. Note that G-Usage does not equal to IoU, since it
+is valid to have guidelines in G∗ that are not part of G.
+User Test: We also conduct user testing (5.3) with profes-
+sional designers and the results are aligned with the FID and
+FD-VG metrics.
+4. Architecture
+Following the image-based latent diffusion model (Rom-
+bach et al., 2022), We formulate PLay as a two-stage model,
+where the first-stage model learns to map layouts from a
+vector graphics space to a latent space, and then the condi-
+tional latent diffusion model learns to generate layouts in
+latent space conditioned on guidelines given by the users.
+Figure 3 illustrates the overview of our model.
+4.1. First-Stage Model
+To map the layout E ∈ RN×D to the latent representation
+z ∈ RN×d, we use a Transformer similar to DETR (Carion
+et al., 2020), with an encoder E(E) and a non-autoregressive
+decoder D(z) modified from DeepSVG (Carlier et al., 2020).
+We also added a small KL-penalty to regularize the latent
+space while keeping the high reconstruction accuracy, which
+is especially critical in the vector graphics space. The reason
+is that unlike pixels, where some artifacts and noises are not
+noticeable, we only have at most 128 elements in a layout,
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 3. Model architecture. After training the first-stage model, we can use it to encode layouts to the latent space for training the latent
+diffusion model. During sampling, it can decode the generated latent representations back to layouts.
+Figure 4. Qualitative results. We can observe that PLay generates reasonable results on complex examples with many elements, whereas
+VTN often struggles in such cases.
+so it will be obvious if any of them is decoded incorrectly,
+such as having a miss-aligned box or an unreasonable class.
+Following VTN, no positional embedding is added to the
+encoder and decoder, as the coordinates already explicitly
+indicate the positions spatially.
+In image-based latent diffusion models (LDMs), the goal of
+the first-stage model is to map the input image to a lower
+dimension while keeping the same perceptual details. How-
+ever, the first-stage model in PLay serves a different purpose:
+to learn a meaningful and continuous latent representation
+for the discrete vector graphic space. We will see in 5.1, that
+the LDM cannot converge with mapping using naive MLP
+layers. In addition, we experiment with encoding the entire
+layout as a single vector using a transformer, but it also fails
+to achieve high reconstruction accuracy.
+4.2. Latent Diffusion Model
+When training the latent diffusion model, the encoded layout
+z is first divided by the standard deviation std of the first
+batch, as suggested by (Rombach et al., 2022), and then the
+scaled z is used as z0 for the forward diffusion process to
+get zt; t = 1...T. For the denoise network ϵθ(zt, τψ(G), t),
+we use a Transformer encoder to replace the U-Net struc-
+ture used in image-based DMs and predict the noise ϵ. The
+discrete time step t is encoded using Feature-wise Linear
+Modulation (Perez et al., 2018) and injected into the Trans-
+former encoder with a feature-wise affine layer. We encode
+the guidelines G using another Transformer encoder τψ, and
+the encoded guidelines, τψ(G) ∈ RM×d, are then fed to ϵθ
+through cross-attention. The loss function can be formulated
+as general LDMs:
+L := EE(x),G,t,ϵ∼N (0,1)
+�
+∥ ϵ − ϵθ(zt, τψ(G), t) ∥2�
+(1)
+We train the LDM as a standard DDPM (Ho et al., 2020)
+with classifier-free guidance, where we randomly drop the
+guideline conditions with the probability pdrop = 0.1.
+4.3. Sampling
+In sampling, we first either sample the number of elements
+N from p(N), which is the element count distribution of the
+dataset, or use n assigned by the user. Then we initialize zT
+and denoise it to get z0 with the given guideline conditions
+using DDPM and CFG, with w = 1.5:
+ˆϵθ(zt, τψ(G), t) = (1 + w)ϵθ(zt, τψ(G), t)
+− wϵθ(zt, τψ(φ), t)
+(2)
+
+(a) Two-Stage Training
+(b) Sampling
+First-Stage
+Latent Diffusion
+V.G. Space
+Latent Space
+oz
+uz 
+clso
+zo
+zn 
+zn 
+clsn
+a°
+am
+ymin
+Ymin
+29
+Forward Proc.
+t
+oQ
+pm
+Screen
+t
+ao
+am
+Screen
+Decoder
+to
+Decoder
+po
+pm
+ymax
+Ymax
+z
+Guideline
+Denoising
+Encoder
+Transfomer
+Guideline
+clso
+clsn
+clso
+clsn
+cross
+Encoder
+x0in
+Xmin
+cross
+Xmin
+Screen
+Denoising
+attn.
+attn.
+ymin
+ymin
+Encoder
+Transfomer
+ymin
+ym
+ymin
+yo
+Backward Proc.
+xmax
+Xmax
+yo
+ym
+ymax
+oz
+en
+ymex
+max
+2%
+uz(a) G.T. from the CLAY dataset
+(b) Samples generated by PLay
+(c) Samples generated by VTNPLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Table 1. Quantitative Results and ablation studies.
+Model
+F.S.
+CFG-W
+FID
+FD-VG
+G-Usage
+VTN
+×
+×
+19.10
+0.352
+n/a
+C-VTN
+×
+×
+16.22
+0.361
+0.819
+PLay
+VAE
+1.25
+12.63
+0.286
+0.970
+VAE
+1.50
+10.59
+0.245
+0.964
+VAE
+1.75
+11.21
+0.269
+0.968
+×
+1.5
+166.7
+4.577
+0.992∗
+VAE
+×
+14.80
+0.375
+n/a
+VTN
+1.50
+14.35
+0.311
+0.835
+VQVAE
+1.50
+11.49
+0.254
+0.937
+∗PLay without the first-stage model generates mostly random,
+unaligned boxes and therefore, has a high guideline usage.
+Table 2. Comparisons across datasets.
+Dataset
+Model
+FID
+FD-VG
+G-Usage
+CLAY
+VTN
+19.10
+0.352
+n/a
+PLay
+10.59
+0.245
+0.964
+RICO-
+VTN
+18.80
+0.415
+n/a
+Semantic
+PLay
+13.00
+0.320
+0.944
+PublayNet
+VTN
+19.80
+0.787
+n/a
+PLay
+13.71
+0.408
+0.971
+We then rescale z0 with the std used in training, and decode
+it back to the vector graphic domain using the first-stage
+decoder: E = D(z).
+Figure 5. User test results. Each cell represents the averaged score
+of the row-column comparison. For example, −0.375 means that
+VTN performs worse than PLay to designers.
+5. Experiments
+5.1. Baseline Comparison and Ablation Study
+In this section, we show how PLay performs compared to the
+baseline and study the effects of different first-stage model
+choices and classifier-free guidance weights. We choose
+VTN (Arroyo et al., 2021) as the baseline model for com-
+parison because it is the most common framework of SoTA
+models for UI layout generation in vector graphic space and
+is easily reproducible. It shares the same architecture with
+Table 3. FID scores in groups of number of elements.
+Model
+All
+1-6
+7-11
+12-18
+19-29
+30+
+VTN
+19.10
+11.83
+16.12
+19.89
+25.76
+30.15
+PLay
+10.59
+9.28
+11.72
+13.23
+13.60
+14.01
+Table 4. Guideline sampling methods.
+Method
+FID
+FD-VG
+G-Usage
+All
+12.11
+0.206
+0.917
+Uniform
+11.89
+0.312
+0.944
+Weight-tiers
+11.70
+0.297
+0.957
+Weighted
+10.59
+0.245
+0.964
+our first-stage model, a variational authoencoder (VAE). We
+also modify VTN to condition on guidelines, called C-VTN
+to ensure a fair comparison. In Table 1, we first try to train
+the diffusion model without the first-stage model. With
+simple MLP layers to encode the class and coordinates, the
+model fails to converge. We then add VAE as the first-stage
+model, and discover the results of latent diffusion training
+outperform the baseline. Adding the classifier-free guid-
+ance further improves the results, with the optimal weight
+w = 1.5 and latent dimension d = 8 for our experiments.
+We also try to use VQVAE as the first-stage model, but it
+falls short on G-usage while having comparable numbers
+in FID and FD-VG. The potential reason can be that the
+learned, frozen codebook is less flexible to exactly match
+the guideline conditions since they are not involved in the
+first-stage training. Additionally, we try to use trained VTN
+as the first-stage model, as it is also a VAE with KL loss
+weight β = 1.0. We achieve worse but still reasonable
+results, which can be explained by the low reconstruction
+accuracy using high β values. The reconstruction accuracy
+plays a crucial role for PLay, and it is not required to use a
+very small β value, e.g., 1e−6 in (Rombach et al., 2022), as
+long as the reconstruction accuracy is good enough. See the
+appendix for the complete table of first-stage model choices.
+PLay also outperforms prior work in all metrics across three
+datasets: CLAY, RICO-Semantic, and PublayNet (Table 2).
+Moreover, for this experiment, we use the same model ar-
+chitecture and hyper-parameters for all datasets. The only
+required change is the input dimension, which demonstrates
+PLay’s ability to generalize to different layout domains.
+We further divide the generated layouts into groups by the
+number of elements in each layout. We discover that PLay
+achieves much better FID scores (Table 3) and is qualita-
+tively better (Figure 4) than VTN in groups with a large
+number of elements. This shows that the advantage of PLay
+over the baseline is more pronounced when a layout is com-
+
+PLay
+VTN
+G.T.
+VTN
+-0.375 -0.313
+PLay
+0.375
+-0.104
+G.T.
+0.313
+0.104PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+plex and involves a large number of elements.
+5.2. Guideline Sampling Methods
+Users usually prefer to specify only the guidelines that can
+represent the main ideas instead of drawing guidelines for
+every element, because at this stage they have not come up
+with all the details for the design yet and want to see the
+potential options. Therefore, we randomly sample a subset
+of guidelines for every example during training. In this way,
+the model can learn to follow the given guidelines as the
+main guidance and create extra details that are not covered
+by the given guidelines.
+We investigate three guideline sampling methods during
+training: uniform, weighted, and weight-tiers. Uniform
+means we uniformly sample a subset of the guidelines. For
+weighted sampling, we compute the weight as how much
+each guideline is obeyed by the elements. If a guideline
+is followed by more elements for alignment, the guideline
+is deemed more important thus has a higher weight. For
+weight-tiers, we further divide the guidelines with different
+ranges of weight into groups and sample the groups as a
+whole. We find that weighted sampling achieves the best
+FID and G-Usage (Table 4). Including all guidelines has
+the best FD-VG score while the G-Usage suffers since it
+becomes a more difficult task to fit all the guidelines. Impor-
+tantly, the model that is trained with all the guidelines can
+only strictly follow the given guidelines and is incapable of
+hallucinating by generating elements beyond the guidelines
+when detailed guidelines are not given by the user, which is
+often the case in layout design.
+5.3. User Test
+We conduct user tests to further examine the quality of gen-
+erated layouts and whether the FID and FD-VG metrics
+align with human evaluation conducted by professional de-
+signers. We follow the method used in (Nauata et al., 2020)
+and (Chang et al., 2021) and invite 15 designers with user
+interface design expertise. The range of the scores is [−1, 1],
+where 1 means a model had won all the comparisons, −1
+for lost all, and 0 means the draw. Our result shows that
+although designers still prefer the ground truth samples over
+both VTN and PLay, the margin between the ground truth
+and PLay is small, whereas PLay wins over VTN with a
+large margin. In other words, professional designers con-
+sider PLay generating more realistic layouts than prior work
+and often favor PLay over the ground truth samples.
+5.4. Conditional Generation and Guideline Editing
+Designers commonly modify existing layouts and build
+upon their earlier designs. Therefore, we develop four ways
+for users to interact with the model using guidelines as
+input conditions. The goal is to create a fast, controllable
+workflow for users to iterate and refine generated layouts.
+5.4.1. GENERATING VARIATIONS FROM GIVEN DESIGN
+As discussed in Section 1, guidelines can represent design
+intentions, rules and element arrangement patterns, and
+therefore, they can be seen as a high-level abstraction or
+skeleton of an existing design, similar to sketches. Based
+on this observation, we develop a method for PLay to cre-
+ate variations from an existing layout, by first extracting
+the guidelines from the given layout and then using the ex-
+tracted guidelines as conditions to generate more layouts. In
+Figure 6, all the generated results share the same high-level
+idea with the original one. By using different numbers of
+guidelines, we can control the levels of similarity to the
+given design. This variation to similarity trade-off is close
+to SDEdit (Meng et al., 2021), but our method can give the
+user explicit and visual control over the level of similarity
+and where it needs to be similar. Note that the number of
+elements is fixed in this experiment.
+Figure 6. Generating variations from a given design using extracted
+guidelines. The results in the top row use all the extracted guide-
+lines and therefore are similar to the original one in detail. The
+results in the bottom row have richer variety but less similar with
+the original since they are less constrained.
+5.4.2. MOVING GUIDELINES
+Besides generating variations, users also need to further edit
+the generated results. One of the intuitive ways to edit the
+design is to allow the user to adjust a guideline by dragging
+it to a new position, expecting the elements around this
+guideline to be adjusted automatically while holding the rest
+parts of the layout intact. We achieve this by recording the
+noise values in diffusion steps, and reuse them to generate a
+new layout with the edited guidelines and the same number
+of elements. In Figure 7a and 7b, we show the results of
+changing guideline positions along the X and Y axes.
+5.4.3. ADDING AND REMOVING GUIDELINES
+Similar to moving guidelines, by fixing the number of el-
+ements and reusing the noise values, we can enable users
+
+Original Design
+Use all
+GLs
+Original Design
+Use GLs
+with high
+weightsPLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 7. Layout editing. Row (a) and (b): drag guidelines along
+the x and y axes. Row (c): change the number of elements while
+keeping the same guidelines. Row (d): gradually draw new guide-
+lines.
+keep adding or removing guidelines and inspect how they af-
+fect the layout arrangements. This feature introduces a new
+experience for layout design similar to (Zhu et al., 2016)
+and (Park et al., 2019), where layouts are being generated
+simultaneously after each of the guidelines is drawn (Fig-
+ure 7d).
+5.4.4. CHANGING THE NUMBER OF ELEMENTS
+During sampling, the number of elements to have in a layout
+can be either sampled from the distribution learned from the
+dataset or assigned by the user, which becomes another way
+to generate variations. Users can specify a different num-
+ber of elements to generate to examine how the generated
+elements fit into the given conditions (Figure 7c).
+5.5. Layout Inpainting
+We report preliminary results for layout inpainting. In Fig-
+ure 8, PLay generates new elements in the cropped area, and
+with the edited guidelines, the results have better element
+alignments in the cropped area compared to the original
+one. We adopt the inpainting method used in image-based
+diffusion models (Saharia et al., 2022a). In this experiment,
+we simply match the number and sequence indices of the
+newly painted elements with the masked elements, which
+imposes a strong constraint and leads to results that are sim-
+ilar to the original design. Further study is needed to create
+a more flexible way to inpaint new elements, such as using
+an auto-regressive decoder to decide where to insert them.
+Figure 8. Layout inpainting. In this example, the inpainting results
+have better element alignments in the cropped area because of the
+guideline conditions.
+5.6. Failure Cases
+We observe three common failure modes in samples gen-
+erated by PLay. The first one is unused guidelines, and it
+can often be found when the number of elements is much
+lower than the number of guidelines, such as Figure 9a.
+The second type is invalid functions. For example, at the
+green dot in Figure 9b, it is unreasonable to have such a thin
+button for users to tap on. The last mode is invalid arrange-
+ments, which can be found when the number of elements
+are too large to be reasonably fit in a layout. Several future
+directions to improve these failure cases are: 1) training a
+generator to sample the number of elements based on guide-
+lines instead of naively sampling from the data distribution,
+and 2) mitigating the imbalance of the number of elements
+distribution in the datasets.
+Figure 9. Failure cases. We can automatically compute if case
+(a) happens in a generated layout, but case (b) and (c) are more
+subjective and require designer’s evaluation.
+6. Conclusion
+We present PLay, a novel parametrically conditioned la-
+tent diffusion model for layout generation, and introduce
+guidelines, a widely used representation by designers, as
+our conditions. We achieve state-of-the-art results across
+three datasets on both qualitative and quantitative metrics,
+including FID, FD-VG, G-Usage, and via user testing with
+designers. We also demonstrate different ways for users to
+control and interact with the generation process using guide-
+lines, including guideline editing, inpainting, and generating
+variations from a given layout with user-controllable simi-
+larities. The code and model will be released at [redacted].
+
+Y offset = -4
+Y offset = 0
+Y offset = 6
+Y offset = 8
+(a)
+X offset = 0
+X offset = 6
+X offset = 8
+X offset = -4
+(b)
+Bbox count = 5
+Bbox count = 15
+Bbox count = 7
+Bbox count = 11
+(c)
+GL count = 5
+GL count = 8
+GL count = 10
+GL count = 11
+(d)Original Design
+To Paint
+Inpainting Results(a) Unused guidelines
+(b) Invalid functions
+() Invalid arrangementsPLay: Parametrically Conditioned Layout Generation using Latent Diffusion
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+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Table 5. Comparing PLay, VTN, and the ground truth layouts on IoU, Overlap, and Alignment. The models are trained on the CLAY
+dataset.
+Model
+IoU
+Overlap
+Alignment
+G.T.
+0.224
+0.764
+0.348
+VTN
+0.233
+0.784
+0.377
+PLay
+0.235
+0.754
+0.302
+A. Datasets
+We can observe the difference in complexity of the three datasets from the number of elements distributions (Figure 10) and
+the visualization of examples (Figure 11, 12, and 13).
+Figure 10. Number of elements distributions in CLAY, RICO-Semantic, and PublayNet.
+Figure 11. Examples from CLAY. Note that black is empty in all examples.
+B. Other Metrics
+In Table 5, We also computed the metrics used in VTN, including IoU, Overlap, and Alignment. PLay performs better than
+VTN on Overlap and Alignment, and its IoU is on par with VTN. However, compared to FID and FD-VG, these metrics
+do not align with the significant difference in our user test results. Moreover, PLay even outperforms the ground truth on
+Overlap and Alignment, which is not reflecting the fact that users still think the ground truth layouts are better than PLay.
+These metrics are also not commonly used for evaluating other design and creativity related generative models. Therefore,
+we choose not to use them as the main metrics.
+
+RICo-Semantic
+0.10 -
+ PublayNet
+CLAY
+0.00
+0
+40
+8
+110
+120
+Number of ElementsPLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 12. Examples from RICO-Semantic. Comparing the RICO-Semantic examples to the CLAY examples in Figure 11, the number of
+elements in each layout is less; the average size of each element is larger; and the design patterns look simpler.
+Figure 13. Examples from PublayNet.
+C. Implementation Details
+The detail of each component in Figure 3 can be found in Figure 14.
+We implemented the proposed architecture in JAX and Flax. We use ADAM optimizer (b1 = 0.9, b2 = 0.98) with 500k
+steps and a batch size of 128. The learning rate is 0.001 with linear warming up to 8k steps. The model is trained using 8
+Google Cloud TPU v4 cores for 47 hours.
+D. User Test Details
+Each participant is presented with 48 randomly generated questions. In each question, there will be a pair of layouts, and the
+user needs to pick the better one (Figure 15. Each layout pair is selected from two of these three groups: ground truth, VTN,
+and PLay. We also ensure the 48 questions equally cover all possible combinations of these groups.
+When answering a question, the group gets +1 score if the user pick the layout that belongs to this group, and gets -1 vice
+versa. Both groups get 0 if the user thinks their layouts are equally good or bad. The final scores are normalized by the
+number of questions.
+We recruit 15 user interface (UI) and user experience (UX) designers with experience in layout design for the user test
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Table 6. List of first-stage models. β is the weight of KL loss, d is the dimension of the latent space for each element, and c is the
+codebook size of VQVAE. In each row, FID-Recon and FD-VG-Recon are computed using the reconstruction samples generated by the
+first-stage model. FID, FD-VG, and G-Usage are computed using PLay trained on this first-stage model.
+First-stage
+β
+d
+c
+FID-Recon
+FD-VG-Recon
+FID
+FD-VG
+G-Usage
+None
+n/a
+8
+n/a
+n/a
+n/a
+167.9
+4.683
+0.991*
+VAE
+1.0
+8
+n/a
+19.75
+2.04e−1
+14.35
+0.311
+0.835
+1e−1
+8
+n/a
+4.010
+1.94e−2
+11.73
+0.279
+0.955
+1e−2
+8
+n/a
+0.612
+3.99e−3
+11.63
+0.262
+0.968
+1e−3
+8
+n/a
+0.215
+1.75e−3
+10.59
+0.245
+0.964
+1e−4
+8
+n/a
+0.142
+1.35e−3
+11.49
+0.252
+0.964
+5e−4
+8
+n/a
+0.138
+1.49e−3
+11.57
+0.236
+0.965
+1e−5
+4
+n/a
+3.805
+2.02e−2
+12.76
+0.281
+0.946
+1e−5
+8
+n/a
+0.115
+1.46e−3
+11.05
+0.253
+0.965
+1e−5
+16
+n/a
+0.053
+1.30e−3
+11.58
+0.257
+0.953
+VQVAE
+n/a
+8
+1024
+10.70
+6.52e−2
+11.34
+0.247
+0.909
+n/a
+8
+4096
+8.68
+5.22e−2
+11.45
+0.244
+0.924
+n/a
+8
+16384
+6.68
+3.70e−2
+11.49
+0.254
+0.937
+∗PLay without the first-stage model generates mostly random, unaligned boxes and therefore, has a high guideline usage.
+sessions. We also give the following criteria and information to the participants:
+• Please evaluate for both aesthetics and functionalities of the layouts. For example, the alignments, proportions, how
+reasonable for the buttons to be put here, etc.
+• Some of the layouts are intentionally not valid nor optimal (many of them are synthesized). Therefore, please do not try
+too hard to justify every layout. Use your intuition and experience as a designer to pick the better one from each pair.
+• Some of the examples that do not look like a full mobile UI screen might still be valid designs. They can represent UI
+cases such as popped windows, opened drawers, or simply without background image.
+• Note that the text elements are often not aligned due to their various lengths.
+E. First-stage Models
+In Table 6, we can see that β values do not have significant effect on the metrics as long as they are small enough (<= 0.1).
+Latent dimension d = 8 seems to be the optimal choice in our experiments. VQVAEs achieve comparable numbers in FID
+and FD-VG but fall short on G-Usage. Although increasing the codebook size improves G-Usage, it is less computationally
+efficient to use c > 16384 compared to a simple VAE.
+F. More Results
+In Figure 16, we decode the output from each denoising step in the sampling process and visualize the rendered results from
+them. We also visualize the generated samples from PLay (Figure 17, 18, and 19) and from VTN (Figure 20).
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 14. Model Components. We use the same Transformer Encoder (a) as the building block for all the components in PLay. The
+decoder (c) is similar to (Carlier et al., 2020), using learned embeddings and adding z in each layer. Note that positional embedding is
+needed in (e), since the encoded elements z does not have explicit positional information (e.g., coordinates). The Feature-wise Linear
+Module is composed by MLP layers to map t to scale and shift for the Feature-wise Affine layer.
+
+(a) Transformer Encoder Block
+Skip
+Skip
+Self
+Layer
+Layer
+input
++
+MLP
++
+indno
+Attention
+Norm
+Norm
+(b) First-Stage Encoder
+MLP
+u
+z
+Transformer
+elements
+Encoder Block
+MLP
+xL
+(c) First-Stage Decoder
+Model Parameters:
+learned
+Transformer
++
+MLP
+elements
+embeddings
+Encoder Block
+L= 4
+L for guideline encoder =- 2
+xL
+num of heads = 8
+qkv dim = 512
+MLP dim = 2048
+(d) Guideline Encoder
+MLP dim for guideline encoder = 1024
+d=8
+Transformer
+guidelines
+encoded guidelines
+Encoder Block
+xL
+pos
+(e) Denoise Network
+x L
+Skip
+Transformer
+Feature-wise
+Layer
+Cross
+MLP
+MLP
+Encoder Block
+Affine
+Norm
+Attention
+00s
+scale
+FiLM
+encoded guidelines
+shiftPLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 15. An example question in the user test.
+
+Question 05:
+Which layout below do you think is better?
+Left
+Right
+Similarly good/bad
+:8:
+Question05
+TEXT
+CONTAINER
+PICTOGRAM
+LIST_ITEM
+BUTTON
+IMAGE
+TOOLBAR
+LABEL
+TEXT_INPUT
+DRAWER
+CHECK_BOX
+NAVIGATION_BAR
+CARD_VIEW
+SPINNER
+SWITCH
+PAGER_INDICATOR
+RADIO_BUTTON
+SLIDER
+PROGRESS_BAR
+NUMBER_STEPPER
+ADVERTISEMENT
+MAP
+DATE_PICKERPLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 16. Decoding the latent diffusion steps back to layouts. Interestingly, although the first 150 steps look noisy and random, the last
+50 steps still demonstrate the nature of the denoising process, forming from low frequency features to high frequency features. This
+coarse-to-fine generation process can be commonly found in imaged-based diffusion models, but it is the first time we observe the same
+process in latent diffusion models for vector graphics.
+
+step=1
+step-11
+step=21
+step=31
+step=41
+step=51
+step=61
+step=71
+ step=81
+step=91
+step=96
+step=101
+step=106
+step-111
+ step=116
+step=121
+step=126
+step=-131
+step-136
+step-141
+step=146
+step=151
+step=161
+ step=156
+step=166
+step=168
+step=170
+step=172
+step=174
+step=176
+step=186
+ step-178
+step-180
+ step-182
+ step-184
+step-188
+step=189
+step=190
+t61-doys
+161=dars
+step=192
+step-193
+step=197
+step=195
+step=196
+step=198
+step=200
+step=199PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 17. Samples generated by PLay trained on CLAY. In each cell, left: input guidelines (blue lines) on top of the generated layout.
+Right: generated layout.
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 18. (continued) Samples generated by PLay trained on CLAY.
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 19. (continued) Samples generated by PLay trained on CLAY.
+
+PLay: Parametrically Conditioned Layout Generation using Latent Diffusion
+Figure 20. Samples generated by VTN trained on CLAY.
+
diff --git a/lNFJT4oBgHgl3EQfZCzN/content/tmp_files/load_file.txt b/lNFJT4oBgHgl3EQfZCzN/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf,len=1162
+page_content='PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Chin-Yi Cheng 1 Forrest Huang 1 Gang Li 1 Yang Li 1  Abstract  Layout design is an important task in various de-  sign fields, including user interfaces, document,  and graphic design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' As this task requires tedious  manual effort by designers, prior works have at-  tempted to automate this process using generative  models, but commonly fell short of providing intu-  itive user controls and achieving design objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In this paper, we build a conditional latent diffu-  sion model, PLay, that generates parametrically  conditioned layouts in vector graphic space from  user-specified guidelines, which are commonly  used by designers for representing their design  intents in current practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Our method outper-  forms prior works across three datasets on metrics  including FID and FD-VG, and in user test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' More-  over, it brings a novel and interactive experience  to professional layout design processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Introduction  Layouts are important artifacts that represent the design and  arrangements of their encapsulated elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' They are used  extensively in creative fields ranging from design to engi-  neering, supporting the authoring processes of numerous  downstream products, such as user interfaces (UIs), docu-  ments, posters, architectural floorplans, and even printed  circuit boards (PCBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Designing a good layout requires  thorough consideration of different aspects, such as the func-  tion, aesthetics, and domain-specific conditions and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Moreover, many of these aspects and objectives cannot be  easily and explicitly evaluated and computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore,  layout design has been a time-consuming, iterative, and  manual process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Several recent works have tried to improve the layout design  process by automating it using generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' However,  these models are typically unconditional, making them dif-  ficult for users to adopt in realistic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' A large  number of unconditionally, randomly generated layouts are  1Google Research, Mountain View, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Corre-  spondence to: Chin-Yi Cheng <cchinyi@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='com>, Yang  Li <liyang@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Copyright 2023 by the author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Sample results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The red lines on the left of each example  represent the guideline conditions, and the generated layout is  on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The generated layouts are also placed under the  guidelines to visualize their alignments to the guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  not useful in practice since it requires users to manually  check each of them on whether the original objectives and  constraints have been met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' More importantly, for users to  enjoy co-designing with generative models and trust the gen-  erated layouts, they need to be able to express their design  ideas and drive the generation process (Louie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Therefore, a model conditioned on user inputs would help  their adoption in users’ workflows, and the type of supported  inputs should represent design intention and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  The key towards achieving our goal of conditional genera-  tion that can be widely adopted is to choose an adequate type  of condition for generating design layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Different types  of conditions have been explored in image-based generative  models, such as text (Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022) and colored semantic segmenta-  tion (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2019), but these might not be ideal choices  for layout generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Text conditions have shown to be  powerful for artistic purposes, but they cannot provide exact  and detailed control critical for design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' On the other hand,  semantic segmentation allows for precise control, but is te-  dious and time-consuming for users to manually author all  segments in the scene, which is equivalent to hand-drawing  the entire layout in our case of layout design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In this work, we investigate layout design workflows in  several domains, including UI, document, and architectural  design, and identify a widely used representation: guide-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='11529v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='LG]  27 Jan 2023  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout and guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' A layout (b) of a UI design  (a) consists of vector graphic elements with types and geometric  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Guidelines (c) can represent the ideas and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', the red lines in Figure 2c), as conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Guide-  lines, also referred to as guides, grids, or partitions, are  a set of lines that serves multiple purposes, for instance:  partitioning the design space for preferred proportions and  alignments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' expressing design ideas such as the two-column  style (top-left example, Figure 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' and showing rules such as  paddings, margins, and gaps between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' However, a  common pain point for designers is that changing guidelines  (which express design thoughts) would require manually  redrawing and adjusting the associated design to conform  to the new guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore, designers currently use  guidelines only as visual references or a way to document  their thoughts and yet cannot benefit from the rich design  information encapsulated by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Hence, using guidelines as the input conditions to our model  can not only provide a high-level yet precise control to users,  but also solve a practical pain point in layout design work-  flows across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We consider the guideline  condition as a type of parametric conditions, borrowing  the definition from parametric design in the fields of com-  putational design and computer-aided design (Monedero,  2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In parametric design, the design artifacts are defined  using algorithms or procedures with parameters as inputs,  and therefore users can easily make design changes by ma-  nipulating the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Our work enables an intuitive  way to instantiate parametric models for layouts by drawing  guidelines, as opposed to manually setting the relationships  and heuristics in traditional parametric design tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We introduce PLay, Parametrically Conditioned Layout  Generation using Guidelines, a two-stage model follow-  ing the idea of latent diffusion models (LDMs) proposed  by (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Compared to (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2022), the main purpose of our first-stage model is to con-  vert the layout from the discrete vector graphics space to  the continuous latent space instead of compressing informa-  tion in original data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' With this continuous latent  representation, the second-stage diffusion model can iter-  atively refine the results similar to image-based diffusion  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' PLay can generate layout designs conditioned on  guidelines across three different datasets with significantly  improved quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We measure the quality by computing the  Fr´echet distances using layouts rendered in both image and  vector graphics domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We further evaluate the results by  conducting user testing on professional designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Using guidelines as a way to express both the high and  low-level ideas, PLay also enables several interactive and  controllable ways for users to create desired layouts: 1)  generating and editing layouts by dragging, adding, and  removing guidelines, similar to drawing a sketch, 2) gener-  ating variations from an existing layout with different levels  of similarity controlled by guidelines, and 3) layout inpaint-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We envision PLay to improve layout design workflows  in various domains, enable different types of conditions  and interactions, and potentially solve more general vector  graphics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In summary, the main contributions of this paper are:  We develop a latent diffusion model in the vector graph-  ics domain for layout generation, achieving a better  performance than prior work in multiple metrics with  large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We introduce guidelines as input conditions for the  latent diffusion model, making the generation process  parametrically controllable and interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We provide a variety of new ways to generate and edit  layouts, including guideline editing, layout inpainting,  and generating variations from an existing layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We propose FID and FD-VG as metrics and conduct  user testing to examine the quality of generated layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout Generation  Several prior works have studied generative models for lay-  outs in the vector graphics domain, and LayoutGAN (Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2019) is among the earliest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' It uses self-attention  layers as the generator and argues that the discriminator  in the image domain can better evaluate the spatial quality,  such as alignments of the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' LayoutVAE (Jyothi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2019), on the other hand, works purely in the vector  graphics domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' It aggregates the information across the  elements using Long Short-Term Memory (LSTM) (Hochre-  iter & Schmidhuber, 1997) and trains variational autoen-  coders (VAEs) (Kingma & Welling, 2013) for generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  A recent work, VTN (Arroyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021), maintains the  VAE architecture in LayoutVAE but replaces the encoder  and decoder by Transformers (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The  results from VTN shows that with Transformers, the model  can learn the proper layout arrangements without mapping  results to the image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Our latent diffusion model in  (a) UI Design  (b) UI Layout  (c) Guidelines  (d) Legend  ##07:27  Invite school  Guideline  Text  Button  Container  Image  Pictogram  Tell us a bit more so that we can invite  your school or daycare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  List_item  Toolbar  Nuddke iddle  Label  Text_input  iddle@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='.com  Drawer  Card_view  School Phone  Send anonymously  Spinner  Check_box  Invite!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Navigation_bar  Switch  Page_indicator  Progress_bar  Number_stepper  Date_picker  口PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  PLay can also be seen as a way to learn a better latent prior  than VTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Various types of conditions for conditional layout genera-  tion have also been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For example, (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020)  adds element attributes such as area and aspect ratio as con-  ditions, and (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020) uses graph neural networks to  constrain inter-element relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In the image domain,  House-GAN (Nauata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020) and House-GAN++ (Nau-  ata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021) also explore graph conditions for layout  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' However, the shared issues of these approaches  are 1) users have to tediously assign the properties of rela-  tionships between each of the elements, and 2) the allowed  number of elements and relationships for conditions are of-  ten too small for a complex layout with more than 20 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  As discussed in 1, the guideline conditions in PLay can over-  come these issues, since one guideline can align multiple  elements, and multiple guidelines can define complex rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Guidelines can also be seen as space partitioning in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For  example, (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021) partitions the 3D space to search  for potential CAD modeling sequences, and (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2021) uses the partitioned space as one of the input condi-  tions for 3D volume generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Both works use partitions  directly as the design or search space, meaning that if the  given partitions are not good enough, the quality of the re-  sults will be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In contrast, PLay can flexibly attend  to various guidelines through cross-attention and generate  elements that do not follow some guidelines if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In addition, some recent works incorporate other modalities  such as images into layout generation: CanvasVAE (Yam-  aguchi, 2021) uses the image information for the elements in  the layout generation process, and CGL-GAN (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2022) generates layouts that fit the given images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Canvas-  VAE has a two-stage encoder-decoder architecture inspired  by DeepSVG (Carlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020), with the first-stage for  the individual elements and the second-stage for the layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In PLay, the element stage is not needed since each element  in our datasets is composed by its class and bounding box  coordinates only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' However, it is worth experimenting with  the element stage for PLay in the future, such as incorporat-  ing hierarchical decoding introduced by (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022)  which can yield better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' This can also augment PLay  to solve more complex tasks, such as CAD layout genera-  tion, where each element can be any shapes or curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Diffusion Models  Recent advances in Diffusion Models (DMs) (Sohl-  Dickstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2015) have shown promising results in  image generation (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Dhari-  wal & Nichol, 2021), where classifier guidance (Dhariwal  & Nichol, 2021) and classifier-free guidance (CFG) (Ho  & Salimans, 2022) methods not only improve the genera-  tion quality but also enable the possibility of developing  conditional diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Most recent works for text-to-  image generation use CFG, including GLIDE (Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2021), DALL·E2 (Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), Imagen (Saharia  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022b), and LDMs (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), and in  addition to text, LDMs explore other conditions such as  bounding boxes and semantic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We also adopt CFG for  the guideline conditions in PLay, as it does not require an  extra classifier and lead to the generation of better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Applying diffusion models outside of the image domain  has also drawn attention from researchers, such as 3D (Lin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), video (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), and music (Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2021) generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' (Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021) converts the discrete  melody tokens into a continuous latent space, and trains the  diffusion model in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' This work inspires us  to convert the discrete layout elements, composed by con-  catenating different types of tokens including their class and  coordinates, to a continuous domain for the diffusion pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' A concurrent work, (Strudel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), applies the  same idea for language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We also follow LDMs  (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022) to add a small KL-penalty to ensure  high layout reconstruction quality and avoid arbitrarily large  variance in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Several recent works explore methods to further control the  diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For instance, Prompt-to-Prompt (Hertz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022) fixes the cross-attention maps to preserve scene  compositions for a new text prompt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' SDEdit (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2021) injects the stroke-based conditions by adding noise  to it and denoising it back to a real image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The level of  added noise becomes a parameter to control the balance  between image realism and faithfulness to the input drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Compared to these works, PLay can provide more explicit  control to generate variations of previously generated layout,  by extracting and editing the guidelines (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout Design  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Datasets  We experiment PLay with three publicly available datasets  for two different domains: UI and document layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  CLAY (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022) contains about 50K UI layouts  with 24 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  RICO-Semantic (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2018) contains about 43K  UI layouts with 13 classes previously used in VTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PublayNet (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2019) contains about 330K  document layouts with 5 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  The layouts in CLAY are more complex and representative  of real UI designs compared to RICO-Semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Although  both of them are extracted and processed from RICO (Deka  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2017), CLAY tries to fix some annotation errors and  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  mismatches between the screenshots and view hierarchies  and introduces new label systems, whereas RICO-Semantic  adds semantic annotations for RICO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We suspect that ei-  ther RICO-Semantics filtered some complex patterns during  post-processing, or the chosen 13 classes from the 25 origi-  nal classes largely reduced the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  To verify that PLay can be applied to other layout domains,  we also train it on PublayNet to generate document layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Compared to CLAY, both RICO-Semantic and PublayNet  have fewer design variations, and their layouts are overall  simpler and have more repetitive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For example,  the complexity of each dataset is reflected by its average  number of elements in each layout: RICO-Semantic=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='79;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PublayNet=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='90;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' CLAY=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore, while we evalu-  ate PLay over all the three datasets, we particularly conduct  in-depth evaluation and analysis of our model on CLAY  with qualitative and quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' See the appendix  for more statistics and examples of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout and Guidelines  The shared data format across the three datasets for a layout  is a sequence of elements: E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', eN}, where  en = {cn, xn  min, yn  min, xn  max, yn  max} and 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The  class of an element, cn, and each coordinate (an integer) of  its bounding box, xn  min, yn  min, xn  max, and yn  max, is repre-  sented as a one-hot vector, which are then concatenated to-  gether as a single vector, and therefore E ∈ {0, 1}N×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We  fix the maximum number of elements per layout: N = 128,  and the layout with fewer elements are padded to the same  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We set the dimension of each layout with width = 36  and height = 64, which result in a fixed D for all layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  The guidelines of a layout is represented as:  G  =  {g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', gM}, where gm = {am, pm} and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Each guideline is composed by its axis am, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', horizontal  versus vertical, and the coordinate position pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We fix the  maximum number of guidelines for each layout: M = 128,  and thus the maximum number of guidelines in each axis is  M/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The representation of layouts involving fewer guide-  lines is padded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' To create the layout-guidelines pairs for  training, for each layout E in the datasets, we can intuitively  obtain the full guidelines Gfull by extending all the bound-  ing box edges of each element and removing the duplicated  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For the model to learn how to synthesize valid details  beyond the given conditions, we also have three different  sampling methods to create random subsets of Gfull during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' More details will be discussed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Metrics  There are no universally established metrics to evaluate lay-  out generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Prior works such as (Arroyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2019) compute several features in both real and  generated layouts such as IoU, overlap, and alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' and  (Yamaguchi, 2021) computes the feature-wise distribution  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Instead of using feature-based methods, we  follow the common practice of evaluating Generative Ad-  versarial Networks (GANs), including several prior layout  generation works (Nauata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=',  2020), to measure the Fr´echet Distance (FD) between two  distributions from latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' To capture various aspects  of layout generation, we compute FD in two ways:  FID (Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2017): we render the layouts into  images with the same aspect ratio and add paddings  if the elements are not fully using the screen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We then feed the images into the pre-trained Incep-  tion (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2017) model to get the activation  vectors, and compute the FD between the real and gen-  erated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  FD-VG: we train a Transformer-based auto-encoder in  the vector graphics domain, use its encoder to encode  the layouts, and compute the FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  G-Usage: We also evaluate if the generated results satisfy  the guideline conditions by computing the G (guideline)-  Usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We first extract the guidelines G∗ from a generated  layout, then we calculate the intersection, Ginter, for G∗  and the given guidelines, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Finally, we obtain the G-Usage  as |Ginter|/|G| and take the average across the generated  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Note that G-Usage does not equal to IoU, since it  is valid to have guidelines in G∗ that are not part of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  User Test: We also conduct user testing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='3) with profes-  sional designers and the results are aligned with the FID and  FD-VG metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Architecture  Following the image-based latent diffusion model (Rom-  bach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), We formulate PLay as a two-stage model,  where the first-stage model learns to map layouts from a  vector graphics space to a latent space, and then the condi-  tional latent diffusion model learns to generate layouts in  latent space conditioned on guidelines given by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 3 illustrates the overview of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' First-Stage Model  To map the layout E ∈ RN×D to the latent representation  z ∈ RN×d, we use a Transformer similar to DETR (Carion  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020), with an encoder E(E) and a non-autoregressive  decoder D(z) modified from DeepSVG (Carlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We also added a small KL-penalty to regularize the latent  space while keeping the high reconstruction accuracy, which  is especially critical in the vector graphics space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The reason  is that unlike pixels, where some artifacts and noises are not  noticeable, we only have at most 128 elements in a layout,  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' After training the first-stage model, we can use it to encode layouts to the latent space for training the latent  diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' During sampling, it can decode the generated latent representations back to layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We can observe that PLay generates reasonable results on complex examples with many elements, whereas  VTN often struggles in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  so it will be obvious if any of them is decoded incorrectly,  such as having a miss-aligned box or an unreasonable class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Following VTN, no positional embedding is added to the  encoder and decoder, as the coordinates already explicitly  indicate the positions spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In image-based latent diffusion models (LDMs), the goal of  the first-stage model is to map the input image to a lower  dimension while keeping the same perceptual details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' How-  ever, the first-stage model in PLay serves a different purpose:  to learn a meaningful and continuous latent representation  for the discrete vector graphic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We will see in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1, that  the LDM cannot converge with mapping using naive MLP  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In addition, we experiment with encoding the entire  layout as a single vector using a transformer, but it also fails  to achieve high reconstruction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Latent Diffusion Model  When training the latent diffusion model, the encoded layout  z is first divided by the standard deviation std of the first  batch, as suggested by (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), and then the  scaled z is used as z0 for the forward diffusion process to  get zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For the denoise network ϵθ(zt, τψ(G), t),  we use a Transformer encoder to replace the U-Net struc-  ture used in image-based DMs and predict the noise ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The  discrete time step t is encoded using Feature-wise Linear  Modulation (Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2018) and injected into the Trans-  former encoder with a feature-wise affine layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We encode  the guidelines G using another Transformer encoder τψ, and  the encoded guidelines, τψ(G) ∈ RM×d, are then fed to ϵθ  through cross-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The loss function can be formulated  as general LDMs:  L := EE(x),G,t,ϵ∼N (0,1)  �  ∥ ϵ − ϵθ(zt, τψ(G), t) ∥2�  (1)  We train the LDM as a standard DDPM (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020)  with classifier-free guidance, where we randomly drop the  guideline conditions with the probability pdrop = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Sampling  In sampling, we first either sample the number of elements  N from p(N), which is the element count distribution of the  dataset, or use n assigned by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Then we initialize zT  and denoise it to get z0 with the given guideline conditions  using DDPM and CFG, with w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='5:  ˆϵθ(zt, τψ(G), t) = (1 + w)ϵθ(zt, τψ(G), t)  − wϵθ(zt, τψ(φ), t)  (2)  (a) Two-Stage Training  (b) Sampling  First-Stage  Latent Diffusion  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Space  Latent Space  oz  uz  clso  zo  zn  zn  clsn  a°  am  ymin  Ymin  29  Forward Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  t  oQ  pm  Screen  t  ao  am  Screen  Decoder  to  Decoder  po  pm  ymax  Ymax  z  Guideline  Denoising  Encoder  Transfomer  Guideline  clso  clsn  clso  clsn  cross  Encoder  x0in  Xmin  cross  Xmin  Screen  Denoising  attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  ymin  ymin  Encoder  Transfomer  ymin  ym  ymin  yo  Backward Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  xmax  Xmax  yo  ym  ymax  oz  en  ymex  max  2%  uz(a) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' from the CLAY dataset  (b) Samples generated by PLay  (c) Samples generated by VTNPLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Quantitative Results and ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Model  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  CFG-W  FID  FD-VG  G-Usage  VTN  ×  ×  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='352  n/a  C-VTN  ×  ×  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='361  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='819  PLay  VAE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='25  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='286  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='970  VAE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='50  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='245  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='964  VAE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='75  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='269  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='968  ×  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='5  166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='577  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='992∗  VAE  ×  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='375  n/a  VTN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='50  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='311  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='835  VQVAE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='50  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='254  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='937  ∗PLay without the first-stage model generates mostly random,  unaligned boxes and therefore, has a high guideline usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Comparisons across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Dataset  Model  FID  FD-VG  G-Usage  CLAY  VTN  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='352  n/a  PLay  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='245  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='964  RICO-  VTN  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='415  n/a  Semantic  PLay  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='320  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='944  PublayNet  VTN  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='787  n/a  PLay  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='408  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='971  We then rescale z0 with the std used in training, and decode  it back to the vector graphic domain using the first-stage  decoder: E = D(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' User test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Each cell represents the averaged score  of the row-column comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For example, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='375 means that  VTN performs worse than PLay to designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Experiments  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Baseline Comparison and Ablation Study  In this section, we show how PLay performs compared to the  baseline and study the effects of different first-stage model  choices and classifier-free guidance weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We choose  VTN (Arroyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021) as the baseline model for com-  parison because it is the most common framework of SoTA  models for UI layout generation in vector graphic space and  is easily reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' It shares the same architecture with  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' FID scores in groups of number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Model  All  1-6  7-11  12-18  19-29  30+  VTN  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='10  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='83  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='12  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='89  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='76  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='15  PLay  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='59  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='28  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='72  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='23  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='60  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='01  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Guideline sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Method  FID  FD-VG  G-Usage  All  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='206  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='917  Uniform  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='312  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='944  Weight-tiers  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='297  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='957  Weighted  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='245  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='964  our first-stage model, a variational authoencoder (VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We  also modify VTN to condition on guidelines, called C-VTN  to ensure a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In Table 1, we first try to train  the diffusion model without the first-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' With  simple MLP layers to encode the class and coordinates, the  model fails to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We then add VAE as the first-stage  model, and discover the results of latent diffusion training  outperform the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Adding the classifier-free guid-  ance further improves the results, with the optimal weight  w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='5 and latent dimension d = 8 for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We also try to use VQVAE as the first-stage model, but it  falls short on G-usage while having comparable numbers  in FID and FD-VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The potential reason can be that the  learned, frozen codebook is less flexible to exactly match  the guideline conditions since they are not involved in the  first-stage training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Additionally, we try to use trained VTN  as the first-stage model, as it is also a VAE with KL loss  weight β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We achieve worse but still reasonable  results, which can be explained by the low reconstruction  accuracy using high β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The reconstruction accuracy  plays a crucial role for PLay, and it is not required to use a  very small β value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 1e−6 in (Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022), as  long as the reconstruction accuracy is good enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' See the  appendix for the complete table of first-stage model choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PLay also outperforms prior work in all metrics across three  datasets: CLAY, RICO-Semantic, and PublayNet (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Moreover, for this experiment, we use the same model ar-  chitecture and hyper-parameters for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The only  required change is the input dimension, which demonstrates  PLay’s ability to generalize to different layout domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We further divide the generated layouts into groups by the  number of elements in each layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We discover that PLay  achieves much better FID scores (Table 3) and is qualita-  tively better (Figure 4) than VTN in groups with a large  number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' This shows that the advantage of PLay  over the baseline is more pronounced when a layout is com-  PLay  VTN  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  VTN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='375 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='313  PLay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='375  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='104  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='313  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='104PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  plex and involves a large number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Guideline Sampling Methods  Users usually prefer to specify only the guidelines that can  represent the main ideas instead of drawing guidelines for  every element, because at this stage they have not come up  with all the details for the design yet and want to see the  potential options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore, we randomly sample a subset  of guidelines for every example during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In this way,  the model can learn to follow the given guidelines as the  main guidance and create extra details that are not covered  by the given guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We investigate three guideline sampling methods during  training: uniform, weighted, and weight-tiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Uniform  means we uniformly sample a subset of the guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For  weighted sampling, we compute the weight as how much  each guideline is obeyed by the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' If a guideline  is followed by more elements for alignment, the guideline  is deemed more important thus has a higher weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For  weight-tiers, we further divide the guidelines with different  ranges of weight into groups and sample the groups as a  whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We find that weighted sampling achieves the best  FID and G-Usage (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Including all guidelines has  the best FD-VG score while the G-Usage suffers since it  becomes a more difficult task to fit all the guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Impor-  tantly, the model that is trained with all the guidelines can  only strictly follow the given guidelines and is incapable of  hallucinating by generating elements beyond the guidelines  when detailed guidelines are not given by the user, which is  often the case in layout design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' User Test  We conduct user tests to further examine the quality of gen-  erated layouts and whether the FID and FD-VG metrics  align with human evaluation conducted by professional de-  signers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We follow the method used in (Nauata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020)  and (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021) and invite 15 designers with user  interface design expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The range of the scores is [−1, 1],  where 1 means a model had won all the comparisons, −1  for lost all, and 0 means the draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Our result shows that  although designers still prefer the ground truth samples over  both VTN and PLay, the margin between the ground truth  and PLay is small, whereas PLay wins over VTN with a  large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In other words, professional designers con-  sider PLay generating more realistic layouts than prior work  and often favor PLay over the ground truth samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Conditional Generation and Guideline Editing  Designers commonly modify existing layouts and build  upon their earlier designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore, we develop four ways  for users to interact with the model using guidelines as  input conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The goal is to create a fast, controllable  workflow for users to iterate and refine generated layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' GENERATING VARIATIONS FROM GIVEN DESIGN  As discussed in Section 1, guidelines can represent design  intentions, rules and element arrangement patterns, and  therefore, they can be seen as a high-level abstraction or  skeleton of an existing design, similar to sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Based  on this observation, we develop a method for PLay to cre-  ate variations from an existing layout, by first extracting  the guidelines from the given layout and then using the ex-  tracted guidelines as conditions to generate more layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In  Figure 6, all the generated results share the same high-level  idea with the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' By using different numbers of  guidelines, we can control the levels of similarity to the  given design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' This variation to similarity trade-off is close  to SDEdit (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2021), but our method can give the  user explicit and visual control over the level of similarity  and where it needs to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Note that the number of  elements is fixed in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Generating variations from a given design using extracted  guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The results in the top row use all the extracted guide-  lines and therefore are similar to the original one in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The  results in the bottom row have richer variety but less similar with  the original since they are less constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' MOVING GUIDELINES  Besides generating variations, users also need to further edit  the generated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' One of the intuitive ways to edit the  design is to allow the user to adjust a guideline by dragging  it to a new position, expecting the elements around this  guideline to be adjusted automatically while holding the rest  parts of the layout intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We achieve this by recording the  noise values in diffusion steps, and reuse them to generate a  new layout with the edited guidelines and the same number  of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In Figure 7a and 7b, we show the results of  changing guideline positions along the X and Y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' ADDING AND REMOVING GUIDELINES  Similar to moving guidelines, by fixing the number of el-  ements and reusing the noise values, we can enable users  Original Design  Use all  GLs  Original Design  Use GLs  with high  weightsPLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Row (a) and (b): drag guidelines along  the x and y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Row (c): change the number of elements while  keeping the same guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Row (d): gradually draw new guide-  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  keep adding or removing guidelines and inspect how they af-  fect the layout arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' This feature introduces a new  experience for layout design similar to (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2016)  and (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2019), where layouts are being generated  simultaneously after each of the guidelines is drawn (Fig-  ure 7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' CHANGING THE NUMBER OF ELEMENTS  During sampling, the number of elements to have in a layout  can be either sampled from the distribution learned from the  dataset or assigned by the user, which becomes another way  to generate variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Users can specify a different num-  ber of elements to generate to examine how the generated  elements fit into the given conditions (Figure 7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout Inpainting  We report preliminary results for layout inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In Fig-  ure 8, PLay generates new elements in the cropped area, and  with the edited guidelines, the results have better element  alignments in the cropped area compared to the original  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We adopt the inpainting method used in image-based  diffusion models (Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In this experiment,  we simply match the number and sequence indices of the  newly painted elements with the masked elements, which  imposes a strong constraint and leads to results that are sim-  ilar to the original design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Further study is needed to create  a more flexible way to inpaint new elements, such as using  an auto-regressive decoder to decide where to insert them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Layout inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In this example, the inpainting results  have better element alignments in the cropped area because of the  guideline conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Failure Cases  We observe three common failure modes in samples gen-  erated by PLay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The first one is unused guidelines, and it  can often be found when the number of elements is much  lower than the number of guidelines, such as Figure 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  The second type is invalid functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For example, at the  green dot in Figure 9b, it is unreasonable to have such a thin  button for users to tap on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The last mode is invalid arrange-  ments, which can be found when the number of elements  are too large to be reasonably fit in a layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Several future  directions to improve these failure cases are: 1) training a  generator to sample the number of elements based on guide-  lines instead of naively sampling from the data distribution,  and 2) mitigating the imbalance of the number of elements  distribution in the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We can automatically compute if case  (a) happens in a generated layout, but case (b) and (c) are more  subjective and require designer’s evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Conclusion  We present PLay, a novel parametrically conditioned la-  tent diffusion model for layout generation, and introduce  guidelines, a widely used representation by designers, as  our conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We achieve state-of-the-art results across  three datasets on both qualitative and quantitative metrics,  including FID, FD-VG, G-Usage, and via user testing with  designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We also demonstrate different ways for users to  control and interact with the generation process using guide-  lines, including guideline editing, inpainting, and generating  variations from a given layout with user-controllable simi-  larities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The code and model will be released at [redacted].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Y offset = -4  Y offset = 0  Y offset = 6  Y offset = 8  (a)  X offset = 0  X offset = 6  X offset = 8  X offset = -4  (b)  Bbox count = 5  Bbox count = 15  Bbox count = 7  Bbox count = 11  (c)  GL count = 5  GL count = 8  GL count = 10  GL count = 11  (d)Original Design  To Paint  Inpainting Results(a) Unused guidelines  (b) Invalid functions  () Invalid arrangementsPLay: Parametrically Conditioned Layout Generation using Latent Diffusion  References  Arroyo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Postels, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', and Tombari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Variational trans-  former networks for layout generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In Proceedings  of the IEEE/CVF Conference on Computer Vision and  Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' 13642–13652, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Carion, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Massa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content=' Advances in neural information  processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Peng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', and Ritchie,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Inferring cad modeling sequences using zone graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  In Proceedings of the IEEE/CVF conference on computer  vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' 6062–6070, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Yamaguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Canvasvae: Learning to generate vector  graphic documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF  International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' 5481–  5489, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Zhong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', and Yepes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Publaynet: largest  dataset ever for document layout analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In 2019 Inter-  national Conference on Document Analysis and Recogni-  tion (ICDAR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' 1015–1022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Ge, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', and Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Composition-aware graphic layout gan for visual-textual  presentation designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='00303,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Kr¨ahenb¨uhl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', Shechtman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', and Efros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Generative visual manipulation on the natural image man-  ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In European conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  597–613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Comparing PLay, VTN, and the ground truth layouts on IoU, Overlap, and Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The models are trained on the CLAY  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Model  IoU  Overlap  Alignment  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='224  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='764  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='348  VTN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='233  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='784  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='377  PLay  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='235  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='754  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='302  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Datasets  We can observe the difference in complexity of the three datasets from the number of elements distributions (Figure 10) and  the visualization of examples (Figure 11, 12, and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Number of elements distributions in CLAY, RICO-Semantic, and PublayNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Examples from CLAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Note that black is empty in all examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Other Metrics  In Table 5, We also computed the metrics used in VTN, including IoU, Overlap, and Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' PLay performs better than  VTN on Overlap and Alignment, and its IoU is on par with VTN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' However, compared to FID and FD-VG, these metrics  do not align with the significant difference in our user test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Moreover, PLay even outperforms the ground truth on  Overlap and Alignment, which is not reflecting the fact that users still think the ground truth layouts are better than PLay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  These metrics are also not commonly used for evaluating other design and creativity related generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore,  we choose not to use them as the main metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  RICo-Semantic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='10 -  PublayNet  CLAY  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='00  0  40  8  110  120  Number of ElementsPLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Examples from RICO-Semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Comparing the RICO-Semantic examples to the CLAY examples in Figure 11, the number of  elements in each layout is less;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' the average size of each element is larger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' and the design patterns look simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Examples from PublayNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Implementation Details  The detail of each component in Figure 3 can be found in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We implemented the proposed architecture in JAX and Flax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We use ADAM optimizer (b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='9, b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='98) with 500k  steps and a batch size of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='001 with linear warming up to 8k steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The model is trained using 8  Google Cloud TPU v4 cores for 47 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' User Test Details  Each participant is presented with 48 randomly generated questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In each question, there will be a pair of layouts, and the  user needs to pick the better one (Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Each layout pair is selected from two of these three groups: ground truth, VTN,  and PLay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We also ensure the 48 questions equally cover all possible combinations of these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  When answering a question, the group gets +1 score if the user pick the layout that belongs to this group, and gets -1 vice  versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Both groups get 0 if the user thinks their layouts are equally good or bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The final scores are normalized by the  number of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  We recruit 15 user interface (UI) and user experience (UX) designers with experience in layout design for the user test  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' List of first-stage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' β is the weight of KL loss, d is the dimension of the latent space for each element, and c is the  codebook size of VQVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In each row, FID-Recon and FD-VG-Recon are computed using the reconstruction samples generated by the  first-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' FID, FD-VG, and G-Usage are computed using PLay trained on this first-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  First-stage  β  d  c  FID-Recon  FD-VG-Recon  FID  FD-VG  G-Usage  None  n/a  8  n/a  n/a  n/a  167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='683  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='991*  VAE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='0  8  n/a  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='04e−1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='937  ∗PLay without the first-stage model generates mostly random, unaligned boxes and therefore, has a high guideline usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We also give the following criteria and information to the participants:  Please evaluate for both aesthetics and functionalities of the layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' For example, the alignments, proportions, how  reasonable for the buttons to be put here, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Some of the layouts are intentionally not valid nor optimal (many of them are synthesized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Therefore, please do not try  too hard to justify every layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Use your intuition and experience as a designer to pick the better one from each pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Some of the examples that do not look like a full mobile UI screen might still be valid designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' They can represent UI  cases such as popped windows, opened drawers, or simply without background image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Note that the text elements are often not aligned due to their various lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' First-stage Models  In Table 6, we can see that β values do not have significant effect on the metrics as long as they are small enough (<= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Latent dimension d = 8 seems to be the optimal choice in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' VQVAEs achieve comparable numbers in FID  and FD-VG but fall short on G-Usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Although increasing the codebook size improves G-Usage, it is less computationally  efficient to use c > 16384 compared to a simple VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' More Results  In Figure 16, we decode the output from each denoising step in the sampling process and visualize the rendered results from  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We also visualize the generated samples from PLay (Figure 17, 18, and 19) and from VTN (Figure 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Model Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' We use the same Transformer Encoder (a) as the building block for all the components in PLay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' The  decoder (c) is similar to (Carlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=', 2020), using learned embeddings and adding z in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Note that positional embedding is  needed in (e), since the encoded elements z does not have explicit positional information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content=' The Feature-wise Linear  Module is composed by MLP layers to map t to scale and shift for the Feature-wise Affine layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='(a) Transformer Encoder Block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Skip  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Skip  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Self  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='indno  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Attention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Norm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Norm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='(b) First-Stage Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='elements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='xL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='Model Parameters:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='elements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='embeddings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Encoder Block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='L= 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='L for guideline encoder =- 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='xL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='num of heads = 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='qkv dim = 512  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='MLP dim = 2048  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='(d) Guideline Encoder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='MLP dim for guideline encoder = 1024  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='d=8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Transformer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='guidelines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='encoded guidelines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Encoder Block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='xL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='pos  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='(e) Denoise Network  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='x L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Skip  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Transformer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Feature-wise  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Layer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Cross  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='MLP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Encoder Block  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Affine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Norm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Attention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='00s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='scale  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='FiLM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='encoded guidelines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='shiftPLay: Parametrically Conditioned Layout Generation using Latent Diffusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' An example question in the user test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Question 05:  Which layout below do you think is better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Left  Right  Similarly good/bad  :8:  Question05  TEXT  CONTAINER  PICTOGRAM  LIST_ITEM  BUTTON  IMAGE  TOOLBAR  LABEL  TEXT_INPUT  DRAWER  CHECK_BOX  NAVIGATION_BAR  CARD_VIEW  SPINNER  SWITCH  PAGER_INDICATOR  RADIO_BUTTON  SLIDER  PROGRESS_BAR  NUMBER_STEPPER  ADVERTISEMENT  MAP  DATE_PICKERPLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Decoding the latent diffusion steps back to layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Interestingly, although the first 150 steps look noisy and random, the last  50 steps still demonstrate the nature of the denoising process, forming from low frequency features to high frequency features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' This  coarse-to-fine generation process can be commonly found in imaged-based diffusion models, but it is the first time we observe the same  process in latent diffusion models for vector graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
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+page_content='step-193  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='step=197  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='step=195  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='step=196  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='step=198  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='step=200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='step=199PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Samples generated by PLay trained on CLAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' In each cell, left: input guidelines (blue lines) on top of the generated layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  Right: generated layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' (continued) Samples generated by PLay trained on CLAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' (continued) Samples generated by PLay trained on CLAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content='  PLay: Parametrically Conditioned Layout Generation using Latent Diffusion  Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
+page_content=' Samples generated by VTN trained on CLAY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFJT4oBgHgl3EQfZCzN/content/2301.11529v1.pdf'}
diff --git a/ltFIT4oBgHgl3EQfsStT/content/tmp_files/2301.11335v1.pdf.txt b/ltFIT4oBgHgl3EQfsStT/content/tmp_files/2301.11335v1.pdf.txt
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@@ -0,0 +1,2693 @@
+arXiv:2301.11335v1  [astro-ph.IM]  26 Jan 2023
+Chapter 1
+Charge exchange in X-ray astrophysics
+Liyi Gu and Chintan Shah
+Abstract Charge exchange is an atomic process that primarily occurs at interfaces
+between the neutral and ionized gas. The study of the process has been carried out on
+three levels: the theoretical calculation of the cross sections, the laboratory measure-
+ments of reaction rates and line strengths, and the observational constraints using
+celestial objects. For a long time in the past, the status of astrophysical observations
+in the X-ray band lagged behind the other two aspects until the discovery of X-ray
+from a comet was made in 1996, which changed the research landscape. Recent
+observational evidence suggests that charge exchange has been seen or can be ex-
+pected from a surprisingly broad range of locations, from the Earth’s exosphere to
+the large-scale structures of the Universe. The rapid development of high-resolution
+X-ray spectroscopy, in particular the non-dispersive micro-calorimeters, is paving
+the way to revolutionary new science possibilities both in the laboratory and astro-
+physics. This chapter summarizes the current knowledge of charge exchange and its
+relevance on astrophysics, especially X-ray spectroscopy.
+1.1 Introduction
+The phenomenon of ion-neutral interaction was studied in early laboratory exper-
+iments. The scattering experiment of alpha particles or bare He ions, against gold
+atoms has led Rutherford [204] to establish the classical model of atomic struc-
+ture. The quantum mechanical treatment of the ion-neutral scattering problem was
+Liyi Gu (�)
+SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, the Nether-
+lands, e-mail: l.gu@sron.nl
+Chintan Shah
+NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771, USA;
+Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany, e-mail:
+chintan.shah@mpi-hd.mpg.de
+1
+
+2
+Liyi Gu and Chintan Shah
+explored by Massey & Smith [167], and Mulliken [178], among others, pioneered
+the theoretical spectral modeling of charge exchange using atomic orbital approxi-
+mation. From 1950s, as the gas-beam technique matured, an increasing number of
+experiments have been put forward to measure directly the charge exchange cross
+sections [71, 227]. Early investigations of this process have been covered in several
+reviews [84, 122, 135].
+It has been well known that the atom-ion charge exchange by far dominates all
+kinds of electron-ion collisions. For a collision of ∼ keV/u, the effective cross sec-
+tion of charge exchange between highly-ionized ions and neutrals are at least two
+orders of magnitude greater than the radiative recombination and electron-impact
+excitation cross sections. From the ion view, charge exchange provides efficient
+recombination on the ionization state. Unlike radiative recombination, charge ex-
+change always ends up in an excited state, producing strong line emissions but zero
+continuum.
+Charge exchange has been well utilised in the research of nuclear fusion. The
+radiative transitions from excited states of impurity ions resulting from charge ex-
+change may constitute a substantial portion of the total energy losses; meanwhile,
+the charge exchange emission can be used as a powerful tool for spectroscopic di-
+agnostics of high-temperature plasmas (e.g., Carolan et al. 42).
+The study of charge exchange has been extended to astrophysics, in particular
+the interaction between solar wind protons with neutral particles near the Sun, in
+the planets, and from the interstellar medium (e.g., Wallis 244). Charge exchange
+has also been found to be the origin of the broad Hα emission lines observed at the
+shock wave of supernova remnants [45]. These lines are emitted from hot hydrogen
+atoms created by electron capture of hot protons in the post-shock plasma.
+An important breakthrough was made in the late 1990s, when the first detection
+of cometary X-rays [154] and the charge exchange interpretation [53] was made.
+Since then, the interest on charge exchange X-ray has been activated in observa-
+tions, but also in theory and in the laboratory. It has been recognized as a primary
+or second emission mechanism in a broad range of astrophysicobjects, as well as a
+time-varying background emission tl X-ray sources. In this review, we provide an
+overview of our current knowledge on charge exchange, with a focus on X-ray as-
+trophysics with spectroscopic utilised In section 1.2 we present the basic physical
+properties of charge exchange, as well as the theoretical and experimental achieve-
+ments. Section 1.3 shows the observational results obtained so far. The review will
+close with conclusions and a short outlook in section 1.4.
+
+1 Charge exchange in X-ray astrophysics
+3
+0.01
+0.1
+1
+10
+100
+1000
+104
+105
+106
+1
+10
+100
+1000
+Langevin
+COB
+Bohr-Lindhard
+MCLZ (kronos)
+Janev 1993
+QMOCC (Nolte)
+Energy (eV/u)
+Cross section (10-16 cm2)
+O7+ + H
+Fig. 1.1: Total cross sections for O7+ + H based on the experimental measure-
+ments of Zhang et al. [267] (black) and Meyer et al. [169] (red). The theoretical
+calculations from Janev et al. [123], multichannel Landau-Zener [176], and quan-
+tum molecular-orbital close-coupling [180] are shown in blue, purple, and cyan.
+The classical Langevin, over-the-barrier, and Bohr-Lindhard models are plotted in
+dashed lines.
+1.2 Plasma modeling
+1.2.1 Physics and classic models
+Charge exchange recombination consists of the transfer of one or more electrons
+from (mostly) the outer atomic shell of target atom/ion M to the outer shell of an
+impinging/projectile atom/ion X, leading to the recombination of the impinging par-
+ticle while the ionization of M. The process can be written as
+Xq+ + M → X(q−m)+ + Mm+,
+(1.1)
+where q is the original charge of the X, and m is the number of electrons transferred.
+m = 1 can be referred as single electron capture (SEC) from the point of view of the
+impinging ion. m = 2 is designated as double electron capture (DEC).
+The total cross section for this process is typically of order of 10−15 - 10−14 cm2,
+greater by a factor of 102 − 103 than the cross sections of typical electron-impact
+radiative recombination and dielectronic recombination. The classical analytic cal-
+
+4
+Liyi Gu and Chintan Shah
+culations, as summarized in Stancil [223], provide order estimate of the total cross
+sections. They include
+•
+Langevin model for < 0.1 eV/u collision, where u stands for atomic mass unit.
+This model predicts an energy dependence E−1/2 of the cross section, which is
+found to be in line with the Landau-Zener approximation shown in Figure 1.1.
+•
+Classical over-the-barrier (COB) model for intermediate collision speed. In this
+approximation, the cross section becomes a constant over energy.
+•
+Bohr-Lindhard model, which predicts a E−7/2 behavior, becomes applicable at
+very high energy.
+These classic treatments for O7+ +H are shown in Figure 1.1 with a comparison
+to various explicit theoretical and experimental results. These simple models seem
+to reproduce the correct general trend as a function of energy, though apparently
+overestimating the absolute values.
+Charge exchange operates in a semi-resonant favor when a subset of the energy
+levels of the projectile ion and the target overlap. If the projectile ion is highly
+charged, its highly excited levels overlap with the neutral’s ground state, making it
+possible to transfer electrons onto high-n levels. The distribution of n with the most
+likelihood has been studied in various ways,
+•
+Simple COB model predicts the dominant n level nmax ≤ q/√2IM, where q is the
+initial charge and IM is the ionization potential of the target.
+•
+Systematic compilation of existing results by Janev & Winter [122] reported a
+scaling nmax ∼ q(1 + (q − 1)/
+�
+(2q))−0.5 ∼ q3/4.
+•
+More recently, Gu et al. [92] suggested that the n distribution is energy depen-
+dent; the capture is peaked on a single n channel at low collision energy, while
+it tends to be shared among several neighbour n channels at the intermediate en-
+ergy range. The analytic form of the transfer is reported as Eq.A.1 of their paper.
+At the very high energy end, the n distributions become narrow again [251].
+For the He-like triplet, the cascade contribution from large n to the formation of
+forbidden lines is substantially larger than that to the resonance lines, yielding a
+large forbidden-to-resonance line ratio.
+To construct an emission spectrum from charge exchange, it is necessary to
+model the population of capture onto states with orbital angular momenta l in the
+range of (0, n-1). For collisions with E ≥ 10 keV/u, it is expected that the l distri-
+bution function W is weighed statistically, Wst = (2l + 1)/n2. Therefore the largest
+possible l = n−1 will be the most populated. As the collision velocity decreases, the
+capture l tends to decrease towards l = 1 or 2 taking one of the following possible
+forms,
+•
+Low energy approximation L1, WL1 = (2l + 1) [(n - 1)!]2 / [(n + l)! (n - l - 1)!],
+•
+Modified low energy approximation or L2, WL2 = l (l + 1) (2l + 1) (n - 1)!(n - 2)!
+/ [(n + l)!(n - l - 1)!],
+•
+Shifted low energy approximation or L3, WL3 = (2l + 3) [(n - 1)!]2 / [(n + l + 1)!
+(n - l - 2)!],
+•
+“Separable”, Wse = (2l + 1)/q e−l(l+1)/q,
+
+1 Charge exchange in X-ray astrophysics
+5
+•
+even or constant over l, Wev = 1 / n.
+For a high-speed collision, shells with large l are significantly populated, resulting
+in more optical and UV lines following the Yrast cascade relaxation. The low-speed
+charge exchange would instead yield more X-ray lines because the shells with small
+l become more populated.
+Commonly the capture on the total spin quantum number is assumed to be sta-
+tistical, and therefore the triplet-to-singlet ratio of He-like ion, produced from the
+H-like ion reaction, is 3. However, bias from the statistical weight is found in recent
+theoretical calculations (e.g., Nolte et al. 180, Wu et al. 258). These works suggested
+that the triplet-to-singlet ratios, and therefore the distribution on the spin quantum
+number, are likely to be energy-dependent.
+Transitions from highly-excited levels with large n and the large forbidden-to-
+resonance line ratio are two basic spectral features of the charge exchange X-ray
+emission. The high-n Rydberg lines are usually stronger in H-like than in He-like
+series because the cascades in the He-like ions are subject to the singlet-to-triplet
+branching ratios. The UV-to-X-ray ratios of charge exchange emission are larger
+than those of the electron-impact emission.
+1.2.2 Theoretical calculations of absolute cross sections
+
+6
+Liyi Gu and Chintan Shah
+Table 1.1: Theoretical data
+Reference
+typea
+ion
+energy (eV/u)
+Savin [206]
+total
+D+ H+, D+ + H
+≤ 6
+Cornelius et al. [51]
+total,nl
+He2+ − Ne10+ + H,Li
+103 - 105
+Stancil & Zygelman [218]
+total
+Li+ H+
+10−4 - 10
+Kimura et al. [128]
+total
+Li+ H+
+10−5 - 103
+Janev et al. [121]
+total,nl
+AA+ + H (5 ≤ A ≤ 74)
+30 - 8×104
+Turner et al. [240]
+total,nl
+B2+ + H
+∼ 10−1 - 104
+Liu & Wang [159]
+total,nl
+B4+ + H
+10−2 - 2×105
+Stancil et al. [220]
+total
+C+ + H
+≤ 103
+Kimura et al. [130]
+total
+C, N, O, Si+ H+
+≤ 103
+Butler et al. [40]
+total
+Cq+, Nq+, Oq+, Neq+ + H (q = 2,3),
+10−1 - 1
+O2+ + He
+Butler & Dalgarno [39]
+total
+Cq+, Nq+, Oq+, Neq+, Mgq+, Siq+, Sq+, Arq+
+10−2 - 1
++H (q = 2,3,4)
+Heil et al. [109]
+nl
+C2+,N3+,O3+,Ne2+,Ne3+ + H
+0.27 - 8.1
+Guevara et al. [99]
+total
+C3+,O3+,Si3+ + H
+∼ 0.1 - 104
+Wu et al. [257]
+total,nl
+C3+ + He
+∼ 10−4 - 103
+Kimura et al. [127]
+total
+Cq+,Nq+,Oq+,Fq+,Neq+,Krq+
+600
++ H (q = 10 − 25 for Kr and 4 − 9 for rest)
+Mullen et al. [177]
+total,nl
+C6+,N7+,O8+,Ne10+,Na11+,Mg12+,
+∼ 10−3 - 105
+Al13+ + H,H2O,CO,CO2,OH,O
+Pindzola & Fogle [192]
+total,nl
+C6+ + H,He
+2.7×103 - 8.3×103
+Cumbee et al. [56]
+total,nl
+C6+ − Al13+ + H,He,H2
+2 ×102 - 5 ×103
+Lin et al. [153]
+total,nl
+N+ H+,N+ + H
+∼ 10−4 - 103
+Barrag´an et al. [15]
+total
+N2+ + H
+2×10−3 - 3 ×105
+Herrero et al. [110]
+total
+N2+ + H
+∼ 10−1 - 102
+Sidky et al. [213]
+total,nl
+N4+ + H
+∼ 50 - 2×104
+Feickert et al. [69]
+total
+N4+ + H
+30 - 105
+Zygelman et al. [272]
+total,nl
+N4+ + H
+∼ 0.1 - 8×103
+Stancil et al. [219]
+total,nl
+N4+ + H
+∼ 10−2 - 6×103
+Shimakura & Kimura [212]
+total,nl
+N5+ + H
+∼ 10−2 - 104
+Hasan et al. [102]
+n
+N7+,O7+ + He,CO,CO2,H2O
+2×103 - 4.67×103
+Chambaud et al. [44]
+total
+O+ H+
+≤ 10−1
+Stancil et al. [221]
+total
+O+ H+, O+ + H
+10−4- 107
+Pichl et al. [189]
+total,nl
+O+ + H2
+1 - 104
+Pichl et al. [190]
+total,nl
+O+ + H2
+∼ 5×102 - 104
+Barrag´an et al. [16]
+total
+O2+ + H,N2+ + H
+∼ 10−3 - 1044
+Cabello et al. [41]
+total
+O2+ + H
+∼ 125 - 3.4 ×103
+Honvault et al. [113]
+total,nl
+O2+ + H
+∼ 10 - 3×104
+Honvault et al. [114]
+total
+O2+ + H
+∼ 3×10−4 - 4
+Wang et al. [246]
+total,nl
+O3+ + H
+0.1 - 103
+Bienstock et al. [26]
+total
+O3+ + H
+0.1 -5×103
+
+1 Charge exchange in X-ray astrophysics
+7
+Reference
+typea
+ion
+energy (eV/u)
+Dalgarna et al. [57]
+nl
+O3+ + H
+0.1 - 1
+Gargaud et al. [83]
+nl
+O3+ + H
+≤ 1
+Roueff & Dalgarno [203]
+total
+O3+ + H
+∼ 1
+Wu et al. [256]
+total,nl
+O3+ + He
+0.01 - 103
+Wang et al. [247]
+total,nl
+O3+ + H2
+0.1 - 104
+Wu et al. [259]
+total,nl
+O6+ + H
+0.1 - 104
+Zhao et al. [271]
+total,nl
+Ne2+ + He
+0.1 - 104
+Lyons et al. [164]
+total,nl
+Ne(8−10)+ + H,Mg(8−12)+ + H
+10−3 - 5 ×104
+Rigazio et al. [201]
+nl,line
+Ne10+ + Ne,He,H2,CO2,H2O
+9
+Croft & Dickinson [54]
+total
+Na+ H+
+≤34
+Allan et al. [6]
+total
+Mg + H+
+≤40
+Fogle & Pindzola [74]
+total,nl
+Mg12+ + H,He
+103 - 5 ×103
+Kimura et al. [129]
+total
+Si+ H+
+≤30
+Suzuki et al. [229]
+total
+Si2+ + He
+∼ 20 - 6 ×103
+Clarke et al. [49]
+total,nl
+Si2+ + H
+∼ 0.03 - 100
+Wang et al. [248]
+total,nl
+Si3+ + H
+0.01 - 106
+Stancil et al. [222]
+total,nl
+Si3+ + He
+∼ 10−4 - 400
+Honvault et al. [115]
+total,nl
+Si3+ + He
+1 - 7 ×103
+Rabli et al. [197]
+total
+Si3+ + He
+∼ 10−4 - 103
+Liu et al. [158]
+total,nl
+Si3+ + H
+0.01 - 2 × 105
+Suzuki et al. [230]
+total
+Si4+ + He
+10−3 - 2.5 ×103
+Wang et al. [245]
+total,nl
+Si4+ + He
+∼ 10−4 - 107
+Vaeck et al. [242]
+total,nl
+Si4+ + He
+∼ 0.01 - 100
+Bacchus-Montabonel et al. [13] total,nl
+P2+ + H
+∼ 100 - 105
+Kimura et al. [131]
+total
+S+ H+
+≤ 103
+Zhao et al. [270]
+total,nl
+S+ H+
+∼ 10−4 - 104
+Bacchus-Montabonel et al. [11] total,nl
+S2+ + H
+100 - 104
+Łabuda et al. [145]
+total,nl
+S3+ + H
+2×103 - 8×103
+Bacchus-Montabonel [12]
+total,nl
+S3+ + He
+2×103 - 5×104
+Stancil et al. [224]
+total,nl
+S4+ + H
+∼ 10−4 - 107
+Mullen et al. [176]
+total,nl
+Fe25+,Fe26+ + H
+10−3 - 105
+Sterling & Stancil [226]
+total,nl
+Geq+,Seq+,Brq+,Krq+,Rbq+,Xeq+
+10−3 - 106
++H (q = 2,3,4,5)
+a: total = total cross section, nl = nl-resolved cross section
+To produce accurate X-ray emission models, it is desirable to have absolute cross
+sections using state-resolved theoretical calculation for all the relevant reactions
+at a broad range of collision velocities. Simple approximations described above,
+e.g., COB, cannot reproduce accurate experiment results (e.g., Hasan et al. 102).
+Furthermore, COB-like models do not have the facility to calculate the l distribution.
+For collisions with energies E ≥ 5 keV/u, the Classical Trajectory Monte Carlo
+(CTMC) method is often considered to be a valid approximation (e.g., Otranto et al.
+183). The CTMC provides a non-perturbative solution to the classical equations of
+
+8
+Liyi Gu and Chintan Shah
+motion in the many body interactions during the collisions. However, this method
+does not give an adequate description of the lowering of l quantum number as the
+collision energy decreases below ∼ 1 keV [17].
+The Multichannel Landau-Zener approach (MCLZ) provides another flexible
+tool for massive calculation. MCLZ considers a slow collision between an ion and
+a neutral particle using a quasi-molecular configuration, therefore, it is mostly used
+for models with E ≤ a few keV/u. It studies the crossing between pre- and post-
+exchange potentials of the configuration and determines the final state of the cap-
+ture. Since the MCLZ requires that crossing distances are fully resolved, while for
+a bare ion collision, the capture l degenerates with the n state, this method cannot
+resolve l distribution for bare ion collision [176]. Empirical l distribution functions
+listed in 1.2.1 must then be applied.
+(Quasi-)quantum mechanical models involving close coupling are known to
+be accurate simulations of atomic collisions. The Atomic-Orbital Close-Coupling
+(AOCC) is found to be applicable between ∼ 0.1 and 100 keV/u. It is based on
+bound state calculation on either center of the two reactors, assuming an infinite
+separation between them. The number of n and l are limited by the finite size of the
+close-coupling expansion.
+The Molecular-Orbital Close-Coupling (MOCC) is another type of close-coupling
+which is optimized for low energy (E ≤ 10 keV/u). It models the union state of the
+two reactors. The molecular wave functions, as well as the coupling terms of the rel-
+evant status, are calculated and fed into the close-coupling scatter calculation. The
+number of n and l are limited in the same way as the atomic-orbital. A full quantum
+mechanical version, QMOCC, has been developed to cope with very low energy
+collision at E < 10 eV/u [223].
+We summarize in Table 1.1 a comprehensive, though not exhaustive, list of the-
+oretical calculations. The reactors, type of the data products (total, n−, and nl−
+resolved cross sections) and the energy range indicated in the references, are all
+recorded. The collision energy determines the range of application; for instance,
+the solar wind ions collide with comet atmosphere at E ∼ 0.1 − 5 keV/u, while for
+the charge exchange component in collisional or photo-ionization equilibrium, the
+relevant collisions are much less energetic with E ∼ 10 eV/u.
+1.2.3 Laboratory measurements
+
+1 Charge exchange in X-ray astrophysics
+9
+Table 1.2: Experimental cross section data
+Reference
+typea
+ion
+energy (eV/u)
+Greenwood et al. [88]
+total
+H+,He+,He2+ + H2O,CO2
+300 - 7.5×103
+Lubinski et al. [163]
+nl
+He2+,N5+ + H2
+5 - 103
+Kusakabe et al. [142]
+total
+He2+ + H2,N2,O2,CO,CO2
+4×103
+Kusakabe et al. [143]
+total
+He2+ + He
+2×102 - 4×103
+Shah et al. [210]
+total
+Liq+ (q=1-3)
+4×103-3.43×105
+Seim et al. [208]
+total
+Liq+ (q=2-3), Nq+ (q=2-5), Neq+ (q=3-5)
+3q×103 - 1.2q×104
+Goffe et al. [86]
+total
+Bq+ (q=1-5), Cq+ (q=1-4)
+q×105 - 2.5q×106
+total
+Cq+ (q=5,6), N7+
+q×105 - 2.5q×106
+McCullough et al. [170]
+total
+B2+, C+, N+, Mg2+
+8×102 - 4×104
+Crandall et al. [52]
+total
+Bq+ (q=2-5), Cq+ (q=3,4), Nq+ (q=3,4), Oq+ (q=5,6)
+4q×103 - 2.5q×104
+Pieksma & Havener [191]
+total
+B4+
+60 - 1.2×103
+Gardner et al. [82]
+total
+Bq+ (q=2-4), Cq+ (q=2-4), Nq+ (q=2-5), Oq+ (q=2-5)
+6q×103 - 2.3q×104
+Meyer et al. [169]
+total
+Bq+ (q=2-5), Cq+ (q=3,4), Nq+ (q=3,4)
+2.5×104 - 2×105
+Phaneuf et al. [187]
+total
+Cq+ (q=1-4), Nq+ (q=1-5), Oq+ (q=1-5), Siq+ (q=2-7) 8.6×103 - 1.65×106
+Nutt et al. [182]
+total
+C2+
+5×102 - 1.4×103
+Havener et al. [107]
+total
+C3+
+0.3 - 3×103
+Greenwood et al. [90]
+total,line
+C,N,O,Neq+ + He,H2,H2O,CO2 (q = 3-9)
+7q×103
+Greenwood et al. [89]
+total,line
+C,N,O,Neq+ + H2O,CO2 (q = 3-10)
+550 - 800 km s−1
+Fogle et al. [73]
+line
+C2+ + H2
+103 - 3×104
+Phaneuf et al. [188]
+total
+Cq+ (q=3,4), Oq+ (q=2-6)
+10 - 1×104
+Lubinski et al. [162]
+nl
+C4+,N5+,O6+ + H2
+5 - 4×103
+Phaneuf et al. [188]
+total
+Cq+ (q=5,6)
+10 - 1×104
+Sant’Anna et al. [205]
+total
+C3+
+1×106 - 3.5×106
+Ciric et al. [48]
+total,nl
+Cq+ (q=3,4), N5+, O6+
+7×102 - 4.6×103
+McCullough et al. [171]
+total,nl
+C3+
+6×102 - 1.8×104
+Panov et al. [186]
+total
+C4+, N5+, O6+, Ne8+
+100 - 1300 km s−1
+total
+Cq+ (q=5,6), Nq+ (q=6,7), Oq+ (q=7,8), Neq+ (q=9,10)
+100 - 1300 km s−1
+Dijkkamp et al. [63]
+total,nl
+Cq+ (q=3,4), N5+, O6+
+2.7 - 13.6
+Gao & Kwong [81]
+total
+Cq+ + CO (q = 3-4)
+5×102
+Fritsch & Lin [77]
+total,nl
+C4+
+100 - 2×104
+Hoekstra et al. [112]
+total,nl
+C4+
+2.4×102 - 4.4×102
+Hoshino et al. [116]
+total
+C4+ + He
+2.4×102 - 4.4×102
+Dragani´c et al. [64]
+total
+C5+ + H
+0.64 - 1.2×104
+Bodewits & Hoekstra [31]
+total, n
+C5+ + H2O
+113 - 3.75×103
+Beiersdorfer et al. [20]
+line
+Cq+,Oq+ + CO2 (q = 5-8)
+several 100 km s−1
+Defay et al. [58]
+line
+C6+ + He
+460 - 3.2×104
+Leung & Kirchner [147]
+total,nl
+C6+ + He,H2
+103 - 2.5×104
+Andrianarijaona et al. [9]
+line
+C6+ + Kr
+320 - 4.6×104
+Leung & Kirchner [148]
+line
+C6+,O8+ + Kr
+500 - 4×104
+Stebbings et al. [225]
+total
+N+, O+
+100 - 4×104
+
+10
+Liyi Gu and Chintan Shah
+Reference
+typea
+ion
+energy (eV/u)
+Folkerts et al. [75]
+total
+N4+
+1-3×102
+Bliek et al. [27]
+total
+N4+ + H , H2
+103-4×103
+Huq et al. [118]
+total
+Nq+ + H (q = 3-5)
+0.9 - 1.4×103
+Zhang et al. [266]
+total
+N7+,O7+ + H
+2 - 1.22×103
+Hasan et al. [102]
+total,n
+N7+,O7+ + He,H2O,CO2,CO
+2×103, 4.67×103
+Fite et al. [72]
+total
+O+
+100 - 3.6×104
+Kusakabe et al. [141]
+total
+O+ + H2,CO2,CO,CH4,C2H2,C2H6,C3H8
+200 - 4.5×103
+Beijers et al. [23]
+nl
+O3+
+45 - 752
+Church & Holzscheiter [47]
+total
+Oq+ (q = 2,3) + H
+1.5×104
+Havener et al. [106]
+total
+Oq+ (q = 3,4) + H(D)
+1-103
+Meyer et al. [169]
+total
+Oq+ (q=3-6), Siq+ (q=4-9), Feq+ (q=4-15)
+2.5×104 - 2×105
+Gao & Kwong [80]
+total
+Oq+ + CO (q = 4,5)
+1.5q×103
+Havener et al. [105]
+total
+O5+
+0.9-800
+Bodewits & Hoekstra [30]
+total,nl
+O6+ + H2O
+100 -7.5×103
+Miller et al. [174]
+line
+O6+ + CO
+3.6×104
+Machacek et al. [165]
+total
+O6+ + CO,H2O,CO2,CH4,N2,NO,N2O,Ar
+1.17×103 - 2.33×103
+Meyer et al. [169]
+total
+Oq+ (q=7,8)
+2.5×104 - 2×105
+Beiersdorfer et al. [18]
+line
+O8+,O7+ + CO2
+15
+Ne10+,Ne9+ + Ne
+9
+Bangsgaard et al. [14]
+total
+Oq+ + He (q = 2-5)
+100 -1.5×103
+Han et al. [100]
+total, n
+O6+ + CO2,CH4,H2,N2
+2.63×103 - 1.95×104
+Otranto et al. [183]
+line
+O8+ + CH4,CO2,N2
+10
+Seely et al. [207]
+line
+O8+ + Kr
+445 - 8.18×103
+Wang & Wang [249]
+total, n
+O8+ + He
+16 - 54
+Mroczkowski et al. [175]
+total
+Ne2+
+139 - 1.49×103
+Seredyuk et al. [209]
+total
+Ne2+ + H(D)
+59-949
+Huber [117]
+total
+Neq+ (q=2-4), Arq+ (q=2-4,6)
+103 - 1.5×104
+Rejoub et al. [200]
+total
+Ne3+ + H
+0.07 -826
+Abu-Haija et al. [3]
+total,n
+Neq+ + CO2,H2O (q = 3-5)
+15q - 500q
+Havener et al. [108]
+total
+Ne4+ + H
+0.1 - 1006
+Kusakabe & Shirai [144]
+total
+Neq+ + Ne,Ar,Kr,Xe
+2q×103
+Abu-Haija et al. [2]
+total
+Ne6+ + CO2,H2O
+450 - 2.4×103
+Zhang et al. [267]
+line
+Ne8+ + He,Kr
+400-900 km s−1
+Leung & Kirchner [149]
+total, n
+Ne8+ + He,H2
+4.6 - 10.9
+Xu et al. [262]
+total, n
+Ne8+,Ne9+ + He,H2
+103 - 2.475×104
+Otranto et al. [184]
+line
+Ne10+,Ar18+,Kr36+ + Ar
+18 - 4×103
+Tawara et al. [237]
+line
+Kr36+,Ar18+,Ne10+ + Ar
+4×103
+Kim et al. [126]
+total
+Siq+ (q=2-7)
+5.1×104 - 2.04×106
+Bruhns et al. [37]
+total
+Si3+
+40 - 2500
+Tawara et al. [233]
+total
+Siq+ + He (q = 3-5)
+∼ 100 - 1000
+Tawara et al. [234]
+total,line
+Si13+,S15+,Ar17+ + Ar,He,CH4
+103-7×104
+Wilson et al. [254]
+total
+S2+ + H , H2
+2×103-8×103
+Shah et al. [211]
+line
+S16+ + CS2
+∼ 10
+
+1 Charge exchange in X-ray astrophysics
+11
+Reference
+typea
+ion
+energy (eV/u)
+McCullough et al. [172]
+total,nl
+Arq+ + H, H2, He (q = 4-6)
+q×103, 2q×103
+Xia et al. [261]
+total, nl
+Ar8+ + He
+1.4×103 - 2×104
+Beiersdorfer et al. [21]
+line
+Ar16+ + H
+4×104
+Allen et al. [7]
+line
+Ar17+,Ar18+ + Ar
+0.25q - 125q
+Allen et al. [8]
+line
+Ar17+,Ar18+ + Ar
+5 - 2×103
+Beiersdorfer et al. [22]
+line
+Feq+ + O2,N2,H2O,CO2 (q = 16 - 23)
+≤ 10-20
+Kumar et al. [140]
+total
+Cuq+ + H2,Ne,Ar,Kr,Xe
+2.5×106
+Betancourt-Martinez et al. [24]
+line
+Ni19+,Ni18+ + He,H2
+10
+Tawara et al. [235]
+total,line
+Krq+ + He,Ne,Ar,N2,CH4 (q = 27-33)
+2×103
+Hasan et al. [103]
+line
+Xe18+,Xe24+ + Na
+500 - 4.5×103
+Tawara et al. [236]
+total,line
+Taq+ + Ar (q = 45-49)
+103 - 4×104
+a: total = total cross section, n = n-resolved cross sections, nl = nl-resolved cross
+sections, line = line emissivities or ratios
+Laboratory measurements provide the absolute calibration of the atomic con-
+stants. The charge exchange process has been measured in the laboratory since
+1950s (e.g., Hasted 104), using a crossed-beam setup in which a beam of ionized
+particles passes through a chamber filled with neutral gas. The storage ring facilities
+are often used for measurements at a high collision energy (E ∼ MeV/u), while the
+electron cyclotron resonances and the electron beam ion sources are applied to mea-
+surements with medium collision energy (E ∼ keV/u). A significant part of the early
+studies with crossed-beam measurements contributed to the fusion research because
+of the charge exchange influence on plasma parameters such as charge state distribu-
+tion and radiative cooling e.g., Seim et al. 208. Recent interest arises for astrophys-
+ical applications with the measurements of highly charged ion collisions, including
+among many others, the total cross sections of bare C, N, O, and Ne charge exchang-
+ing with H2O, He, H2, CO2 measured at Jet Propulsion Laboratory [88, 89], and the
+K-shell charge exchange between He2+ and H2O using the crossed-beam facility at
+Kernfysisch Versneller Instituut in Groningen [28, 112].
+The combination of charge state determination and spectroscopic measurements
+becomes more available lately, providing reliable measurement on the velocity-
+dependent total and state-resolved cross sections. Some such measurements has
+been made with the cold target recoil-ion momentum spectroscopy (COLTRIMS),
+which has been developing since 1990s (e.g., Abdallah et al. 1). The n states, some-
+times even l levels of the captured electrons, are inferred from the measurement
+of the ion kinetic energy loss (the “Q” value, e.g., Fischer et al. 70, Knoop et al.
+134, Xue et al. 263, Zhang et al. 265). A few most recent efforts are carried out in
+the COLTRIMS facilities in the Institute of Modern Physics of China on the Ne8+
+and Ne9+ collision with He and H2 [262], and at the COLTRIMS in combination
+with simultaneous X-ray spectroscopic experiments at the electron cyclotron reso-
+nance source of the University of Nevada, Reno [5].
+One potential issue with beam-gas approach is that the radiative cascade from the
+metastable levels of the product ions might take longer than the dynamical timescale
+
+12
+Liyi Gu and Chintan Shah
+1
+10 100
+1
+10 100
+1
+10 100
+1
+10
+100
+1000
+104
+105
+106
+1
+10 100
+O8+ + H
+Panov+83
+Meyer+79
+Wu+94 (+ref)
+Meyer+79     Greenwood+01
+Panov+83
+Greenwood+00
+O8+ + He
+O8+ + H2
+O8+ + CO2
+energy (eV/u)
+Cross section (10-16 cm2)
+0.55
+0.65
+0.75
+0.85
+Lyα
+Lyβ
+Lyγ
+Lyδ
+X-ray energy (keV)
+count rate
+0.018
+0.036
+0.016
+0.032
+Lyβ/Lyα
+Lyγ/Lyα
+Lyδ/Lyα
+0.3
+0.1
+0.15
+0.05
+0.3
+0.1
+500
+1000
+line ratio
+velocity (km/s)
+Gu+22
+Seely+17
+(a)
+(b)
+(c)
+293 km/s
+414 km/s
+Fig. 1.2: (a) Total cross sections as a function of collision energy for O8+ interact-
+ing with atomic hydrogen, atomic helium, molecular hydrogen, and CO2. The solid
+lines are the model values from the calculations with the MCLZ (black) and Janev et
+al. [123]. Data points are experimental results from sources indicated in the labels.
+(b) X-ray spectra from Seely et al. [207] on O8+ + Kr reaction for two different ve-
+locities. (c) Comparison of experimental and theoretical line ratios for O8+ charge
+exchange. The experimental data are taken from Seely et al. [207], and the predic-
+tions from Gu et al. [92], MCLZ, and Janev et al. [123] are shown in black, red, and
+blue curves.
+in the interaction region, making it complicated to study the X-ray spectra observed.
+The electron beam ion trap (EBIT) [152] devices provide a solution, as the ions are
+radially confined in a quasi-stationary state by the space charge potential of the
+electron beam and axially trapped by the electrostatic potential to the cylindrical
+drift tube electrodes. A useful technique for measuring charge exchange with the
+EBIT is the magnetic trap mode, in which the electron beam is turned off when the
+proper charge states are reached. The ions expand to form a larger cloud, but remain
+confined in the radial direction by the strong magnetic field and the potential applied
+to the trap electrodes. Typically, the trap potentials limit the energy of the ions stored
+in the EBIT. Therefore, the collision energy for the CX reaction is usually very
+low, less than 10–20 eV/u [17]. However, the advantage of the magnetic trapping
+mode is that the electron-ion collision can be safely eliminated, while the ion-neutral
+collision remains efficient and produces the pure charge exchange emission.
+The experiments using EBITs do not measure the absolute cross sections; in-
+stead, they often record the detailed X-ray spectra of the reactions. Earlier EBIT
+
+1 Charge exchange in X-ray astrophysics
+13
+experiments with limited spectral resolution used the parameter known as the hard-
+ness ratio, i.e., the intensity ratio of X-ray line emission from levels with n ≥ 3
+relative to n = 2 K-shell emission, to study the dependence on collision ener-
+gies [8, 17, 22, 211]. Recent high-resolution (resolved to a few eV) X-ray spectral
+measurements have become possible with the introduction of X-ray microcalorime-
+ters [19]. The high resolution is particularly important for charge exchange, which
+emits lines from high n levels, often in a narrow energy range. The NASA Goddard
+Space Flight Center X-ray Spectrometer (XRS) and EBIT Calorimeter Spectrometer
+(ECS) [195, 196] have been used for a number of charge exchange measurements
+performed at the Lawrence Livermore National Laboratory EBIT (e.g., Frankel et
+al. 76, Leutenegger et al. 150).
+The n-resolved line diagnostics are made possible with the EBIT plus high-
+resolution spectrometer, while the l-distribution of the capture, which is sensitive to
+the collision velocity (see § 1.2.1), remains challenging since the ion-neutral colli-
+sions in the magnetic trapping mode are always mild (≤ 10 eV). A possible solution
+is to measure the charge exchange photons simultaneously at longer wavelengths,
+e.g., in the UV and optical bands, where the radiative cascades of the Rydberg levels
+might provide additional constraints on the l-state distribution. In addition, efforts
+are needed to combine EBIT and high-resolution calorimeters with the COLTRIMS
+setup to fully investigate the kinematics of charge-exchangecollision and multi elec-
+tron capture processes [5]. Furthermore, the development of a suitable atomic hy-
+drogen target for EBIT measurements is ideal so that only single-electron capture
+capture can take place [151].
+In Figure 1.2, we show a set of historical and state-of-the-art measurements of
+the O8+ charge exchange. The total cross sections of O8+ with H, He, H2, and CO2
+targets are measured in crossed-beam in the 1970s to 2000s. In panel b, the X-ray
+quantum microcalorimeter spectra taken with the beam-gas measurements at Oak
+Ridge on O8+ with Kr charge exchange are shown for two different velocities. The
+comparison of the observed and calculated line ratios is plotted in panel c. A more
+comprehensive list of laboratory measurement efforts made in the past can be found
+in Table 1.2.
+1.3 Observations
+1.3.1 Ionization balance
+Charge exchange is relevant for determining the ion concentration as it serves as a
+recombination term for ion Xq+ with
+Xq+ + H(He) → X(q−1)+ + H+(He+),
+(1.2)
+but also in some conditions an ionization term with
+
+14
+Liyi Gu and Chintan Shah
+10−3
+0.01
+5×10−4
+2×10−3
+5×10−3
+0
+0.5
+1
+temperature (keV)
+temperature (keV)
+ion concentration 
+O I    O II
+O III  O IV
+N I    N II
+N III  
+10−3
+0.01
+5×10−4
+2×10−3
+5×10−3
+0
+0.5
+1
+Fig. 1.3: Charge state distributions of N (left) and O (right) as a function of equi-
+librium temperature for the CIE plasma. The dashed lines show the calculations
+when the charge exchange recombination rates are changed by 50%, while the other
+ionization and recombination data are kept the same.
+X(q−1)+ + H+(He+) → Xq+ + H(He).
+(1.3)
+These terms are most effective for photoionized and non-equilibrium collisional
+ionization plasmas where highly charged ions naturally co-exist with the neutral hy-
+drogen and helium. As shown in Gu et al. [96], the charge exchange contribution
+could even be effective for the collisional ionization equilibrium at a low tempera-
+ture (∼ a few eV, Fig. 1.3). The collision induced by the thermal motion of ions has
+a typical velocity of km s−1.
+Sets of recombination and ionization rates by charge exchange are tabulated in
+Arnaud & Rothenflug [10] and Kingdon & Ferland [133]. These tables have been
+used broadly in the ion concentration calculation, however, they lacks data of highly
+charged ions with q > 4. Moreover, a significant part of the rates are derived from
+theoretical calculations with classical approximations, e.g., Landau-Zener method.
+A systematic review and update are then needed to expand the q range as well as to
+improve the quality of charge exchange data for the ionization balance calculation.
+In table 1.3 we list a set of astrophysical observations made so far. For each
+observation, we feature the object, instrument, relevant emission features, together
+with a few remarks on the quality of the detection (e.g, using image only, with
+timing, and using spectrum with normal or high spectral resolution).
+
+1 Charge exchange in X-ray astrophysics
+15
+Table 1.3: Observations of charge exchange in X-ray
+Reference
+detector
+sources
+relevant ion(s)
+note
+Lisse et al. [154]
+ROSAT
+C/Hyakutake 1996 B2
+−
+image
+Owens et al. [185]
+BeppoSAX
+C/1995 O1 (Hale-Bopp)
+−
+image, spectrum
+Lisse et al. [155]
+ROSAT, EUVE
+P/Encke 1997
+−
+image
+Dennerl [59]
+Chandra
+Mars
+−
+spectrum, low S/N
+Krasnopolsky et al. [125]
+Chandra
+C/1999 T1
+−
+image, spectrum
+Wargelin et al. [250]
+Chandra
+Earth geocorona
+O VII, O VIII
+spectrum
+Snowden et al. [216]
+XMM-Newton
+Heliosphere
+C, O, Ne, Mg
+spectrum
+Lisse et al. [156]
+Chandra
+2P/Encke 2003
+−
+image, spectrum
+Dennerl et al. [60]
+XMM-Newton
+Mars
+C, N, O, Ne
+high res. spectrum
+Willingale et al. [253]
+Swift
+9P/Tempel 1 (deep impact)
+−
+timing, spectrum
+Krasnopolsky [139]
+Chandra
+sample of 4 comets
+−
+spectrum
+Fujimoto et al. [78]
+Suzaku
+Earth geocorona
+−
+spectrum
+Dennerl [61]
+Chandra
+Venus
+−
+spectrum, low S/N
+Bodewits et al. [29]
+Chandra
+sample of 8 comets
+−
+spectrum
+Snowden et al. [217]
+XMM-Newton
+Earth geocorona
+O VII
+timing, spectrum
+Wolk et al. [255]
+Chandra
+73P/2006, C/1999 S4
+−
+spectrum
+Christian et al. [46]
+Chandra
+8p/Tuttle, 17p/Holmes
+−
+spectrum
+Ezoe et al. [67]
+Suzaku
+Earth geocorona
+O VII variation
+timing, spectrum
+Koutroumpa et al. [138]
+XMM-Newton
+Heliosphere
+O VII, O VIII
+spectrum
+Katsuda et al. [124]
+Suzaku
+Cygnus Loop SNR
+O VII at 0.7 keV
+spectrum
+Liu et al. [160]
+XMM-Newton
+M82
+O VII, Ne IX, Mg XI high res. spectrum
+Konami et al. [136]
+Suzaku
+M82
+O VIII, Ne X
+spectrum
+Liu et al. [161]
+XMM-Newton
+9 star-forming galaxies
+O VII
+high res. spectrum
+Townsley et al. [238]
+Chandra
+Carina Nebula
+−
+spectrum
+Pollock [194]
+XMM-Newton
+colliding-wind stars
+Mg XI & XII, Si XIII high res. spectrum
+Carter et al. [43]
+Swift
+C/2007 N3 (Lulin)
+−
+image, spectrum
+Guerrero et al. [98]
+Chandra, XMM
+Pl. Nebula A30
+C VI
+spectrum, low S/N
+Ewing et al. [66]
+Chandra
+5 comets
+Mg, Si
+spectrum
+Lisse et al. [157]
+Chandra
+103P/Hartley 2
+−
+image, spectrum
+Ishikawa et al. [119]
+Suzaku
+Earth geocorona
+−
+spectrum
+Roberts & Wang [202]
+Suzaku
+Cygus Loop SNR
+O VII triplet
+spectrum
+Snios et al. [215]
+Chandra
+2 comets
+−
+image, spectrum
+Cumbee et al. [55]
+Suzaku
+M82
+Ne X
+spectrum
+Gu et al. [91]
+XMM-Newton
+sample of clusters
+S XVI at 3.5 keV
+spectrum
+Gu et al. [94]
+Hubble
+NGC 1275 AGN
+Ne X, S XV
+spectrum, 2σ
+Gu et al. [92]
+XMM-Newton
+C/2000 WM1
+−
+high res. spectrum
+Mullen et al. [177]
+XMM-Newton
+C/2000 WM1
+−
+high res. spectrum
+Gu et al. [95]
+XMM-Newton
+sample of clusters
+O VIII
+high res. spectrum
+Zhang et al. [269]
+XMM-Newton
+Earth geocorona
+−
+spectrum
+Mao et al. [166]
+XMM-Newton
+NGC 5548 AGN
+N VII
+high res. spectrum
+Gu et al. [97]
+XMM-Newton
+NGC 5548 AGN
+O VI
+high res. spectrum
+
+16
+Liyi Gu and Chintan Shah
+Reference
+detector
+source(s)
+relevant ion(s)
+note
+Yang et al. [264]
+XMM-Newton
+M51
+O VII, O VIII
+high res. spectrum
+Walker et al. [243]
+Chandra
+Perseus cluster
+−
+spectrum
+Aharonian et al. [4]
+Hitomi
+Perseus cluster
+S XVI
+high res. spectrum
+Uchida et al. [241]
+XMM-Newton
+Cygnus Loop SNR
+−
+high res. spectrum
+Iwakiri et al. [120]
+RXTE
+EXO 1745-248 burst
+Ti, Cr, Fe, Co
+spectrum
+Hitomi Col. et al. [111]
+Hitomi
+Perseus cluster
+S XVI, Fe XVII
+high res. spectrum
+Tanaka et al. [231]
+XMM-Newton
+G296.1-0.5 SNR
+−
+high res. spectrum
+Suzuki et al. [228]
+XMM-Newton
+N132D SNR
+−
+low significance
+Koshiba et al. [137]
+XMM-Newton
+J0453.6-6829 SNR
+O VII
+high res. spectrum
+Rasmussen et al. [199]
+XMM-Newton
+1E 0102.2-7219 SNR
+−
+high res. spectrum
+Pinto et al. [193]
+XMM-Newton
+NGC 4636
+O VII
+high res. spectrum
+Branduardi-Raymont et al. [35] XMM-Newton
+Jupiter
+−
+high res. spectrum
+Smith et al. [214]
+Chandra
+SWCX
+O VII, O VIII
+spectrum
+Wulf et al. [260]
+XQC
+SWCX
+C, N, O
+high res. spectrum
+Zhang et al. [268]
+XMM-Newton
+M82
+N, O, Ne
+high res. spectrum
+1.3.2 Solar system objects
+On 25 March 1996, comet C/1996 B2 (Hyakutake) had a close encounter with Earth
+at a distance of ∼ 0.1 AU. Rosat took the opportunity to search for X-ray signal from
+this unusual event. While astrnomers expected very low luminosity because comets
+are too cold to emit X-ray, the Rosat image showed a surprisingly bright source from
+a crescent-shaped region on the sunward side of comet [154]. Follow-up studies of
+Rosat archival data proved that X-ray emission is common in comets. One year
+later after the Rosat discovery, Cravens [53] solved the mystery: the X-ray emission
+originates from the charge exchange between highly charged heavy solar wind ions
+and neutral gas around the comet atmosphere. The spatial and spectral properties
+predicted by the charge exchange model has been found to be consistent with Rosat
+data of other comets, as well as the later Chandra and XMM-Newton observations
+with better spatial and spectral resolutions [62].
+That the 1996 Rosat data established the charge exchange study of solar system
+objects may seem odd, as a number of earlier studies (e.g., Gombosi et al. 87, Harwit
+& Hoyle 101) already pointed out that charge exchange is non-negligible in the solar
+wind interaction. However, the early works tended to underestimate or overlook
+the charge exchange contribution in the X-ray band. The Rosat discovery greatly
+expanded the field, leading to a number of dedicated astrophysical observations,
+but also stimulating relevant laboratory measurements and theoretical calculations.
+A new focus of the laboratory and theoretical studies is to reproduce the X-ray
+observations, with highly-charged ions (C, N, O, Ne, etc) and cometary neutrals
+colliding at a speed of several hundred kilometer per second [19].
+
+1 Charge exchange in X-ray astrophysics
+17
+1.3.2.1 Comets
+In the context of solar wind charge exchange, comets are often used as natural
+probes of the latitude- and time-dependent properties of the solar wind ions, which
+produce emission features that can be recognized in the X-ray spectra. The solar
+wind study with comets benefits from the facts that the comets approach the Sun in
+a much broader range of latitudes than the planets, and their occurrence in a proper
+distance range is quasi-constant on year basis. The representative X-ray observa-
+tions of comets are summarized in Table 1.3.
+The solar winds are found to be largely bimodal at solar minimum: the fast
+wind with velocity ∼ 700 km s−1, characterized by low density and low ionization,
+launched probably from the open magnetic fields regions at high solar latitudes. The
+slow wind, blowing at v ∼ 400 km s−1 from the solar equatorial region with a closed
+magnetic field, is often found to be denser and more ionized. When the Sun moves
+towards its maximum, the fast and slow winds become more mixed up.
+Line diagnostics with the cometary X-ray spectra provide useful constraints on
+the speed and the chemical composition of the solar wind (e.g., Bodewits et al.
+29, Krasnopolsky 139). In addition, the X-ray morphology of comets has been used
+to infer the shape and the spatial structure of bow shocks [252]. Finally, Mullen et
+al. [177] derived a constraint on the neutral composition of the cometary atmosphere
+using a global fit to the XMM-Newton grating spectrum.
+Two typical comet spectra are shown in Figure 1.4. The Chandra observation
+of C/1999 S4 reveals a solar wind composition of C, N, and O in fully striped and
+hydrogen-like states. The XMM-Newton grating spectrum of C/2000 WM1 (linear)
+further resolves the strong individual C, N, and O lines, posing even tighter con-
+straints on the solar wind abundance and even the chemistry of the cometary atmo-
+sphere. These data showcase the cometary X-ray as a powerful solar wind probe.
+However, most of the X-ray spectra are taken with a low CCD-like resolution of
+∼ 100 eV, making it impossible to study the solar wind dynamics using comets.
+A further advance in the X-ray study of comets requires observations using high-
+resolution X-ray spectrometers, in particular the microcalorimeters onboard XRISM
+[232] and Athena [179] missions.
+1.3.2.2 Planets
+Jupiter is known as a bright X-ray source from the early Einstein observations [168].
+The origin of the X-ray has been settled just recently: the emission from the equa-
+torial region is likely the scattering of solar X-ray in the dense planetary atmo-
+sphere, while the X-ray in the aurorae can be a mixture of two components, the
+bremsstrahlung radiation dominating > 2 keV by energetic electrons precipitating
+from the magnetosphere, and the charge exchange in soft X-ray between the in-
+falling ions and the atmosphere c neutrals [25, 33, 34, 35, 36, 65, 85, 132, 181]. The
+ion diagnostics using charge exchange provides an opportunity to understand where
+the ions come from: the X-ray spectra would be C- and O- rich if the solar wind is
+
+18
+Liyi Gu and Chintan Shah
+the main source of the ions, and S- and O is rich if the bulk population originates
+from the Io plasma torus falling into the Jovian magnetosphere.
+The XMM-Newton reflection grating spectrometer (RGS) spectrum of Jupiter in
+2003 shown in Figure 1.5 show so-far the best example of the charge exchange
+diagnostics with X-ray spectroscopy of planets. It shows a clear separation of the
+charge exchange emission, mostly in oxygen lines, from the low-latitude disk com-
+ponent with Fe and O lines. The charge exchange lines are broadened by a velocity
+of ∼ 5000 km s−1, implying that we are witnessing a bulk of accelerated particles
+precipitating into the Jovian polar area.
+Mars and Venus are possible charge exchange emitters, even though the detection
+was made at a lower significance than Jupiter [60, 61]. The scatter of solar X-rays
+in the planetary atmosphere is constantly bright in, even dominating, the overall soft
+X-ray band, making it challenging for charge exchange observation. No significant
+charge exchange signal has yet been detected in other non-Earth planets.
+1.3.2.3 Geocorona and heliosphere
+Charge exchange emission is generated when the highly ionized solar wind inter-
+acts with neutral or near-neutral gas in the Earth’s exosphere and the heliosphere.
+This component is commonly known as the solar wind charge exchange (SWCX),
+which can affect any X-ray observation so far. Therefore, it is treated as a trouble-
+some background, consisting of multiple emission lines (e.g., C, N, and O) in the
+soft X-ray band. Concerning its temporal, spectral, and spatial variations, it is often
+challenging to fully eliminate the SWCX from the extrasolar sources.
+Chandra observation of the dark moon has been considered direct evidence for
+the nearby SWCX since it shows charge exchange-type spectrum [250]. Similarly,
+Smith et al. [214] reported an enhancement of O VII and O VIII lines from SWCX in
+the Chandra observation of molecular cloud MBM 12, and Fujimoto et al. [78] ob-
+served a transient increase of SWCX flux by comparing several spectra of a Suzaku
+blank field in the direction of the north ecliptic pole.
+In a CCD resolution, the X-ray spectra from the geocoronal and heliospheric
+SWCX could sometimes resemble those from the local hot bubble around the solar
+system, making it difficult to isolate one from another in the spectrum. One solution
+is to separate them based on directional variation: the local hot bubble is isotropic
+while the SWCX might increase in the focusing cone of heliospheric helium caused
+by the inflow of interstellar medium. By scanning the sky through the helium cone
+using the diffuse X-rays from the local galaxy (DXL) mission, Galeazzi et al. [79]
+separated the SWCX and the local hot bubble emission based on their spatial dis-
+tributions. Their results showed that the SWCX emission contributes about 40% of
+the 1/4 keV X-ray flux in the Galactic plane.
+A second approach is to distinguish SWCX using high-resolution X-ray spec-
+troscopy. The X-ray quantum calorimeter (XQC, ∼ 6 eV FWHM), onboard the
+University of Wisconsin-Madison/Goddard Space Flight Center sounding rocket,
+has measured several times the diffuse X-ray background below 1 keV [260]. As
+
+1 Charge exchange in X-ray astrophysics
+19
+shown in Figure 1.6, the high-resolution spectroscopy allows the separation of the
+SWCX from the 0.1 keV local hot bubble and the 0.2 keV Milky Way halo compo-
+nents. It could be used to further isolate the charge exchange from fast solar wind
+and slow wind components.
+1.3.3 Astrophysical objects
+While the charge exchange from the solar system objects is well established, the de-
+tection in extrasolar astrophysical objects remains somewhat controversial though
+evidence is accumulating in particular in the past decade. A general issue is that
+many claims are made with the CCDs on Chandra, XMM-Newton, and Suzaku
+with 50 − 100 eV resolutions, or with dispersive gratings (e.g., RGS) of better but
+source-dependent resolutions. A revisit with non-dispersive high resolution X-ray
+spectrometers (e.g., XRISM and Athena) is needed to verify these features.
+1.3.3.1 Stars and supernova remnants
+Pollock [194] discussed a scenario that colliding wind ejected from binary stars
+might lead to ion-ion collision and charge exchange. The ion-ion charge exchange
+can be realized only when the collisions occur at a large velocity (≥ 2000 km s−1),
+which indeed resembles some of the observed terminal shock speeds of the collid-
+ing stellar winds. Pollock [194] suggested that the charge exchange recombination
+might explain the low Mg XII/Mg XI ratios observed in systems including WR22,
+WR140, and ζ Pup.
+The wind interaction condition might apply to other types of stellar objects.
+Iwakiri et al. [120] reported the detection of a 6.6 keV emission feature in the spec-
+trum of a neutron star X-ray binary EXO 1745-248 40 hours after its superburst. It
+is a broad feature with an equivalent width of 4.3 keV, which is much larger than
+that of the standard ionized Fe line formed by the reflection in the disk. A spectral
+analysis with the RXTE data implied that the feature can be a set of charge exchange
+lines, likely emitted from the interaction of a failed wind launched by the superburst
+with the neutrals in the neutron star disk.
+The interaction between supernova remnants and ambient neutral clouds might
+generate charge exchange emission. It is expected that this emission would be vis-
+ible around the edge of the remnants, where interstellar neutrals pass through the
+shock front and collide with the post-shock hot ions. Katsuda et al. [124] reported a
+detection of charge exchange emission at the outer shell of the Cygnus Loop, which
+shows an enhanced feature at the energy of highly excited O VII lines in the Suzaku
+spectrum [202]. As plot in Figure 1.7, Uchida et al. [241] took an XMM-Newton
+RGS spectrum of a bright X-ray spot on the outer shell, and showed that the charge
+exchange component, together with an ionized absorber, are indeed required to ex-
+plain the forbidden-to-resonance line ratio of O VII. Similarly, the RGS spectra of
+
+20
+Liyi Gu and Chintan Shah
+G296.1-0.5 [231] and J0453.6-6829 [228] show high forbidden-to-resonance ratios
+that can be explained by charge exchange.
+The limited imaging capability of XMM-Newton RGS allows brief mapping of
+the ion-neutral interaction region in the supernova remnants. Uchida et al. [241]
+pioneered this method by differentiating the O VII forbidden line and resonance
+line monochromatic images, and discovered that the charge exchange lines can be
+emitted from a compact region where a dense neutral cloud is also found.
+1.3.3.2 Interstellar medium and star forming regions
+The north polar spur is a large ridge of X-ray and radio-emitting gas in the north-
+ern sky. Similar to some of the supernova remnant spectra discussed above, the
+north polar spur also shows large forbidden-to-resonance ratios for He-like triplets
+of O and Ne [146, 173], which might potentially indicate a charge exchange origin.
+However, this scenario is far from conclusive since the forbidden-to-resonance ratio
+could also be explained by others atomic physics including an ionized absorber. As
+suggested in Gu et al. [93], the ionized absorption might be responsible for most of
+the observed line ratios while the charge exchange contributes no more than 10% to
+the forbidden lines.
+The star-forming regions in normal or starburst galaxies are another locations
+where interactions between hot gas and cold clouds are expected. By modeling the
+Chandra spectrum of the Carina star-forming complex with multiple thermal com-
+ponents, Townsley et al. [238] reported several line-like excesses in the residual in
+0.5 − 2.0 keV. The distribution of these line residuals is spatially correlated with
+the cold star-forming filaments, where the gas-filament charge exchange might be
+responsible for these line emissions.
+There have been claims for the detection of charge exchange X-ray in several
+starburst galaxies. M82 is the best-studied object in this category. The X-ray spec-
+troscopic works using XMM-Newton and Suzaku data showed that the thermal model
+is insufficient in particular for the O VII and O VIII lines, while adding an additional
+charge exchange component could fix most of the problem [55, 136, 160, 239, 268?
+]. A recent study using the RGS spectrum is plotted in Figure 1.8. Liu et al. [161]
+presented a sample study of nine star-forming galaxies with XMM-Newton RGS,
+and showed that most of these galaxies are undergoing charge exchange as indi-
+cated by the forbidden-to-resonance line ratios. Therefore charge exchange can be
+a class property of star-forming regions and galaxies. Naturally, the ion-neutral in-
+teraction could occur between the starburst-driven outflows and the cold interstellar
+medium.
+In the quiescent galaxies, charge exchange might still occur at various locations
+including the interfaces between the nucleus-driven outflows and the circumnuclear
+neutral clouds. Yang et al. [264] reported a possible detection of charge exchange
+component which contributes significantly to the N, O, and Ne lines, in particular
+the He-like forbidden lines, in the RGS spectrum of M51 nucleus. The interaction
+between the radio jet and the cold interstellar medium might be responsible for the
+
+1 Charge exchange in X-ray astrophysics
+21
+emission. Yet another possibility is that these lines originate from photoionization
+by the AGN outbursts in the past.
+1.3.3.3 Active galactic nuclei
+AGNs host multiple gas inflows and outflows of a broad range of ionization states,
+making it another potential target of the charge exchange search. Gu et al. [97] re-
+ported a detection of unidentified features at 18.4 ˚A in the grating spectra of Seyfert
+I AGN NGC 5548. By stacking all observations, this feature is seen at > 5σ sig-
+nificance taking into account the look-elsewhere effect. As indicated in Figure 1.9,
+the wavelength of the anomaly coincides with the high-n transitions from He- and
+Li-like oxygen, therefore, it is likely a charge exchange line. The authors suggested
+that the interaction could happen within the same outflow by mixing the partially
+ionized and neutral layers, or between the outflow with the neutral close environ-
+ment. Another report by Gu et al. [94] showed that the Hubble STIS spectrum of the
+radio galaxy NGC 1275 can be modeled by including three weak lines at 1223.6 ˚A,
+1242.4 ˚A, and 1244.0 ˚A, each with a significance of 2 − 3σ. These features can be
+explained by a mixture of charge exchange between highly ionized hydrogen, neon,
+and sulfur with neutral matter, indicating for an outflow with v ∼ 3400 km s−1.
+1.3.3.4 Clusters of galaxies
+Though the hot plasmas are the dominating baryonic component in clusters of galax-
+ies, cold gas clouds do exist, and they are often observed in and near the central
+galaxies, as well as in the wake of member galaxies during their infall to the center
+[50]. These neutral structures are immersed in the giant pool of hot, highly ionized
+plasma, and their interfaces are candidates for charge exchange. Fabian et al. [68]
+and Walker et al. [243] suggested that the charge exchange can account for a part of
+the Hα and soft X-ray emission from bright filaments in the Perseus cluster.
+During its brief lifetime, the micro-calorimeter onboard Hitomi observed the
+Perseus cluster with a ∼ 5 eV resolution in the 2 − 10 keV band. As shown in Fig-
+ure 1.10, there are indeed hints for charge exchange in the Hitomi spectrum. The
+high-n Rydberg transitions are found to be 1.6σ and 2.4σ significance for S XVI
+and Fe XXV. The detection remains challenging even for Hitomi, since the charge
+exchange in Perseus cluster is overwhelmed by the thermal emission by two orders
+of magnitude in flux. The S XVI charge exchange is in particular interesting since
+it nearly overlaps in energy with the mysterious 3.5 keV line detected in a large
+sample of clusters by Bulbul et al. [38] and Boyarsky et al. [32]. The 3.5 keV line
+was originally proposed as an evidence for the radiative decay of sterile neutrino, a
+theoretical form of dark matter. As shown in Aharonian et al. [4], the Hitomi spec-
+trum of the Perseus cluster prefers the S XVI charge exchange scenario, although
+the data quality is insufficient to rule out the dark matter line. In addition, Gu et al.
+[95] reported a 2.8σ detection of a line feature at 14.82 ˚A, which is likely a O VIII
+
+22
+Liyi Gu and Chintan Shah
+charge exchange line. The line flux measured with the RGS is in good agreement
+with the possible Hitomi detection.
+1.4 Ending remarks
+The introduction of the charge exchange process to general X-ray astronomy was
+made in 1996 by the discovery of X-ray from a comet. It has been broadly under-
+stood now that, in parallel to the electron impact and photon induced X-ray, the
+ion-neutral interaction can also efficiently generate X-ray in astrophysical objects.
+In the past two decades after the cometary discovery, X-ray astronomers have spot-
+ted charge exchange-like X-rays in a variety of objects, ranging from the neighbour
+planets and heliosphere to the interstellar medium, galaxies, supermassive black
+holes, and clusters of galaxies on the cosmological scales. Charge exchange emis-
+sion is no doubt the best tool for detecting an interface, because the normal electron-
+impact X-ray flux emitted from the physically-thin interface is likely much dimmer.
+It has the potential to change the research landscape on the interactions of (1) AGNs
+and the host galaxies; (2) galactic outflow and interstellar medium; (3) infalling
+galaxies and the cluster.
+Charge exchange produces only line emission, making it a science case tailored
+for high-resolution spectroscopy. With low-resolution spectra it often remains chal-
+lenging to disentangle charge exchange from a thermal or photoionized component.
+Once fully resolved, charge exchange lines provide diagnostics on (1) the collision
+speed between ions and neutrals; (2) the chemical composition and ionization state
+of the hot ions; (3) relation with the Hα-emitting region; and (4) the neutral species.
+Spatially-resolved spectroscopy can further constrain the interaction region.
+New astrophysical discoveries could feedback to the theoretical and laboratory
+studies of charge exchange. Substantial work remains to remove the tension between
+the theoretical calculations of cross sections and the corresponding laboratory mea-
+surements. The Li-like and ion species with more than three electrons are still poorly
+covered in the current calculations and experiments. The l- and S- distributions of
+the electron capture are crucial for an accurate line model, but they remain less cer-
+tain than the n- distribution. Finally, the multi-electron charge exchange remain to
+be addressed. A consistent and continuous effort will be required to ensure that the
+atomic data are ready for the next generation of high-resolution X-ray data to be
+obtained with XRISM and Athena.
+Acknowledgements SRON is supported financially by NWO, the Netherlands Organization for
+Scientific Research. C.S. acknowledges support from NASA under award number 80GSFC21M0002
+and Max-Planck-Gesellschaft (MPG).
+
+1 Charge exchange in X-ray astrophysics
+23
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+doi:10.1103/PhysRevA.56.457
+
+36
+Liyi Gu and Chintan Shah
+OVIII Lyα
+OVII 
+Heα
+f
+ir
+OVII Heβ
+NVII Lyα
+NVII Lyβ
+NVII Lyγ + Lyδ
+NVI 
+Heα
+f
+ir
+CVI 
+Lyα
+CVI Lyβ
+CVI Lyγ
+CVI Lyδ
+Si XI
+Si XI
+CV Lyβ
+CV Lyγ
+H target
+H2O target
+C4+
+C5+
+N5+
+N6+
+O6+
+O7+
+0.2
+0.2
+0.6
+1.0
+1.4
+0.4
+0.6
+photon energy (keV)
+Intensity
+0.8
+Comet C/1999 S4
+Chandra ACIS
+Comet C/2000 WM1
+XMM-Newton RGS
+Fig. 1.4: (Upper) Comet C/1999 S4 Chandra CCD spectrum taken on 14 July 2000,
+compared with the model of C, N, O ions charge exchanging with CO2 measured at
+EBIT [20]. (Lower) Comet C/2000 WM1 XMM-Newton grating spectrum fit with
+the SPEX cx model taking into account the hydrogen and water targets.
+
+1 Charge exchange in X-ray astrophysics
+37
+wavelength (Å)
+Intensity
+Jupiter, XMM-Newton RGS
+Fig. 1.5: The XMM-Newton grating spectrum of Jupiter. A phenomenological fit
+to the spectrum is plotted in red, and the disk component is shown in green. The
+emission from X-ray aurorae is the difference between the two.
+Fig. 1.6: (Left) X-ray quantum calorimeter spectrum of the diffuse soft X-ray back-
+ground. The model is shown in the middle, and the residual is plotted at the bottom.
+The effective area of the detector is shown in the dashed line. (Right) The model
+is divided into components: LHB - local hot bubble; Fast/slow SWCX - fast/slow
+solar wind charge exchange; Halo - Milky way gas halo; and CXRB - cosmic X-ray
+background.
+
+Total
+LHB
+Aax
+FastSwcx
+-as)
+Counts
+Slow Swcx
+10
+Halo
+5
+15
+10
+CXRB
+5
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+1100
+Energy (eV)kev-1)
+25
+Data
+15
+10
+Counts
+5
+(sec-lkev-1)
+Model
+NVI+CVI
+Fexvill
+Fexvill+Ovill
+FexIX+Nelx
+25
+CVI
+CVI
+NVII
+OVIII
+FexVIl
+0.5
+20
+15
+0.3
+Counts
+10
+5
+0.1
+Residual
++2
+n
+0
+Ln
+LL
+20
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+1100
+Energy (ev)38
+Liyi Gu and Chintan Shah
+Cygnus Loop SNR (XMM-Newton RGS)
+Fig. 1.7: XMM-Newton grating spectrum of a bright knot in the Cygnus Loop super-
+nova remnant, fit with the charge exchange plus non-equilibrium collisional ioniza-
+tion model. The neutral and ionized absorbers are taken into account.
+
+(b1)
+O VII Heα
+0.15
+Total: Neutral Abs. × Ionized Abs. Isolar x (NEI + CX)
+NEI component
+0.125
+CX component
+0.1
+Fe XVII (3d-2p)
+Counts
+ N Lyα + N Heβ
+0.075
+C Lya.
+Ne Ix Heaα
+Lyα
+Heβ
+0.05
+N vI Heα
+Lyy.
+O Lyβ
+LyY
+0.025
+Mg Heα
+0
+Error
+0.05
+T
+(b2)
+0
+ Error Rel.i
+-0.05
+0.05
+(b3)
+0
+Rel.
+-0.05
+10
+20
+30
+Wavelength (A)1 Charge exchange in X-ray astrophysics
+39
+18
+19
+20
+21
+22
+23
+0
+5 10
+10
+ O VII
+ O VII
+ O VII
+ O VII
+ O VIII
+(r)
+(i)
+(f)
+wavelength (Å)
+Intensity
+M82 XMM-Newton RGS
+Fig. 1.8: Part of the XMM-Newton grating spectrum of the M82 star-forming galaxy.
+The charge exchange lines from O VII and O VIII are plotted in colors indicated by
+the label.
+
+40
+Liyi Gu and Chintan Shah
+16
+18
+20
+22
+24
+0
+2
+4
+6
+Wavelength (Å) 
+Counts / m2 / s / Å
+O VI Sat. 1s-6p 
+O VI Sat. 1s-4p 
+O VII 1s-4p 1P1
+O VII 1s-3p 1P1
+O VII 1s-2p
+O VIII 1s-2p
+O VII 1s-5p 1P1
+NGC 5548 
+RGS 
+2021
+Fig. 1.9: XMM-Newton RGS spectrum of NGC 5548 AGN taken in January 2021
+fit with the photoionization model. A set of excess features are seen around 18.4 ˚A
+indicating the possible charge exchange emission.
+
+1 Charge exchange in X-ray astrophysics
+41
+Perseus cluster Hitomi/SXS
+residuals at S and Fe bands
+Fig. 1.10: (Left) Hitomi spectrum of the Perseus cluster fit with a sum of thermal,
+AGN, and charge exchange components. (Right) Residuals of the fit with the ther-
+mal plus AGN model. The red curve in each panel shows the model change by
+including the charge exchange component.
+
diff --git a/ltFIT4oBgHgl3EQfsStT/content/tmp_files/load_file.txt b/ltFIT4oBgHgl3EQfsStT/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f6766b279b3deeeb9cb60e2527468ebaa6272a57
--- /dev/null
+++ b/ltFIT4oBgHgl3EQfsStT/content/tmp_files/load_file.txt
@@ -0,0 +1,3127 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf,len=3126
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='11335v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='IM]  26 Jan 2023  Chapter 1  Charge exchange in X-ray astrophysics  Liyi Gu and Chintan Shah  Abstract Charge exchange is an atomic process that primarily occurs at interfaces  between the neutral and ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The study of the process has been carried out on  three levels: the theoretical calculation of the cross sections, the laboratory measure-  ments of reaction rates and line strengths, and the observational constraints using  celestial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' For a long time in the past, the status of astrophysical observations  in the X-ray band lagged behind the other two aspects until the discovery of X-ray  from a comet was made in 1996, which changed the research landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Recent  observational evidence suggests that charge exchange has been seen or can be ex-  pected from a surprisingly broad range of locations, from the Earth’s exosphere to  the large-scale structures of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The rapid development of high-resolution  X-ray spectroscopy, in particular the non-dispersive micro-calorimeters, is paving  the way to revolutionary new science possibilities both in the laboratory and astro-  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' This chapter summarizes the current knowledge of charge exchange and its  relevance on astrophysics, especially X-ray spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 Introduction  The phenomenon of ion-neutral interaction was studied in early laboratory exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The scattering experiment of alpha particles or bare He ions, against gold  atoms has led Rutherford [204] to establish the classical model of atomic struc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The quantum mechanical treatment of the ion-neutral scattering problem was  Liyi Gu (�)  SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, the Nether-  lands, e-mail: l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='gu@sron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='nl  Chintan Shah  NASA Goddard Space Flight Center, 8800 Greenbelt Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Greenbelt, MD 20771, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany, e-mail:  chintan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='shah@mpi-hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='de  1  2  Liyi Gu and Chintan Shah  explored by Massey & Smith [167], and Mulliken [178], among others, pioneered  the theoretical spectral modeling of charge exchange using atomic orbital approxi-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' From 1950s, as the gas-beam technique matured, an increasing number of  experiments have been put forward to measure directly the charge exchange cross  sections [71, 227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Early investigations of this process have been covered in several  reviews [84, 122, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  It has been well known that the atom-ion charge exchange by far dominates all  kinds of electron-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' For a collision of ∼ keV/u, the effective cross sec-  tion of charge exchange between highly-ionized ions and neutrals are at least two  orders of magnitude greater than the radiative recombination and electron-impact  excitation cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' From the ion view, charge exchange provides efficient  recombination on the ionization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Unlike radiative recombination, charge ex-  change always ends up in an excited state, producing strong line emissions but zero  continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Charge exchange has been well utilised in the research of nuclear fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The  radiative transitions from excited states of impurity ions resulting from charge ex-  change may constitute a substantial portion of the total energy losses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' meanwhile,  the charge exchange emission can be used as a powerful tool for spectroscopic di-  agnostics of high-temperature plasmas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Carolan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The study of charge exchange has been extended to astrophysics, in particular  the interaction between solar wind protons with neutral particles near the Sun, in  the planets, and from the interstellar medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Wallis 244).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Charge exchange  has also been found to be the origin of the broad Hα emission lines observed at the  shock wave of supernova remnants [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These lines are emitted from hot hydrogen  atoms created by electron capture of hot protons in the post-shock plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  An important breakthrough was made in the late 1990s, when the first detection  of cometary X-rays [154] and the charge exchange interpretation [53] was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Since then, the interest on charge exchange X-ray has been activated in observa-  tions, but also in theory and in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It has been recognized as a primary  or second emission mechanism in a broad range of astrophysicobjects, as well as a  time-varying background emission tl X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' In this review, we provide an  overview of our current knowledge on charge exchange, with a focus on X-ray as-  trophysics with spectroscopic utilised In section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 we present the basic physical  properties of charge exchange, as well as the theoretical and experimental achieve-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 shows the observational results obtained so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The review will  close with conclusions and a short outlook in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1 Charge exchange in X-ray astrophysics  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1  1  10  100  1000  104  105  106  1  10  100  1000  Langevin  COB  Bohr-Lindhard  MCLZ (kronos)  Janev 1993  QMOCC (Nolte)  Energy (eV/u)  Cross section (10-16 cm2)  O7+ + H  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1: Total cross sections for O7+ + H based on the experimental measure-  ments of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [267] (black) and Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [169] (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The theoretical  calculations from Janev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [123], multichannel Landau-Zener [176], and quan-  tum molecular-orbital close-coupling [180] are shown in blue, purple, and cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The classical Langevin, over-the-barrier, and Bohr-Lindhard models are plotted in  dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 Plasma modeling  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 Physics and classic models  Charge exchange recombination consists of the transfer of one or more electrons  from (mostly) the outer atomic shell of target atom/ion M to the outer shell of an  impinging/projectile atom/ion X, leading to the recombination of the impinging par-  ticle while the ionization of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The process can be written as  Xq+ + M → X(q−m)+ + Mm+,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1)  where q is the original charge of the X, and m is the number of electrons transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  m = 1 can be referred as single electron capture (SEC) from the point of view of the  impinging ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' m = 2 is designated as double electron capture (DEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The total cross section for this process is typically of order of 10−15 - 10−14 cm2,  greater by a factor of 102 − 103 than the cross sections of typical electron-impact  radiative recombination and dielectronic recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The classical analytic cal-  4  Liyi Gu and Chintan Shah  culations, as summarized in Stancil [223], provide order estimate of the total cross  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' They include Langevin model for < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 eV/u collision, where u stands for atomic mass unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  This model predicts an energy dependence E−1/2 of the cross section, which is  found to be in line with the Landau-Zener approximation shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Classical over-the-barrier (COB) model for intermediate collision speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' In this  approximation, the cross section becomes a constant over energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Bohr-Lindhard model, which predicts a E−7/2 behavior, becomes applicable at  very high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  These classic treatments for O7+ +H are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 with a comparison  to various explicit theoretical and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These simple models seem  to reproduce the correct general trend as a function of energy, though apparently  overestimating the absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Charge exchange operates in a semi-resonant favor when a subset of the energy  levels of the projectile ion and the target overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' If the projectile ion is highly  charged, its highly excited levels overlap with the neutral’s ground state, making it  possible to transfer electrons onto high-n levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The distribution of n with the most  likelihood has been studied in various ways, Simple COB model predicts the dominant n level nmax ≤ q/√2IM, where q is the  initial charge and IM is the ionization potential of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Systematic compilation of existing results by Janev & Winter [122] reported a  scaling nmax ∼ q(1 + (q − 1)/  �  (2q))−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 ∼ q3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' More recently, Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [92] suggested that the n distribution is energy depen-  dent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' the capture is peaked on a single n channel at low collision energy, while  it tends to be shared among several neighbour n channels at the intermediate en-  ergy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The analytic form of the transfer is reported as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 of their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  At the very high energy end, the n distributions become narrow again [251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  For the He-like triplet, the cascade contribution from large n to the formation of  forbidden lines is substantially larger than that to the resonance lines, yielding a  large forbidden-to-resonance line ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  To construct an emission spectrum from charge exchange, it is necessary to  model the population of capture onto states with orbital angular momenta l in the  range of (0, n-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' For collisions with E ≥ 10 keV/u, it is expected that the l distri-  bution function W is weighed statistically, Wst = (2l + 1)/n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Therefore the largest  possible l = n−1 will be the most populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As the collision velocity decreases, the  capture l tends to decrease towards l = 1 or 2 taking one of the following possible  forms, Low energy approximation L1, WL1 = (2l + 1) [(n - 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' ]2 / [(n + l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (n - l - 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' ], Modified low energy approximation or L2, WL2 = l (l + 1) (2l + 1) (n - 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (n - 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  / [(n + l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (n - l - 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' ], Shifted low energy approximation or L3, WL3 = (2l + 3) [(n - 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' ]2 / [(n + l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  (n - l - 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' ], “Separable”, Wse = (2l + 1)/q e−l(l+1)/q,  1 Charge exchange in X-ray astrophysics  5 even or constant over l, Wev = 1 / n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  For a high-speed collision, shells with large l are significantly populated, resulting  in more optical and UV lines following the Yrast cascade relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The low-speed  charge exchange would instead yield more X-ray lines because the shells with small  l become more populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Commonly the capture on the total spin quantum number is assumed to be sta-  tistical, and therefore the triplet-to-singlet ratio of He-like ion, produced from the  H-like ion reaction, is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' However, bias from the statistical weight is found in recent  theoretical calculations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Nolte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 180, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 258).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These works suggested  that the triplet-to-singlet ratios, and therefore the distribution on the spin quantum  number, are likely to be energy-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Transitions from highly-excited levels with large n and the large forbidden-to-  resonance line ratio are two basic spectral features of the charge exchange X-ray  emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The high-n Rydberg lines are usually stronger in H-like than in He-like  series because the cascades in the He-like ions are subject to the singlet-to-triplet  branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The UV-to-X-ray ratios of charge exchange emission are larger  than those of the electron-impact emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 Theoretical calculations of absolute cross sections  6  Liyi Gu and Chintan Shah  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1: Theoretical data  Reference  typea  ion  energy (eV/u)  Savin [206]  total  D+ H+, D+ + H  ≤ 6  Cornelius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [51]  total,nl  He2+ − Ne10+ + H,Li  103 - 105  Stancil & Zygelman [218]  total  Li+ H+  10−4 - 10  Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [128]  total  Li+ H+  10−5 - 103  Janev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [121]  total,nl  AA+ + H (5 ≤ A ≤ 74)  30 - 8×104  Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [240]  total,nl  B2+ + H  ∼ 10−1 - 104  Liu & Wang [159]  total,nl  B4+ + H  10−2 - 2×105  Stancil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [220]  total  C+ + H  ≤ 103  Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [130]  total  C, N, O, Si+ H+  ≤ 103  Butler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [40]  total  Cq+, Nq+, Oq+, Neq+ + H (q = 2,3),  10−1 - 1  O2+ + He  Butler & Dalgarno [39]  total  Cq+, Nq+, Oq+, Neq+, Mgq+, Siq+, Sq+, Arq+  10−2 - 1  +H (q = 2,3,4)  Heil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [109]  nl  C2+,N3+,O3+,Ne2+,Ne3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='27 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1  Guevara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [99]  total  C3+,O3+,Si3+ + H  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 104  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [257]  total,nl  C3+ + He  ∼ 10−4 - 103  Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [127]  total  Cq+,Nq+,Oq+,Fq+,Neq+,Krq+  600  + H (q = 10 − 25 for Kr and 4 − 9 for rest)  Mullen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [177]  total,nl  C6+,N7+,O8+,Ne10+,Na11+,Mg12+,  ∼ 10−3 - 105  Al13+ + H,H2O,CO,CO2,OH,O  Pindzola & Fogle [192]  total,nl  C6+ + H,He  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='7×103 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3×103  Cumbee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [56]  total,nl  C6+ − Al13+ + H,He,H2  2 ×102 - 5 ×103  Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [153]  total,nl  N+ H+,N+ + H  ∼ 10−4 - 103  Barrag´an et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [15]  total  N2+ + H  2×10−3 - 3 ×105  Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [110]  total  N2+ + H  ∼ 10−1 - 102  Sidky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [213]  total,nl  N4+ + H  ∼ 50 - 2×104  Feickert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [69]  total  N4+ + H  30 - 105  Zygelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [272]  total,nl  N4+ + H  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 8×103  Stancil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [219]  total,nl  N4+ + H  ∼ 10−2 - 6×103  Shimakura & Kimura [212]  total,nl  N5+ + H  ∼ 10−2 - 104  Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [102]  n  N7+,O7+ + He,CO,CO2,H2O  2×103 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='67×103  Chambaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [44]  total  O+ H+  ≤ 10−1  Stancil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [221]  total  O+ H+, O+ + H  10−4- 107  Pichl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [189]  total,nl  O+ + H2  1 - 104  Pichl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [190]  total,nl  O+ + H2  ∼ 5×102 - 104  Barrag´an et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [16]  total  O2+ + H,N2+ + H  ∼ 10−3 - 1044  Cabello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [41]  total  O2+ + H  ∼ 125 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4 ×103  Honvault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [113]  total,nl  O2+ + H  ∼ 10 - 3×104  Honvault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [114]  total  O2+ + H  ∼ 3×10−4 - 4  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [246]  total,nl  O3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 103  Bienstock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [26]  total  O3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 -5×103  1 Charge exchange in X-ray astrophysics  7  Reference  typea  ion  energy (eV/u)  Dalgarna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [57]  nl  O3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 1  Gargaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [83]  nl  O3+ + H  ≤ 1  Roueff & Dalgarno [203]  total  O3+ + H  ∼ 1  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [256]  total,nl  O3+ + He  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01 - 103  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [247]  total,nl  O3+ + H2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 104  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [259]  total,nl  O6+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 104  Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [271]  total,nl  Ne2+ + He  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 104  Lyons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [164]  total,nl  Ne(8−10)+ + H,Mg(8−12)+ + H  10−3 - 5 ×104  Rigazio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [201]  nl,line  Ne10+ + Ne,He,H2,CO2,H2O  9  Croft & Dickinson [54]  total  Na+ H+  ≤34  Allan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [6]  total  Mg + H+  ≤40  Fogle & Pindzola [74]  total,nl  Mg12+ + H,He  103 - 5 ×103  Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [129]  total  Si+ H+  ≤30  Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [229]  total  Si2+ + He  ∼ 20 - 6 ×103  Clarke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [49]  total,nl  Si2+ + H  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='03 - 100  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [248]  total,nl  Si3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01 - 106  Stancil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [222]  total,nl  Si3+ + He  ∼ 10−4 - 400  Honvault et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [115]  total,nl  Si3+ + He  1 - 7 ×103  Rabli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [197]  total  Si3+ + He  ∼ 10−4 - 103  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [158]  total,nl  Si3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01 - 2 × 105  Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [230]  total  Si4+ + He  10−3 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 ×103  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [245]  total,nl  Si4+ + He  ∼ 10−4 - 107  Vaeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [242]  total,nl  Si4+ + He  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01 - 100  Bacchus-Montabonel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [13] total,nl  P2+ + H  ∼ 100 - 105  Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [131]  total  S+ H+  ≤ 103  Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [270]  total,nl  S+ H+  ∼ 10−4 - 104  Bacchus-Montabonel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [11] total,nl  S2+ + H  100 - 104  Łabuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [145]  total,nl  S3+ + H  2×103 - 8×103  Bacchus-Montabonel [12]  total,nl  S3+ + He  2×103 - 5×104  Stancil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [224]  total,nl  S4+ + H  ∼ 10−4 - 107  Mullen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [176]  total,nl  Fe25+,Fe26+ + H  10−3 - 105  Sterling & Stancil [226]  total,nl  Geq+,Seq+,Brq+,Krq+,Rbq+,Xeq+  10−3 - 106  +H (q = 2,3,4,5)  a: total = total cross section, nl = nl-resolved cross section  To produce accurate X-ray emission models, it is desirable to have absolute cross  sections using state-resolved theoretical calculation for all the relevant reactions  at a broad range of collision velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Simple approximations described above,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', COB, cannot reproduce accurate experiment results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Furthermore, COB-like models do not have the facility to calculate the l distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  For collisions with energies E ≥ 5 keV/u, the Classical Trajectory Monte Carlo  (CTMC) method is often considered to be a valid approximation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Otranto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  183).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The CTMC provides a non-perturbative solution to the classical equations of  8  Liyi Gu and Chintan Shah  motion in the many body interactions during the collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' However, this method  does not give an adequate description of the lowering of l quantum number as the  collision energy decreases below ∼ 1 keV [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The Multichannel Landau-Zener approach (MCLZ) provides another flexible  tool for massive calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' MCLZ considers a slow collision between an ion and  a neutral particle using a quasi-molecular configuration, therefore, it is mostly used  for models with E ≤ a few keV/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It studies the crossing between pre- and post-  exchange potentials of the configuration and determines the final state of the cap-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Since the MCLZ requires that crossing distances are fully resolved, while for  a bare ion collision, the capture l degenerates with the n state, this method cannot  resolve l distribution for bare ion collision [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Empirical l distribution functions  listed in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 must then be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  (Quasi-)quantum mechanical models involving close coupling are known to  be accurate simulations of atomic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The Atomic-Orbital Close-Coupling  (AOCC) is found to be applicable between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 and 100 keV/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It is based on  bound state calculation on either center of the two reactors, assuming an infinite  separation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The number of n and l are limited by the finite size of the  close-coupling expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The Molecular-Orbital Close-Coupling (MOCC) is another type of close-coupling  which is optimized for low energy (E ≤ 10 keV/u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It models the union state of the  two reactors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The molecular wave functions, as well as the coupling terms of the rel-  evant status, are calculated and fed into the close-coupling scatter calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The  number of n and l are limited in the same way as the atomic-orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A full quantum  mechanical version, QMOCC, has been developed to cope with very low energy  collision at E < 10 eV/u [223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  We summarize in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 a comprehensive, though not exhaustive, list of the-  oretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The reactors, type of the data products (total, n−, and nl−  resolved cross sections) and the energy range indicated in the references, are all  recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The collision energy determines the range of application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' for instance,  the solar wind ions collide with comet atmosphere at E ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 − 5 keV/u, while for  the charge exchange component in collisional or photo-ionization equilibrium, the  relevant collisions are much less energetic with E ∼ 10 eV/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 Laboratory measurements  1 Charge exchange in X-ray astrophysics  9  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2: Experimental cross section data  Reference  typea  ion  energy (eV/u)  Greenwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [88]  total  H+,He+,He2+ + H2O,CO2  300 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×103  Lubinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [163]  nl  He2+,N5+ + H2  5 - 103  Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [142]  total  He2+ + H2,N2,O2,CO,CO2  4×103  Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [143]  total  He2+ + He  2×102 - 4×103  Shah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [210]  total  Liq+ (q=1-3)  4×103-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='43×105  Seim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [208]  total  Liq+ (q=2-3), Nq+ (q=2-5), Neq+ (q=3-5)  3q×103 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2q×104  Goffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [86]  total  Bq+ (q=1-5), Cq+ (q=1-4)  q×105 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5q×106  total  Cq+ (q=5,6), N7+  q×105 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5q×106  McCullough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [170]  total  B2+, C+, N+, Mg2+  8×102 - 4×104  Crandall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [52]  total  Bq+ (q=2-5), Cq+ (q=3,4), Nq+ (q=3,4), Oq+ (q=5,6)  4q×103 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5q×104  Pieksma & Havener [191]  total  B4+  60 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2×103  Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [82]  total  Bq+ (q=2-4), Cq+ (q=2-4), Nq+ (q=2-5), Oq+ (q=2-5)  6q×103 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3q×104  Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [169]  total  Bq+ (q=2-5), Cq+ (q=3,4), Nq+ (q=3,4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×104 - 2×105  Phaneuf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [187]  total  Cq+ (q=1-4), Nq+ (q=1-5), Oq+ (q=1-5), Siq+ (q=2-7) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6×103 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='65×106  Nutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [182]  total  C2+  5×102 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×103  Havener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [107]  total  C3+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 - 3×103  Greenwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [90]  total,line  C,N,O,Neq+ + He,H2,H2O,CO2 (q = 3-9)  7q×103  Greenwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [89]  total,line  C,N,O,Neq+ + H2O,CO2 (q = 3-10)  550 - 800 km s−1  Fogle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [73]  line  C2+ + H2  103 - 3×104  Phaneuf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [188]  total  Cq+ (q=3,4), Oq+ (q=2-6)  10 - 1×104  Lubinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [162]  nl  C4+,N5+,O6+ + H2  5 - 4×103  Phaneuf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [188]  total  Cq+ (q=5,6)  10 - 1×104  Sant’Anna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [205]  total  C3+  1×106 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×106  Ciric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [48]  total,nl  Cq+ (q=3,4), N5+, O6+  7×102 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6×103  McCullough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [171]  total,nl  C3+  6×102 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='8×104  Panov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [186]  total  C4+, N5+, O6+, Ne8+  100 - 1300 km s−1  total  Cq+ (q=5,6), Nq+ (q=6,7), Oq+ (q=7,8), Neq+ (q=9,10)  100 - 1300 km s−1  Dijkkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [63]  total,nl  Cq+ (q=3,4), N5+, O6+  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='7 - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6  Gao & Kwong [81]  total  Cq+ + CO (q = 3-4)  5×102  Fritsch & Lin [77]  total,nl  C4+  100 - 2×104  Hoekstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [112]  total,nl  C4+  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×102 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×102  Hoshino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [116]  total  C4+ + He  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×102 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×102  Dragani´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [64]  total  C5+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='64 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2×104  Bodewits & Hoekstra [31]  total, n  C5+ + H2O  113 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='75×103  Beiersdorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [20]  line  Cq+,Oq+ + CO2 (q = 5-8)  several 100 km s−1  Defay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [58]  line  C6+ + He  460 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2×104  Leung & Kirchner [147]  total,nl  C6+ + He,H2  103 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×104  Andrianarijaona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [9]  line  C6+ + Kr  320 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6×104  Leung & Kirchner [148]  line  C6+,O8+ + Kr  500 - 4×104  Stebbings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [225]  total  N+, O+  100 - 4×104  10  Liyi Gu and Chintan Shah  Reference  typea  ion  energy (eV/u)  Folkerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [75]  total  N4+  1-3×102  Bliek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [27]  total  N4+ + H , H2  103-4×103  Huq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [118]  total  Nq+ + H (q = 3-5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='9 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×103  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [266]  total  N7+,O7+ + H  2 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='22×103  Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [102]  total,n  N7+,O7+ + He,H2O,CO2,CO  2×103, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='67×103  Fite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [72]  total  O+  100 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6×104  Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [141]  total  O+ + H2,CO2,CO,CH4,C2H2,C2H6,C3H8  200 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×103  Beijers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [23]  nl  O3+  45 - 752  Church & Holzscheiter [47]  total  Oq+ (q = 2,3) + H  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×104  Havener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [106]  total  Oq+ (q = 3,4) + H(D)  1-103  Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [169]  total  Oq+ (q=3-6), Siq+ (q=4-9), Feq+ (q=4-15)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×104 - 2×105  Gao & Kwong [80]  total  Oq+ + CO (q = 4,5)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5q×103  Havener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [105]  total  O5+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='9-800  Bodewits & Hoekstra [30]  total,nl  O6+ + H2O  100 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×103  Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [174]  line  O6+ + CO  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6×104  Machacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [165]  total  O6+ + CO,H2O,CO2,CH4,N2,NO,N2O,Ar  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='17×103 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='33×103  Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [169]  total  Oq+ (q=7,8)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×104 - 2×105  Beiersdorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [18]  line  O8+,O7+ + CO2  15  Ne10+,Ne9+ + Ne  9  Bangsgaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [14]  total  Oq+ + He (q = 2-5)  100 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×103  Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [100]  total, n  O6+ + CO2,CH4,H2,N2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='63×103 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='95×104  Otranto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [183]  line  O8+ + CH4,CO2,N2  10  Seely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [207]  line  O8+ + Kr  445 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='18×103  Wang & Wang [249]  total, n  O8+ + He  16 - 54  Mroczkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [175]  total  Ne2+  139 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='49×103  Seredyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [209]  total  Ne2+ + H(D)  59-949  Huber [117]  total  Neq+ (q=2-4), Arq+ (q=2-4,6)  103 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×104  Rejoub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [200]  total  Ne3+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='07 -826  Abu-Haija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [3]  total,n  Neq+ + CO2,H2O (q = 3-5)  15q - 500q  Havener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [108]  total  Ne4+ + H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 - 1006  Kusakabe & Shirai [144]  total  Neq+ + Ne,Ar,Kr,Xe  2q×103  Abu-Haija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [2]  total  Ne6+ + CO2,H2O  450 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×103  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [267]  line  Ne8+ + He,Kr  400-900 km s−1  Leung & Kirchner [149]  total, n  Ne8+ + He,H2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6 - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='9  Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [262]  total, n  Ne8+,Ne9+ + He,H2  103 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='475×104  Otranto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [184]  line  Ne10+,Ar18+,Kr36+ + Ar  18 - 4×103  Tawara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [237]  line  Kr36+,Ar18+,Ne10+ + Ar  4×103  Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [126]  total  Siq+ (q=2-7)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1×104 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='04×106  Bruhns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [37]  total  Si3+  40 - 2500  Tawara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [233]  total  Siq+ + He (q = 3-5)  ∼ 100 - 1000  Tawara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [234]  total,line  Si13+,S15+,Ar17+ + Ar,He,CH4  103-7×104  Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [254]  total  S2+ + H , H2  2×103-8×103  Shah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [211]  line  S16+ + CS2  ∼ 10  1 Charge exchange in X-ray astrophysics  11  Reference  typea  ion  energy (eV/u)  McCullough et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [172]  total,nl  Arq+ + H, H2, He (q = 4-6)  q×103, 2q×103  Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [261]  total, nl  Ar8+ + He  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4×103 - 2×104  Beiersdorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [21]  line  Ar16+ + H  4×104  Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [7]  line  Ar17+,Ar18+ + Ar  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='25q - 125q  Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [8]  line  Ar17+,Ar18+ + Ar  5 - 2×103  Beiersdorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [22]  line  Feq+ + O2,N2,H2O,CO2 (q = 16 - 23)  ≤ 10-20  Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [140]  total  Cuq+ + H2,Ne,Ar,Kr,Xe  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×106  Betancourt-Martinez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [24]  line  Ni19+,Ni18+ + He,H2  10  Tawara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [235]  total,line  Krq+ + He,Ne,Ar,N2,CH4 (q = 27-33)  2×103  Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [103]  line  Xe18+,Xe24+ + Na  500 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5×103  Tawara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [236]  total,line  Taq+ + Ar (q = 45-49)  103 - 4×104  a: total = total cross section, n = n-resolved cross sections, nl = nl-resolved cross  sections, line = line emissivities or ratios  Laboratory measurements provide the absolute calibration of the atomic con-  stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The charge exchange process has been measured in the laboratory since  1950s (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Hasted 104), using a crossed-beam setup in which a beam of ionized  particles passes through a chamber filled with neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The storage ring facilities  are often used for measurements at a high collision energy (E ∼ MeV/u), while the  electron cyclotron resonances and the electron beam ion sources are applied to mea-  surements with medium collision energy (E ∼ keV/u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A significant part of the early  studies with crossed-beam measurements contributed to the fusion research because  of the charge exchange influence on plasma parameters such as charge state distribu-  tion and radiative cooling e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Seim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Recent interest arises for astrophys-  ical applications with the measurements of highly charged ion collisions, including  among many others, the total cross sections of bare C, N, O, and Ne charge exchang-  ing with H2O, He, H2, CO2 measured at Jet Propulsion Laboratory [88, 89], and the  K-shell charge exchange between He2+ and H2O using the crossed-beam facility at  Kernfysisch Versneller Instituut in Groningen [28, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The combination of charge state determination and spectroscopic measurements  becomes more available lately, providing reliable measurement on the velocity-  dependent total and state-resolved cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Some such measurements has  been made with the cold target recoil-ion momentum spectroscopy (COLTRIMS),  which has been developing since 1990s (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Abdallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The n states, some-  times even l levels of the captured electrons, are inferred from the measurement  of the ion kinetic energy loss (the “Q” value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 70, Knoop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  134, Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 263, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A few most recent efforts are carried out in  the COLTRIMS facilities in the Institute of Modern Physics of China on the Ne8+  and Ne9+ collision with He and H2 [262], and at the COLTRIMS in combination  with simultaneous X-ray spectroscopic experiments at the electron cyclotron reso-  nance source of the University of Nevada, Reno [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  One potential issue with beam-gas approach is that the radiative cascade from the  metastable levels of the product ions might take longer than the dynamical timescale  12  Liyi Gu and Chintan Shah  1  10 100  1  10 100  1  10 100  1  10  100  1000  104  105  106  1  10 100  O8+ + H  Panov+83  Meyer+79  Wu+94 (+ref)  Meyer+79     Greenwood+01  Panov+83  Greenwood+00  O8+ + He  O8+ + H2  O8+ + CO2  energy (eV/u)  Cross section (10-16 cm2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='85  Lyα  Lyβ  Lyγ  Lyδ  X-ray energy (keV)  count rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='018  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='016  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='032  Lyβ/Lyα  Lyγ/Lyα  Lyδ/Lyα  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1  500  1000  line ratio  velocity (km/s)  Gu+22  Seely+17  (a)  (b)  (c)  293 km/s  414 km/s  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2: (a) Total cross sections as a function of collision energy for O8+ interact-  ing with atomic hydrogen, atomic helium, molecular hydrogen, and CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The solid  lines are the model values from the calculations with the MCLZ (black) and Janev et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Data points are experimental results from sources indicated in the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  (b) X-ray spectra from Seely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [207] on O8+ + Kr reaction for two different ve-  locities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (c) Comparison of experimental and theoretical line ratios for O8+ charge  exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The experimental data are taken from Seely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [207], and the predic-  tions from Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [92], MCLZ, and Janev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [123] are shown in black, red, and  blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  in the interaction region, making it complicated to study the X-ray spectra observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The electron beam ion trap (EBIT) [152] devices provide a solution, as the ions are  radially confined in a quasi-stationary state by the space charge potential of the  electron beam and axially trapped by the electrostatic potential to the cylindrical  drift tube electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A useful technique for measuring charge exchange with the  EBIT is the magnetic trap mode, in which the electron beam is turned off when the  proper charge states are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The ions expand to form a larger cloud, but remain  confined in the radial direction by the strong magnetic field and the potential applied  to the trap electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Typically, the trap potentials limit the energy of the ions stored  in the EBIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Therefore, the collision energy for the CX reaction is usually very  low, less than 10–20 eV/u [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' However, the advantage of the magnetic trapping  mode is that the electron-ion collision can be safely eliminated, while the ion-neutral  collision remains efficient and produces the pure charge exchange emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The experiments using EBITs do not measure the absolute cross sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' in-  stead, they often record the detailed X-ray spectra of the reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Earlier EBIT  1 Charge exchange in X-ray astrophysics  13  experiments with limited spectral resolution used the parameter known as the hard-  ness ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', the intensity ratio of X-ray line emission from levels with n ≥ 3  relative to n = 2 K-shell emission, to study the dependence on collision ener-  gies [8, 17, 22, 211].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Recent high-resolution (resolved to a few eV) X-ray spectral  measurements have become possible with the introduction of X-ray microcalorime-  ters [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The high resolution is particularly important for charge exchange, which  emits lines from high n levels, often in a narrow energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The NASA Goddard  Space Flight Center X-ray Spectrometer (XRS) and EBIT Calorimeter Spectrometer  (ECS) [195, 196] have been used for a number of charge exchange measurements  performed at the Lawrence Livermore National Laboratory EBIT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Frankel et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 76, Leutenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The n-resolved line diagnostics are made possible with the EBIT plus high-  resolution spectrometer, while the l-distribution of the capture, which is sensitive to  the collision velocity (see § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1), remains challenging since the ion-neutral colli-  sions in the magnetic trapping mode are always mild (≤ 10 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A possible solution  is to measure the charge exchange photons simultaneously at longer wavelengths,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', in the UV and optical bands, where the radiative cascades of the Rydberg levels  might provide additional constraints on the l-state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' In addition, efforts  are needed to combine EBIT and high-resolution calorimeters with the COLTRIMS  setup to fully investigate the kinematics of charge-exchangecollision and multi elec-  tron capture processes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Furthermore, the development of a suitable atomic hy-  drogen target for EBIT measurements is ideal so that only single-electron capture  capture can take place [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  In Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2, we show a set of historical and state-of-the-art measurements of  the O8+ charge exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The total cross sections of O8+ with H, He, H2, and CO2  targets are measured in crossed-beam in the 1970s to 2000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' In panel b, the X-ray  quantum microcalorimeter spectra taken with the beam-gas measurements at Oak  Ridge on O8+ with Kr charge exchange are shown for two different velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The  comparison of the observed and calculated line ratios is plotted in panel c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A more  comprehensive list of laboratory measurement efforts made in the past can be found  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 Observations  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 Ionization balance  Charge exchange is relevant for determining the ion concentration as it serves as a  recombination term for ion Xq+ with  Xq+ + H(He) → X(q−1)+ + H+(He+),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2)  but also in some conditions an ionization term with  14  Liyi Gu and Chintan Shah  10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01  5×10−4  2×10−3  5×10−3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5  1  temperature (keV)  temperature (keV)  ion concentration  O I    O II  O III  O IV  N I    N II  N III  10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='01  5×10−4  2×10−3  5×10−3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5  1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3: Charge state distributions of N (left) and O (right) as a function of equi-  librium temperature for the CIE plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The dashed lines show the calculations  when the charge exchange recombination rates are changed by 50%, while the other  ionization and recombination data are kept the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  X(q−1)+ + H+(He+) → Xq+ + H(He).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3)  These terms are most effective for photoionized and non-equilibrium collisional  ionization plasmas where highly charged ions naturally co-exist with the neutral hy-  drogen and helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As shown in Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [96], the charge exchange contribution  could even be effective for the collisional ionization equilibrium at a low tempera-  ture (∼ a few eV, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The collision induced by the thermal motion of ions has  a typical velocity of km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Sets of recombination and ionization rates by charge exchange are tabulated in  Arnaud & Rothenflug [10] and Kingdon & Ferland [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These tables have been  used broadly in the ion concentration calculation, however, they lacks data of highly  charged ions with q > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Moreover, a significant part of the rates are derived from  theoretical calculations with classical approximations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Landau-Zener method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  A systematic review and update are then needed to expand the q range as well as to  improve the quality of charge exchange data for the ionization balance calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  In table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 we list a set of astrophysical observations made so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' For each  observation, we feature the object, instrument, relevant emission features, together  with a few remarks on the quality of the detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g, using image only, with  timing, and using spectrum with normal or high spectral resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1 Charge exchange in X-ray astrophysics  15  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3: Observations of charge exchange in X-ray  Reference  detector  sources  relevant ion(s)  note  Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [154]  ROSAT  C/Hyakutake 1996 B2  −  image  Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [185]  BeppoSAX  C/1995 O1 (Hale-Bopp)  −  image, spectrum  Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [155]  ROSAT, EUVE  P/Encke 1997  −  image  Dennerl [59]  Chandra  Mars  −  spectrum, low S/N  Krasnopolsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [125]  Chandra  C/1999 T1  −  image, spectrum  Wargelin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [250]  Chandra  Earth geocorona  O VII, O VIII  spectrum  Snowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [216]  XMM-Newton  Heliosphere  C, O, Ne, Mg  spectrum  Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [156]  Chandra  2P/Encke 2003  −  image, spectrum  Dennerl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [60]  XMM-Newton  Mars  C, N, O, Ne  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Willingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [253]  Swift  9P/Tempel 1 (deep impact)  −  timing, spectrum  Krasnopolsky [139]  Chandra  sample of 4 comets  −  spectrum  Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [78]  Suzaku  Earth geocorona  −  spectrum  Dennerl [61]  Chandra  Venus  −  spectrum, low S/N  Bodewits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [29]  Chandra  sample of 8 comets  −  spectrum  Snowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [217]  XMM-Newton  Earth geocorona  O VII  timing, spectrum  Wolk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [255]  Chandra  73P/2006, C/1999 S4  −  spectrum  Christian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [46]  Chandra  8p/Tuttle, 17p/Holmes  −  spectrum  Ezoe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [67]  Suzaku  Earth geocorona  O VII variation  timing, spectrum  Koutroumpa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [138]  XMM-Newton  Heliosphere  O VII, O VIII  spectrum  Katsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [124]  Suzaku  Cygnus Loop SNR  O VII at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='7 keV  spectrum  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [160]  XMM-Newton  M82  O VII, Ne IX, Mg XI high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Konami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [136]  Suzaku  M82  O VIII, Ne X  spectrum  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [161]  XMM-Newton  9 star-forming galaxies  O VII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Townsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [238]  Chandra  Carina Nebula  −  spectrum  Pollock [194]  XMM-Newton  colliding-wind stars  Mg XI & XII, Si XIII high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [43]  Swift  C/2007 N3 (Lulin)  −  image, spectrum  Guerrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [98]  Chandra, XMM  Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Nebula A30  C VI  spectrum, low S/N  Ewing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [66]  Chandra  5 comets  Mg, Si  spectrum  Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [157]  Chandra  103P/Hartley 2  −  image, spectrum  Ishikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [119]  Suzaku  Earth geocorona  −  spectrum  Roberts & Wang [202]  Suzaku  Cygus Loop SNR  O VII triplet  spectrum  Snios et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [215]  Chandra  2 comets  −  image, spectrum  Cumbee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [55]  Suzaku  M82  Ne X  spectrum  Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [91]  XMM-Newton  sample of clusters  S XVI at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 keV  spectrum  Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [94]  Hubble  NGC 1275 AGN  Ne X, S XV  spectrum, 2σ  Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [92]  XMM-Newton  C/2000 WM1  −  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Mullen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [177]  XMM-Newton  C/2000 WM1  −  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [95]  XMM-Newton  sample of clusters  O VIII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [269]  XMM-Newton  Earth geocorona  −  spectrum  Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [166]  XMM-Newton  NGC 5548 AGN  N VII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [97]  XMM-Newton  NGC 5548 AGN  O VI  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  16  Liyi Gu and Chintan Shah  Reference  detector  source(s)  relevant ion(s)  note  Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [264]  XMM-Newton  M51  O VII, O VIII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [243]  Chandra  Perseus cluster  −  spectrum  Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [4]  Hitomi  Perseus cluster  S XVI  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [241]  XMM-Newton  Cygnus Loop SNR  −  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Iwakiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [120]  RXTE  EXO 1745-248 burst  Ti, Cr, Fe, Co  spectrum  Hitomi Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [111]  Hitomi  Perseus cluster  S XVI, Fe XVII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [231]  XMM-Newton  G296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 SNR  −  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [228]  XMM-Newton  N132D SNR  −  low significance  Koshiba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [137]  XMM-Newton  J0453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6-6829 SNR  O VII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Rasmussen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [199]  XMM-Newton  1E 0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2-7219 SNR  −  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Pinto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [193]  XMM-Newton  NGC 4636  O VII  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Branduardi-Raymont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [35] XMM-Newton  Jupiter  −  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [214]  Chandra  SWCX  O VII, O VIII  spectrum  Wulf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [260]  XQC  SWCX  C, N, O  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [268]  XMM-Newton  M82  N, O, Ne  high res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' spectrum  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 Solar system objects  On 25 March 1996, comet C/1996 B2 (Hyakutake) had a close encounter with Earth  at a distance of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Rosat took the opportunity to search for X-ray signal from  this unusual event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' While astrnomers expected very low luminosity because comets  are too cold to emit X-ray, the Rosat image showed a surprisingly bright source from  a crescent-shaped region on the sunward side of comet [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Follow-up studies of  Rosat archival data proved that X-ray emission is common in comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' One year  later after the Rosat discovery, Cravens [53] solved the mystery: the X-ray emission  originates from the charge exchange between highly charged heavy solar wind ions  and neutral gas around the comet atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The spatial and spectral properties  predicted by the charge exchange model has been found to be consistent with Rosat  data of other comets, as well as the later Chandra and XMM-Newton observations  with better spatial and spectral resolutions [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  That the 1996 Rosat data established the charge exchange study of solar system  objects may seem odd, as a number of earlier studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Gombosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 87, Harwit  & Hoyle 101) already pointed out that charge exchange is non-negligible in the solar  wind interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' However, the early works tended to underestimate or overlook  the charge exchange contribution in the X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The Rosat discovery greatly  expanded the field, leading to a number of dedicated astrophysical observations,  but also stimulating relevant laboratory measurements and theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  A new focus of the laboratory and theoretical studies is to reproduce the X-ray  observations, with highly-charged ions (C, N, O, Ne, etc) and cometary neutrals  colliding at a speed of several hundred kilometer per second [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1 Charge exchange in X-ray astrophysics  17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 Comets  In the context of solar wind charge exchange, comets are often used as natural  probes of the latitude- and time-dependent properties of the solar wind ions, which  produce emission features that can be recognized in the X-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The solar  wind study with comets benefits from the facts that the comets approach the Sun in  a much broader range of latitudes than the planets, and their occurrence in a proper  distance range is quasi-constant on year basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The representative X-ray observa-  tions of comets are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The solar winds are found to be largely bimodal at solar minimum: the fast  wind with velocity ∼ 700 km s−1, characterized by low density and low ionization,  launched probably from the open magnetic fields regions at high solar latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The  slow wind, blowing at v ∼ 400 km s−1 from the solar equatorial region with a closed  magnetic field, is often found to be denser and more ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' When the Sun moves  towards its maximum, the fast and slow winds become more mixed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Line diagnostics with the cometary X-ray spectra provide useful constraints on  the speed and the chemical composition of the solar wind (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', Bodewits et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  29, Krasnopolsky 139).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' In addition, the X-ray morphology of comets has been used  to infer the shape and the spatial structure of bow shocks [252].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Finally, Mullen et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [177] derived a constraint on the neutral composition of the cometary atmosphere  using a global fit to the XMM-Newton grating spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Two typical comet spectra are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The Chandra observation  of C/1999 S4 reveals a solar wind composition of C, N, and O in fully striped and  hydrogen-like states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The XMM-Newton grating spectrum of C/2000 WM1 (linear)  further resolves the strong individual C, N, and O lines, posing even tighter con-  straints on the solar wind abundance and even the chemistry of the cometary atmo-  sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These data showcase the cometary X-ray as a powerful solar wind probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  However, most of the X-ray spectra are taken with a low CCD-like resolution of  ∼ 100 eV, making it impossible to study the solar wind dynamics using comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  A further advance in the X-ray study of comets requires observations using high-  resolution X-ray spectrometers, in particular the microcalorimeters onboard XRISM  [232] and Athena [179] missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 Planets  Jupiter is known as a bright X-ray source from the early Einstein observations [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The origin of the X-ray has been settled just recently: the emission from the equa-  torial region is likely the scattering of solar X-ray in the dense planetary atmo-  sphere, while the X-ray in the aurorae can be a mixture of two components, the  bremsstrahlung radiation dominating > 2 keV by energetic electrons precipitating  from the magnetosphere, and the charge exchange in soft X-ray between the in-  falling ions and the atmosphere c neutrals [25, 33, 34, 35, 36, 65, 85, 132, 181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The  ion diagnostics using charge exchange provides an opportunity to understand where  the ions come from: the X-ray spectra would be C- and O- rich if the solar wind is  18  Liyi Gu and Chintan Shah  the main source of the ions, and S- and O is rich if the bulk population originates  from the Io plasma torus falling into the Jovian magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The XMM-Newton reflection grating spectrometer (RGS) spectrum of Jupiter in  2003 shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 show so-far the best example of the charge exchange  diagnostics with X-ray spectroscopy of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It shows a clear separation of the  charge exchange emission, mostly in oxygen lines, from the low-latitude disk com-  ponent with Fe and O lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The charge exchange lines are broadened by a velocity  of ∼ 5000 km s−1, implying that we are witnessing a bulk of accelerated particles  precipitating into the Jovian polar area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Mars and Venus are possible charge exchange emitters, even though the detection  was made at a lower significance than Jupiter [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The scatter of solar X-rays  in the planetary atmosphere is constantly bright in, even dominating, the overall soft  X-ray band, making it challenging for charge exchange observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' No significant  charge exchange signal has yet been detected in other non-Earth planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 Geocorona and heliosphere  Charge exchange emission is generated when the highly ionized solar wind inter-  acts with neutral or near-neutral gas in the Earth’s exosphere and the heliosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  This component is commonly known as the solar wind charge exchange (SWCX),  which can affect any X-ray observation so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Therefore, it is treated as a trouble-  some background, consisting of multiple emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', C, N, and O) in the  soft X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Concerning its temporal, spectral, and spatial variations, it is often  challenging to fully eliminate the SWCX from the extrasolar sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Chandra observation of the dark moon has been considered direct evidence for  the nearby SWCX since it shows charge exchange-type spectrum [250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Similarly,  Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [214] reported an enhancement of O VII and O VIII lines from SWCX in  the Chandra observation of molecular cloud MBM 12, and Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [78] ob-  served a transient increase of SWCX flux by comparing several spectra of a Suzaku  blank field in the direction of the north ecliptic pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  In a CCD resolution, the X-ray spectra from the geocoronal and heliospheric  SWCX could sometimes resemble those from the local hot bubble around the solar  system, making it difficult to isolate one from another in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' One solution  is to separate them based on directional variation: the local hot bubble is isotropic  while the SWCX might increase in the focusing cone of heliospheric helium caused  by the inflow of interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' By scanning the sky through the helium cone  using the diffuse X-rays from the local galaxy (DXL) mission, Galeazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [79]  separated the SWCX and the local hot bubble emission based on their spatial dis-  tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Their results showed that the SWCX emission contributes about 40% of  the 1/4 keV X-ray flux in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  A second approach is to distinguish SWCX using high-resolution X-ray spec-  troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The X-ray quantum calorimeter (XQC, ∼ 6 eV FWHM), onboard the  University of Wisconsin-Madison/Goddard Space Flight Center sounding rocket,  has measured several times the diffuse X-ray background below 1 keV [260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As  1 Charge exchange in X-ray astrophysics  19  shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6, the high-resolution spectroscopy allows the separation of the  SWCX from the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 keV local hot bubble and the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 keV Milky Way halo compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It could be used to further isolate the charge exchange from fast solar wind  and slow wind components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 Astrophysical objects  While the charge exchange from the solar system objects is well established, the de-  tection in extrasolar astrophysical objects remains somewhat controversial though  evidence is accumulating in particular in the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A general issue is that  many claims are made with the CCDs on Chandra, XMM-Newton, and Suzaku  with 50 − 100 eV resolutions, or with dispersive gratings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', RGS) of better but  source-dependent resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A revisit with non-dispersive high resolution X-ray  spectrometers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=', XRISM and Athena) is needed to verify these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1 Stars and supernova remnants  Pollock [194] discussed a scenario that colliding wind ejected from binary stars  might lead to ion-ion collision and charge exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The ion-ion charge exchange  can be realized only when the collisions occur at a large velocity (≥ 2000 km s−1),  which indeed resembles some of the observed terminal shock speeds of the collid-  ing stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Pollock [194] suggested that the charge exchange recombination  might explain the low Mg XII/Mg XI ratios observed in systems including WR22,  WR140, and ζ Pup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The wind interaction condition might apply to other types of stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Iwakiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [120] reported the detection of a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6 keV emission feature in the spec-  trum of a neutron star X-ray binary EXO 1745-248 40 hours after its superburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It  is a broad feature with an equivalent width of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 keV, which is much larger than  that of the standard ionized Fe line formed by the reflection in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A spectral  analysis with the RXTE data implied that the feature can be a set of charge exchange  lines, likely emitted from the interaction of a failed wind launched by the superburst  with the neutrals in the neutron star disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The interaction between supernova remnants and ambient neutral clouds might  generate charge exchange emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It is expected that this emission would be vis-  ible around the edge of the remnants, where interstellar neutrals pass through the  shock front and collide with the post-shock hot ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Katsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [124] reported a  detection of charge exchange emission at the outer shell of the Cygnus Loop, which  shows an enhanced feature at the energy of highly excited O VII lines in the Suzaku  spectrum [202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As plot in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='7, Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [241] took an XMM-Newton  RGS spectrum of a bright X-ray spot on the outer shell, and showed that the charge  exchange component, together with an ionized absorber, are indeed required to ex-  plain the forbidden-to-resonance line ratio of O VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Similarly, the RGS spectra of  20  Liyi Gu and Chintan Shah  G296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 [231] and J0453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6-6829 [228] show high forbidden-to-resonance ratios  that can be explained by charge exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The limited imaging capability of XMM-Newton RGS allows brief mapping of  the ion-neutral interaction region in the supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [241]  pioneered this method by differentiating the O VII forbidden line and resonance  line monochromatic images, and discovered that the charge exchange lines can be  emitted from a compact region where a dense neutral cloud is also found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='2 Interstellar medium and star forming regions  The north polar spur is a large ridge of X-ray and radio-emitting gas in the north-  ern sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Similar to some of the supernova remnant spectra discussed above, the  north polar spur also shows large forbidden-to-resonance ratios for He-like triplets  of O and Ne [146, 173], which might potentially indicate a charge exchange origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  However, this scenario is far from conclusive since the forbidden-to-resonance ratio  could also be explained by others atomic physics including an ionized absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As  suggested in Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [93], the ionized absorption might be responsible for most of  the observed line ratios while the charge exchange contributes no more than 10% to  the forbidden lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The star-forming regions in normal or starburst galaxies are another locations  where interactions between hot gas and cold clouds are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' By modeling the  Chandra spectrum of the Carina star-forming complex with multiple thermal com-  ponents, Townsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [238] reported several line-like excesses in the residual in  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='0 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The distribution of these line residuals is spatially correlated with  the cold star-forming filaments, where the gas-filament charge exchange might be  responsible for these line emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  There have been claims for the detection of charge exchange X-ray in several  starburst galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' M82 is the best-studied object in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The X-ray spec-  troscopic works using XMM-Newton and Suzaku data showed that the thermal model  is insufficient in particular for the O VII and O VIII lines, while adding an additional  charge exchange component could fix most of the problem [55, 136, 160, 239, 268?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A recent study using the RGS spectrum is plotted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [161]  presented a sample study of nine star-forming galaxies with XMM-Newton RGS,  and showed that most of these galaxies are undergoing charge exchange as indi-  cated by the forbidden-to-resonance line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Therefore charge exchange can be  a class property of star-forming regions and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Naturally, the ion-neutral in-  teraction could occur between the starburst-driven outflows and the cold interstellar  medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  In the quiescent galaxies, charge exchange might still occur at various locations  including the interfaces between the nucleus-driven outflows and the circumnuclear  neutral clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [264] reported a possible detection of charge exchange  component which contributes significantly to the N, O, and Ne lines, in particular  the He-like forbidden lines, in the RGS spectrum of M51 nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The interaction  between the radio jet and the cold interstellar medium might be responsible for the  1 Charge exchange in X-ray astrophysics  21  emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Yet another possibility is that these lines originate from photoionization  by the AGN outbursts in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3 Active galactic nuclei  AGNs host multiple gas inflows and outflows of a broad range of ionization states,  making it another potential target of the charge exchange search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [97] re-  ported a detection of unidentified features at 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4 ˚A in the grating spectra of Seyfert  I AGN NGC 5548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' By stacking all observations, this feature is seen at > 5σ sig-  nificance taking into account the look-elsewhere effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As indicated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='9,  the wavelength of the anomaly coincides with the high-n transitions from He- and  Li-like oxygen, therefore, it is likely a charge exchange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The authors suggested  that the interaction could happen within the same outflow by mixing the partially  ionized and neutral layers, or between the outflow with the neutral close environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Another report by Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [94] showed that the Hubble STIS spectrum of the  radio galaxy NGC 1275 can be modeled by including three weak lines at 1223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6 ˚A,  1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4 ˚A, and 1244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='0 ˚A, each with a significance of 2 − 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These features can be  explained by a mixture of charge exchange between highly ionized hydrogen, neon,  and sulfur with neutral matter, indicating for an outflow with v ∼ 3400 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4 Clusters of galaxies  Though the hot plasmas are the dominating baryonic component in clusters of galax-  ies, cold gas clouds do exist, and they are often observed in and near the central  galaxies, as well as in the wake of member galaxies during their infall to the center  [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' These neutral structures are immersed in the giant pool of hot, highly ionized  plasma, and their interfaces are candidates for charge exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Fabian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [68]  and Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [243] suggested that the charge exchange can account for a part of  the Hα and soft X-ray emission from bright filaments in the Perseus cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  During its brief lifetime, the micro-calorimeter onboard Hitomi observed the  Perseus cluster with a ∼ 5 eV resolution in the 2 − 10 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As shown in Fig-  ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='10, there are indeed hints for charge exchange in the Hitomi spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The  high-n Rydberg transitions are found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6σ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4σ significance for S XVI  and Fe XXV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The detection remains challenging even for Hitomi, since the charge  exchange in Perseus cluster is overwhelmed by the thermal emission by two orders  of magnitude in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The S XVI charge exchange is in particular interesting since  it nearly overlaps in energy with the mysterious 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 keV line detected in a large  sample of clusters by Bulbul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [38] and Boyarsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5 keV line  was originally proposed as an evidence for the radiative decay of sterile neutrino, a  theoretical form of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' As shown in Aharonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' [4], the Hitomi spec-  trum of the Perseus cluster prefers the S XVI charge exchange scenario, although  the data quality is insufficient to rule out the dark matter line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' In addition, Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  [95] reported a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='8σ detection of a line feature at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='82 ˚A, which is likely a O VIII  22  Liyi Gu and Chintan Shah  charge exchange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The line flux measured with the RGS is in good agreement  with the possible Hitomi detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4 Ending remarks  The introduction of the charge exchange process to general X-ray astronomy was  made in 1996 by the discovery of X-ray from a comet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' It has been broadly under-  stood now that, in parallel to the electron impact and photon induced X-ray, the  ion-neutral interaction can also efficiently generate X-ray in astrophysical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  In the past two decades after the cometary discovery, X-ray astronomers have spot-  ted charge exchange-like X-rays in a variety of objects, ranging from the neighbour  planets and heliosphere to the interstellar medium, galaxies, supermassive black  holes, and clusters of galaxies on the cosmological scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Charge exchange emis-  sion is no doubt the best tool for detecting an interface, because the normal electron-  impact X-ray flux emitted from the physically-thin interface is likely much dimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  It has the potential to change the research landscape on the interactions of (1) AGNs  and the host galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (2) galactic outflow and interstellar medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (3) infalling  galaxies and the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Charge exchange produces only line emission, making it a science case tailored  for high-resolution spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' With low-resolution spectra it often remains chal-  lenging to disentangle charge exchange from a thermal or photoionized component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Once fully resolved, charge exchange lines provide diagnostics on (1) the collision  speed between ions and neutrals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (2) the chemical composition and ionization state  of the hot ions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (3) relation with the Hα-emitting region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' and (4) the neutral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Spatially-resolved spectroscopy can further constrain the interaction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  New astrophysical discoveries could feedback to the theoretical and laboratory  studies of charge exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Substantial work remains to remove the tension between  the theoretical calculations of cross sections and the corresponding laboratory mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The Li-like and ion species with more than three electrons are still poorly  covered in the current calculations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The l- and S- distributions of  the electron capture are crucial for an accurate line model, but they remain less cer-  tain than the n- distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Finally, the multi-electron charge exchange remain to  be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A consistent and continuous effort will be required to ensure that the  atomic data are ready for the next generation of high-resolution X-ray data to be  obtained with XRISM and Athena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Acknowledgements SRON is supported financially by NWO, the Netherlands Organization for  Scientific Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' acknowledges support from NASA under award number 80GSFC21M0002  and Max-Planck-Gesellschaft (MPG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content='1088/0953-4075/39/24/012  1 Charge exchange in X-ray astrophysics  35  [272] Zygelman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content='457  36  Liyi Gu and Chintan Shah  OVIII Lyα  OVII  Heα  f  ir  OVII Heβ  NVII Lyα  NVII Lyβ  NVII Lyγ + Lyδ  NVI  Heα  f  ir  CVI  Lyα  CVI Lyβ  CVI Lyγ  CVI Lyδ  Si XI  Si XI  CV Lyβ  CV Lyγ  H target  H2O target  C4+  C5+  N5+  N6+  O6+  O7+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content='6  photon energy (keV)  Intensity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='8  Comet C/1999 S4  Chandra ACIS  Comet C/2000 WM1  XMM-Newton RGS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4: (Upper) Comet C/1999 S4 Chandra CCD spectrum taken on 14 July 2000,  compared with the model of C, N, O ions charge exchanging with CO2 measured at  EBIT [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (Lower) Comet C/2000 WM1 XMM-Newton grating spectrum fit with  the SPEX cx model taking into account the hydrogen and water targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1 Charge exchange in X-ray astrophysics  37  wavelength (Å)  Intensity  Jupiter, XMM-Newton RGS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5: The XMM-Newton grating spectrum of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+page_content=' The  emission from X-ray aurorae is the difference between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='6: (Left) X-ray quantum calorimeter spectrum of the diffuse soft X-ray back-  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The model is shown in the middle, and the residual is plotted at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The effective area of the detector is shown in the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (Right) The model  is divided into components: LHB - local hot bubble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Fast/slow SWCX - fast/slow  solar wind charge exchange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Halo - Milky way gas halo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' and CXRB - cosmic X-ray  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Total  LHB  Aax  FastSwcx  as)  Counts  Slow Swcx  10  Halo  5  15  10  CXRB  5  100  200  300  400  500  600  700  800  900  1000  1100  Energy (eV)kev-1)  25  Data  15  10  Counts  5  (sec-lkev-1)  Model  NVI+CVI  Fexvill  Fexvill+Ovill  FexIX+Nelx  25  CVI  CVI  NVII  OVIII  FexVIl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='5  20  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='3  Counts  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1  Residual  +2  n  0  Ln  LL  20  100  200  300  400  500  600  700  800  900  1000  1100  Energy (ev)38  Liyi Gu and Chintan Shah  Cygnus Loop SNR (XMM-Newton RGS)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='7: XMM-Newton grating spectrum of a bright knot in the Cygnus Loop super-  nova remnant, fit with the charge exchange plus non-equilibrium collisional ioniza-  tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The neutral and ionized absorbers are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  (b1)  O VII Heα  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='15  Total: Neutral Abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' × Ionized Abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' Isolar x (NEI + CX)  NEI component  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='125  CX component  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='1  Fe XVII (3d-2p)  Counts  N Lyα + N Heβ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='075  C Lya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  Ne Ix Heaα  Lyα  Heβ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='05  N vI Heα  Lyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  O Lyβ  LyY  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='025  Mg Heα  0  Error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='05  T  (b2)  0  Error Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='05  (b3)  0  Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='05  10  20  30  Wavelength (A)1 Charge exchange in X-ray astrophysics  39  18  19  20  21  22  23  0  5 10  10  O VII  O VII  O VII  O VII  O VIII  (r)  (i)  (f)  wavelength (Å)  Intensity  M82 XMM-Newton RGS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='8: Part of the XMM-Newton grating spectrum of the M82 star-forming galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  The charge exchange lines from O VII and O VIII are plotted in colors indicated by  the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  40  Liyi Gu and Chintan Shah  16  18  20  22  24  0  2  4  6  Wavelength (Å)  Counts / m2 / s / Å  O VI Sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1s-6p  O VI Sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1s-4p  O VII 1s-4p 1P1  O VII 1s-3p 1P1  O VII 1s-2p  O VIII 1s-2p  O VII 1s-5p 1P1  NGC 5548  RGS  2021  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='9: XMM-Newton RGS spectrum of NGC 5548 AGN taken in January 2021  fit with the photoionization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' A set of excess features are seen around 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='4 ˚A  indicating the possible charge exchange emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='  1 Charge exchange in X-ray astrophysics  41  Perseus cluster Hitomi/SXS  residuals at S and Fe bands  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content='10: (Left) Hitomi spectrum of the Perseus cluster fit with a sum of thermal,  AGN, and charge exchange components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' (Right) Residuals of the fit with the ther-  mal plus AGN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
+page_content=' The red curve in each panel shows the model change by  including the charge exchange component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFIT4oBgHgl3EQfsStT/content/2301.11335v1.pdf'}
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+Duet: Creating Harmony between Processors
+and Embedded FPGAs
+Ang Li
+Princeton University
+Princeton, NJ 08544, USA
+angl@princeton.edu
+August Ning
+Princeton University
+Princeton, NJ 08544, USA
+aning@princeton.edu
+David Wentzlaff
+Princeton University
+Princeton, NJ 08544, USA
+wentzlaf@princeton.edu
+Abstract—The demise of Moore’s Law has led to the rise
+of hardware acceleration. However, the focus on accelerating
+stable algorithms in their entirety neglects the abundant fine-
+grained acceleration opportunities available in broader domains
+and squanders host processors’ compute power.
+This paper presents Duet, a scalable, manycore-FPGA archi-
+tecture that promotes embedded FPGAs (eFPGA) to be equal
+peers with processors through non-intrusive, bi-directionally
+cache-coherent integration. In contrast to existing CPU-FPGA
+hybrid systems in which the processors play a supportive role,
+Duet unleashes the full potential of both the processors and the
+eFPGAs with two classes of post-fabrication enhancements: fine-
+grained acceleration, which partitions an application into small
+tasks and offloads the frequently-invoked, compute-intensive ones
+onto various small accelerators, leveraging the processors to
+handle dynamic control flow and less accelerable tasks; hardware
+augmentation, which employs eFPGA-emulated hardware widgets
+to improve processor efficiency or mitigate software overheads
+in certain execution models.
+An RTL-level implementation of Duet is developed to evaluate
+the architecture with high fidelity. Experiments using synthetic
+benchmarks show that Duet can reduce the processor-accelerator
+communication latency by up to 82% and increase the bandwidth
+by up to 9.5x. The RTL implementation is further evaluated with
+seven application benchmarks, achieving 1.5-24.9x speedup.
+I. INTRODUCTION
+The demise of Moore’s Law [33] and the stagnation of
+processor performance growth [44] have given rise to hardware
+acceleration [12], [22], [34]. Since ASIC-based accelerators
+suffer from high non-recurring engineering costs and low ver-
+satility, field-programmable gate arrays (FPGAs) are gaining
+steam due to their post-fabrication, gate-level reconfigurability
+and close-to-ASIC performance [16], [20], [8]. FPGAs can
+be integrated either as standalone devices (Fig. 1a) or into
+field-programmable system-on-chips (FPSoC) (Fig. 1b). The
+conventional, FPGA-based acceleration paradigm is coarse-
+grained acceleration (Fig. 2b), which offloads an algorithm
+in its entirety onto the FPGA. Despite offering substantial
+improvements in performance and energy efficiency, coarse-
+grained acceleration is only practically applicable to limited
+algorithms that are both stable (to justify the accelerator design
+costs) and sufficiently large (to justify the control and data
+transfer overheads).
+This work proposes Duet (Fig. 1c, Sec. II), a novel,
+cache-coherent, manycore-FPGA architecture that is tailored
+to enable two novel paradigms of hardware acceleration.
+FPSoC (e.g. Versal)
+Processor 
+Subsystem
+Embedded
+FPGA
+PCIe Ctrl
+Snooping Bus
+Core
+Core
+Cache
+Coherence 
+Crossbar
+ACP
+ACE
+AXI Switch
+Ethernet Ctrl
+GPIO
+ACP
+ACE
+AXI
+NoC
+DRAM Ctrl
+Scratchpad 
+Memory
+On-Chip Devices
+e.g. AI Engines
+L2$
+L2$
+LLC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+Duet
+(This Work)
+Logic-rich
+eFPGA
+Memory-rich
+eFPGA
+LLC
+Shard 
+NoC
+Duet 
+Adapter
+LLC
+Shard 
+NoC
+Duet 
+Adapter
+LLC
+Shard 
+NoC
+Duet 
+Adapter
+LLC
+Shard 
+L2$
+Core
+NoC
+Multicore/Manycore Processor
+L2$
+Core
+Other 
+I/O
+DRAM 
+Ctrl
+PCIe 
+Ctrl
+PCIe 
+Switches
+FPGA
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+(a) Multicore/Manycore Processor and Standalone FPGA
+L2$=Private cache; NoC=Network-on-chip; LLC=Last-level cache
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+Duet
+(This Work)
+Logic-rich
+eFPGA
+Memory-rich
+eFPGA
+LLC
+Shard 
+NoC
+Duet 
+Adapter
+LLC
+Shard 
+NoC
+Duet 
+Adapter
+LLC
+Shard 
+NoC
+Duet 
+Adapter
+LLC
+Shard 
+L2$
+Core
+NoC
+Multicore/Manycore Processor
+L2$
+Core
+Other 
+I/O
+DRAM 
+Ctrl
+PCIe 
+Ctrl
+PCIe 
+Switches
+FPGA
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+ACP
+ACE
+ACP
+ACE
+AXI
+L2$
+L2$
+LLC
+Core
+Core
+Snooping Bus
+Processor 
+Subsystem
+FPSoC (e.g. Versal)
+Scratchpad 
+Memory
+AXI Switch
+Cache
+Coherence 
+Crossbar
+On-Chip Devices
+e.g. AI Engines
+NoC
+DRAM Ctrl
+PCIe Ctrl
+Ethernet Ctrl
+GPIO
+Embedded
+FPGA
+(b) Field-Programmable System-on-Chip (FPSoC), e.g. Versal [19]
+AXI [3]: non-coherent interconnect; ACP [3]: uni-directional
+coherent interconnect; ACE [3]: fully coherent interconnect.
+Multicore/Manycore Processor
+L2$
+Core
+Other 
+I/O
+DRAM 
+Ctrl
+PCIe 
+Ctrl
+PCIe 
+Switches
+FPGA
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+L2$
+Core
+NoC
+LLC
+Shard 
+ACP
+ACE
+ACP
+ACE
+AXI
+L2$
+L2$
+LLC
+Core
+Core
+Snooping Bus
+Processor 
+Subsystem
+FPSoC (e.g. Versal)
+Scratchpad 
+Memory
+AXI Switch
+Cache
+Coherence 
+Crossbar
+On-Chip Devices
+e.g. AI Engines
+NoC
+DRAM Ctrl
+PCIe Ctrl
+Ethernet Ctrl
+GPIO
+Embedded
+FPGA
+L2$
+Core
+Duet
+(This Work)
+LLC
+Shard 
+NoC
+L2$
+Core
+LLC
+Shard 
+NoC
+L2$
+Core
+LLC
+Shard 
+NoC
+L2$
+Core
+LLC
+Shard 
+NoC
+L2$
+Core
+LLC
+Shard 
+NoC
+L2$
+Core
+LLC
+Shard 
+NoC
+NoC
+LLC
+Shard 
+Duet 
+Adapter
+NoC
+LLC
+Shard 
+Duet 
+Adapter
+NoC
+LLC
+Shard 
+Duet 
+Adapter
+Memory-rich
+eFPGA
+Logic-rich
+eFPGA
+(c) Duet (This Work)
+Each Duet Adapter can be configured independently to support
+uni- or bi-directional coherence in addition to a memory-mapped
+non-coherent interface. Duet enables scalable integration of: 1) any
+number of processors; 2) multiple independent embedded FPGAs
+(eFPGAs); 3) multiple NoC access points per eFPGA.
+Fig. 1: CPU-FPGA Systems
+Fine-grained acceleration (Fig. 2c, Sec. III-A) partitions an
+algorithm into smaller tasks and offloads only the frequently-
+invoked, compute-intensive ones onto a variety of small ac-
+celerators. Processors still play a critical role by handling dy-
+namic control flow, memory/IO-bound tasks, or any other less
+accelerable computations. For example, fine-grained acceler-
+ators can be used for special instructions like tangent, basic
+algorithms like sorting, or inner loop bodies like the compute
+payload per node/edge during a graph traversal. Hardware
+1
+arXiv:2301.02785v1  [cs.AR]  7 Jan 2023
+
+Time
+B+
+Communication 
+Overhead
+Accelerated 
+Function
+Critical Section
+(e)
+Manycore
+Baseline
+...
+Coarse-Grained
+Acceleration
+...
+...
+...
+DMA
+DMA &
+Interrupt
+Hardware 
+Augmentation
+(This work)
+...
+...
+Fine-Grained
+Acceleration
+(This Work)
+(a)
+(b)
+(c)
+(d)
+CPU Execution
+A
+B
+B
+B
+B
+C
+C
+C
+C
+B
+D
+A
+C
+D
+B
+D
+A
+B+ B+
+B+
+C+ C+
+C+
+C+
+A
+D
+B
+B
+B
+B
+C+
+C+
+C+
+C+
+B
+A
+D
+B
+B+
+B+
+B+
+B+
+C
+C
+C
+C
+B+
+Fig. 2: Accelerating a Hypothetical Program
+(a-c) Execution time of a manycore baseline and different
+acceleration paradigms; (d) An example of hardware augmentation
+in which the embedded FPGA emulates a lock-free task scheduler;
+(e) Control flow graph of the program.
+augmentation (Fig. 2d, Sec. III-B) takes an application-
+agnostic approach — it employs FPGA-emulated hardware
+widgets to reduce processor idle time or mitigate software
+overhead in certain execution models. For example, hardware-
+implemented, lock-free data structures can reduce synchro-
+nization overhead in shared-memory, multi-thread programs;
+hardware task schedulers enable task-level parallelism with
+lower overhead than software schedulers.
+Existing CPU-FPGA systems are inefficient for fine-grained
+acceleration and hardware augmentation due to two reasons:
+first, the centralized, bandwidth-optimized CPU-FPGA inter-
+connect performs poorly when lots of cores and accelerators
+communicate via short, frequent messages; second, even on
+state-of-the-art FPSoCs which support bi-directional cache
+coherence between the processors and the FPGA, cacheline-
+granular memory sharing incurs non-trivial overhead. The
+overhead is justifiable for coarse-grained acceleration due to
+extremely high compute-to-memory ratio, but becomes critical
+for small, frequently-invoked, fine-grained accelerators.
+Duet is tailored to the distinct architecture requirements
+posed by fine-grained acceleration and hardware augmen-
+tation. In essence, Duet promotes eFPGAs to be equal
+peers with processors and integrates them as first-class
+citizens on the network-on-chip (NoC) through the novel,
+lightweight, Duet Adapters. In particular, Duet contains the
+following novelties:
+• Scalability: Duet enables tight integration of one to mul-
+tiple eFPGAs of various resource compositions into a scalable
+manycore system. Each eFPGA may be connected to multiple
+Duet Adapters for higher aggregate memory bandwidth.
+• Hybrid Cache Coherence: Duet adopts a hybrid scheme
+to coherently integrate the eFPGAs. Each Duet Adapter con-
+tains one to multiple private, local, hardware Proxy Caches
+(Sec. II-C) that translate the platform-dependent, cache coher-
+ence protocols into simple memory interfaces for the eFPGA.
+Furthermore, each Proxy Cache can be configured at eFPGA
+programming time to support an optional, bi-directionally
+coherent, soft cache built out of eFPGA resources. The use
+of soft caches improves accelerator performance by exploiting
+data locality while coherently sharing data with the processors.
+• Non-Intrusive Integration: For fine-grained acceleration
+and hardware augmentation, a sizable amount of compute
+is still run on the processors and is critical to the overall
+performance. However, the direct participation of eFPGA-
+emulated, soft caches in cache coherence may slow down
+the cache system because eFPGAs run at a lower clock
+frequency and suffer from clock-domain-crossing overheads.
+Addressing this challenge, the Duet Adapters operate in the
+processors’ (fast) clock domain and move FPGA-side coher-
+ence maintenance off of the critical path. As a result, the cache
+system performs equally fast as if each Duet Adapter was an
+additional processor-owned private cache.
+• Plug-and-Play Integration: Duet requires little to no
+hardware changes to existing manycore systems because the
+Duet Adapters transduce between the eFPGAs and the NoC.
+Such decoupling is critical because modifying a mature pro-
+cessor design often leads to performance degradation or even
+hardware bugs, especially as design complexity and verifica-
+tion costs skyrocket on advanced technology nodes [23].
+To evaluate Duet with high fidelity, we build Dolly, a
+prototype instance of Duet at the RTL level (Verilog &
+SystemVerilog). In addition, we assemble a full toolchain
+for software development and accelerator design, leveraging
+an array of open-source projects, including OpenPiton [6],
+BYOC [5], PRGA [31], Yosys [54], and VTR [38]. To
+encourage further research in tightly-integrated, hard-
+ware cache-coherent CPU-eFPGA systems, we have open-
+sourced Dolly and its toolchain, available at https://github.
+com/PrincetonUniversity/Duet.
+Experiments show that the Duet Adapter introduces neg-
+ligible hardware overhead for the CPU-eFPGA integration
+(Sec. V-B). The Proxy Cache reduces the latency of processor-
+accelerator communication up to 82% and increases the band-
+width up to 9.5x compared to having the eFPGA participating
+directly in the coherence protocol (Sec. V-C). The latency
+reduction and bandwidth increase are stable across different
+eFPGA clock frequencies. On selected benchmarks (Sec. V-D),
+Dolly improves the overall performance up to 24.9x in com-
+parison to corresponding processor-only baselines and up to
+4x in comparison to other CPU-FPGA systems.
+The major contributions of this work are:
+• Presenting Duet, a manycore-FPGA architecture which
+employs novel Duet Adapters to integrate embedded FPGAs
+in a scalable, non-intrusive, cache-coherent manner.
+• Identifying and demonstrating with examples two novel
+paradigms of hardware acceleration enabled by Duet, namely
+fine-grained acceleration and hardware augmentation.
+• Presenting Dolly, an RTL-level prototype of Duet and a
+full toolchain for software development and accelerator design.
+• Evaluating Dolly’s silicon area consumption and perfor-
+mance with synthetic and application benchmarks.
+• Releasing Dolly and its toolchain for open-source access.
+2
+
+Shared LLC*
+DRAM Ctrl
+PCIe Ctrl
+Other I/O
+Network-on-Chip*
+L2$
+Core
+L2$
+Core
+L2$
+Core
+Control Hub (Sec. II-E)
+FPGA Manager
+Soft Register 
+Interface
+Programmable 
+Clock Generator
+Programming 
+Engine
+Exception 
+Handler
+Feature 
+Switches
+Shadow 
+Registers 
+(Sec. II-F)
+Async
+FIFO
+Memory Hub
+(Sec. II-B)
+Exception 
+Handler
+Feature 
+Switches
+TLB
+(Sec. II-D)
+Proxy
+Cache
+(Sec. II-C)
+Async
+FIFO
+Duet 
+Adapter
+More
+Memory 
+Hubs
+More
+Duet 
+Adapters
+More
+eFPGAs
+eFPGA
+Configuration 
+Memory
+FPGA (Slow) 
+Clock
+Domain
+Soft Accelerator
+Soft Controller
+Soft Cache (Sec. II-C)
+More Soft Caches
+Non-Coherent Memory
+Fig. 3: Duet Architecture and an Emulated Soft Accelerator
+* For simplicity, the figure shows a bus-based NoC and a centralized LLC. Duet can be adapted to other
+NoC topology and distributed LLC. For example, Dolly (Sec. IV) uses a 2D-mesh and a distributed LLC.
+II. ARCHITECTURE
+Before diving into the architecture of Duet, we want to
+clarify several terms that are used throughout this paper:
+software refers to the program running on the processors;
+hardware, hard cache, etc. refer to the components that are
+fixed at fabrication time, in contrast to the soft components that
+are emulated with the reconfigurable resources in the eFPGAs.
+In addition, we use the term soft accelerator to refer to both
+fine-grained accelerators and hardware augmentation widgets.
+A. Overview
+Fig. 3 shows an overview of the Duet architecture. Duet
+integrates eFPGAs as first-class citizens on the NoC through
+novel, hardware Duet Adapters. Each Duet Adapter contains
+one to several Memory Hubs and one Control Hub. The
+Memory Hubs enable bi-directionally coherent memory ac-
+cesses of the soft accelerators and transduce between the mem-
+ory interfaces of the eFPGAs and the NoC. The Control Hubs
+present the eFPGAs as on-chip devices that are accessible via
+memory-mapped I/Os (MMIOs).
+The fact that eFPGA-emulated soft accelerators typically
+run at a much slower clock than the rest of the system poses a
+challenge to minimizing processor-accelerator communication
+overhead. In particular, any traffic entering or leaving the slow
+clock domain must pay for the clock-domain-crossing (CDC)
+overhead, and every slow clock cycle significantly penalizes
+the total communication time (Fig. 5a & Fig. 6a). A few dozen
+cycles seem negligible, but since the accelerated tasks are short
+and invoked frequently, the accumulated overhead can chip
+away a large fraction of the effective speedup. Addressing this
+challenge, the Duet Adapter adopts various strategies to move
+the logic in the slow clock domain off of the critical path.
+B. Memory Hub
+Each Duet Adapter may contain multiple Memory Hubs,
+each attached to the NoC using an independent connection.
+The soft accelerator may use all or any subset of them
+to access the memory. Each Memory Hub consists of an
+Soft-Only
+Clock 
+Boundary
+eFPGA
+Soft Cache
+eFPGA
+eFPGA
+Soft Cache
+Shared LLC
+Shared LLC
+Shared LLC
+Hard-Only
+Hybrid
+Accelerator
+NoC
+Accelerator
+Accelerator
+Hard Cache
+Hard Cache
+NoC
+NoC
+Fig. 4: FPGA-Side Cache Organization Options
+exception handler, a set of feature switches, and a Proxy
+Cache (Sec. II-C), all implemented in hardware. Besides the
+hardware Proxy Cache, each memory hub can support one
+optional, bi-directionally coherent, Soft Cache built out of
+eFPGA resources. The exception handler as well as all the
+feature switches can be configured by the processors via on-
+chip MMIOs.
+The exception handler employs timeout and parity checks
+to monitor eFPGA outputs. When an exception is detected,
+e.g., due to an RTL or software bug, it asserts an error
+code and deactivates all Memory Hubs in the same Duet
+Adapter. Once deactivated, the Memory Hubs stop accepting
+any memory requests from the eFPGA, but the Proxy Caches
+remain functional, so that any in-flight coherence messages
+are processed properly. This mechanism prevents accelerator
+bugs from halting the system at the micro-architecture level.
+The feature switches allow the processors to configure the
+Memory Hubs according to the state of the eFPGA and the
+specifications of the soft accelerator. For example, the Memory
+Hubs should be deactivated during eFPGA reconfiguration; if
+soft caches are used, the Memory Hubs should be configured
+to forward invalidation requests into the eFPGA.
+C. Proxy Cache
+A key challenge in building coherently-integrated CPU-
+FPGA systems is how to organize the FPGA-side caches.
+Fig. 4 shows the options:
+3
+
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+NoC Req
+NoC Resp
+FIFO (Proxy→FPGA)
+FIFO (FPGA→Proxy)
+Inv
+Inv
+Soft Cache
+Inv
+Ack
+Clock Boundary
+Acc Datapath
+Load
+Hit
+Data
+Crit. Path
+Ack
+Inv
+Ack
+Crit. Path
+eFPGA Clock
+Processor Clock
+(a) Cache Operations with Only Soft Caches
+Inv=Invalidation from a remote cache; Ack=Acknowledgement
+NoC Req
+NoC Resp
+eFPGA Clock
+Processor Clock
+FIFO (Proxy→FPGA)
+FIFO (FPGA→Proxy)
+0
+1
+2
+3
+4
+5
+6
+Ack
+Clock Boundary
+Hard Cache
+Inv
+Acc Datapath
+Load
+7
+8
+9
+Ld
+Hit
+Data
+Data
+Data
+10
+11
+Load
+Crit. Path
+Inv
+Crit. Path
+(b) Cache Operations with Only Hard Caches
+Crit. Path
+NoC Req
+NoC Resp
+FIFO (Proxy→FPGA)
+FIFO (FPGA→Proxy)
+0
+1
+2
+3
+4
+5
+6
+Inv
+Soft Cache
+Inv
+Ack
+Clock Boundary
+Proxy Cache
+Inv
+Acc Datapath
+Load
+Hit
+Data
+7
+Crit. Path
+8
+9
+Inv
+10
+11
+Inv
+eFPGA Clock
+Processor Clock
+(c) Cache Operations with Duet (This Work)
+Fig. 5: Cache Operations with Different Cache Organizations
+For simplicity, this figure shows single-stage async FIFOs, and the
+FPGA clock runs at half of the system clock frequency. On real
+hardware, async FIFOs typically take two to four stages, and the
+FPGA clock could be as slow as 1/10 of the system clock.
+Soft-Only The eFPGA is directly connected to the NoC. The
+soft cache, if used, must implement the system-
+wide cache coherence protocol.
+Hard-Only A hardware, private cache is added between the
+eFPGA and the NoC. Cache coherence is hidden
+from the eFPGA, so soft caches are not supported.
+Hybrid A hardware, private cache participates in the
+system-wide cache coherence on behalf of the
+eFPGA and supports the use of soft caches with
+a local cache coherence protocol.
+Option 1, soft-only, has three main drawbacks. First, it
+becomes the accelerator developers’ burden to design a cache
+in compliance with a platform-dependent, complex cache
+coherence protocol. The sophisticated control logic consumes
+more logic resources, has higher access latency, and is rarely
+reusable across devices — a soft cache designed for ACE [3]
+would hardly work with CHI [4]. Second, since the soft caches
+have access to the micro-architectural state (e.g., can block a
+NoC message indefinitely), the system cannot fully contain
+faulty or malicious behaviors of the soft caches. Third, as
+shown in Fig. 5a, the soft caches may slow down the hardware
+cache system due to CDC overheads and the slow cycles
+spent in the FPGA clock domain. Nonetheless, soft-only is the
+de facto design on most commodity FPSoCs because these
+FPGA-centric architectures are designed for coarse-grained
+acceleration. Specifically, coherent memory sharing is rare and
+mainly offered to simplify programming; soft cache IPs are
+licensable from the vendors; and accelerator bugs are expected
+to fail the entire system.
+Option 2, hard-only, addresses all the drawbacks of the
+soft-only approach but has other limitations. First, the CDC
+overhead is now imposed on the accelerator datapaths (5b).
+Second, the cache implementation is fixed at fabrication time
+and may be suboptimal for certain accelerators.
+Given the downsides of the soft-only and hard-only options,
+Duet takes a hybrid approach. A private, local, hardware Proxy
+Cache implements the platform-dependent, system-wide cache
+coherence protocol and provides a simple memory interface
+to the eFPGA. Each Proxy Cache can be configured through
+feature switches to support an eFPGA-emulated, soft cache
+which can be tightly integrated into the accelerator datapaths.
+Moreover, the Proxy Cache contains two key novelties when
+compared to a naïve implementation of the hybrid approach:
+First, the Proxy Cache neither requires nor accepts any
+acknowledgements from the soft cache. As a result, the
+Proxy Cache always responds to coherence messages promptly
+(Fig. 5c), insulating the cache system from the slow eFPGA
+clock. To maintain coherence, the Proxy Cache requires the
+soft cache to be write-through but allows write buffering (the
+Proxy Cache itself can be write-back or write-through, de-
+pending on the coherence protocol of the LLC). Furthermore,
+because the asynchronous FIFOs deliver messages in order, the
+soft cache always receives invalidations, line fills, and write
+acks in the same order as they are sent by the Proxy Cache.
+Rigid proof of the correctness of such protocols is out of the
+scope of this paper, but the PCX protocol in OpenSPARC
+T1 [41] and its extension, the TRI protocol in BYOC [5], are
+two examples that meet the above requirements.
+Second, the protocol is simple yet flexible. In the com-
+mon case, the soft cache only needs to support two request
+types (Load and Store) and three response types (LoadAck,
+StoreAck and Invalidation). It is up to the accelerator designer
+whether to use a write buffer, how many entries the write
+buffer has, and whether read-after-write forwarding is com-
+patible with the consistency assumptions of the application.
+The Proxy Cache can be configured to work with either a
+write-allocate or a write-no-allocate soft cache, as well as be
+configured to enable atomic operations which require the soft
+cache to support incrementally more message types.
+D. Memory Protection and Virtualization
+Another key challenge for CPU-FPGA systems is memory
+protection and virtualization. The Proxy Cache is physically-
+indexed, physically-tagged because it is implemented in hard-
+ware and is closer to the LLC. However, it depends on the
+type of soft accelerator whether the eFPGA should use virtual
+or physical addresses. Hardware augmentation widgets may
+be trusted firmware and can be granted access to physical
+memory. On the contrary, application-specific, fine-grained
+accelerators are like user programs and can be faulty or
+malicious, so they are better restricted to virtual addresses.
+To enable virtual memory accesses of the soft accelerator,
+Duet adds a translation look-aside buffer (TLB) to each
+4
+
+Memory Hub. The TLB can be disabled if the soft accelerator
+is granted access to the physical memory space; otherwise,
+accelerator-initiated memory accesses must be translated by
+the TLB while being speculatively processed by the Proxy
+Cache. On a page fault, the TLB sends an interrupt to a
+processor, then the kernel-level interrupt handler either updates
+the TLB using MMIOs or kills the accelerator if the page
+access is deemed invalid.
+One special case is when soft caches are used but the
+accelerator only has access to virtual addresses, i.e., the
+soft caches are virtually-indexed, virtually-tagged. To enable
+reverse-mapping from physical address to virtual address when
+the Proxy Cache forwards an invalidation into the soft cache,
+the Proxy Cache stores the virtual page number beside the
+physical tag of each cacheline. This also rules out the coexis-
+tence of synonym aliases (different virtual addresses mapping
+to the same physical address) in the soft cache. In particular,
+the Proxy Cache can invalidate the existing virtual address
+before responding to a load request for the same physical
+cacheline through a different virtual address.
+E. Control Hub
+The Control Hub consists of two submodules: the FPGA
+Manager and the Soft Register Interface. The FPGA Man-
+ager provides necessary hardware support for programming
+and monitoring the eFPGA. The programming engine loads
+the bitstream into the configuration memory, and performs
+integrity checks to detect data corruption. The programmable
+clock generator either divides the system clock, or integrates
+a separate PLL for finer control over the generation of the
+FPGA clock. The exception handler and the feature switches
+are similar to those in the Memory Hubs. In particular, the
+feature switches can be used to set the timeout limit, reset the
+soft accelerator, or clear previously-logged error codes.
+The Soft Register Interface enables the soft accelerator to
+implement a soft "device controller" similar to those com-
+monly seen on off-chip peripheral devices. The soft registers
+emulated by the eFPGA are accessible via on-chip MMIOs
+and often have additional effects rather than simply holding a
+value. For example, reading a soft register may dequeue from
+a FIFO inside the accelerator so that repetitive reads return
+different values. When the Control Hub is deactivated, the Soft
+Register Interface returns bogus data to all processor accesses
+so that the system is not halted.
+F. Shadow Registers
+To prevent unwanted side effects, MMIOs typically adhere
+to a strict memory ordering model, e.g., I/O ordering. Unfortu-
+nately, as stressed in Sec. II-A, these strictly ordered accesses
+may stall the processors’ pipelines because the eFPGA runs at
+a slower clock (Fig. 6a). Addressing this inefficiency, the Soft
+Register Interface is augmented with several types of Shadow
+Registers residing in the fast clock domain (Fig. 6b). When
+a processor writes to a shadowed soft register, the Shadow
+Register acknowledges the request before forwarding the write
+into the eFPGA. Conversely, the soft accelerator actively
+Clock Boundary
+NoC Req
+NoC Resp
+Soft Reg.
+FIFO (NoC→FPGA)
+FIFO (FPGA→NoC)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+WR
+Ack
+Ack
+WR
+1
+Crit. Path
+0
+eFPGA Clock
+Processor Clock
+(a) Accesses to a Normal Soft Register
+Clock Boundary
+NoC Req
+NoC Resp
+Soft Reg.
+FIFO (NoC→FPGA)
+FIFO (FPGA→NoC)
+Shadow Reg.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+WR
+RD
+RD
+Ack
+Ack
+Sync
+Sync
+0
+1
+1
+2
+0
+1
+2
+Crit. Path
+1
+Ack
+eFPGA Clock
+Processor Clock
+Accelerator 
+Write
+(b) Accesses to a Shadowed Soft Register (This Work)
+Clock Boundary
+NoC Req
+NoC Resp
+FIFO (NoC→FPGA)
+FIFO (FPGA→NoC)
+Shadow Reg. B
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+WR:A
+WR:A
+Soft Reg. A
+WR:B (stalled)
+0
+1
+Ack:A
+0
+eFPGA Clock
+Processor Clock
+Ack:A
+Sync:B
+Ack:B
+1
+Soft Reg. B
+0
+1
+(c) Strict Ordering of Two Soft Register Writes (This Work)
+Fig. 6: Accessing Soft Registers and Shadow Registers
+synchronizes shadowed soft registers over the asynchronous
+FIFO, so that the Shadow Registers can respond to processor
+reads immediately without notifying the eFPGA.
+Duet supports four types of Shadow Registers: plain, FPGA-
+bound FIFO, CPU-bound FIFO, and token FIFO. Plain
+Shadow Registers only keep the last value of multiple writes,
+which are ideal for passing constant parameters. FPGA-bound
+and CPU-bound FIFOs, as their names suggest, record all
+writes from one side and allow the other side to read in
+order. CPU-bound FIFO is blocking, i.e., a processor read
+is stalled until the soft accelerator pushes into the FIFO or
+the request times out. In contrast, CPU-bound token FIFO
+is a dataless, non-blocking FIFO that consumes a token or
+returns "empty" in response to a processor read. Token FIFO
+is designed particularly to emulate the non-blocking try_join
+semantic in parallel programming models.
+Note that normal soft registers are still available in case the
+software requires non-bufferable accesses that must be sent
+all the way to the endpoint. For example, a soft register can
+be dedicated as a barrier for synchronizing the processor and
+the eFPGA. The processor signals its arrival at the barrier
+by reading the soft register, while the eFPGA signals its
+arrival at the barrier by acknowledging the read. To maintain
+I/O ordering, Shadow Register accesses are processed and
+responded to in order with respect to other shadowed or normal
+register accesses, as shown in Fig. 6c.
+III. APPLICATIONS
+Duet enables two novel paradigms of hardware acceleration,
+namely fine-grained acceleration and hardware augmentation.
+In this section, we illustrate both paradigms with examples
+and discuss why Duet is the ideal architecture for them.
+5
+
+Listing 1: The key compute loop of the Barnes-Hut simulation
+algorithm [7]. The highlighted functions are static, compute-
+intensive, and ideal for fine-grained acceleration.
+// Calculate the net force on particle "p"
+void CalculateNetForce (BHTreeNode node, Particle p):
+if (Distance (node, p) > node.radius * THRESHOLD)
+// "p" is far enough from the particles in the subtree of
+// "node", so we approximate the net force with a low-order
+// multipole expansion and stop traversing the subtree
+p.force += ApproxForce (node, p);
+else if (IsLeafNode (node))
+for (Particle q : node.children)
+p.force += CalcForce (q, p);
+else
+// traverse the subtree via recursive calls
+for (BHTreeNode child : node.children)
+CalculateNetForce (child, p);
+// Calculate the net forces on all particles in parallel
+void CalculateNetForceOnAllParticles (
+BHTreeNode root, vector<Particle> allParticles):
+parallel_for (Particle p : allParticles)
+CalculateNetForce (root, p);
+A. Fine-Grained Acceleration
+1) Overview: At its core, fine-grained acceleration and
+coarse-grained acceleration are similar because they both
+offload computation onto application-specific accelerators. In
+fact, there is a spectrum rather than a clear boundary between
+the two paradigms — the shorter the accelerated function is, or
+the more frequent the accelerator interacts with the processors,
+the closer it is to the fine-grained end of the spectrum.
+Fine-grained acceleration has the following advantages:
+first, it fully utilizes the compute power of the processors by
+running the less accelerable fractions of an algorithm on them,
+for example dynamic control flow, memory/IO-bound tasks,
+managing complex data structures, etc.; second, it reduces
+software and accelerator design costs by moving less logic
+onto the FPGA; third, in comparison to a monolithic acceler-
+ator, a collection of small accelerators are more composable,
+reusable, and more resilient to software updates.
+2) Example: Barnes-Hut (BH) algorithm [7] is an approxi-
+mation algorithm for simulating a dynamic system of particles
+interacting via certain physical forces such as gravity. In each
+time step, the simulation volume is divided into hierarchical,
+cubic cells stored in an octree (for three-dimensional space), so
+that the total force exerted by particles in distant cells can be
+approximated through a center-of-mass abstraction or a low-
+order multipole expansion. Listing 1 shows the key functions
+that calculate the net force on all particles.
+Prior work [17] has proposed an FPGA-based, coarse-
+grained BH accelerator which contains replicated pipelines to
+parallelize force calculation on multiple particles. However,
+despite the use of a special traversal cache that facilitates
+data reuse between the pipelines, control flow divergence often
+leads to low pipeline utilization. Furthermore, the accelerator
+relies on the host processor to serialize the BH-tree and preload
+the node pointers into the traversal cache, making it inappli-
+cable if the dataset exceeds the FPGA’s BRAM capacity. In
+summary, building a coarse-grained BH accelerator can be an
+intimidating task without a satisfactory result.
+Fig. 7 shows the timeline of BH running on a dual-
+CPU
+Thread #1
+CPU
+Thread #2
+Accelerator #2
+CalcForce
+Accelerator #1
+ApproxForce
+Visit Node
+Invoke ApproxForce
+Visit Node
+Invoke CalcForce
+Visit Node
+Invoke CalcForce
+Invoke CalcForce
+Invoke CalcForce
+1
+2
+3
+Fig. 7: Multi-Threaded BH with Fine-grained Acceleration
+1⃝ Loop-carry dependencies and dynamic control flow are handled
+by the processors; 2⃝ Processors and accelerators run in parallel
+through software pipelining; 3⃝ Accelerators are time-multiplexed
+by multiple CPU threads to increase utilization.
+core system with two fine-grained accelerators. Besides being
+easy to program, this software-hardware co-design is scal-
+able and flexible: multiple accelerators of each type can be
+instantiated to increase parallelism; if THRESHOLD is high,
+i.e., ApproxForce is called for less times, the system can
+allocate more eFPGA resources to implement more and/or
+faster CalcForce accelerators, and vice versa if THRESHOLD
+is low.
+The fine-grained BH accelerators would be inefficient with-
+out Duet’s architectural support. First, the accelerators access
+just a few cachelines at random memory locations per ex-
+ecution, resulting in poor utilization of the block-oriented,
+bandwidth-optimized memory system in most existing CPU-
+FPGA architectures. In contrast, Duet’s hybrid cache organiza-
+tion minimizes accelerators’ memory access latency, especially
+for random accesses at cacheline granularity. Second, the
+accelerators are invoked frequently and time-multiplexed by
+multiple CPU threads. A centralized CPU-FPGA interconnect
+may be congested by the MMIOs issued by the processors to
+invoke the accelerators. Duet’s scalable integration makes it
+possible to split the workload across multiple eFPGAs, and
+the shadow registers can further minimize MMIO latency.
+B. Hardware Augmentation
+1) Overview: Hardware augmentation allows application
+developers to add various hardware widgets to the system
+after chip fabrication. These widgets help improve proces-
+sor efficiency and/or mitigate software overheads in certain
+execution models, providing application-agnostic acceleration.
+Different hardware augmentation widgets often exhibit unique
+behaviors and provide distinct software APIs. For example,
+decoupled access-execute engines [49] hide the latency of
+memory redirects by dereferencing software-specified pointers
+and collect the data into a hardware queue; hardware garbage
+collectors [32] run autonomously in the background and
+alleviate software overhead of the runtime environment. Since
+hardware augmentation widgets collaborate closely with the
+processors, Duet’s scalable, tight, cache-coherent integration
+is critical to maximize performance.
+2) Example: Discrete event simulation (DES) is a well-
+known challenge for parallel execution [18]. All the simu-
+6
+
+o
++Ariane
+L2
+Cache
+L2
+Cache
+L2
+Cache
+L2
+Cache
+Coherent 
+Mem Intf
+Coherent 
+Mem Intf
+1
+2
+3
+L3 Cache 
+Shard
+L3 Cache 
+Shard
+L3 Cache 
+Shard
+L3 Cache 
+Shard
+NoC 
+Router
+NoC 
+Router
+NoC 
+Router
+NoC 
+Router
+Ariane
+P-Mesh
+Socket
+P-Mesh
+Socket
+P-Tile
+P-Tile
+C-Tile
+M-Tile
+P-Mesh
+Socket
+P-Mesh
+Socket
+Soft Reg Intf
+FPGA Manager
+Duet Adapter
+FPGA
+Config 
+Memory
+Soft Accelerator
+Soft Cache Soft Cache
+Fig. 8: Architecture of Dolly-P2M2
+Dolly-P2M2 has 2 processors, 1 eFPGA, and 2 Memory Hubs.
+P-tile, C-tile and M-tile are physical tiles in a 2D mesh network.
+P-Mesh Socket is a physical wrapper for common components in
+all physical tiles, including an L2 cache, a NoC router, and a shard
+of the shared L3 cache. 1⃝ is the Control hub (Sec. II-E). 2⃝ and
+3⃝ are two Memory Hubs (Sec. II-B). Note that 1⃝ and 2⃝ reside
+in the same physical C-tile.
+lating threads must either progress in lock step (conservative
+method), which suffers from high synchronization overheads,
+or exploit speculative execution (optimistic method), which re-
+quires a carefully designed software runtime or an architecture
+that supports thread-level speculation, e.g., SWARM [27].
+Duet opens up another possibility through hardware aug-
+mentation: an eFPGA-emulated task scheduler can be loaded
+as part of the simulator software while offering hardware-level
+performance. Consider a possible implementation of such a
+task scheduler. Processors schedule new events by pushing
+memory pointers to the events into a FPGA-bound FIFO,
+after which the task scheduler fetches the event data from
+shared memory and adds the pointer into the proper event
+queue. Once certain events are ready to be processed, the
+task scheduler pushes the pointers into an CPU-bound FIFO
+so that the processors can streamline event processing with
+minimal communication latency with the scheduler. The task
+scheduler can support task speculation by fetching the cache-
+lines that may be modified by a speculative event and storing
+versioned copies of them in its non-coherent memory. On a
+mis-speculation, the task scheduler rolls back the cachelines to
+the most up-to-date, non-speculative versions, then reschedules
+the mis-speculated events.
+IV. DOLLY
+To evaluate Duet with high fidelity, we built Dolly1, a
+prototype system at the RTL level (Verilog/SystemVerilog)
+with a full toolchain for software development and accelerator
+design. Dolly is configurable in the number of processors, the
+logic capacity of the eFPGA, the number of memory hubs, etc.
+For simplicity, we name each Dolly instance PpMm, where
+p specifies the number of processors, and m specifies the
+1Named after Dolly Suite, Op. 56, a collection of pieces for piano duet
+composed by Gabriel Fauré.
+number of memory hubs available to the eFPGA. For example,
+Fig. 8 shows the architecture of Dolly-P2M2. To encourage
+further research in tightly-integrated, hardware cache-
+coherent CPU-eFPGA systems, we have open-sourced
+Dolly and its toolchain, available at https://github.com/
+PrincetonUniversity/Duet.
+Dolly is based on the OpenPiton P-Mesh cache coherence
+system [6] in a 2D mesh configuration. The cache system
+is organized in three levels: the L1 caches are tightly inter-
+woven into the processors; one private, write-back, 8KB L2
+cache per physical tile interfaces with the corresponding L1
+cache through the transaction-response interface (TRI) [5];
+the shared L3 cache is distributed among all physical tiles,
+64KB per shard, and runs a directory-based MESI protocol
+together with the private L2 caches. The NoC offers point-
+to-point ordering of message delivery and supports additional
+message types besides the coherence messages, enabling on-
+chip MMIOs required by Dolly.
+Dolly contains three types of physical tiles and an eFPGA.
+Each P-Tile hosts an Ariane [57] processor, a 6-stage, single-
+issue, in-order CPU which implements the 64-bit RISC-V
+instruction set. Each Ariane processor has a private hardware
+FPU, an 8KB L1 instruction (L1I) cache, and an 8KB L1 data
+(L1D) cache. C-Tiles and M-Tiles compose the Duet Adapter:
+each C-Tile contains one Control Hub (Sec. II-E) and one
+Memory Hub (Sec. II-B); each M-Tile includes one Memory
+Hub. To demonstrate Duet’s adaptability, Dolly implements
+the Proxy Cache by adding a coherent memory interface to
+the unmodified P-Mesh L2 cache. The L2 cache, L3 cache
+shard and NoC router are the same across all physical tile
+types, so we wrap them into one module, the P-Mesh socket.
+The eFPGA is built with PRGA [31] and employs a
+standard island-style architecture composed of logic blocks
+and routing blocks. The eFPGA includes all of the common
+logic resources available on modern FPGAs, including look-
+up tables (LUT), bypassable flip-flops, hard adder chains,
+Block RAMs (BRAM), and hard multipliers. Dolly employs
+a non-synthesizable clock generator to generate the eFPGA
+clock whose frequency can be specified in software. All the
+asynchronous FIFOs are implemented with dual-clock RAMs
+and Gray-coded, 2-stage synchronizers.
+V. EVALUATION
+A. Overview
+We conduct our evaluation of Duet through RTL simulations
+of Dolly instances and focus on three perspectives: first of all,
+we report the area consumption and the typical frequency of
+the hard components of Dolly (Sec. V-B); second, we run syn-
+thetic benchmarks to measure the CPU-FPGA communication
+latency and bandwidth achieved by the Duet Adapter under
+different conditions (Sec. V-C); third, we run seven application
+benchmarks and compare their performance against processor-
+only and FPSoC baselines (Sec. V-D).
+To show the lower bound of Duet-enabled improvements,
+we give the processors various advantages throughout our
+evaluation. Specifically, we boost the clock frequency of the
+7
+
+TABLE I: Area and Typical Frequency of Dolly Components
+Component
+Device
+Area
+Freq. Scaled Area* Scaled Freq.*
+Technology
+(mm2) (MHz)
+(mm2)
+(MHz)
+Ariane [58]
+GlobalFoundries
+0.39
+910
+1.56
+455†
+22nm FDX
+P-Mesh
+IBM 32nm SOI
+0.55
+1000
+1.1*
+711*†
+Socket [6]
+FPGA Mgr +
+FreePDK45
+0.21
+925
+0.21*
+925*
+Soft Reg Intf
+Coherent
+0.04
+1250
+0.04*
+1250*
+Memory Intf
+* Scaled to 45nm with a linear MOSFET scaling model
+† To emulate a higher-performance multi-core, our evaluation sim-
+ulates the Ariane cores and the hardware cache system at 1GHz
+Ariane processors and the P-Mesh cache system to 1GHz,
+despite that previous works reported their maximum frequen-
+cies (after scaling to 45nm) to be 455MHz and 711MHz,
+respectively. For the application benchmarks, processor-only
+baselines always start with a warm cache, while the soft
+accelerators always start with a cold cache.
+B. Area and Typical Frequency of the Hard Components
+Table I summarizes the area and typical frequency of each
+hardware component of Dolly. The numbers for Ariane and
+P-Mesh components are retrieved from previous works. The
+submodules of the Control Hub and the Memory Hub are
+synthesized using Synopsys Design Compiler and Silvaco’s
+Open-Cell Library [48] based on the NCSU FreePDK45
+process design kit [50] using an area utilization rate of 70%.
+In summary, the Control Hub and the Memory Hub require
+minimal hardware resources. The area consumption of the
+eFPGA and the clock frequency of the emulated accelerator
+are reported on a per-benchmark basis in Sec. V-D.
+C. CPU-eFPGA Communication Latency and Bandwidth
+In this section, we evaluate the peak performance of Duet
+with a synthetic benchmark. The eFPGA emulates a simple
+scratchpad memory and a processor uses different mechanisms
+to access it. To show that the eFPGA’s clock frequency has
+a significant impact on performance, we sweep the eFPGA
+clock from 20MHz to 500MHz while fixing the system clock
+at 1GHz. In particular, we study the following CPU-eFPGA
+communication mechanisms and show how Duet improves
+them over existing CPU-FPGA systems:
+Soft registers are memory-mapped, eFPGA-emulated regis-
+ters which the processor can read and write via non-coherent
+MMIOs. These soft registers can be implemented inside the
+eFPGA (Normal Registers), in which case processor ac-
+cesses must pay the Clock-Domain-Crossing (CDC) overhead.
+Shadow Registers address this inefficiency. In this study, we
+use FPGA-bound FIFOs to handle processor writes and CPU-
+bound FIFOs for processor reads.
+Alternatively, the processor can store the data in shared
+memory and pass the memory address to the eFPGA via a
+soft register write. The soft register write not only signals
+the eFPGA to load the data (eFPGA Pull), but also works
+as a synchronization to ensure that all processor stores are
+committed into the cache system before the eFPGA starts
+0
+50
+100
+150
+200
+250
+300
+Roundtrip Latency (ns)
+100
+200
+500
+100
+200
+500
+100
+200
+500
+100
+200
+500
+100
+200
+500
+100
+200
+500
+eFPGA Frequency (MHz)
+20
+20
+20
+20
+20
+20
+17
+17
+17
+17
+17
+17
+26
+26
+26
+26
+26
+26
+33
+33
+33
+47
+47
+47
+25
+25
+25
+142
+70
+28
+100
+51
+20
+105
+57
+27
+103
+56
+27
+104
+57
+27
+104
+56
+26
+300
+180
+108
+170
+123
+94
+229
+133
+72
+42
+42
+42
+130
+82
+52
+26
+26
+26
+NoC Latency
+Cache Logic (Fast Clock Domain)
+Cache Logic (Slow Clock Domain)
+Clock-Domain-Crossing Overhead
+Shadow Reg.
+(This Work)
+Normal Reg.
+CPU Pull w/
+Proxy Cache
+(This Work)
+CPU Pull w/
+Slow Cache
+eFPGA Pull w/
+Proxy Cache
+(This Work)
+eFPGA Pull w/
+Slow Cache
+Communication Mechanism
+Fig. 9: CPU-eFPGA Communication Latency
+(Single processor; Single transaction; Lower is better)
+loading. The eFPGA can send data to the processor in a
+similar way (CPU Pull). As described in Sec. II-C, commodity
+FPSoCs typically emulate FPGA-side caches using eFPGA
+resources (Slow Cache), while Duet employs the novel Proxy
+Cache to improve cache performance.
+Latency Study
+We first measure the minimum round-trip latency of the
+aforementioned communication mechanisms on Dolly-P1M1.
+Fig. 9 shows the breakdown of the CPU-eFPGA communi-
+cation latency into four parts: NoC latency, cache processing
+time in the fast clock domain, cache processing time in the
+slow clock domain, and the CDC overhead. All numbers are
+collected in an ideal scenario, that is, single processor (no
+contention or NoC congestion), single transaction (no buffer
+clogging). Note that the eFPGA pulls and CPU pulls are
+guaranteed to miss in the requesting cache and to hit in the
+other party’s private cache in a modified state. Both the NoC
+latency and the cache processing time in the fast clock domain
+include the time for the distributed directory to send and
+process the secondary write-back requests.
+As we can see, the CDC overhead and the slow clock
+penalty on cache processing constitute over half of the round-
+trip latency for a slow cache, even when the eFPGA runs
+at 50% of the CPU clock frequency. For eFPGA pulls, the
+Proxy Cache can reduce the latency by 13% to 43%, and
+the reduction increases as the eFPGA runs slower. For CPU
+pulls, the Proxy Cache achieves a constant latency regardless
+of the eFPGA clock frequency, reducing the latency by 42% to
+82%. The Shadow Registers also have a fixed latency, reducing
+CPU-eFPGA communication latency by 50% to 80%. Note
+that the Proxy Cache and the Shadow Registers only move the
+eFPGA off of the critical path, and the eFPGA still needs time
+to issue memory accesses and to handle soft register accesses.
+Single-Processor Bandwidth Study
+Next, we study the maximum bandwidth of single-processor
+communications on Dolly-P1M1. In this study, we configure
+the synthetic benchmark to pass 512 quad-word (8 Bytes)
+integers from one processor to the eFPGA and then fetch
+them back. With the soft registers, this is done by having
+the processor execute a loop that writes/reads one integer
+8
+
+20 (2%)
+50 (5%)
+100 (10%)
+200 (20%)
+500 (50%)
+eFPGA Frequency (MHz) and Ratio to CPU Frequency (1GHz)
+0
+100
+200
+300
+400
+500
+600
+700
+Bandwidth (MBytes/s)
+Normal Reg.
+CPU Pull w/ Slow Cache
+eFPGA Pull w/ Slow Cache
+Shadow Reg. (This work)
+CPU Pull w/ Proxy Cache (This work)
+eFPGA Pull w/ Proxy Cache (This work)
+Fig. 10: Processor-eFPGA Communication Bandwidth vs.
+eFPGA Clock Frequency (Single processor; Higher is better)
+per iteration. When using shared memory, the processor first
+allocates two 4KB buffers in the memory and passes the base
+addresses to the eFPGA using two plain shadow registers.
+Then, the processor stores all the integers into one of the
+two buffers and sends a read request to another normal
+soft register, awakening the eFPGA and blocking itself. The
+eFPGA proactively loads the entire array into its scratchpad
+memory, stores it back to the other buffer, and then unblocks
+the processor by acknowledging the processor’s read request.
+Finally, the processor loads the array from shared memory.
+Fig. 10 shows the results of this study. The Proxy Cache
+delivers the highest bandwidth across all eFPGA frequencies.
+In fact, the Proxy Cache reaches the peak bandwidth for
+eFPGA pulls (558MB/s) when the eFPGA runs faster than
+100MHz (10% of the CPU clock frequency); for CPU pulls,
+the peak bandwidth (201MB/s) is reached at 50MHz. In
+comparison, the slow cache can only achieve 287MB/s for
+eFPGA pulls and 144MB/s for CPU pulls even when the
+eFPGA runs at 500MHz. The largest bandwidth gap is 9.5x
+when the eFPGA runs at 100MHz. The upper bounds of cache-
+based communications are determined by the NoC bandwidth
+and the number of concurrent, in-flight, memory requests
+supported by the Proxy Cache. Note that the upper bounds
+of CPU pulls are much lower than those of eFPGA pulls.
+This is because the cache line size is 16 Bytes and the
+eFPGA can load up to one line per cycle, but the L2 cache in
+Dolly only supports stores up to 8 Bytes, so the eFPGA must
+send two requests to store one cacheline, under-utilizing the
+asynchronous FIFOs.
+The Shadow Registers also achieve a stable bandwidth
+(213MB/s) once the eFPGA clock frequency exceeds 10% of
+the CPU clock frequency. In comparison, normal registers can
+only reach 121MB/s at 500MHz. The upper bound of soft
+register accesses is limited by the strict consistency model of
+MMIOs. If the I/O consistency model is relaxed, the upper
+bound is then limited by the number of concurrent, in-flight
+MMIOs supported by the load-store queue of the processor.
+Multi-Processor Scalability Study
+One key difference between Duet and existing CPU-FPGA
+systems is the scalability. At the architecture level, Duet has no
+constraint on the number of processors, eFPGA fabrics, or the
+1
+2
+4
+8
+16
+Number of Processors Accessing the Registers
+101
+102
+103
+Bandwidth (MBytes/s, log-scale)
+Normal Reg. Write
+Normal Reg. Read
+Shadow Reg. Write (This Work)
+Shadow Reg. Read (This Work)
+Fig. 11: Processor-eFPGA Communication Bandwidth (Per
+Processor) vs. Number of Contending Processors
+number of Memory Hubs per eFPGA. In this section, we study
+the effect of multi-processor contention on the soft registers. In
+particular, we run the synthetic benchmark on multiple Dolly
+instances with different number of processors which constantly
+writes/reads the same soft register. The eFPGA clock is fixed
+at 500MHz (50% of the CPU clock frequency).
+Fig. 11 shows the results. The Shadow Registers can stably
+support up to 8 processors before the per-processor bandwidth
+starts to drop, while the normal registers can only support
+up to 2 processors. Considering that the processors need
+to perform other tasks between soft register accesses, the
+Shadow Registers can support even more processors in real
+applications.
+Summary
+In summary, the significant reduction in latency and the
+stable increase in bandwidth clearly justify the promotion of
+soft registers and private caches into the fast clock domain.
+D. Application Benchmarks
+In this section, we evaluate Duet with seven application
+benchmarks. For each benchmark, we first run a C imple-
+mentation as the processor-only baseline and record the total
+runtime. Note that all benchmarks are run in bare metal due
+to limited simulation speed at the RTL level. We then design
+the soft accelerators for each benchmark, either directly at the
+RTL level, or by synthesizing an alternative C implementation
+with Catapult HLS [47]. Each accelerator is synthesized,
+placed and routed with the PRGA [31] workflow to get the
+accurate eFPGA clock frequency and an area estimation of the
+utilized FPGA resources. In particular, we map each bench-
+mark onto the flagship FPGA model provided by VTR [38]
+(k6_frac_N10_frac_chain_mem32K_40nm) which resembles an Altera
+Stratix IV FPGA. With the accurate eFPGA clock frequency,
+we rerun the benchmarks on various Dolly instances and an
+FPSoC-like architecture, recording the total runtime which
+includes all the overhead in CPU-FPGA communication and
+synchronization. In particular, the FPSoC model moves the
+P-Mesh L2 cache into the eFPGA’s (slow) clock domain and
+downgrades all shadowed soft registers to normal registers. We
+then compute the speedup over processor-only baselines and
+compare the results achieved by Dolly and the FPSoC model.
+9
+
+0
+2
+4
+6
+8
+Normalized Speedup
+(Higher is better)
+9.8
+12.9
+16.2
+15.1
+24.9
+2.14
+4.53
+CPU
+FPSoC
+Duet (This Work)
+tangent
+popcount
+sort/32
+sort/64
+sort/128
+dijkstra barnes-hut pdes/4
+pdes/8
+pdes/16
+bfs/4
+bfs/8
+bfs/16
+geomean
+Benchmark
+0
+1
+2
+3
+Normalized ADP
+(Lower is better)
+1.23
+0.61
+Fig. 12: Normalized Speedup and ADP of Application Benchmarks
+sort/N indicates the array size of the sorting accelerator; pdes/N and bfs/N indicates the number of cores sharing the accelerator.
+We use Area-Delay-Product (ADP) to quantify and com-
+pare area efficiency. When calculating area consumption, the
+processor-only baseline only counts the processors and the
+hardware cache system. The FPSoC-like architecture adds the
+silicon area of the FPGA on top of the processor-only baseline,
+and Dolly further includes the Duet Adapters.
+The seven benchmarks, their corresponding Dolly instance,
+and the acceleration paradigm (FG for fine-grained accelera-
+tion, HA for hardware augmentation) are the following:
+• Tangent (P1M0, FG): A floating-point Tangent accelerator
+is implemented with Catapult HLS using a piece-wise linear
+approximation algorithm with a maximum error rate of 0.3%
+compared to the C math library (libm). An FPGA-bound FIFO
+is used to pass the argument to the accelerator and invoke it.
+Results are returned through an CPU-bound FIFO.
+• Popcount (P1M1, FG): Popcount counts the number of ones
+in a long bit vector (512 bits). Since the Ariane processor
+does not support the RISC-V BitManip Extension, we use a
+byte look-up algorithm for the processor-only baseline. The
+accelerator is hand-written in Verilog and uses one Memory
+Hub to load the bit vector from coherent memory.
+• Sort (P1M2, FG): We use the SPIRAL Project [60] to
+generate 3 sorting networks in Verilog for sorting 32, 64,
+128 double-word (4-Byte) integers. The accelerator uses two
+memory hubs, one for reading the input array from coherent
+memory and one for writing the sorted array back, so that the
+accelerator can be pipelined to sort fixed-length slices of a
+larger array which can then be merge-sorted by the processor.
+The processor-only baseline runs quicksort on the entire array.
+• Dijkstra (P1M1, FG): We implement an accelerator for
+Dijkstra’s Shortest Path algorithm with Catapult HLS and use
+a soft cache to exploit data locality between consecutive calls
+to the accelerator.
+• Barnes-Hut (P4M1, FG): As explained in Sec. III-A2, the
+two key kernels in the Barnes-Hut algorithm are implemented
+as soft accelerators using Catapult HLS. Both accelerators
+are pipelined and time-multiplexed by four processors. The
+processor-only baseline also parallelizes force calculation for
+different particles across four processors.
+• PDES
+(P4M1/P8M1/P16M1,
+HA):
+As
+described
+in
+TABLE II: Clock Frequency and Area of Soft Accelerators
+Benchmark
+Max. Freq
+Norm.
+Resource Utilization
+(MHz)
+Area *
+CLB †
+BRAM
+Ariane +
+1000
+1.0
+-
+-
+P-Mesh Socket
+Tangent
+282
+0.47
+0.84
+0
+Popcount
+189
+2.77
+0.83
+0.56
+Sort (32)
+228
+6.29
+0.30
+0.76
+Sort (64)
+234
+8.10
+0.27
+0.92
+Sort (128)
+228
+10.27
+0.27
+0.92
+Dijkstra
+127
+1.94
+0.96
+0.31
+Barnes-Hut
+85
+14.22
+0.99
+0.05
+BFS
+208
+1.24
+0.61
+0.75
+PDES
+126
+2.77
+0.47
+0.56
+* Total silicon area needed by the eFPGA to implement the accelerator,
+normalized to 1x Ariane + 1x P-Mesh Socket
+† Configurable Logic Block, including LUTs and flip-flops
+Sec. III-B2, a non-speculative, hardware task scheduler is
+designed in Verilog to accelerate Parallel Discrete Event
+Simulation (PDES) of a digital circuit. The processor-only
+baseline uses MCS locks [35] to arbitrate accesses to the
+shared event queue, and the lock contention can be severe as
+the number of cores increases.
+• BFS (P4M0/P8M0/P16M0, HA): We implement multiple
+hardware, lock-free queues in Verilog to alleviate the synchro-
+nization overhead in parallel Breadth-First Search (BFS). The
+processors traverse the graph in barrier-synchronized steps and
+use the queues to store the current and next search frontiers.
+Similar to the PDES benchmark, the processor-only baseline
+suffers from synchronization bottlenecks.
+Table. II lists the maximum clock frequency and eFPGA
+resource utilization of the soft accelerators. Note that the soft
+accelerators can only run at 8% - 28% of the processors’
+clock frequency. As studied in Sec. V-C, Duet achieves peak
+bandwidth in this frequency range and outperforms the con-
+ventional FPSoC communication mechanisms by at least 2x.
+Fig. 12 shows the normalized speedup and ADP of the appli-
+cation benchmarks. Duet outperforms FPSoC across all bench-
+marks and achieves a geometric mean of 4.53x speedup over
+processor-only baselines, in comparison to 2.14x achieved by
+10
+
+FPSoC using the same accelerators. The sorting accelerators
+provide up to 16.2x speedup on Duet but only 4.0x on FPSoC,
+because the FPGA-side caches in FPSoC are penalized by the
+slow clock cycles. Some may argue that the FPGA-side caches
+are unnecessary because sorted arrays are consecutive blocks
+for which DMAs are efficient. However, due to the small array
+sizes (128-512B), even LLC-coherent DMA incurs too much
+control overhead in cache flushing. The lock-free queues used
+in BFS provide up to 24.9x speedup on Duet but only 7.8x
+on FPSoC, because the processor-accelerator communication
+latency on FPSoC is much higher without the shadow registers.
+Note that the BFS benchmark shows a superlinear speedup
+when scaling from a 4-core system (Dolly-P4M0) to an 8-
+core system (Dolly-P8M0). This is because the processor-only
+baseline sees a performance decrease when scaling from 4
+cores to 8 cores due to intensifying lock contention.
+In terms of area efficiency, Duet achieves a geometric mean
+of 0.39x lower ADP than the processor-only baseline (lower
+is better), while the FPSoC is 0.23x higher. Duet outperforms
+FPSoC on most benchmarks except for Dijkstra. This is
+because the FPSoC model hardens the FPGA-side cache in the
+slow clock domain, so that soft caches become unnecessary
+and can be removed to save eFPGA resources.
+In summary, Dolly achieves superior speedup with the same
+eFPGA-emulated accelerators than FPSoCs. Therefore, we can
+conclude that Duet provides better support for fine-grained
+acceleration and hardware augmentation.
+VI. RELATED WORK
+A. Standalone FPGA-Based Accelerators
+FPGAs are being used in a fast-growing range of application
+domains due to their programmability and close-to-ASIC
+performance. On a smaller scale, standalone FPGAs are used
+to accelerate database queries [51], image processing [14],
+network filtering [52], and most popularly, deep learning
+applications [21], [46], [59]. On the other end of the spectrum,
+FPGAs are massively deployed in the Cloud, either offered di-
+rectly as an on-demand computing service such as the Amazon
+AWS F1 instances [2], or used to build cloud-scale accelerators
+such as Microsoft Catapult [10] and BrainWave [15]. In these
+systems, the FPGAs are often integrated as PCIe peripherals
+or separate machines on the datacenter network. As explained
+in Sec. I, such systems excel in accelerating at the application
+level, but are insufficient for fine-grained acceleration due to
+high CPU-FPGA communication overhead.
+B. Same-Package CPU-FPGA Hybrids and FPSoCs
+Addressing the CPU-FPGA communication bottleneck,
+academia and industry have explored tighter integration of
+the two. Intel Harp [13] combines an Intel Xeon multi-core
+CPU and an Altera FPGA in one package, and bridges the
+two with QPI, a point-to-point interconnect with cache coher-
+ence support. Field-Programmable System-on-Chips (FPSoC)
+achieves even tighter integration by integrating processors and
+FPGAs on the same chip, similar to Duet at a high level. Many
+commodity FPSoCs [26], [36], [37], [42], [55] and academic
+FPSoCs [28], [45], [53] support full or partial cache coherence.
+For example, the Xilinx Zynq-7000 employs the AXI4 ACP
+interface [3] which supports uni-directional cache coherence
+(I/O coherency). The Xilinx Zynq UltraScale+ MPSoC [56]
+and Versal ACAP [19] provide full coherence support with
+the ACE protocol [3].
+There are two key differences between Duet and FPSoCs.
+First, as shown in Fig. 1b, the processor subsystem and the
+eFPGA in FPSoCs are separated by a centralized interconnect,
+and the hardware cache hierarchy resides in the processor sub-
+system. Such organization is reasonable for an FPGA-centric,
+dual-core architecture, but cannot scale to a larger number of
+cores. Second, in the absence of FPGA-side hardware caches,
+it becomes the accelerator designers’ burden to design and
+verify soft caches that comply with the coherence protocol,
+for example ACE [3]. Besides increasing design complexity,
+the soft caches’ direct participation in cache coherence slows
+down the entire cache system because they run in the slow
+clock domain. Moreover, since the soft caches have access to
+the micro-architecture states of the NoC, the system cannot
+restrain faulty or malicious behaviors of the soft caches.
+C. Near- and In-Processor Reconfigurable Computing
+Dating back to the early days of reconfigurable computing,
+computer architects have proposed to integrate Reconfigurable
+Fabrics (RF) very close to or even into processors. Garp [9]
+places an RF between a processor and its private cache,
+making the two compute units share the entire memory sys-
+tem. Chimaera [24] and PRISC [43] embed Reconfigurable
+Functional Units (RFUs) directly into a processor’s datap-
+ath, enabling post-fabrication customization of the Instruction
+Set Architecture (ISA). The Post-Fabrication Microarchitec-
+ture [29] couples an RF with a superscalar core and allows
+the RF to observe and microarchitecturally intervene at key
+pipeline stages. These systems have their advantages but con-
+flict with our plug-and-play integration goal as they mandate
+the redesign of the processors.
+D. Coherence Protocols for Accelerators
+Hardware cache coherence has been widely adopted in
+multi-processor systems as they provide not only programma-
+bility but also performance. In the context of hardware accel-
+eration, recent studies have also shown benefits of employing
+hardware cache coherence and have proposed a variety of
+solutions. FUSION [30] is a hierarchical, timestamp-based
+coherence protocol emphasizing data transfer between accel-
+erators; Crossing Guard [39] proposes the use of a hard cache
+transducer to decouple the accelerator coherence protocol
+and the processor (host) protocol, as well as to incorporate
+fault-tolerance and memory translation; Spandex [1] offers
+flexible write strategy and granularity, adapting to highly het-
+erogeneous architectures. While all of these proposals achieve
+hardware cache coherence, they are not optimized for FPGA-
+based accelerators in that they all require direct participation of
+the accelerators. However, through the use of the Proxy Cache,
+Duet can be adapted to these coherence protocols, pretending
+11
+
+to be a "fast" accelerator to the NoC and hiding the slow clock
+domain from the coherence protocol.
+Accelerator coherence standards and interconnects are also
+emerging in industry, e.g., CCIX [11], OpenCAPI [40], and
+CXL [25]. These proposals target the board level or above and
+are really optimized for coarse-grained acceleration.
+VII. CONCLUSION
+In this work, we present Duet, a scalable, manycore-FPGA
+architecture with non-intrusive, coherently-integrated, embed-
+ded FPGAs that is optimized for fine-grained acceleration and
+hardware augmentation in the broad general-purpose domains.
+By promoting the eFPGA to a first-class citizen on chip, Duet
+enables the eFPGA to access the NoC and to participate in bi-
+directional cache coherence just as any other processor. The
+novel, lightweight, Duet Adapters reduce critical communi-
+cation overheads by placing the Proxy Caches and Shadow
+Registers in the faster processor clock domain. Our evaluation
+shows that Dolly, an RTL-level implementation of Duet, can
+reduce the latency of processor-accelerator communications by
+up to 82% and increase the bandwidth by up to 9.5x, stably
+across a wide range of eFPGA clock frequencies. Selected
+benchmarks leveraging Duet-enabled soft accelerators show up
+to 24.9x speedup over processor-only baselines and up to 4x
+over FPSoCs. Dolly and its toolchain have been open-sourced
+and available at https://github.com/PrincetonUniversity/Duet.
+ACKNOWLEDGEMENTS
+This material is based on research sponsored by the Air
+Force Research Laboratory (AFRL) and Defense Advanced
+Research Projects Agency (DARPA) under agreement No.
+FA8650-18-2-7852. This material is based upon work sup-
+ported by the National Science Foundation under Grant No.
+CNS-1823222 and the National Science Foundation Graduate
+Research Fellowship Program under Grant No. DGE-2039656.
+The U.S. Government is authorized to reproduce and distribute
+reprints for Governmental purposes notwithstanding any copy-
+right notation thereon. The views and conclusions contained
+herein are those of the authors and should not be interpreted as
+necessarily representing the official policies or endorsements,
+either expressed or implied, of Air Force Research Laboratory
+(AFRL) and Defense Advanced Research Projects Agency
+(DARPA) or the U.S. Government. Any opinions, findings, and
+conclusions or recommendations expressed in this material are
+those of the author(s) and do not necessarily reflect the views
+of the National Science Foundation.
+12
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+page_content='Duet: Creating Harmony between Processors  and Embedded FPGAs  Ang Li  Princeton University  Princeton, NJ 08544, USA  angl@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='edu  August Ning  Princeton University  Princeton, NJ 08544, USA  aning@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='edu  David Wentzlaff  Princeton University  Princeton, NJ 08544, USA  wentzlaf@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='edu  Abstract—The demise of Moore’s Law has led to the rise  of hardware acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' However, the focus on accelerating  stable algorithms in their entirety neglects the abundant fine-  grained acceleration opportunities available in broader domains  and squanders host processors’ compute power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  This paper presents Duet, a scalable, manycore-FPGA archi-  tecture that promotes embedded FPGAs (eFPGA) to be equal  peers with processors through non-intrusive, bi-directionally  cache-coherent integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In contrast to existing CPU-FPGA  hybrid systems in which the processors play a supportive role,  Duet unleashes the full potential of both the processors and the  eFPGAs with two classes of post-fabrication enhancements: fine-  grained acceleration, which partitions an application into small  tasks and offloads the frequently-invoked, compute-intensive ones  onto various small accelerators, leveraging the processors to  handle dynamic control flow and less accelerable tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' hardware  augmentation, which employs eFPGA-emulated hardware widgets  to improve processor efficiency or mitigate software overheads  in certain execution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  An RTL-level implementation of Duet is developed to evaluate  the architecture with high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Experiments using synthetic  benchmarks show that Duet can reduce the processor-accelerator  communication latency by up to 82% and increase the bandwidth  by up to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='5x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The RTL implementation is further evaluated with  seven application benchmarks, achieving 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='5-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='9x speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' INTRODUCTION  The demise of Moore’s Law [33] and the stagnation of  processor performance growth [44] have given rise to hardware  acceleration [12], [22], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Since ASIC-based accelerators  suffer from high non-recurring engineering costs and low ver-  satility, field-programmable gate arrays (FPGAs) are gaining  steam due to their post-fabrication, gate-level reconfigurability  and close-to-ASIC performance [16], [20], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' FPGAs can  be integrated either as standalone devices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 1a) or into  field-programmable system-on-chips (FPSoC) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  conventional, FPGA-based acceleration paradigm is coarse-  grained acceleration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2b), which offloads an algorithm  in its entirety onto the FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Despite offering substantial  improvements in performance and energy efficiency, coarse-  grained acceleration is only practically applicable to limited  algorithms that are both stable (to justify the accelerator design  costs) and sufficiently large (to justify the control and data  transfer overheads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  This work proposes Duet (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 1c, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II), a novel,  cache-coherent, manycore-FPGA architecture that is tailored  to enable two novel paradigms of hardware acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  FPSoC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Versal)  Processor  Subsystem  Embedded  FPGA  PCIe Ctrl  Snooping Bus  Core  Core  Cache  Coherence  Crossbar  ACP  ACE  AXI Switch  Ethernet Ctrl  GPIO  ACP  ACE  AXI  NoC  DRAM Ctrl  Scratchpad  Memory  On-Chip Devices  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' AI Engines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content=' Versal [19]  AXI [3]: non-coherent interconnect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' ACP [3]: uni-directional  coherent interconnect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' ACE [3]: fully coherent interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Multicore/Manycore Processor  L2$  Core  Other  I/O  DRAM  Ctrl  PCIe  Ctrl  PCIe  Switches  FPGA  NoC  LLC  Shard  L2$  Core  NoC  LLC  Shard  L2$  Core  NoC  LLC  Shard  L2$  Core  NoC  LLC  Shard  L2$  Core  NoC  LLC  Shard  L2$  Core  NoC  LLC  Shard  ACP  ACE  ACP  ACE  AXI  L2$  L2$  LLC  Core  Core  Snooping Bus  Processor  Subsystem  FPSoC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Versal)  Scratchpad  Memory  AXI Switch  Cache  Coherence  Crossbar  On-Chip Devices  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' AI Engines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='NoC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='DRAM Ctrl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='PCIe Ctrl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Ethernet Ctrl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='GPIO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Embedded  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='FPGA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='L2$  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Core  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Duet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='(This Work)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='LLC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Shard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='Core  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='Core  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='NoC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='NoC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='Duet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Adapter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='NoC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='LLC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Shard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Duet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Adapter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='NoC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='LLC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Shard  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Duet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Adapter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Memory-rich  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='eFPGA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Logic-rich  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='eFPGA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='(c) Duet (This Work)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Each Duet Adapter can be configured independently to support  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='uni- or bi-directional coherence in addition to a memory-mapped  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='non-coherent interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Duet enables scalable integration of: 1) any  number of processors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2) multiple independent embedded FPGAs  (eFPGAs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 3) multiple NoC access points per eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 1: CPU-FPGA Systems  Fine-grained acceleration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2c, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' III-A) partitions an  algorithm into smaller tasks and offloads only the frequently-  invoked, compute-intensive ones onto a variety of small ac-  celerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Processors still play a critical role by handling dy-  namic control flow, memory/IO-bound tasks, or any other less  accelerable computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example, fine-grained acceler-  ators can be used for special instructions like tangent, basic  algorithms like sorting, or inner loop bodies like the compute  payload per node/edge during a graph traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Hardware  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='02785v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='AR]  7 Jan 2023  Time  B+  Communication  Overhead  Accelerated  Function  Critical Section  (e)  Manycore  Baseline  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Coarse-Grained  Acceleration  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  DMA  DMA &  Interrupt  Hardware  Augmentation  (This work)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fine-Grained  Acceleration  (This Work)  (a)  (b)  (c)  (d)  CPU Execution  A  B  B  B  B  C  C  C  C  B  D  A  C  D  B  D  A  B+ B+  B+  C+ C+  C+  C+  A  D  B  B  B  B  C+  C+  C+  C+  B  A  D  B  B+  B+  B+  B+  C  C  C  C  B+  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2: Accelerating a Hypothetical Program  (a-c) Execution time of a manycore baseline and different  acceleration paradigms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' (d) An example of hardware augmentation  in which the embedded FPGA emulates a lock-free task scheduler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  (e) Control flow graph of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  augmentation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2d, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' III-B) takes an application-  agnostic approach — it employs FPGA-emulated hardware  widgets to reduce processor idle time or mitigate software  overhead in certain execution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example, hardware-  implemented, lock-free data structures can reduce synchro-  nization overhead in shared-memory, multi-thread programs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  hardware task schedulers enable task-level parallelism with  lower overhead than software schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Existing CPU-FPGA systems are inefficient for fine-grained  acceleration and hardware augmentation due to two reasons:  first, the centralized, bandwidth-optimized CPU-FPGA inter-  connect performs poorly when lots of cores and accelerators  communicate via short, frequent messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' second, even on  state-of-the-art FPSoCs which support bi-directional cache  coherence between the processors and the FPGA, cacheline-  granular memory sharing incurs non-trivial overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  overhead is justifiable for coarse-grained acceleration due to  extremely high compute-to-memory ratio, but becomes critical  for small, frequently-invoked, fine-grained accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Duet is tailored to the distinct architecture requirements  posed by fine-grained acceleration and hardware augmen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In essence, Duet promotes eFPGAs to be equal  peers with processors and integrates them as first-class  citizens on the network-on-chip (NoC) through the novel,  lightweight, Duet Adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular, Duet contains the  following novelties:  Scalability: Duet enables tight integration of one to mul-  tiple eFPGAs of various resource compositions into a scalable  manycore system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each eFPGA may be connected to multiple  Duet Adapters for higher aggregate memory bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Hybrid Cache Coherence: Duet adopts a hybrid scheme  to coherently integrate the eFPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each Duet Adapter con-  tains one to multiple private, local, hardware Proxy Caches  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-C) that translate the platform-dependent, cache coher-  ence protocols into simple memory interfaces for the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Furthermore, each Proxy Cache can be configured at eFPGA  programming time to support an optional, bi-directionally  coherent, soft cache built out of eFPGA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The use  of soft caches improves accelerator performance by exploiting  data locality while coherently sharing data with the processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Non-Intrusive Integration: For fine-grained acceleration  and hardware augmentation, a sizable amount of compute  is still run on the processors and is critical to the overall  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' However, the direct participation of eFPGA-  emulated, soft caches in cache coherence may slow down  the cache system because eFPGAs run at a lower clock  frequency and suffer from clock-domain-crossing overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Addressing this challenge, the Duet Adapters operate in the  processors’ (fast) clock domain and move FPGA-side coher-  ence maintenance off of the critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' As a result, the cache  system performs equally fast as if each Duet Adapter was an  additional processor-owned private cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Plug-and-Play Integration: Duet requires little to no  hardware changes to existing manycore systems because the  Duet Adapters transduce between the eFPGAs and the NoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Such decoupling is critical because modifying a mature pro-  cessor design often leads to performance degradation or even  hardware bugs, especially as design complexity and verifica-  tion costs skyrocket on advanced technology nodes [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  To evaluate Duet with high fidelity, we build Dolly, a  prototype instance of Duet at the RTL level (Verilog &  SystemVerilog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In addition, we assemble a full toolchain  for software development and accelerator design, leveraging  an array of open-source projects, including OpenPiton [6],  BYOC [5], PRGA [31], Yosys [54], and VTR [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To  encourage further research in tightly-integrated, hard-  ware cache-coherent CPU-eFPGA systems, we have open-  sourced Dolly and its toolchain, available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  com/PrincetonUniversity/Duet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Experiments show that the Duet Adapter introduces neg-  ligible hardware overhead for the CPU-eFPGA integration  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Proxy Cache reduces the latency of processor-  accelerator communication up to 82% and increases the band-  width up to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='5x compared to having the eFPGA participating  directly in the coherence protocol (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The latency  reduction and bandwidth increase are stable across different  eFPGA clock frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On selected benchmarks (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-D),  Dolly improves the overall performance up to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='9x in com-  parison to corresponding processor-only baselines and up to  4x in comparison to other CPU-FPGA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The major contributions of this work are:  Presenting Duet, a manycore-FPGA architecture which  employs novel Duet Adapters to integrate embedded FPGAs  in a scalable, non-intrusive, cache-coherent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Identifying and demonstrating with examples two novel  paradigms of hardware acceleration enabled by Duet, namely  fine-grained acceleration and hardware augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Presenting Dolly, an RTL-level prototype of Duet and a  full toolchain for software development and accelerator design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Evaluating Dolly’s silicon area consumption and perfor-  mance with synthetic and application benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Releasing Dolly and its toolchain for open-source access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  2  Shared LLC*  DRAM Ctrl  PCIe Ctrl  Other I/O  Network-on-Chip*  L2$  Core  L2$  Core  L2$  Core  Control Hub (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-E)  FPGA Manager  Soft Register  Interface  Programmable  Clock Generator  Programming  Engine  Exception  Handler  Feature  Switches  Shadow  Registers  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-F)  Async  FIFO  Memory Hub  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-B)  Exception  Handler  Feature  Switches  TLB  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-D)  Proxy  Cache  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-C)  Async  FIFO  Duet  Adapter  More  Memory  Hubs  More  Duet  Adapters  More  eFPGAs  eFPGA  Configuration  Memory  FPGA (Slow)  Clock  Domain  Soft Accelerator  Soft Controller  Soft Cache (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-C)  More Soft Caches  Non-Coherent Memory  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 3: Duet Architecture and an Emulated Soft Accelerator  For simplicity, the figure shows a bus-based NoC and a centralized LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Duet can be adapted to other  NoC topology and distributed LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example, Dolly (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' IV) uses a 2D-mesh and a distributed LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' ARCHITECTURE  Before diving into the architecture of Duet, we want to  clarify several terms that are used throughout this paper:  software refers to the program running on the processors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  hardware, hard cache, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' refer to the components that are  fixed at fabrication time, in contrast to the soft components that  are emulated with the reconfigurable resources in the eFPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  In addition, we use the term soft accelerator to refer to both  fine-grained accelerators and hardware augmentation widgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Overview  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 3 shows an overview of the Duet architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Duet  integrates eFPGAs as first-class citizens on the NoC through  novel, hardware Duet Adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each Duet Adapter contains  one to several Memory Hubs and one Control Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  Memory Hubs enable bi-directionally coherent memory ac-  cesses of the soft accelerators and transduce between the mem-  ory interfaces of the eFPGAs and the NoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Control Hubs  present the eFPGAs as on-chip devices that are accessible via  memory-mapped I/Os (MMIOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The fact that eFPGA-emulated soft accelerators typically  run at a much slower clock than the rest of the system poses a  challenge to minimizing processor-accelerator communication  overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular, any traffic entering or leaving the slow  clock domain must pay for the clock-domain-crossing (CDC)  overhead, and every slow clock cycle significantly penalizes  the total communication time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 5a & Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' A few dozen  cycles seem negligible, but since the accelerated tasks are short  and invoked frequently, the accumulated overhead can chip  away a large fraction of the effective speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Addressing this  challenge, the Duet Adapter adopts various strategies to move  the logic in the slow clock domain off of the critical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Memory Hub  Each Duet Adapter may contain multiple Memory Hubs,  each attached to the NoC using an independent connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The soft accelerator may use all or any subset of them  to access the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each Memory Hub consists of an  Soft-Only  Clock  Boundary  eFPGA  Soft Cache  eFPGA  eFPGA  Soft Cache  Shared LLC  Shared LLC  Shared LLC  Hard-Only  Hybrid  Accelerator  NoC  Accelerator  Accelerator  Hard Cache  Hard Cache  NoC  NoC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 4: FPGA-Side Cache Organization Options  exception handler, a set of feature switches, and a Proxy  Cache (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-C), all implemented in hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Besides the  hardware Proxy Cache, each memory hub can support one  optional, bi-directionally coherent, Soft Cache built out of  eFPGA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The exception handler as well as all the  feature switches can be configured by the processors via on-  chip MMIOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The exception handler employs timeout and parity checks  to monitor eFPGA outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' When an exception is detected,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', due to an RTL or software bug, it asserts an error  code and deactivates all Memory Hubs in the same Duet  Adapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Once deactivated, the Memory Hubs stop accepting  any memory requests from the eFPGA, but the Proxy Caches  remain functional, so that any in-flight coherence messages  are processed properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' This mechanism prevents accelerator  bugs from halting the system at the micro-architecture level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The feature switches allow the processors to configure the  Memory Hubs according to the state of the eFPGA and the  specifications of the soft accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example, the Memory  Hubs should be deactivated during eFPGA reconfiguration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' if  soft caches are used, the Memory Hubs should be configured  to forward invalidation requests into the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Proxy Cache  A key challenge in building coherently-integrated CPU-  FPGA systems is how to organize the FPGA-side caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 4 shows the options:  3  0  1  2  3  4  5  6  7  8  9  10  11  NoC Req  NoC Resp  FIFO (Proxy→FPGA)  FIFO (FPGA→Proxy)  Inv  Inv  Soft Cache  Inv  Ack  Clock Boundary  Acc Datapath  Load  Hit  Data  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  Ack  Inv  Ack  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  eFPGA Clock  Processor Clock  (a) Cache Operations with Only Soft Caches  Inv=Invalidation from a remote cache;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Ack=Acknowledgement  NoC Req  NoC Resp  eFPGA Clock  Processor Clock  FIFO (Proxy→FPGA)  FIFO (FPGA→Proxy)  0  1  2  3  4  5  6  Ack  Clock Boundary  Hard Cache  Inv  Acc Datapath  Load  7  8  9  Ld  Hit  Data  Data  Data  10  11  Load  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  Inv  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  (b) Cache Operations with Only Hard Caches  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  NoC Req  NoC Resp  FIFO (Proxy→FPGA)  FIFO (FPGA→Proxy)  0  1  2  3  4  5  6  Inv  Soft Cache  Inv  Ack  Clock Boundary  Proxy Cache  Inv  Acc Datapath  Load  Hit  Data  7  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  8  9  Inv  10  11  Inv  eFPGA Clock  Processor Clock  (c) Cache Operations with Duet (This Work)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 5: Cache Operations with Different Cache Organizations  For simplicity, this figure shows single-stage async FIFOs, and the  FPGA clock runs at half of the system clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On real  hardware, async FIFOs typically take two to four stages, and the  FPGA clock could be as slow as 1/10 of the system clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Soft-Only The eFPGA is directly connected to the NoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  soft cache, if used, must implement the system-  wide cache coherence protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Hard-Only A hardware, private cache is added between the  eFPGA and the NoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Cache coherence is hidden  from the eFPGA, so soft caches are not supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Hybrid A hardware, private cache participates in the  system-wide cache coherence on behalf of the  eFPGA and supports the use of soft caches with  a local cache coherence protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Option 1, soft-only, has three main drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' First, it  becomes the accelerator developers’ burden to design a cache  in compliance with a platform-dependent, complex cache  coherence protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The sophisticated control logic consumes  more logic resources, has higher access latency, and is rarely  reusable across devices — a soft cache designed for ACE [3]  would hardly work with CHI [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Second, since the soft caches  have access to the micro-architectural state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', can block a  NoC message indefinitely), the system cannot fully contain  faulty or malicious behaviors of the soft caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Third, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 5a, the soft caches may slow down the hardware  cache system due to CDC overheads and the slow cycles  spent in the FPGA clock domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Nonetheless, soft-only is the  de facto design on most commodity FPSoCs because these  FPGA-centric architectures are designed for coarse-grained  acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Specifically, coherent memory sharing is rare and  mainly offered to simplify programming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' soft cache IPs are  licensable from the vendors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' and accelerator bugs are expected  to fail the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Option 2, hard-only, addresses all the drawbacks of the  soft-only approach but has other limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' First, the CDC  overhead is now imposed on the accelerator datapaths (5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Second, the cache implementation is fixed at fabrication time  and may be suboptimal for certain accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Given the downsides of the soft-only and hard-only options,  Duet takes a hybrid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' A private, local, hardware Proxy  Cache implements the platform-dependent, system-wide cache  coherence protocol and provides a simple memory interface  to the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each Proxy Cache can be configured through  feature switches to support an eFPGA-emulated, soft cache  which can be tightly integrated into the accelerator datapaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Moreover, the Proxy Cache contains two key novelties when  compared to a naïve implementation of the hybrid approach:  First, the Proxy Cache neither requires nor accepts any  acknowledgements from the soft cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' As a result, the  Proxy Cache always responds to coherence messages promptly  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 5c), insulating the cache system from the slow eFPGA  clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To maintain coherence, the Proxy Cache requires the  soft cache to be write-through but allows write buffering (the  Proxy Cache itself can be write-back or write-through, de-  pending on the coherence protocol of the LLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Furthermore,  because the asynchronous FIFOs deliver messages in order, the  soft cache always receives invalidations, line fills, and write  acks in the same order as they are sent by the Proxy Cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Rigid proof of the correctness of such protocols is out of the  scope of this paper, but the PCX protocol in OpenSPARC  T1 [41] and its extension, the TRI protocol in BYOC [5], are  two examples that meet the above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Second, the protocol is simple yet flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In the com-  mon case, the soft cache only needs to support two request  types (Load and Store) and three response types (LoadAck,  StoreAck and Invalidation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' It is up to the accelerator designer  whether to use a write buffer, how many entries the write  buffer has, and whether read-after-write forwarding is com-  patible with the consistency assumptions of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The Proxy Cache can be configured to work with either a  write-allocate or a write-no-allocate soft cache, as well as be  configured to enable atomic operations which require the soft  cache to support incrementally more message types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Memory Protection and Virtualization  Another key challenge for CPU-FPGA systems is memory  protection and virtualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Proxy Cache is physically-  indexed, physically-tagged because it is implemented in hard-  ware and is closer to the LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' However, it depends on the  type of soft accelerator whether the eFPGA should use virtual  or physical addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Hardware augmentation widgets may  be trusted firmware and can be granted access to physical  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On the contrary, application-specific, fine-grained  accelerators are like user programs and can be faulty or  malicious, so they are better restricted to virtual addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  To enable virtual memory accesses of the soft accelerator,  Duet adds a translation look-aside buffer (TLB) to each  4  Memory Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The TLB can be disabled if the soft accelerator  is granted access to the physical memory space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' otherwise,  accelerator-initiated memory accesses must be translated by  the TLB while being speculatively processed by the Proxy  Cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On a page fault, the TLB sends an interrupt to a  processor, then the kernel-level interrupt handler either updates  the TLB using MMIOs or kills the accelerator if the page  access is deemed invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  One special case is when soft caches are used but the  accelerator only has access to virtual addresses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', the  soft caches are virtually-indexed, virtually-tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To enable  reverse-mapping from physical address to virtual address when  the Proxy Cache forwards an invalidation into the soft cache,  the Proxy Cache stores the virtual page number beside the  physical tag of each cacheline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' This also rules out the coexis-  tence of synonym aliases (different virtual addresses mapping  to the same physical address) in the soft cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular,  the Proxy Cache can invalidate the existing virtual address  before responding to a load request for the same physical  cacheline through a different virtual address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Control Hub  The Control Hub consists of two submodules: the FPGA  Manager and the Soft Register Interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The FPGA Man-  ager provides necessary hardware support for programming  and monitoring the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The programming engine loads  the bitstream into the configuration memory, and performs  integrity checks to detect data corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The programmable  clock generator either divides the system clock, or integrates  a separate PLL for finer control over the generation of the  FPGA clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The exception handler and the feature switches  are similar to those in the Memory Hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular, the  feature switches can be used to set the timeout limit, reset the  soft accelerator, or clear previously-logged error codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The Soft Register Interface enables the soft accelerator to  implement a soft "device controller" similar to those com-  monly seen on off-chip peripheral devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The soft registers  emulated by the eFPGA are accessible via on-chip MMIOs  and often have additional effects rather than simply holding a  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example, reading a soft register may dequeue from  a FIFO inside the accelerator so that repetitive reads return  different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' When the Control Hub is deactivated, the Soft  Register Interface returns bogus data to all processor accesses  so that the system is not halted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Shadow Registers  To prevent unwanted side effects, MMIOs typically adhere  to a strict memory ordering model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', I/O ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Unfortu-  nately, as stressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-A, these strictly ordered accesses  may stall the processors’ pipelines because the eFPGA runs at  a slower clock (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Addressing this inefficiency, the Soft  Register Interface is augmented with several types of Shadow  Registers residing in the fast clock domain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' When  a processor writes to a shadowed soft register, the Shadow  Register acknowledges the request before forwarding the write  into the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Conversely, the soft accelerator actively  Clock Boundary  NoC Req  NoC Resp  Soft Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  FIFO (NoC→FPGA)  FIFO (FPGA→NoC)  0  1  2  3  4  5  6  7  8  9  10  11  WR  Ack  Ack  WR  1  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  0  eFPGA Clock  Processor Clock  (a) Accesses to a Normal Soft Register  Clock Boundary  NoC Req  NoC Resp  Soft Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  FIFO (NoC→FPGA)  FIFO (FPGA→NoC)  Shadow Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  0  1  2  3  4  5  6  7  8  9  10  11  WR  RD  RD  Ack  Ack  Sync  Sync  0  1  1  2  0  1  2  Crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Path  1  Ack  eFPGA Clock  Processor Clock  Accelerator  Write  (b) Accesses to a Shadowed Soft Register (This Work)  Clock Boundary  NoC Req  NoC Resp  FIFO (NoC→FPGA)  FIFO (FPGA→NoC)  Shadow Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' B  0  1  2  3  4  5  6  7  8  9  WR:A  WR:A  Soft Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' A  WR:B (stalled)  0  1  Ack:A  0  eFPGA Clock  Processor Clock  Ack:A  Sync:B  Ack:B  1  Soft Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' B  0  1  (c) Strict Ordering of Two Soft Register Writes (This Work)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 6: Accessing Soft Registers and Shadow Registers  synchronizes shadowed soft registers over the asynchronous  FIFO, so that the Shadow Registers can respond to processor  reads immediately without notifying the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Duet supports four types of Shadow Registers: plain, FPGA-  bound FIFO, CPU-bound FIFO, and token FIFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Plain  Shadow Registers only keep the last value of multiple writes,  which are ideal for passing constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' FPGA-bound  and CPU-bound FIFOs, as their names suggest, record all  writes from one side and allow the other side to read in  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' CPU-bound FIFO is blocking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', a processor read  is stalled until the soft accelerator pushes into the FIFO or  the request times out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In contrast, CPU-bound token FIFO  is a dataless, non-blocking FIFO that consumes a token or  returns "empty" in response to a processor read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Token FIFO  is designed particularly to emulate the non-blocking try_join  semantic in parallel programming models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Note that normal soft registers are still available in case the  software requires non-bufferable accesses that must be sent  all the way to the endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example, a soft register can  be dedicated as a barrier for synchronizing the processor and  the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The processor signals its arrival at the barrier  by reading the soft register, while the eFPGA signals its  arrival at the barrier by acknowledging the read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To maintain  I/O ordering, Shadow Register accesses are processed and  responded to in order with respect to other shadowed or normal  register accesses, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' APPLICATIONS  Duet enables two novel paradigms of hardware acceleration,  namely fine-grained acceleration and hardware augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  In this section, we illustrate both paradigms with examples  and discuss why Duet is the ideal architecture for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  5  Listing 1: The key compute loop of the Barnes-Hut simulation  algorithm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The highlighted functions are static, compute-  intensive, and ideal for fine-grained acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  // Calculate the net force on particle "p"  void CalculateNetForce (BHTreeNode node, Particle p):  if (Distance (node, p) > node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='radius * THRESHOLD)  // "p" is far enough from the particles in the subtree of  // "node", so we approximate the net force with a low-order  // multipole expansion and stop traversing the subtree  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='force += ApproxForce (node, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  else if (IsLeafNode (node))  for (Particle q : node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='children)  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='force += CalcForce (q, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  else  // traverse the subtree via recursive calls  for (BHTreeNode child : node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='children)  CalculateNetForce (child, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  // Calculate the net forces on all particles in parallel  void CalculateNetForceOnAllParticles (  BHTreeNode root, vector<Particle> allParticles):  parallel_for (Particle p : allParticles)  CalculateNetForce (root, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Fine-Grained Acceleration  1) Overview: At its core, fine-grained acceleration and  coarse-grained acceleration are similar because they both  offload computation onto application-specific accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In  fact, there is a spectrum rather than a clear boundary between  the two paradigms — the shorter the accelerated function is, or  the more frequent the accelerator interacts with the processors,  the closer it is to the fine-grained end of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fine-grained acceleration has the following advantages:  first, it fully utilizes the compute power of the processors by  running the less accelerable fractions of an algorithm on them,  for example dynamic control flow, memory/IO-bound tasks,  managing complex data structures, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' second, it reduces  software and accelerator design costs by moving less logic  onto the FPGA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' third, in comparison to a monolithic acceler-  ator, a collection of small accelerators are more composable,  reusable, and more resilient to software updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  2) Example: Barnes-Hut (BH) algorithm [7] is an approxi-  mation algorithm for simulating a dynamic system of particles  interacting via certain physical forces such as gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In each  time step, the simulation volume is divided into hierarchical,  cubic cells stored in an octree (for three-dimensional space), so  that the total force exerted by particles in distant cells can be  approximated through a center-of-mass abstraction or a low-  order multipole expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Listing 1 shows the key functions  that calculate the net force on all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Prior work [17] has proposed an FPGA-based, coarse-  grained BH accelerator which contains replicated pipelines to  parallelize force calculation on multiple particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' However,  despite the use of a special traversal cache that facilitates  data reuse between the pipelines, control flow divergence often  leads to low pipeline utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Furthermore, the accelerator  relies on the host processor to serialize the BH-tree and preload  the node pointers into the traversal cache, making it inappli-  cable if the dataset exceeds the FPGA’s BRAM capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In  summary, building a coarse-grained BH accelerator can be an  intimidating task without a satisfactory result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 7 shows the timeline of BH running on a dual-  CPU  Thread #1  CPU  Thread #2  Accelerator #2  CalcForce  Accelerator #1  ApproxForce  Visit Node  Invoke ApproxForce  Visit Node  Invoke CalcForce  Visit Node  Invoke CalcForce  Invoke CalcForce  Invoke CalcForce  1  2  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 7: Multi-Threaded BH with Fine-grained Acceleration  1⃝ Loop-carry dependencies and dynamic control flow are handled  by the processors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2⃝ Processors and accelerators run in parallel  through software pipelining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 3⃝ Accelerators are time-multiplexed  by multiple CPU threads to increase utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  core system with two fine-grained accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Besides being  easy to program, this software-hardware co-design is scal-  able and flexible: multiple accelerators of each type can be  instantiated to increase parallelism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' if THRESHOLD is high,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', ApproxForce is called for less times, the system can  allocate more eFPGA resources to implement more and/or  faster CalcForce accelerators, and vice versa if THRESHOLD  is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The fine-grained BH accelerators would be inefficient with-  out Duet’s architectural support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' First, the accelerators access  just a few cachelines at random memory locations per ex-  ecution, resulting in poor utilization of the block-oriented,  bandwidth-optimized memory system in most existing CPU-  FPGA architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In contrast, Duet’s hybrid cache organiza-  tion minimizes accelerators’ memory access latency, especially  for random accesses at cacheline granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Second, the  accelerators are invoked frequently and time-multiplexed by  multiple CPU threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' A centralized CPU-FPGA interconnect  may be congested by the MMIOs issued by the processors to  invoke the accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Duet’s scalable integration makes it  possible to split the workload across multiple eFPGAs, and  the shadow registers can further minimize MMIO latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Hardware Augmentation  1) Overview: Hardware augmentation allows application  developers to add various hardware widgets to the system  after chip fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' These widgets help improve proces-  sor efficiency and/or mitigate software overheads in certain  execution models, providing application-agnostic acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Different hardware augmentation widgets often exhibit unique  behaviors and provide distinct software APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example,  decoupled access-execute engines [49] hide the latency of  memory redirects by dereferencing software-specified pointers  and collect the data into a hardware queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' hardware garbage  collectors [32] run autonomously in the background and  alleviate software overhead of the runtime environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Since  hardware augmentation widgets collaborate closely with the  processors, Duet’s scalable, tight, cache-coherent integration  is critical to maximize performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  2) Example: Discrete event simulation (DES) is a well-  known challenge for parallel execution [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' All the simu-  6  o  +Ariane  L2  Cache  L2  Cache  L2  Cache  L2  Cache  Coherent  Mem Intf  Coherent  Mem Intf  1  2  3  L3 Cache  Shard  L3 Cache  Shard  L3 Cache  Shard  L3 Cache  Shard  NoC  Router  NoC  Router  NoC  Router  NoC  Router  Ariane  P-Mesh  Socket  P-Mesh  Socket  P-Tile  P-Tile  C-Tile  M-Tile  P-Mesh  Socket  P-Mesh  Socket  Soft Reg Intf  FPGA Manager  Duet Adapter  FPGA  Config  Memory  Soft Accelerator  Soft Cache Soft Cache  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 8: Architecture of Dolly-P2M2  Dolly-P2M2 has 2 processors, 1 eFPGA, and 2 Memory Hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  P-tile, C-tile and M-tile are physical tiles in a 2D mesh network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  P-Mesh Socket is a physical wrapper for common components in  all physical tiles, including an L2 cache, a NoC router, and a shard  of the shared L3 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 1⃝ is the Control hub (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 2⃝ and  3⃝ are two Memory Hubs (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Note that 1⃝ and 2⃝ reside  in the same physical C-tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  lating threads must either progress in lock step (conservative  method), which suffers from high synchronization overheads,  or exploit speculative execution (optimistic method), which re-  quires a carefully designed software runtime or an architecture  that supports thread-level speculation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', SWARM [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Duet opens up another possibility through hardware aug-  mentation: an eFPGA-emulated task scheduler can be loaded  as part of the simulator software while offering hardware-level  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Consider a possible implementation of such a  task scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Processors schedule new events by pushing  memory pointers to the events into a FPGA-bound FIFO,  after which the task scheduler fetches the event data from  shared memory and adds the pointer into the proper event  queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Once certain events are ready to be processed, the  task scheduler pushes the pointers into an CPU-bound FIFO  so that the processors can streamline event processing with  minimal communication latency with the scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The task  scheduler can support task speculation by fetching the cache-  lines that may be modified by a speculative event and storing  versioned copies of them in its non-coherent memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On a  mis-speculation, the task scheduler rolls back the cachelines to  the most up-to-date, non-speculative versions, then reschedules  the mis-speculated events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' DOLLY  To evaluate Duet with high fidelity, we built Dolly1, a  prototype system at the RTL level (Verilog/SystemVerilog)  with a full toolchain for software development and accelerator  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Dolly is configurable in the number of processors, the  logic capacity of the eFPGA, the number of memory hubs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  For simplicity, we name each Dolly instance PpMm, where  p specifies the number of processors, and m specifies the  1Named after Dolly Suite, Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 56, a collection of pieces for piano duet  composed by Gabriel Fauré.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  number of memory hubs available to the eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For example,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 8 shows the architecture of Dolly-P2M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To encourage  further research in tightly-integrated, hardware cache-  coherent CPU-eFPGA systems, we have open-sourced  Dolly and its toolchain, available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='com/  PrincetonUniversity/Duet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Dolly is based on the OpenPiton P-Mesh cache coherence  system [6] in a 2D mesh configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The cache system  is organized in three levels: the L1 caches are tightly inter-  woven into the processors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' one private, write-back, 8KB L2  cache per physical tile interfaces with the corresponding L1  cache through the transaction-response interface (TRI) [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  the shared L3 cache is distributed among all physical tiles,  64KB per shard, and runs a directory-based MESI protocol  together with the private L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The NoC offers point-  to-point ordering of message delivery and supports additional  message types besides the coherence messages, enabling on-  chip MMIOs required by Dolly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Dolly contains three types of physical tiles and an eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Each P-Tile hosts an Ariane [57] processor, a 6-stage, single-  issue, in-order CPU which implements the 64-bit RISC-V  instruction set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each Ariane processor has a private hardware  FPU, an 8KB L1 instruction (L1I) cache, and an 8KB L1 data  (L1D) cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' C-Tiles and M-Tiles compose the Duet Adapter:  each C-Tile contains one Control Hub (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-E) and one  Memory Hub (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' each M-Tile includes one Memory  Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To demonstrate Duet’s adaptability, Dolly implements  the Proxy Cache by adding a coherent memory interface to  the unmodified P-Mesh L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The L2 cache, L3 cache  shard and NoC router are the same across all physical tile  types, so we wrap them into one module, the P-Mesh socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The eFPGA is built with PRGA [31] and employs a  standard island-style architecture composed of logic blocks  and routing blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The eFPGA includes all of the common  logic resources available on modern FPGAs, including look-  up tables (LUT), bypassable flip-flops, hard adder chains,  Block RAMs (BRAM), and hard multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Dolly employs  a non-synthesizable clock generator to generate the eFPGA  clock whose frequency can be specified in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' All the  asynchronous FIFOs are implemented with dual-clock RAMs  and Gray-coded, 2-stage synchronizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Overview  We conduct our evaluation of Duet through RTL simulations  of Dolly instances and focus on three perspectives: first of all,  we report the area consumption and the typical frequency of  the hard components of Dolly (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' second, we run syn-  thetic benchmarks to measure the CPU-FPGA communication  latency and bandwidth achieved by the Duet Adapter under  different conditions (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' third, we run seven application  benchmarks and compare their performance against processor-  only and FPSoC baselines (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  To show the lower bound of Duet-enabled improvements,  we give the processors various advantages throughout our  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Specifically, we boost the clock frequency of the  7  TABLE I: Area and Typical Frequency of Dolly Components  Component  Device  Area  Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Scaled Area* Scaled Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' *  Technology  (mm2) (MHz)  (mm2)  (MHz)  Ariane [58]  GlobalFoundries  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='39  910  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='56  455†  22nm FDX  P-Mesh  IBM 32nm SOI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='55  1000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='1*  711*†  Socket [6]  FPGA Mgr +  FreePDK45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='21  925  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='21*  925*  Soft Reg Intf  Coherent  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='04  1250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='04*  1250*  Memory Intf  Scaled to 45nm with a linear MOSFET scaling model  † To emulate a higher-performance multi-core, our evaluation sim-  ulates the Ariane cores and the hardware cache system at 1GHz  Ariane processors and the P-Mesh cache system to 1GHz,  despite that previous works reported their maximum frequen-  cies (after scaling to 45nm) to be 455MHz and 711MHz,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For the application benchmarks, processor-only  baselines always start with a warm cache, while the soft  accelerators always start with a cold cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Area and Typical Frequency of the Hard Components  Table I summarizes the area and typical frequency of each  hardware component of Dolly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The numbers for Ariane and  P-Mesh components are retrieved from previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  submodules of the Control Hub and the Memory Hub are  synthesized using Synopsys Design Compiler and Silvaco’s  Open-Cell Library [48] based on the NCSU FreePDK45  process design kit [50] using an area utilization rate of 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  In summary, the Control Hub and the Memory Hub require  minimal hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The area consumption of the  eFPGA and the clock frequency of the emulated accelerator  are reported on a per-benchmark basis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' CPU-eFPGA Communication Latency and Bandwidth  In this section, we evaluate the peak performance of Duet  with a synthetic benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The eFPGA emulates a simple  scratchpad memory and a processor uses different mechanisms  to access it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' To show that the eFPGA’s clock frequency has  a significant impact on performance, we sweep the eFPGA  clock from 20MHz to 500MHz while fixing the system clock  at 1GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular, we study the following CPU-eFPGA  communication mechanisms and show how Duet improves  them over existing CPU-FPGA systems:  Soft registers are memory-mapped, eFPGA-emulated regis-  ters which the processor can read and write via non-coherent  MMIOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' These soft registers can be implemented inside the  eFPGA (Normal Registers), in which case processor ac-  cesses must pay the Clock-Domain-Crossing (CDC) overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Shadow Registers address this inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In this study, we  use FPGA-bound FIFOs to handle processor writes and CPU-  bound FIFOs for processor reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Alternatively, the processor can store the data in shared  memory and pass the memory address to the eFPGA via a  soft register write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The soft register write not only signals  the eFPGA to load the data (eFPGA Pull),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' but also works  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='as a synchronization to ensure that all processor stores are  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='committed into the cache system before the eFPGA starts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='Roundtrip Latency (ns)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='eFPGA Frequency (MHz)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+page_content='42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='130  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='82  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='52  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='NoC Latency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Cache Logic (Fast Clock Domain)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Cache Logic (Slow Clock Domain)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Clock-Domain-Crossing Overhead  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='Shadow Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  (This Work)  Normal Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  CPU Pull w/  Proxy Cache  (This Work)  CPU Pull w/  Slow Cache  eFPGA Pull w/  Proxy Cache  (This Work)  eFPGA Pull w/  Slow Cache  Communication Mechanism  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 9: CPU-eFPGA Communication Latency  (Single processor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Single transaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Lower is better)  loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The eFPGA can send data to the processor in a  similar way (CPU Pull).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II-C, commodity  FPSoCs typically emulate FPGA-side caches using eFPGA  resources (Slow Cache), while Duet employs the novel Proxy  Cache to improve cache performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Latency Study  We first measure the minimum round-trip latency of the  aforementioned communication mechanisms on Dolly-P1M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 9 shows the breakdown of the CPU-eFPGA communi-  cation latency into four parts: NoC latency, cache processing  time in the fast clock domain, cache processing time in the  slow clock domain, and the CDC overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' All numbers are  collected in an ideal scenario, that is, single processor (no  contention or NoC congestion), single transaction (no buffer  clogging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Note that the eFPGA pulls and CPU pulls are  guaranteed to miss in the requesting cache and to hit in the  other party’s private cache in a modified state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Both the NoC  latency and the cache processing time in the fast clock domain  include the time for the distributed directory to send and  process the secondary write-back requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  As we can see, the CDC overhead and the slow clock  penalty on cache processing constitute over half of the round-  trip latency for a slow cache, even when the eFPGA runs  at 50% of the CPU clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For eFPGA pulls, the  Proxy Cache can reduce the latency by 13% to 43%, and  the reduction increases as the eFPGA runs slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For CPU  pulls, the Proxy Cache achieves a constant latency regardless  of the eFPGA clock frequency, reducing the latency by 42% to  82%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Shadow Registers also have a fixed latency, reducing  CPU-eFPGA communication latency by 50% to 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Note  that the Proxy Cache and the Shadow Registers only move the  eFPGA off of the critical path, and the eFPGA still needs time  to issue memory accesses and to handle soft register accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Single-Processor Bandwidth Study  Next, we study the maximum bandwidth of single-processor  communications on Dolly-P1M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In this study, we configure  the synthetic benchmark to pass 512 quad-word (8 Bytes)  integers from one processor to the eFPGA and then fetch  them back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' With the soft registers, this is done by having  the processor execute a loop that writes/reads one integer  8  20 (2%)  50 (5%)  100 (10%)  200 (20%)  500 (50%)  eFPGA Frequency (MHz) and Ratio to CPU Frequency (1GHz)  0  100  200  300  400  500  600  700  Bandwidth (MBytes/s)  Normal Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  CPU Pull w/ Slow Cache  eFPGA Pull w/ Slow Cache  Shadow Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' (This work)  CPU Pull w/ Proxy Cache (This work)  eFPGA Pull w/ Proxy Cache (This work)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 10: Processor-eFPGA Communication Bandwidth vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  eFPGA Clock Frequency (Single processor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Higher is better)  per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' When using shared memory, the processor first  allocates two 4KB buffers in the memory and passes the base  addresses to the eFPGA using two plain shadow registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Then, the processor stores all the integers into one of the  two buffers and sends a read request to another normal  soft register, awakening the eFPGA and blocking itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  eFPGA proactively loads the entire array into its scratchpad  memory, stores it back to the other buffer, and then unblocks  the processor by acknowledging the processor’s read request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Finally, the processor loads the array from shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 10 shows the results of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Proxy Cache  delivers the highest bandwidth across all eFPGA frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  In fact, the Proxy Cache reaches the peak bandwidth for  eFPGA pulls (558MB/s) when the eFPGA runs faster than  100MHz (10% of the CPU clock frequency);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' for CPU pulls,  the peak bandwidth (201MB/s) is reached at 50MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In  comparison, the slow cache can only achieve 287MB/s for  eFPGA pulls and 144MB/s for CPU pulls even when the  eFPGA runs at 500MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The largest bandwidth gap is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='5x  when the eFPGA runs at 100MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The upper bounds of cache-  based communications are determined by the NoC bandwidth  and the number of concurrent, in-flight, memory requests  supported by the Proxy Cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Note that the upper bounds  of CPU pulls are much lower than those of eFPGA pulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  This is because the cache line size is 16 Bytes and the  eFPGA can load up to one line per cycle, but the L2 cache in  Dolly only supports stores up to 8 Bytes, so the eFPGA must  send two requests to store one cacheline, under-utilizing the  asynchronous FIFOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The Shadow Registers also achieve a stable bandwidth  (213MB/s) once the eFPGA clock frequency exceeds 10% of  the CPU clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In comparison, normal registers can  only reach 121MB/s at 500MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The upper bound of soft  register accesses is limited by the strict consistency model of  MMIOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' If the I/O consistency model is relaxed, the upper  bound is then limited by the number of concurrent, in-flight  MMIOs supported by the load-store queue of the processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Multi-Processor Scalability Study  One key difference between Duet and existing CPU-FPGA  systems is the scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' At the architecture level, Duet has no  constraint on the number of processors, eFPGA fabrics, or the  1  2  4  8  16  Number of Processors Accessing the Registers  101  102  103  Bandwidth (MBytes/s, log-scale)  Normal Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Write  Normal Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Read  Shadow Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Write (This Work)  Shadow Reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Read (This Work)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 11: Processor-eFPGA Communication Bandwidth (Per  Processor) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Number of Contending Processors  number of Memory Hubs per eFPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In this section, we study  the effect of multi-processor contention on the soft registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In  particular, we run the synthetic benchmark on multiple Dolly  instances with different number of processors which constantly  writes/reads the same soft register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The eFPGA clock is fixed  at 500MHz (50% of the CPU clock frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 11 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Shadow Registers can stably  support up to 8 processors before the per-processor bandwidth  starts to drop, while the normal registers can only support  up to 2 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Considering that the processors need  to perform other tasks between soft register accesses, the  Shadow Registers can support even more processors in real  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Summary  In summary, the significant reduction in latency and the  stable increase in bandwidth clearly justify the promotion of  soft registers and private caches into the fast clock domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Application Benchmarks  In this section, we evaluate Duet with seven application  benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' For each benchmark, we first run a C imple-  mentation as the processor-only baseline and record the total  runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Note that all benchmarks are run in bare metal due  to limited simulation speed at the RTL level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' We then design  the soft accelerators for each benchmark, either directly at the  RTL level, or by synthesizing an alternative C implementation  with Catapult HLS [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Each accelerator is synthesized,  placed and routed with the PRGA [31] workflow to get the  accurate eFPGA clock frequency and an area estimation of the  utilized FPGA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular, we map each bench-  mark onto the flagship FPGA model provided by VTR [38]  (k6_frac_N10_frac_chain_mem32K_40nm) which resembles an Altera  Stratix IV FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' With the accurate eFPGA clock frequency,  we rerun the benchmarks on various Dolly instances and an  FPSoC-like architecture, recording the total runtime which  includes all the overhead in CPU-FPGA communication and  synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In particular, the FPSoC model moves the  P-Mesh L2 cache into the eFPGA’s (slow) clock domain and  downgrades all shadowed soft registers to normal registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' We  then compute the speedup over processor-only baselines and  compare the results achieved by Dolly and the FPSoC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  9  0  2  4  6  8  Normalized Speedup  (Higher is better)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='9  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='2  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='53  CPU  FPSoC  Duet (This Work)  tangent  popcount  sort/32  sort/64  sort/128  dijkstra barnes-hut pdes/4  pdes/8  pdes/16  bfs/4  bfs/8  bfs/16  geomean  Benchmark  0  1  2  3  Normalized ADP  (Lower is better)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='61  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 12: Normalized Speedup and ADP of Application Benchmarks  sort/N indicates the array size of the sorting accelerator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' pdes/N and bfs/N indicates the number of cores sharing the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  We use Area-Delay-Product (ADP) to quantify and com-  pare area efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' When calculating area consumption, the  processor-only baseline only counts the processors and the  hardware cache system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The FPSoC-like architecture adds the  silicon area of the FPGA on top of the processor-only baseline,  and Dolly further includes the Duet Adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The seven benchmarks, their corresponding Dolly instance,  and the acceleration paradigm (FG for fine-grained accelera-  tion, HA for hardware augmentation) are the following:  Tangent (P1M0, FG): A floating-point Tangent accelerator  is implemented with Catapult HLS using a piece-wise linear  approximation algorithm with a maximum error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='3%  compared to the C math library (libm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' An FPGA-bound FIFO  is used to pass the argument to the accelerator and invoke it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Results are returned through an CPU-bound FIFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Popcount (P1M1, FG): Popcount counts the number of ones  in a long bit vector (512 bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Since the Ariane processor  does not support the RISC-V BitManip Extension, we use a  byte look-up algorithm for the processor-only baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  accelerator is hand-written in Verilog and uses one Memory  Hub to load the bit vector from coherent memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Sort (P1M2, FG): We use the SPIRAL Project [60] to  generate 3 sorting networks in Verilog for sorting 32, 64,  128 double-word (4-Byte) integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The accelerator uses two  memory hubs, one for reading the input array from coherent  memory and one for writing the sorted array back, so that the  accelerator can be pipelined to sort fixed-length slices of a  larger array which can then be merge-sorted by the processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The processor-only baseline runs quicksort on the entire array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Dijkstra (P1M1, FG): We implement an accelerator for  Dijkstra’s Shortest Path algorithm with Catapult HLS and use  a soft cache to exploit data locality between consecutive calls  to the accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Barnes-Hut (P4M1, FG): As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' III-A2, the  two key kernels in the Barnes-Hut algorithm are implemented  as soft accelerators using Catapult HLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Both accelerators  are pipelined and time-multiplexed by four processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  processor-only baseline also parallelizes force calculation for  different particles across four processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  PDES  (P4M1/P8M1/P16M1,  HA):  As  described  in  TABLE II: Clock Frequency and Area of Soft Accelerators  Benchmark  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Freq  Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Resource Utilization  (MHz)  Area *  CLB †  BRAM  Ariane +  1000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='0 P-Mesh Socket  Tangent  282  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='84  0  Popcount  189  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='56  Sort (32)  228  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='76  Sort (64)  234  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='92  Sort (128)  228  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='92  Dijkstra  127  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='31  Barnes-Hut  85  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='05  BFS  208  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='75  PDES  126  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='56  Total silicon area needed by the eFPGA to implement the accelerator,  normalized to 1x Ariane + 1x P-Mesh Socket  † Configurable Logic Block, including LUTs and flip-flops  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' III-B2, a non-speculative, hardware task scheduler is  designed in Verilog to accelerate Parallel Discrete Event  Simulation (PDES) of a digital circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The processor-only  baseline uses MCS locks [35] to arbitrate accesses to the  shared event queue, and the lock contention can be severe as  the number of cores increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  BFS (P4M0/P8M0/P16M0, HA): We implement multiple  hardware, lock-free queues in Verilog to alleviate the synchro-  nization overhead in parallel Breadth-First Search (BFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  processors traverse the graph in barrier-synchronized steps and  use the queues to store the current and next search frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Similar to the PDES benchmark, the processor-only baseline  suffers from synchronization bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' II lists the maximum clock frequency and eFPGA  resource utilization of the soft accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Note that the soft  accelerators can only run at 8% - 28% of the processors’  clock frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' As studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' V-C, Duet achieves peak  bandwidth in this frequency range and outperforms the con-  ventional FPSoC communication mechanisms by at least 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 12 shows the normalized speedup and ADP of the appli-  cation benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Duet outperforms FPSoC across all bench-  marks and achieves a geometric mean of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='53x speedup over  processor-only baselines, in comparison to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='14x achieved by  10  FPSoC using the same accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The sorting accelerators  provide up to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='2x speedup on Duet but only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='0x on FPSoC,  because the FPGA-side caches in FPSoC are penalized by the  slow clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Some may argue that the FPGA-side caches  are unnecessary because sorted arrays are consecutive blocks  for which DMAs are efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' However, due to the small array  sizes (128-512B), even LLC-coherent DMA incurs too much  control overhead in cache flushing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The lock-free queues used  in BFS provide up to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='9x speedup on Duet but only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='8x  on FPSoC, because the processor-accelerator communication  latency on FPSoC is much higher without the shadow registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Note that the BFS benchmark shows a superlinear speedup  when scaling from a 4-core system (Dolly-P4M0) to an 8-  core system (Dolly-P8M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' This is because the processor-only  baseline sees a performance decrease when scaling from 4  cores to 8 cores due to intensifying lock contention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  In terms of area efficiency, Duet achieves a geometric mean  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='39x lower ADP than the processor-only baseline (lower  is better), while the FPSoC is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='23x higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Duet outperforms  FPSoC on most benchmarks except for Dijkstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' This is  because the FPSoC model hardens the FPGA-side cache in the  slow clock domain, so that soft caches become unnecessary  and can be removed to save eFPGA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  In summary, Dolly achieves superior speedup with the same  eFPGA-emulated accelerators than FPSoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Therefore, we can  conclude that Duet provides better support for fine-grained  acceleration and hardware augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Standalone FPGA-Based Accelerators  FPGAs are being used in a fast-growing range of application  domains due to their programmability and close-to-ASIC  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On a smaller scale, standalone FPGAs are used  to accelerate database queries [51], image processing [14],  network filtering [52], and most popularly, deep learning  applications [21], [46], [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' On the other end of the spectrum,  FPGAs are massively deployed in the Cloud, either offered di-  rectly as an on-demand computing service such as the Amazon  AWS F1 instances [2], or used to build cloud-scale accelerators  such as Microsoft Catapult [10] and BrainWave [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In these  systems, the FPGAs are often integrated as PCIe peripherals  or separate machines on the datacenter network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' As explained  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' I, such systems excel in accelerating at the application  level, but are insufficient for fine-grained acceleration due to  high CPU-FPGA communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Same-Package CPU-FPGA Hybrids and FPSoCs  Addressing the CPU-FPGA communication bottleneck,  academia and industry have explored tighter integration of  the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Intel Harp [13] combines an Intel Xeon multi-core  CPU and an Altera FPGA in one package, and bridges the  two with QPI, a point-to-point interconnect with cache coher-  ence support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Field-Programmable System-on-Chips (FPSoC)  achieves even tighter integration by integrating processors and  FPGAs on the same chip, similar to Duet at a high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Many  commodity FPSoCs [26], [36], [37], [42], [55] and academic  FPSoCs [28], [45], [53] support full or partial cache coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  For example, the Xilinx Zynq-7000 employs the AXI4 ACP  interface [3] which supports uni-directional cache coherence  (I/O coherency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Xilinx Zynq UltraScale+ MPSoC [56]  and Versal ACAP [19] provide full coherence support with  the ACE protocol [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  There are two key differences between Duet and FPSoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  First, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' 1b, the processor subsystem and the  eFPGA in FPSoCs are separated by a centralized interconnect,  and the hardware cache hierarchy resides in the processor sub-  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Such organization is reasonable for an FPGA-centric,  dual-core architecture, but cannot scale to a larger number of  cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Second, in the absence of FPGA-side hardware caches,  it becomes the accelerator designers’ burden to design and  verify soft caches that comply with the coherence protocol,  for example ACE [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Besides increasing design complexity,  the soft caches’ direct participation in cache coherence slows  down the entire cache system because they run in the slow  clock domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Moreover, since the soft caches have access to  the micro-architecture states of the NoC, the system cannot  restrain faulty or malicious behaviors of the soft caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Near- and In-Processor Reconfigurable Computing  Dating back to the early days of reconfigurable computing,  computer architects have proposed to integrate Reconfigurable  Fabrics (RF) very close to or even into processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Garp [9]  places an RF between a processor and its private cache,  making the two compute units share the entire memory sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Chimaera [24] and PRISC [43] embed Reconfigurable  Functional Units (RFUs) directly into a processor’s datap-  ath, enabling post-fabrication customization of the Instruction  Set Architecture (ISA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The Post-Fabrication Microarchitec-  ture [29] couples an RF with a superscalar core and allows  the RF to observe and microarchitecturally intervene at key  pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' These systems have their advantages but con-  flict with our plug-and-play integration goal as they mandate  the redesign of the processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Coherence Protocols for Accelerators  Hardware cache coherence has been widely adopted in  multi-processor systems as they provide not only programma-  bility but also performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' In the context of hardware accel-  eration, recent studies have also shown benefits of employing  hardware cache coherence and have proposed a variety of  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' FUSION [30] is a hierarchical, timestamp-based  coherence protocol emphasizing data transfer between accel-  erators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Crossing Guard [39] proposes the use of a hard cache  transducer to decouple the accelerator coherence protocol  and the processor (host) protocol, as well as to incorporate  fault-tolerance and memory translation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Spandex [1] offers  flexible write strategy and granularity, adapting to highly het-  erogeneous architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' While all of these proposals achieve  hardware cache coherence, they are not optimized for FPGA-  based accelerators in that they all require direct participation of  the accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' However, through the use of the Proxy Cache,  Duet can be adapted to these coherence protocols, pretending  11  to be a "fast" accelerator to the NoC and hiding the slow clock  domain from the coherence protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  Accelerator coherence standards and interconnects are also  emerging in industry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=', CCIX [11], OpenCAPI [40], and  CXL [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' These proposals target the board level or above and  are really optimized for coarse-grained acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' CONCLUSION  In this work, we present Duet, a scalable, manycore-FPGA  architecture with non-intrusive, coherently-integrated, embed-  ded FPGAs that is optimized for fine-grained acceleration and  hardware augmentation in the broad general-purpose domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  By promoting the eFPGA to a first-class citizen on chip, Duet  enables the eFPGA to access the NoC and to participate in bi-  directional cache coherence just as any other processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The  novel, lightweight, Duet Adapters reduce critical communi-  cation overheads by placing the Proxy Caches and Shadow  Registers in the faster processor clock domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Our evaluation  shows that Dolly, an RTL-level implementation of Duet, can  reduce the latency of processor-accelerator communications by  up to 82% and increase the bandwidth by up to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='5x, stably  across a wide range of eFPGA clock frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Selected  benchmarks leveraging Duet-enabled soft accelerators show up  to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='9x speedup over processor-only baselines and up to 4x  over FPSoCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Dolly and its toolchain have been open-sourced  and available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='com/PrincetonUniversity/Duet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This material is based on research sponsored by the Air  Force Research Laboratory (AFRL) and Defense Advanced  Research Projects Agency (DARPA) under agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  FA8650-18-2-7852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' This material is based upon work sup-  ported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  CNS-1823222 and the National Science Foundation Graduate  Research Fellowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' DGE-2039656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='  The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Government is authorized to reproduce and distribute  reprints for Governmental purposes notwithstanding any copy-  right notation thereon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' The views and conclusions contained  herein are those of the authors and should not be interpreted as  necessarily representing the official policies or endorsements,  either expressed or implied, of Air Force Research Laboratory  (AFRL) and Defense Advanced Research Projects Agency  (DARPA) or the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
+page_content=' Any opinions, findings, and  conclusions or recommendations expressed in this material are  those of the author(s) and do not necessarily reflect the views  of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE0T4oBgHgl3EQf8QI3/content/2301.02785v1.pdf'}
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+Preprint 30 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Constraining cosmological vorticity modes with CMB
+secondary anisotropies
+William R. Coulton,1 Kazuyuki Akitsu,2 Masahiro Takada3
+1Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
+2School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
+3Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS,
+The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
+30 January 2023
+ABSTRACT
+Observational searches for large-scale vorticity modes in the late time Universe are underexplored. Within the stan-
+dard ΛCDM model, this is well motivated given the observed properties of the cosmic microwave background (CMB).
+However, this means that searches for cosmic vorticity modes can serve as a powerful consistency test of our cos-
+mological model. We show that through combining CMB measurements of the kinetic Sunyaev-Zel’dovich and the
+moving lens effects with galaxy survey data we can constrain vorticity fields independently from the large scale cos-
+mic velocity field. This approach can provide stringent constraints on the largest scale modes and can be achieved
+by a simple change in the standard estimators. Alternatively if one assumes there are no cosmic vorticity modes,
+this estimator can be used to test for systematic biases in existing analyses of kinetic Sunyaev-Zel’dovich effect in a
+manner analogous to curl-lensing.
+1 INTRODUCTION
+Consistent measurements of the properties of the Universe
+across 13 billion years has established the ΛCDM cosmologi-
+cal model (Aghanim et al. 2020; Abbott et al. 2022; D’Amico
+et al. 2021; Kobayashi et al. 2022; Philcox & Ivanov 2022;
+Brout et al. 2022). However, we still lack insight into the
+two of the key components, dark matter and dark energy,
+and there are now hints of potential cracks in this model –
+in the form of tensions between probes of different eras (see
+Abdalla et al. 2022; Sch¨oneberg et al. 2022, for an overview
+of the ‘Hubble’ and ‘σ8’ tensions). Testing the ΛCDM model
+through ever more powerful and varied tests is thus essential
+if we are to advance our understanding on these fundamental
+issues.
+Perturbations to the homogeneous, isotropic background
+Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) Universe
+can be decomposed into three types of perturbations: scalars,
+vectors and tensors. At the linear level, these three are de-
+coupled and do not mix (see e.g. Dodelson & Schmidt 2020;
+Weinberg 2008; Mukhanov 2005). Through measurements of
+the cosmic microwave background (CMB), we have discov-
+ered that the early Universe is dominated by small scalar
+perturbations. As linear vector and tensor decay within the
+standard ΛCDM model, these high redshift measurements
+place tight bounds on the expected level of vector and ten-
+sor modes (Bennett et al. 2003; Ichiki et al. 2012; Saga et al.
+2014). Whilst the evolution of the small, scalar perturbations
+is non-linear, these processes do not generate large levels of
+large-scale vector and tensor perturbations (Baumann et al.
+2007; Lu et al. 2009; Pueblas & Scoccimarro 2009; Hahn
+et al. 2015; Jelic-Cizmek et al. 2018). Thus an important
+prediction of our current model is the absence of late-time,
+large-scale, cosmic vorticity, velocity fields that cannot be ex-
+pressed as the gradient of a scalar and are a type of vector
+perturbation. To date, there have been only a few analyses
+of vorticity/vector modes in the late time Universe - Motloch
+et al. (2021) search for vorticity modes in galaxy spins, whilst
+Namikawa et al. (2013) use the CMB lensing curl mode.
+Recently numerous authors have shown how measurements
+of the kinetic Sunyaev-Zel’dovich effect (kSZ) or moving lens-
+ing (ML) effect, two CMB secondary anisotropies imprinted
+on the CMB as photons propagate to the observer from the
+surface of last scattering, can be used to measure the large
+scale velocity field (Li et al. 2014; Alonso et al. 2016; Ter-
+rana et al. 2017; Smith et al. 2018; Yasini et al. 2019; Hotinli
+et al. 2019; McCarthy & Johnson 2020). This achieved via a
+technique called “kSZ/ML tomography” where high resolu-
+tion, low noise measurements of the CMB are combined with
+measurements of the large scale structure of the Universe,
+such as galaxy surveys. This technique is a highly promising
+avenue for studying large scale velocity fields with upcom-
+ing surveys predicted to make O(100)s signal-to-noise ratio
+(SNR) measurements of velocity fields on k ≲ 10−3 hMpc−1
+(Smith et al. 2018; M¨unchmeyer et al. 2019).
+In this work we explore whether kSZ and ML measure-
+ments can be used to study late-time vorticity modes. Such
+measurements would provide strong tests of the standard
+model expectation that these modes are small, constrain
+novel cosmological models that can produce large, late-time
+vorticity modes or, if one assumes there are no vorticity
+modes, act as a systematics-null test for kSZ measurements.
+This work complements recent work by Bonvin et al. (2018),
+which explores how redshift space distortions can be used to
+constrain vorticity modes, and we find that our approach of-
+fers a significantly more powerful means of constraining the
+largest scales.
+© 0000 The Authors
+arXiv:2301.11344v1  [astro-ph.CO]  26 Jan 2023
+
+2
+W. R. Coulton et al.
+Throughout this paper, we work in the ‘snapshot’ geome-
+try described in Smith et al. (2018) where we compute our
+observables by treating the universe as a periodic box at a
+distance χ∗ from the observer. This setup allows the intu-
+ition to be clearly developed and the results can be extend to
+include light-cone effects and a treatment of the curved sky
+in an analogous way to Terrana et al. (2017). In Appendix A
+we outline how this idea can be framed in the framework of
+Terrana et al. (2017).
+This paper is strucutred as follows: in Section 2 we review
+vorticity modes. In Sections 3 and 4 we show how vorticity
+modes source kSZ and ML anisotropies. In Section 5, we de-
+scribe how to use these effects to reconstruct the vorticity
+perturbations and present a forecast of this approach in Sec-
+tion 6. Our conclusions and outlook are presented in Section
+7.
+2 VORTICITY MODES
+An arbitrary vector can be decomposed with Helmholtz de-
+composition into a part that is curl-free, u, and a part that
+is divergence-free, ω. Thus, we can decompose the Fourier
+space velocity perturbations into divergence, u(k), and curl
+(vorticity), ωi(k) contributions as
+vi = iki
+k u(k) + iωi(k),
+(1)
+where the curl component satisfies ωiki = 0. To proceed it is
+useful to represent the divergence-free mode with the polar-
+ization vectors
+ωi(k) =
+�
+s
+ϵs
+i(ˆk)ω(s)(k),
+(2)
+where ϵs
+i are the polarization basis vector and s denotes the
+two polarization states (+1, −1). It is convenient to work in
+a rotated basis defined as
+ω+ =
+1
+√
+2
+�
+ω(+1) + ω(−1)�
+and ω× =
+1
+√
+2
+�
+ω(+1) − ω(−1)�
+.
+(3)
+In linear cosmological perturbation theory curl modes are not
+sourced by density (scalar) modes and so are expected to van-
+ish. This means that these curl modes are a powerful probe of
+new physics. As an example consider cosmological vector per-
+turbations. In the synchronous gauge, the perturbed FLRW
+metric can be written as
+ds2 = a(τ)2×
+�
+−dτ 2 +
+�
+(1 − 2φ)δij + 2(D,ij + Ω(i,j) + hT T
+ij )
+�
+dxidxj�
+,
+(4)
+where φ(x, t) and D(x, t) are scalar perturbations, Ωi(x, t) is
+the transverse vector perturbation and hT T
+ij (x, t) is the trans-
+verse, traceless tensor perturbation. The vector modes source
+vorticity perturbations as
+ωi(x, τ) = −
+∇2Ω′
+i
+16πGa2(ϵ0 + p0),
+(5)
+where a is the scale factor, G is the Gravitational constant, ′
+denotes the conformal time derivative, and ϵ0 and p0 are the
+energy density and pressure. Note that, in matter domination
+and without a source, the vector perturbations, Ωi, decay as
+a− 5
+2 and so their induced vorticity decays as, a−1.
+To have observable vector modes they thus need to be
+sourced. In general vector modes can be sourced whenever
+there is non-vanishing anisotropic stress such as neutrino ve-
+locity isocurvature modes (Lewis 2004b) or primordial mag-
+netic field (Lewis 2004a). More exotic sources include modifi-
+cations to general relativity (Battye et al. 2017), topological
+defects (Seljak et al. 1997; Durrer et al. 2002; Daverio et al.
+2016), a global rotation (Carneiro 2000) and some dark en-
+ergy models (Battye & Moss 2006, 2007, 2009).
+3 KINETIC SUNYAEV-ZEL’DOVICH EFFECT
+The kinetic Sunyaev-Zel’dovich (kSZ) effect occurs when
+CMB photons are Thomson scattering off electrons that are
+moving with bulk motions, such as in galaxy groups and clus-
+ters (Sunyaev & Zeldovich 1980). This scattering generates
+anisotropies in the CMB, without altering the CMB black-
+body spectrum, that depend on the electron’s properties via
+∆T kSZ(ˆn)
+TCMB
+= −
+�
+dχa(χ)σT ne(χ)v(χˆn) · ˆne−τ(χ),
+(6)
+where v is the electron velocity, ne is the electron density, χ
+is the comoving distance, σT is the Thomson scattering cross-
+section and τ is the optical depth. In our simplified geometry
+the line-of-sight integral integrates across our box and the
+observed angle n describes points of the surface of the box at
+location (x, y) = χn.
+In Fourier space we can see that the kSZ anisotropies are
+given by
+˜T kSZ(ℓ) =iaσT ¯nee−τ
+χ2∗
+�
+dk3
+1
+(2π)3
+dk3
+2
+(2π)3 (2π)3δ(3)
+D
+� ℓ
+χ∗ − k1 − k2
+�
+× δe(k1)
+�
+cos θk2u(k2) − sin θk2ω+(k2)
+�
+,
+(7)
+where δ(3)
+D (x) is the 3D Dirac delta function and δe(k) is the
+Fourier transform of the electron density fluctuation field.
+The scalar and vector sources contribute via different angular
+dependencies, implying that these effects can be separated.
+Further note that the kSZ effect is only sensitive to the ω+
+modes.
+4 THE MOVING LENS EFFECT
+The moving lens effect is a second-order effect that generates
+anisotropies in the CMB. This effect can be understood from
+two equivalent perspectives either as part of the Rees-Scamia,
+or non-linear integrated Sachs-Wolfe effect, or as lensing by
+a moving cluster (Rees & Sciama 1968; Birkinshaw & Gull
+1983; Gurvits & Mitrofanov 1986; Lewis & Challinor 2006;
+Yasini et al. 2019; Hotinli et al. 2019). The first perspec-
+tive describes how the energy of photons is altered as they
+enter the potential of clusters moving perpendicular to the
+line of sight: if they pass in-front of the cluster they will be
+blueshifted as they fall into clusters however as they leave
+they are redshifted by a larger amount as the cluster has
+moved closer to the line of sight and deepened the potential.
+Note that there is no additional blue-shifting from the change
+in the potential as the cluster’s velocity is perpendicular to
+MNRAS 000, 000–000 (0000)
+
+3
+the photon trajectory. The opposite effect occurs for photons
+passing behind the cluster. The second perspective can be
+understood by considering photon trajectories as viewed in
+the cluster rest frame. In this frame a dynamical dipole is
+seen and the dipolar distorted photons are deflected by the
+cluster’s mass. Boosting back to the CMB rest frame removes
+the initial, unlensed dipole leaving a residual signal.
+The size of the induced anisotropies is given by
+∆T ML(n)
+TCMB
+= −2
+�
+dχv⊥ · ∇⊥Φ,
+(8)
+where Φ is the gravitational potential. Inserting our expres-
+sions for the velocities into this, we find that the Fourier space
+ML anisotropies are given by
+˜T ML(ℓ) = 2
+χ2∗
+�
+d3k
+(2π)3 Φ
+� ℓ
+χ∗ − k
+�
+×
+�
+u(k)
+� ℓ
+χ∗ sin θk cos(φk − φℓ) − k sin2 θk
+�
++ i sin(φk − φℓ)ω×(k)
++ ω+(k)
+� ℓ
+χ∗ cos θk cos(φk − φℓ) − k sin θk cos θk
+� �
+.
+(9)
+There are several interesting features here: first the standard
+scalar velocity modes have a different angular dependence
+than the kSZ effect (this is expected as it probes transverse
+modes), which is essential for breaking the degeneracies with
+the vector kSZ modes. Second the vorticity modes likewise
+have a different angular dependence, and so in principle could
+be separated out from the other ML contributions. Finally
+the ML effect is also sensitive to the cross component (ω×)
+of the vector modes. However, this term is geometrically sup-
+pressed and as it is thus likely unmeasurable we do not con-
+sider it further here.
+As was shown in Hotinli et al. (2019) the moving-lens
+anisotropies are generally smaller than the kSZ anisotropies
+– their contirubtion to the power spectrum at ℓ ∼ 3000 is an
+order of magnitude lower.
+5 VELOCITY RECONSTRUCTION
+Recent work (Hand et al. 2012; Terrana et al. 2017; Smith
+et al. 2018; Cayuso et al. 2021) has shown that by combining
+kSZ or ML anisotropies with galaxy position measurements
+large scale scalar velocity modes, u(k), can be reconstructed.
+The reconstruction uses a quadratic estimator
+�uX(K) =
+� d3kd2ℓ
+(2π)2 W(k, l) ˜T X(ℓ)δg(k)δ(3)
+D
+�
+K − k − ℓ
+χ∗
+�
+,
+(10)
+where δg(k) is the 3D Fourier transform of the galaxy den-
+sity field, ˜T X(ℓ) is the 2D Fourier transform of the kSZ
+or ML anisotropies and W(k, l) is a set of weights. This
+method is akin to CMB lensing where the lensing poten-
+tial is reconstructed from measurements of the primary CMB
+anisotropies. Usually the weights in this estimator are cho-
+sen to obtain a minimum variance, unbiased estimator of
+the scalar velocity field i.e. ⟨�u(k)u∗(k′)⟩ = (2π)3δ(3)(k −
+k′)Pu(k), where Pu(k) is the velocity power spectrum. Given
+these requirements, the general case of reconstructing field
+X, the weights are (Darwish et al. 2020)
+W(k, l) = BXδgT ∗(−k − ℓ/χ∗, k, ℓ/χ∗)
+P tot
+gg (k)Ctot
+ℓ
+PXX(−k − ℓ/χ∗)×
+��
+d2ℓ
+(2π)2
+BXδgT ∗(−k − ℓ/χ∗, k, ℓ)BXδgT(−k − ℓ/χ∗, k, ℓ)
+P tot
+gg (k)Ctot
+ℓ
+P 2
+XX(−k − ℓ/χ∗)
+�−1
+,
+(11)
+where BXδgT ∗(−k−ℓ/χ∗, k, ℓ/χ∗) is the bispectrum between
+the true field, X, and the two fields in the quadratic esti-
+mator (δg and T), and P tot
+gg (k), PXX(k) and Ctot
+ℓ
+are the
+power spectra where the superscript “tot” denotes that the
+power spectra include all contributions: signal, instrument
+noise, foregrounds and shot-noise – as appropriate. For re-
+constructing scalar modes from the kSZ, this is ⟨uδgT⟩ and
+is (Smith et al. 2018)
+BuδgT −kSZ
+�
+K, k, ℓ
+χ∗
+�
+=
+aσT ¯nee−τ
+χ2∗
+i cos θK
+�
+Puu(K)Pge(k) − K
+k Pue(K)Pug(k)
+�
+.
+(12)
+where ¯ne is the mean electron density at the scattering red-
+shift, Pge, Pug and Pue are the galaxy-electron, velocity-
+galaxy and velocity-electron power spectra. Similarly for
+scalar sources with the ML effect we have
+BuδgT −ML
+�
+K, k, ℓ
+χ∗
+�
+= 2
+χ2∗
+�Puu(K)
+K
+PgΦ(k) + PuΦ(K)Pgu(k)
+k
+�
+×
+�
+K ℓ
+χ∗ sin θK cos(φK − φℓ) − K2 sin2 θK
+�
+.
+(13)
+It is trivial to form a combined estimator by using the com-
+bined bispectra, i.e. BuδgT −Comb = BuδgT −kSZ + BuδgT −ML.
+Extending these estimators to reconstruct the vorticity
+modes is also straightforward. A quadratic estimator, ˆω+,
+can be formed in an identical manner to Eq. 10 with the vor-
+ticity bispectra used instead. For a reconstruction from kSZ
+anisotropies, this is
+Bω+δgT −kSZ
+�
+K, k, ℓ
+χ∗
+�
+= −aσT ¯nee−τ
+χ2∗
+i sin θKPeg(k)Pωω(K),
+(14)
+where Pωω is the vorticity power spectrum and for recon-
+struction from ML anisotropies this is
+Bω+δgT −ML
+�
+K, k, ℓ
+χ∗
+�
+= PΦg(k)Pωω(K)
+×
+� ℓ
+χ∗ cos θK cos(φK − φℓ) − K sin θK cos θK
+�
+.
+(15)
+As for the scalar case a combined estimator can be formed
+by using the sum of the bispectra.
+However, these vorticity estimators are not orthogonal to
+the scalar velocity estimators. This is problematic as mea-
+surements of the vorticity modes will likely be biased by the
+scalar modes, which are expected to be much larger than the
+vorticity terms. To avoid these biases we can construct vortic-
+ity estimators which are orthogonal, i.e. have zero response,
+to the scalar modes.
+MNRAS 000, 000–000 (0000)
+
+4
+W. R. Coulton et al.
+The quadratic estimator that is unbiased, minimum vari-
+ance and orthogonal to the scalar modes is given by
+ˆω+(K) =
+�
+d2ℓ
+(2π)2 T(ℓ)δg
+�
+K − ℓ
+χ∗
+�
+1
+P tot
+gg (K −
+ℓ
+χ∗ )Ctot
+ℓ
+×
+1
+AωωAuu − AωuAuω
+�
+Auu ¯B∗ωgT − Auω ¯B∗ugT �
+, (16)
+where
+AXY (K) =
+�
+d2ℓ
+(2π)2
+¯BXgT ¯B∗Y gT
+P(K −
+ℓ
+χ∗ )Cℓ
+,
+(17)
+and ¯BXgT = BXgT /P XX. Note that the vector power spec-
+trum in the estimator completely cancels with equivalent
+terms implicit in the bispectra, so the estimator is indepen-
+dent of the shape of the velocity power spectrum.
+6 FORECAST CONSTRAINTS
+Using the formalism from Smith et al. (2018) we investigate
+the constraining power on this method. Specifically the noise
+power spectrum per mode for the standard estimator is given
+by
+N kSZ−scalar(K) =
+1
+Auu(K),
+(18)
+and for the orthogonal vorticity estimator
+N vorticity(K) =
+Auu(K)
+Aωω(K)Auu(K) − Auω(K)Aωu(K),
+(19)
+– note that the noise is highly anisotropic. Then the uncer-
+tainity on the signal power spectrum, with an inverse noise
+weighting is
+1
+σ2(P XX(K)) = V
+� K2dK sin θdθKdφK
+(2π)3
+1
+N XX(K).
+(20)
+We consider how well a spectroscopic galaxy survey, like DESI
+(DESI Collaboration et al. 2016), in combination with a next
+generation CMB survey, such as CMB-S4 (Abazajian et al.
+2016), can measure these modes. For the DESI-like survey
+we use the experimental setup from Smith et al. (2018) (also
+see Coulton et al. 2020), thus we assume a mean redshift
+of z = 0.75, a survey volume of 116 (h−1Gpc)3, a galaxy
+number density of 1.7 × 10−4 Mpc−3 and a galaxy bias of
+bg = 1.51. We use the CMB-S4 configuration described in
+Abazajian et al. (2019) and compute the noise curves after
+component separation with the ILC method (Tegmark & Ef-
+stathiou 1996).
+First we consider what can be learnt from only studying the
+kSZ anisotropies. The expected constraining power is shown
+in Fig. 1. Enforcing that the vector estimator has no response
+to the standard, scalar kSZ contribution comes at a huge
+noise cost. As stated earlier, this condition is necessary as the
+scalar modes are expected to be significantly larger than any
+vector contribution and without it vorticity measurements
+can be biased. The reason for the large noise cost can be seen
+from Eq. 7 – whilst in principle the two sources have different
+angular dependencies and so could be separated, the region
+of phase space where there are purely scalar or vector sources
+is vanishing. Thus the vector modes constraints come from
+modes that are close to perpendicular to the line of sight. If
+the scalar perturbations have an anisotropic power spectrum
+of the form ∼ sin θK/ cos θKPuu(|K|), their kSZ signature
+from this region of phase space would be highly similar to
+the vector modes. Our estimator makes no assumptions on
+the properties of the scalar perturbations and the large noise
+cost arises from accounting for the possibility of anisotropic
+scalar modes.
+For sufficiently low noise measurements it is possible to
+disentangle the two contributions with kSZ alone. Though a
+further complication arises as this separation requires mod-
+elling the bispectrum beyond the squeezed limit and, whilst
+the squeezed limit can be modeled with high accuracy (Giri &
+Smith 2022), accurately modeling the full bispectrum can be
+challenging. The ML-effect on its own can similarly be used to
+constrain vorticity modes, however it suffers the same issues
+and has even larger noise than kSZ based estimator (similar
+to that found for scalar velocity modes (Hotinli et al. 2019)).
+To alleviate these issues we consider using a combined esti-
+mator from the kSZ and ML effects. The ML and kSZ effects
+are sensitive to orthogonal velocities which is the precisely
+what is required to completely break the degeneracy. The re-
+sult of the combined estimator is shown in Fig. 1. The com-
+bined estimator provides powerful constraints on the large
+scale vorticity that are comparable to those of the scalar ve-
+locity mode. Note that this is achieved despite the low con-
+straining power of the ML estimator; the contribution of the
+estimator is removal of the possibility of anisotropic scalar
+modes - as these would lead to a very large ML signal.
+7 CONCLUSIONS
+Large-scale, cosmic vorticity modes in the late time Universe
+are an interesting cosmological observable. In the standard
+cosmological model these modes should be absent. Thus,
+searching for these modes is a powerful consistency test of
+our model and a discovery of these modes would represent
+the detection of new physics.
+In this work we show that the kSZ and ML effects, in combi-
+nation with a galaxy survey, are an ideal method to search for
+these signals. Using tomography with galaxy surveys, we are
+able to separate vorticity modes from velocity perturbations
+from scalar, density modes. Tomography enables the differen-
+tiation of fluctuations in the velocity field that are parallel to
+the velocity vector (density-source velocity modes) from those
+perpendicular to the velocity vector, vorticity modes. This
+joint method leads to almost white noise up to the largest
+scales accessible with the surveys, allowing for powerful con-
+straints on the largest scales. Through combining galaxy and
+CMB measurements in a quadratic estimator to reconstruct
+the large scales, the method should have fewer large scale
+systematic effects. Especially as the non-trivial parity of the
+signal (see e.g. Smith et al. 2018) provides an extra means to
+suppress systematic biases.
+As a demonstration of the power of this approach, we con-
+sider constraining topological defects, a well known source of
+vorticity. Within this model the vorticity power spectrum is
+given as
+Pωω(k) = 14(Gµ)2
+k3
+�
+kτ/12
+1 + (kτ/12)3.13
+�
+(21)
+where Gµ characterizes the amplitude of the topical defects
+(Urrestilla et al. 2008). Using our formalism we find that we
+MNRAS 000, 000–000 (0000)
+
+5
+10
+3
+10
+2
+10
+1
+100
+101
+k h/Mpc
+10
+8
+10
+6
+10
+4
+10
+2
+100
+102
+(P(k)) Mpc3/h3
+Scalar - kSZ
+Vector - kSZ
+Vector - kSZ+ML
+Density-sourced Velocity
+Topological Defects
+Figure 1. A Fisher forecast for how well the DESI and CMB-S4 experiments can constrain the vorticity power spectrum. We show the
+constraints obtained using the kinetic Sunyaev-Zel’dovich effect alone and combining it with the moving lens effect. The joint analysis
+of these two effects allows the vorticity modes to be constrained independently of the scalar velocity modes. For comparison we show
+the constraints obtainable on the scalar velocity power spectrum, the expected ΛCDM scalar velocity power spectrum (dotted) and the
+topological defect spectrum given in Eq. 21. We use a logarithimic k-binning of width ∆ ln k = 0.2
+can constraint (Gµ)2 ≤ 1.6 ×10−10. Whilst topological de-
+fects leave strong and distinct signatures on the CMB, so
+are constrained significantly better by CMB measurements
+(Dunkley et al. 2011; Lizarraga et al. 2016; Planck Collab-
+oration et al. 2014, 2016), constraints obtainable with this
+kSZ formalism are more than two orders of magnitude tighter
+than those obtainable redshift space distortions (Bonvin et al.
+2018). This approach provides comparable constraints to
+those from CMB curl lensing (Namikawa et al. 2013) and
+by probing vector modes on a redshift slice, rather than the
+integrated over the lensing kernel, is highly complementary.
+This highlights the potential of this approach to probe new
+regimes in the late-time Universe.
+Here we have only discussed one way to separate these
+scalar and vorticity modes, by combing the kSZ and ML esti-
+mators, when in practice many methods would be explored to
+ensure robust results. These could include using direct mea-
+surements of the density perturbations, such as through clus-
+tering measurements, to remove the scalar velocity modes
+(that are linearly related to density perturbations). We fo-
+cused on the combined method as it is always necessary to
+account for both effects simultaneously to avoid biases from
+scalar modes from the other effect. This is necessary as sepa-
+rating the two effects in CMB maps is not possible with stan-
+dard component separation methods, as both effects produce
+anisotropies with the same spectrum – the same as the pri-
+mary temperature anisotropies. The key other possible con-
+taminants to this measurement are the thermal-kinetic SZ
+effect (Coulton et al. 2020), which can be removed due to its
+distinct frequency spectrum, and gravitational lensing, which
+can be removed with established delensing techniques (Seljak
+& Hirata 2004; Smith et al. 2012; Green et al. 2017).
+A second use of the methods outlined here is as a null-test
+for the measurement of scalar velocity modes. A key goal of
+upcoming kSZ measurements is to search for primordial non-
+Gaussianity (e.g. M¨unchmeyer et al. 2019) and one of the
+biggest challenges for such analyses is mitigating systematic
+biases (Pullen & Hirata 2013; Rezaie et al. 2021). If we as-
+sume there is no new physics, then the vector-mode velocity
+estimator can be used as a null-test for these analyses. This
+is analogous to how curl lensing (Namikawa et al. 2012) is
+ubiquitously used to test for systematic effects in weak lens-
+ing analyses.
+MNRAS 000, 000–000 (0000)
+
+6
+W. R. Coulton et al.
+APPENDIX A: LIGHT CONE GEOMETRY
+In this work we focused on discussing this analysis in the
+‘snapshot’ geometry picture, however equivalent expressions
+can be derived for the ‘light cone’ picture. From this perspec-
+tive we no longer work in a periodic box at a single redshift
+but instead in terms of spherical harmonic coefficients on the
+full sky and account for a more complex line of sight.
+In Terrana et al. (2017); Deutsch et al. (2018) an optimal,
+quadratic estimator is derived for the remote dipole field,
+veff, from galaxy data in redshifts bins, denoted by α. This
+estimator is
+ˆ
+veff
+α,ℓm =
+Nαℓ
+�
+ℓ1m1ℓ2m2
+(−1)mΓkSZ
+ℓ1ℓ2ℓα
+� ℓ1
+ℓ2
+ℓ
+m1
+m2
+−m
+� aT
+ℓ1m1δα
+g,ℓ2m2
+CT T
+ℓ
+C
+δgδg
+αℓ2
+(A1)
+where
+ˆ
+veff
+α,ℓm is the estimated dipole spherical harmonic coeffi-
+cients in a given redshift bin, Nαℓ is a normalization constant,
+ΓkSZ
+ℓ1ℓ2ℓα is a coupling matrix which combined with the Wigner
+3j symbol is the equivalent of the bispectrum in Eqs. 10-11,
+aT
+ℓ1m1 and δα
+g,ℓ2m2 are the spherical harmonic coefficients of
+the CMB map and galaxy density map and CXX
+ℓ
+are the
+power spectra of those maps. This can be thought of as an
+equivalent expression to Eq. 10.
+The expected contribution to this estimator from a scalar
+mode is given by
+veff
+α,ℓm =
+�
+dχW α(χ)
+�
+d3k
+(2π)3
+4πiℓ
+2ℓ + 1Φ(k)Y ∗
+ℓm(ˆk)
+× [ℓjℓ−1(kχ) − (ℓ + 1)jℓ+1(kχ)]
+(A2)
+where jℓ(x) are the spherical Bessel functions and W α(χ)
+defines the redshift bin. Computing the equivalent expression
+for vector modes gives
+veff
+α,ℓm =
+�
+dχW α(χ)
+�
+d3k
+(2π)3
+4πiℓ+1
+2ℓ + 1 ωs(k)−sY ∗
+ℓm(ˆk)
+×
+�
+ℓ(ℓ + 1)
+2
+[jℓ−1(kχ) + jℓ+1(kχ)] .
+(A3)
+The key difference between the two contributions is in the
+spherical Bessel projections – one plus the other minus – that
+is equivalent to the sin and cos terms in the vector/scalar kSZ
+estimator. Thus the scalar and vector modes can be differen-
+tiated by combining observations at a range of redshift bins
+with different weightings.
+ACKNOWLEDGMENTS
+The authors are grateful for fruitful discussions with Colin
+Hill, David Spergel, Emmanuel Schaan, Anthony Challinor
+and Oliver Philcox. This work was supported in part by
+World Premier International Research Center Initiative (WPI
+Initiative), MEXT, Japan, and JSPS KAKENHI Grants
+No. JP20J22055, JP20H05850, JP20H05855, JP19H00677,
+and by Basic Research Grant (Super AI) of Institute for AI
+and Beyond of the University of Tokyo. KA is supported by
+JSPS Overseas Research Fellowships.
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diff --git a/n9FIT4oBgHgl3EQfuivw/content/tmp_files/load_file.txt b/n9FIT4oBgHgl3EQfuivw/content/tmp_files/load_file.txt
new file mode 100644
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+page_content='0  Constraining cosmological vorticity modes with CMB  secondary anisotropies  William R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Coulton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='1 Kazuyuki Akitsu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='2 Masahiro Takada3  1Center for Computational Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
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+page_content=' Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 1 Einstein Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
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+page_content=' USA  3Kavli Institute for the Physics and Mathematics of the Universe (WPI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' UTIAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 5-1-5 Kashiwanoha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Kashiwa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Chiba 277-8583,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Japan  30 January 2023  ABSTRACT  Observational searches for large-scale vorticity modes in the late time Universe are underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Within the stan-  dard ΛCDM model, this is well motivated given the observed properties of the cosmic microwave background (CMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  However, this means that searches for cosmic vorticity modes can serve as a powerful consistency test of our cos-  mological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' We show that through combining CMB measurements of the kinetic Sunyaev-Zel’dovich and the  moving lens effects with galaxy survey data we can constrain vorticity fields independently from the large scale cos-  mic velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This approach can provide stringent constraints on the largest scale modes and can be achieved  by a simple change in the standard estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Alternatively if one assumes there are no cosmic vorticity modes,  this estimator can be used to test for systematic biases in existing analyses of kinetic Sunyaev-Zel’dovich effect in a  manner analogous to curl-lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  1 INTRODUCTION  Consistent measurements of the properties of the Universe  across 13 billion years has established the ΛCDM cosmologi-  cal model (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' D’Amico  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Philcox & Ivanov 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Brout et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' However, we still lack insight into the  two of the key components, dark matter and dark energy,  and there are now hints of potential cracks in this model –  in the form of tensions between probes of different eras (see  Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Sch¨oneberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2022, for an overview  of the ‘Hubble’ and ‘σ8’ tensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Testing the ΛCDM model  through ever more powerful and varied tests is thus essential  if we are to advance our understanding on these fundamental  issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Perturbations to the homogeneous, isotropic background  Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) Universe  can be decomposed into three types of perturbations: scalars,  vectors and tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' At the linear level, these three are de-  coupled and do not mix (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Dodelson & Schmidt 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Weinberg 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Mukhanov 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Through measurements of  the cosmic microwave background (CMB), we have discov-  ered that the early Universe is dominated by small scalar  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' As linear vector and tensor decay within the  standard ΛCDM model, these high redshift measurements  place tight bounds on the expected level of vector and ten-  sor modes (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Ichiki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Saga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Whilst the evolution of the small, scalar perturbations  is non-linear, these processes do not generate large levels of  large-scale vector and tensor perturbations (Baumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Pueblas & Scoccimarro 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Hahn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Jelic-Cizmek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Thus an important  prediction of our current model is the absence of late-time,  large-scale, cosmic vorticity, velocity fields that cannot be ex-  pressed as the gradient of a scalar and are a type of vector  perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' To date, there have been only a few analyses  of vorticity/vector modes in the late time Universe - Motloch  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2021) search for vorticity modes in galaxy spins, whilst  Namikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2013) use the CMB lensing curl mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Recently numerous authors have shown how measurements  of the kinetic Sunyaev-Zel’dovich effect (kSZ) or moving lens-  ing (ML) effect, two CMB secondary anisotropies imprinted  on the CMB as photons propagate to the observer from the  surface of last scattering, can be used to measure the large  scale velocity field (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Ter-  rana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Yasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Hotinli  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' McCarthy & Johnson 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This achieved via a  technique called “kSZ/ML tomography” where high resolu-  tion, low noise measurements of the CMB are combined with  measurements of the large scale structure of the Universe,  such as galaxy surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This technique is a highly promising  avenue for studying large scale velocity fields with upcom-  ing surveys predicted to make O(100)s signal-to-noise ratio  (SNR) measurements of velocity fields on k ≲ 10−3 hMpc−1  (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' M¨unchmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  In this work we explore whether kSZ and ML measure-  ments can be used to study late-time vorticity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Such  measurements would provide strong tests of the standard  model expectation that these modes are small, constrain  novel cosmological models that can produce large, late-time  vorticity modes or, if one assumes there are no vorticity  modes, act as a systematics-null test for kSZ measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  This work complements recent work by Bonvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2018),  which explores how redshift space distortions can be used to  constrain vorticity modes, and we find that our approach of-  fers a significantly more powerful means of constraining the  largest scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  © 0000 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='11344v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='CO]  26 Jan 2023  2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Coulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Throughout this paper, we work in the ‘snapshot’ geome-  try described in Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2018) where we compute our  observables by treating the universe as a periodic box at a  distance χ∗ from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This setup allows the intu-  ition to be clearly developed and the results can be extend to  include light-cone effects and a treatment of the curved sky  in an analogous way to Terrana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In Appendix A  we outline how this idea can be framed in the framework of  Terrana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  This paper is strucutred as follows: in Section 2 we review  vorticity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In Sections 3 and 4 we show how vorticity  modes source kSZ and ML anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In Section 5, we de-  scribe how to use these effects to reconstruct the vorticity  perturbations and present a forecast of this approach in Sec-  tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Our conclusions and outlook are presented in Section  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  2 VORTICITY MODES  An arbitrary vector can be decomposed with Helmholtz de-  composition into a part that is curl-free, u, and a part that  is divergence-free, ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Thus, we can decompose the Fourier  space velocity perturbations into divergence, u(k), and curl  (vorticity), ωi(k) contributions as  vi = iki  k u(k) + iωi(k),  (1)  where the curl component satisfies ωiki = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' To proceed it is  useful to represent the divergence-free mode with the polar-  ization vectors  ωi(k) =  �  s  ϵs  i(ˆk)ω(s)(k),  (2)  where ϵs  i are the polarization basis vector and s denotes the  two polarization states (+1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' It is convenient to work in  a rotated basis defined as  ω+ =  1  √  2  �  ω(+1) + ω(−1)�  and ω× =  1  √  2  �  ω(+1) − ω(−1)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (3)  In linear cosmological perturbation theory curl modes are not  sourced by density (scalar) modes and so are expected to van-  ish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This means that these curl modes are a powerful probe of  new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' As an example consider cosmological vector per-  turbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In the synchronous gauge, the perturbed FLRW  metric can be written as  ds2 = a(τ)2×  �  −dτ 2 +  �  (1 − 2φ)δij + 2(D,ij + Ω(i,j) + hT T  ij )  �  dxidxj�  ,  (4)  where φ(x, t) and D(x, t) are scalar perturbations, Ωi(x, t) is  the transverse vector perturbation and hT T  ij (x, t) is the trans-  verse, traceless tensor perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The vector modes source  vorticity perturbations as  ωi(x, τ) = −  ∇2Ω′  i  16πGa2(ϵ0 + p0),  (5)  where a is the scale factor, G is the Gravitational constant, ′  denotes the conformal time derivative, and ϵ0 and p0 are the  energy density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Note that, in matter domination  and without a source, the vector perturbations, Ωi, decay as  a− 5  2 and so their induced vorticity decays as, a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  To have observable vector modes they thus need to be  sourced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In general vector modes can be sourced whenever  there is non-vanishing anisotropic stress such as neutrino ve-  locity isocurvature modes (Lewis 2004b) or primordial mag-  netic field (Lewis 2004a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' More exotic sources include modifi-  cations to general relativity (Battye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2017), topological  defects (Seljak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Durrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Daverio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  2016), a global rotation (Carneiro 2000) and some dark en-  ergy models (Battye & Moss 2006, 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  3 KINETIC SUNYAEV-ZEL’DOVICH EFFECT  The kinetic Sunyaev-Zel’dovich (kSZ) effect occurs when  CMB photons are Thomson scattering off electrons that are  moving with bulk motions, such as in galaxy groups and clus-  ters (Sunyaev & Zeldovich 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This scattering generates  anisotropies in the CMB, without altering the CMB black-  body spectrum, that depend on the electron’s properties via  ∆T kSZ(ˆn)  TCMB  = −  �  dχa(χ)σT ne(χ)v(χˆn) · ˆne−τ(χ),  (6)  where v is the electron velocity, ne is the electron density, χ  is the comoving distance, σT is the Thomson scattering cross-  section and τ is the optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In our simplified geometry  the line-of-sight integral integrates across our box and the  observed angle n describes points of the surface of the box at  location (x, y) = χn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  In Fourier space we can see that the kSZ anisotropies are  given by  ˜T kSZ(ℓ) =iaσT ¯nee−τ  χ2∗  �  dk3  1  (2π)3  dk3  2  (2π)3 (2π)3δ(3)  D  � ℓ  χ∗ − k1 − k2  �  × δe(k1)  �  cos θk2u(k2) − sin θk2ω+(k2)  �  ,  (7)  where δ(3)  D (x) is the 3D Dirac delta function and δe(k) is the  Fourier transform of the electron density fluctuation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  The scalar and vector sources contribute via different angular  dependencies, implying that these effects can be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Further note that the kSZ effect is only sensitive to the ω+  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  4 THE MOVING LENS EFFECT  The moving lens effect is a second-order effect that generates  anisotropies in the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This effect can be understood from  two equivalent perspectives either as part of the Rees-Scamia,  or non-linear integrated Sachs-Wolfe effect, or as lensing by  a moving cluster (Rees & Sciama 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Birkinshaw & Gull  1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Gurvits & Mitrofanov 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Lewis & Challinor 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Yasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Hotinli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The first perspec-  tive describes how the energy of photons is altered as they  enter the potential of clusters moving perpendicular to the  line of sight: if they pass in-front of the cluster they will be  blueshifted as they fall into clusters however as they leave  they are redshifted by a larger amount as the cluster has  moved closer to the line of sight and deepened the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Note that there is no additional blue-shifting from the change  in the potential as the cluster’s velocity is perpendicular to  MNRAS 000, 000–000 (0000)  3  the photon trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The opposite effect occurs for photons  passing behind the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The second perspective can be  understood by considering photon trajectories as viewed in  the cluster rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In this frame a dynamical dipole is  seen and the dipolar distorted photons are deflected by the  cluster’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Boosting back to the CMB rest frame removes  the initial, unlensed dipole leaving a residual signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  The size of the induced anisotropies is given by  ∆T ML(n)  TCMB  = −2  �  dχv⊥ · ∇⊥Φ,  (8)  where Φ is the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Inserting our expres-  sions for the velocities into this, we find that the Fourier space  ML anisotropies are given by  ˜T ML(ℓ) = 2  χ2∗  �  d3k  (2π)3 Φ  � ℓ  χ∗ − k  �  ×  �  u(k)  � ℓ  χ∗ sin θk cos(φk − φℓ) − k sin2 θk  �  + i sin(φk − φℓ)ω×(k)  + ω+(k)  � ℓ  χ∗ cos θk cos(φk − φℓ) − k sin θk cos θk  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (9)  There are several interesting features here: first the standard  scalar velocity modes have a different angular dependence  than the kSZ effect (this is expected as it probes transverse  modes), which is essential for breaking the degeneracies with  the vector kSZ modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Second the vorticity modes likewise  have a different angular dependence, and so in principle could  be separated out from the other ML contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Finally  the ML effect is also sensitive to the cross component (ω×)  of the vector modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' However, this term is geometrically sup-  pressed and as it is thus likely unmeasurable we do not con-  sider it further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  As was shown in Hotinli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2019) the moving-lens  anisotropies are generally smaller than the kSZ anisotropies  – their contirubtion to the power spectrum at ℓ ∼ 3000 is an  order of magnitude lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  5 VELOCITY RECONSTRUCTION  Recent work (Hand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Terrana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Smith  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Cayuso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2021) has shown that by combining  kSZ or ML anisotropies with galaxy position measurements  large scale scalar velocity modes, u(k), can be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  The reconstruction uses a quadratic estimator  �uX(K) =  � d3kd2ℓ  (2π)2 W(k, l) ˜T X(ℓ)δg(k)δ(3)  D  �  K − k − ℓ  χ∗  �  ,  (10)  where δg(k) is the 3D Fourier transform of the galaxy den-  sity field, ˜T X(ℓ) is the 2D Fourier transform of the kSZ  or ML anisotropies and W(k, l) is a set of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This  method is akin to CMB lensing where the lensing poten-  tial is reconstructed from measurements of the primary CMB  anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Usually the weights in this estimator are cho-  sen to obtain a minimum variance, unbiased estimator of  the scalar velocity field i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' ⟨�u(k)u∗(k′)⟩ = (2π)3δ(3)(k −  k′)Pu(k), where Pu(k) is the velocity power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Given  these requirements, the general case of reconstructing field  X, the weights are (Darwish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2020)  W(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' l) = BXδgT ∗(−k − ℓ/χ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' ℓ/χ∗)  P tot  gg (k)Ctot  ℓ  PXX(−k − ℓ/χ∗)×  ��  d2ℓ  (2π)2  BXδgT ∗(−k − ℓ/χ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' ℓ)BXδgT(−k − ℓ/χ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' ℓ)  P tot  gg (k)Ctot  ℓ  P 2  XX(−k − ℓ/χ∗)  �−1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (11)  where BXδgT ∗(−k−ℓ/χ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' ℓ/χ∗) is the bispectrum between  the true field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' and the two fields in the quadratic esti-  mator (δg and T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' and P tot  gg (k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' PXX(k) and Ctot  ℓ  are the  power spectra where the superscript “tot” denotes that the  power spectra include all contributions: signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' instrument  noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' foregrounds and shot-noise – as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' For re-  constructing scalar modes from the kSZ, this is ⟨uδgT⟩ and  is (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2018)  BuδgT −kSZ  �  K, k, ℓ  χ∗  �  =  aσT ¯nee−τ  χ2∗  i cos θK  �  Puu(K)Pge(k) − K  k Pue(K)Pug(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (12)  where ¯ne is the mean electron density at the scattering red-  shift, Pge, Pug and Pue are the galaxy-electron, velocity-  galaxy and velocity-electron power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Similarly for  scalar sources with the ML effect we have  BuδgT −ML  �  K, k, ℓ  χ∗  �  = 2  χ2∗  �Puu(K)  K  PgΦ(k) + PuΦ(K)Pgu(k)  k  �  ×  �  K ℓ  χ∗ sin θK cos(φK − φℓ) − K2 sin2 θK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (13)  It is trivial to form a combined estimator by using the com-  bined bispectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' BuδgT −Comb = BuδgT −kSZ + BuδgT −ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Extending these estimators to reconstruct the vorticity  modes is also straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' A quadratic estimator, ˆω+,  can be formed in an identical manner to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 10 with the vor-  ticity bispectra used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' For a reconstruction from kSZ  anisotropies, this is  Bω+δgT −kSZ  �  K, k, ℓ  χ∗  �  = −aσT ¯nee−τ  χ2∗  i sin θKPeg(k)Pωω(K),  (14)  where Pωω is the vorticity power spectrum and for recon-  struction from ML anisotropies this is  Bω+δgT −ML  �  K, k, ℓ  χ∗  �  = PΦg(k)Pωω(K)  ×  � ℓ  χ∗ cos θK cos(φK − φℓ) − K sin θK cos θK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (15)  As for the scalar case a combined estimator can be formed  by using the sum of the bispectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  However, these vorticity estimators are not orthogonal to  the scalar velocity estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This is problematic as mea-  surements of the vorticity modes will likely be biased by the  scalar modes, which are expected to be much larger than the  vorticity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' To avoid these biases we can construct vortic-  ity estimators which are orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' have zero response,  to the scalar modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  MNRAS 000, 000–000 (0000)  4  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Coulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  The quadratic estimator that is unbiased, minimum vari-  ance and orthogonal to the scalar modes is given by  ˆω+(K) =  �  d2ℓ  (2π)2 T(ℓ)δg  �  K − ℓ  χ∗  �  1  P tot  gg (K −  ℓ  χ∗ )Ctot  ℓ  ×  1  AωωAuu − AωuAuω  �  Auu ¯B∗ωgT − Auω ¯B∗ugT �  , (16)  where  AXY (K) =  �  d2ℓ  (2π)2  ¯BXgT ¯B∗Y gT  P(K −  ℓ  χ∗ )Cℓ  ,  (17)  and ¯BXgT = BXgT /P XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Note that the vector power spec-  trum in the estimator completely cancels with equivalent  terms implicit in the bispectra, so the estimator is indepen-  dent of the shape of the velocity power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  6 FORECAST CONSTRAINTS  Using the formalism from Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2018) we investigate  the constraining power on this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Specifically the noise  power spectrum per mode for the standard estimator is given  by  N kSZ−scalar(K) =  1  Auu(K),  (18)  and for the orthogonal vorticity estimator  N vorticity(K) =  Auu(K)  Aωω(K)Auu(K) − Auω(K)Aωu(K),  (19)  – note that the noise is highly anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Then the uncer-  tainity on the signal power spectrum, with an inverse noise  weighting is  1  σ2(P XX(K)) = V  � K2dK sin θdθKdφK  (2π)3  1  N XX(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (20)  We consider how well a spectroscopic galaxy survey, like DESI  (DESI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2016), in combination with a next  generation CMB survey, such as CMB-S4 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  2016), can measure these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' For the DESI-like survey  we use the experimental setup from Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2018) (also  see Coulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2020), thus we assume a mean redshift  of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='75, a survey volume of 116 (h−1Gpc)3, a galaxy  number density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='7 × 10−4 Mpc−3 and a galaxy bias of  bg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' We use the CMB-S4 configuration described in  Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2019) and compute the noise curves after  component separation with the ILC method (Tegmark & Ef-  stathiou 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  First we consider what can be learnt from only studying the  kSZ anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The expected constraining power is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Enforcing that the vector estimator has no response  to the standard, scalar kSZ contribution comes at a huge  noise cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' As stated earlier, this condition is necessary as the  scalar modes are expected to be significantly larger than any  vector contribution and without it vorticity measurements  can be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The reason for the large noise cost can be seen  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 7 – whilst in principle the two sources have different  angular dependencies and so could be separated, the region  of phase space where there are purely scalar or vector sources  is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Thus the vector modes constraints come from  modes that are close to perpendicular to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' If  the scalar perturbations have an anisotropic power spectrum  of the form ∼ sin θK/ cos θKPuu(|K|), their kSZ signature  from this region of phase space would be highly similar to  the vector modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Our estimator makes no assumptions on  the properties of the scalar perturbations and the large noise  cost arises from accounting for the possibility of anisotropic  scalar modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  For sufficiently low noise measurements it is possible to  disentangle the two contributions with kSZ alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Though a  further complication arises as this separation requires mod-  elling the bispectrum beyond the squeezed limit and, whilst  the squeezed limit can be modeled with high accuracy (Giri &  Smith 2022), accurately modeling the full bispectrum can be  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The ML-effect on its own can similarly be used to  constrain vorticity modes, however it suffers the same issues  and has even larger noise than kSZ based estimator (similar  to that found for scalar velocity modes (Hotinli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  To alleviate these issues we consider using a combined esti-  mator from the kSZ and ML effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The ML and kSZ effects  are sensitive to orthogonal velocities which is the precisely  what is required to completely break the degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The re-  sult of the combined estimator is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The com-  bined estimator provides powerful constraints on the large  scale vorticity that are comparable to those of the scalar ve-  locity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Note that this is achieved despite the low con-  straining power of the ML estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' the contribution of the  estimator is removal of the possibility of anisotropic scalar  modes - as these would lead to a very large ML signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  7 CONCLUSIONS  Large-scale, cosmic vorticity modes in the late time Universe  are an interesting cosmological observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' In the standard  cosmological model these modes should be absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Thus,  searching for these modes is a powerful consistency test of  our model and a discovery of these modes would represent  the detection of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  In this work we show that the kSZ and ML effects, in combi-  nation with a galaxy survey, are an ideal method to search for  these signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Using tomography with galaxy surveys, we are  able to separate vorticity modes from velocity perturbations  from scalar, density modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Tomography enables the differen-  tiation of fluctuations in the velocity field that are parallel to  the velocity vector (density-source velocity modes) from those  perpendicular to the velocity vector, vorticity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This  joint method leads to almost white noise up to the largest  scales accessible with the surveys, allowing for powerful con-  straints on the largest scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Through combining galaxy and  CMB measurements in a quadratic estimator to reconstruct  the large scales, the method should have fewer large scale  systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Especially as the non-trivial parity of the  signal (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2018) provides an extra means to  suppress systematic biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  As a demonstration of the power of this approach, we con-  sider constraining topological defects, a well known source of  vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Within this model the vorticity power spectrum is  given as  Pωω(k) = 14(Gµ)2  k3  �  kτ/12  1 + (kτ/12)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='13  �  (21)  where Gµ characterizes the amplitude of the topical defects  (Urrestilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Using our formalism we find that we  MNRAS 000, 000–000 (0000)  5  10  3  10  2  10  1  100  101  k h/Mpc  10  8  10  6  10  4  10  2  100  102  (P(k)) Mpc3/h3  Scalar - kSZ  Vector - kSZ  Vector - kSZ+ML  Density-sourced Velocity  Topological Defects  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' A Fisher forecast for how well the DESI and CMB-S4 experiments can constrain the vorticity power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' We show the  constraints obtained using the kinetic Sunyaev-Zel’dovich effect alone and combining it with the moving lens effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The joint analysis  of these two effects allows the vorticity modes to be constrained independently of the scalar velocity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' For comparison we show  the constraints obtainable on the scalar velocity power spectrum, the expected ΛCDM scalar velocity power spectrum (dotted) and the  topological defect spectrum given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' We use a logarithimic k-binning of width ∆ ln k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='2  can constraint (Gµ)2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='6 ×10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Whilst topological de-  fects leave strong and distinct signatures on the CMB, so  are constrained significantly better by CMB measurements  (Dunkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Lizarraga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Planck Collab-  oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2014, 2016), constraints obtainable with this  kSZ formalism are more than two orders of magnitude tighter  than those obtainable redshift space distortions (Bonvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This approach provides comparable constraints to  those from CMB curl lensing (Namikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2013) and  by probing vector modes on a redshift slice, rather than the  integrated over the lensing kernel, is highly complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  This highlights the potential of this approach to probe new  regimes in the late-time Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  Here we have only discussed one way to separate these  scalar and vorticity modes, by combing the kSZ and ML esti-  mators, when in practice many methods would be explored to  ensure robust results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' These could include using direct mea-  surements of the density perturbations, such as through clus-  tering measurements, to remove the scalar velocity modes  (that are linearly related to density perturbations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' We fo-  cused on the combined method as it is always necessary to  account for both effects simultaneously to avoid biases from  scalar modes from the other effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This is necessary as sepa-  rating the two effects in CMB maps is not possible with stan-  dard component separation methods, as both effects produce  anisotropies with the same spectrum – the same as the pri-  mary temperature anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' The key other possible con-  taminants to this measurement are the thermal-kinetic SZ  effect (Coulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2020), which can be removed due to its  distinct frequency spectrum, and gravitational lensing, which  can be removed with established delensing techniques (Seljak  & Hirata 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  A second use of the methods outlined here is as a null-test  for the measurement of scalar velocity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' A key goal of  upcoming kSZ measurements is to search for primordial non-  Gaussianity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' M¨unchmeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2019) and one of the  biggest challenges for such analyses is mitigating systematic  biases (Pullen & Hirata 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Rezaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' If we as-  sume there is no new physics, then the vector-mode velocity  estimator can be used as a null-test for these analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This  is analogous to how curl lensing (Namikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 2012) is  ubiquitously used to test for systematic effects in weak lens-  ing analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  MNRAS 000, 000–000 (0000)  6  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Coulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  APPENDIX A: LIGHT CONE GEOMETRY  In this work we focused on discussing this analysis in the  ‘snapshot’ geometry picture, however equivalent expressions  can be derived for the ‘light cone’ picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' From this perspec-  tive we no longer work in a periodic box at a single redshift  but instead in terms of spherical harmonic coefficients on the  full sky and account for a more complex line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  In Terrana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' (2018) an optimal,  quadratic estimator is derived for the remote dipole field,  veff, from galaxy data in redshifts bins, denoted by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This  estimator is  ˆ  veff  α,ℓm =  Nαℓ  �  ℓ1m1ℓ2m2  (−1)mΓkSZ  ℓ1ℓ2ℓα  � ℓ1  ℓ2  ℓ  m1  m2  −m  � aT  ℓ1m1δα  g,ℓ2m2  CT T  ℓ  C  δgδg  αℓ2  (A1)  where  ˆ  veff  α,ℓm is the estimated dipole spherical harmonic coeffi-  cients in a given redshift bin, Nαℓ is a normalization constant,  ΓkSZ  ℓ1ℓ2ℓα is a coupling matrix which combined with the Wigner  3j symbol is the equivalent of the bispectrum in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 10-11,  aT  ℓ1m1 and δα  g,ℓ2m2 are the spherical harmonic coefficients of  the CMB map and galaxy density map and CXX  ℓ  are the  power spectra of those maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This can be thought of as an  equivalent expression to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  The expected contribution to this estimator from a scalar  mode is given by  veff  α,ℓm =  �  dχW α(χ)  �  d3k  (2π)3  4πiℓ  2ℓ + 1Φ(k)Y ∗  ℓm(ˆk)  × [ℓjℓ−1(kχ) − (ℓ + 1)jℓ+1(kχ)]  (A2)  where jℓ(x) are the spherical Bessel functions and W α(χ)  defines the redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Computing the equivalent expression  for vector modes gives  veff  α,ℓm =  �  dχW α(χ)  �  d3k  (2π)3  4πiℓ+1  2ℓ + 1 ωs(k)−sY ∗  ℓm(ˆk)  ×  �  ℓ(ℓ + 1)  2  [jℓ−1(kχ) + jℓ+1(kχ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  (A3)  The key difference between the two contributions is in the  spherical Bessel projections – one plus the other minus – that  is equivalent to the sin and cos terms in the vector/scalar kSZ  estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' Thus the scalar and vector modes can be differen-  tiated by combining observations at a range of redshift bins  with different weightings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors are grateful for fruitful discussions with Colin  Hill, David Spergel, Emmanuel Schaan, Anthony Challinor  and Oliver Philcox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' This work was supported in part by  World Premier International Research Center Initiative (WPI  Initiative), MEXT, Japan, and JSPS KAKENHI Grants  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' JP20J22055, JP20H05850, JP20H05855, JP19H00677,  and by Basic Research Grant (Super AI) of Institute for AI  and Beyond of the University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
+page_content=' KA is supported by  JSPS Overseas Research Fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FIT4oBgHgl3EQfuivw/content/2301.11344v1.pdf'}
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+Springer Nature 2021 LATEX template
+Geometric Ergodicity in Modified Variations
+of Riemannian Manifold and Lagrangian
+Monte Carlo
+James A. Brofos1*, Vivekananda Roy2 and Roy R. Lederman1
+1Department of Statistics and Data Science, Yale University, 24
+Hillhouse Ave., New Haven, 06510, Connecticut, USA.
+2Department of Statistics, Iowa State University, 2438 Osborn
+Drive, Ames, 50011, Iowa, USA.
+*Corresponding author(s). E-mail(s): james.brofos@yale.edu;
+Contributing authors: vroy@iastate.edu; roy.lederman@yale.edu;
+Abstract
+Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte
+Carlo (LMC) have emerged as powerful methods of Bayesian infer-
+ence. Unlike Euclidean Hamiltonian Monte Carlo (EHMC) and the
+Metropolis-adjusted Langevin algorithm (MALA), the geometric ergod-
+icity of these Riemannian algorithms has not been extensively studied.
+On the other hand, the manifold Metropolis-adjusted Langevin algo-
+rithm (MMALA) has recently been shown to exhibit geometric ergodicity
+under certain conditions. This work investigates the mixture of the LMC
+and RMHMC transition kernels with MMALA in order to equip the
+resulting method with an “inherited” geometric ergodicity theory. We
+motivate this mixture kernel based on an analogy between single-step
+HMC and MALA. We then proceed to evaluate the original and mod-
+ified transition kernels on several benchmark Bayesian inference tasks.
+Keywords: Markov chain Monte Carlo, Riemannian Manifold Hamiltonian
+Monte Carlo, Statistical Computing, Lagrangian Monte Carlo
+1
+arXiv:2301.01409v1  [stat.ME]  4 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+1 Introduction
+Bayesian inference seeks to combine subjective sources of information with
+observational data. By specifying one’s prior beliefs and correctly capturing
+sources of uncertainty in a stochastic system, one may employ the Bayesian
+approach in order to capture and reason about uncertainty under the posterior
+distribution. Formally, the posterior distribution is a probability distribution
+Π : B(Rm) → [0, 1] with density π : Rm → R+. As a practical matter in
+Bayesian inference, one is concerned with the computation of expectations with
+respect to Π of integrable functions h : Rm → R; that is, we wish to compute
+E
+q∼Π[h(q)]. Since π is generally intractable, these expectations are not available
+in closed form and Monte Carlo methods based on samples from π are often
+used to approximate these means. For instance, one may compute the mean
+and variance of the posterior by employing appropriate choices of expectation.
+Therefore, a central problem in Bayesian inference is the generation of samples
+from the posterior distribution.
+One popular Monte Carlo method is based on approximate samples from
+the posterior distribution generated via the technique of Markov chain Monte
+Carlo (MCMC). In MCMC, given some initial state of the chain, q0, a sequence
+of random variables (q1, q2, . . .) is generated inductively: the conditional dis-
+tribution qn|qn−1, . . . , q0 obeys the Markov property. Under certain desirable
+regularity conditions, the random variables qn will have a distribution which,
+asymptotically, converges to Π. Establishing the rate of convergence, or at
+least an upper bound on that rate, serves to quantify how quickly the chain
+(q1, q2, . . .) mixes toward the target distribution. In the case where the rate of
+convergence is geometrically fast, the Markov chain enjoys a strong stability
+property in that the estimator n−1 �n
+i=1 h(qi) of
+E
+q∼Π[h(q)] can be equipped
+with a central limit theorem under additional technical assumptions. This cen-
+tral limit theorem has important practical implications in that it allows the use
+of asymptotically valid standard errors of MCMC estimates, which, in turn,
+can be used to decide how long to run the Markov chains [Roy, 2020].
+One of the most popular MCMC methods is Euclidean Hamiltonian Monte
+Carlo (EHMC) [Neal, 2010b]. In EHMC, one computes approximate solutions
+to Hamilton’s equations of motion using numerical integrators, taking care to
+ensure that the resulting Markov chain satisfies detailed balance in a phase-
+space consisting of the position variables q ∈ Rm and auxiliary momentum
+variables p ∈ Rm. The rate of convergence of HMC was established by Liv-
+ingstone et al. [2016] in the presence of conditions on the log-density [see
+also Durmus et al., 2020]. Traditional EHMC, however, struggles in distri-
+butions that exhibit multiple spatial scales [Pourzanjani and Petzold, 2019,
+Betancourt, 2012]. This observation led to the development of geometric meth-
+ods of HMC, specifically the Riemannian manifold Hamiltonian Monte Carlo
+(RMHMC) method of Girolami and Calderhead [2011].
+Unlike EHMC, RMHMC adapts to the second-order structure of the poste-
+rior, which allows it to align its proposals in the direction of the posterior that
+
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+Modified Riemannian Manifold and Lagrangian Monte Carlo
+3
+exhibits the greatest local variation. However, the sophisticated form of the
+Hamiltonian employed in RMHMC necessitates the use of complex numerical
+integrators that are significantly more expensive than the numerical integrator
+employed in EHMC. This concern was partially alleviated in the introduction
+of Lagrangian Monte Carlo (LMC) in Lan et al. [2015], which was able to con-
+struct a MCMC method with a more efficient numerical integration procedure,
+while simultaneously continuing to take advantage of second-order geometric
+knowledge in order to efficiently traverse the posterior. Both RMHMC and
+LMC can be viewed as generalizations of the Euclidean Hamiltonian Monte
+Carlo algorithm, which incorporate second-order geometric information about
+the posterior into the Markov chain transition kernel. Incorporating second-
+order information allows these geometric methods of MCMC to explore the
+typical region of the target distribution more efficiently.
+However, neither RMHMC nor LMC have, as of yet, been equipped with
+a geometric ergodicity theory. The focus in this work is in establishing the
+geometric ergodicity of modified versions of RMHMC and LMC. In imple-
+mentations of RMHMC and LMC it is common to randomize the number of
+integration steps, and to take one-step with positive probability. In the case of
+Euclidean HMC, the equivalence between single-step HMC and the Metropolis-
+adjusted Langevin algorithm (MALA) is critical for establishing geometric
+ergodicity of HMC in Livingstone et al. [2016]. Our approach is two-fold. First,
+we propose a simple modification to both RMHMC and LMC which is moti-
+vated by a unique correspondence between single-step Euclidean HMC and
+MALA. In particular, we propose that instead of applying the transition kernel
+corresponding to RMHMC or LMC with a single integration step, one instead
+applies the transition kernel of the manifold Metropolis-adjusted Langevin
+algorithm (MMALA). Recent work in Roy and Zhang [2023] gave conditions
+under which MMALA is geometrically ergodic. We may imbue these modi-
+fied variations of RMHMC and LMC with a geometric ergodicity theory via
+inheritance once one establishes that the RMHMC and LMC transition kernels
+(with a fixed, non-random number of integration steps) are reversible in the
+required sense. This construction is described in section 3. In section 4 we pro-
+ceed to a numerical evaluation of the proposed modification of RMHMC and
+LMC, with special attention given to the probability of applying the MMALA
+transition kernel. We begin, however, in section 2 with an overview of required
+mathematical concepts from numerical integration and Markov chains.
+2 Preliminaries
+In section 2.1 we review the generalized leapfrog and Lagrangian leapfrog
+employed in geometric MCMC methods for performing Bayesian inference.
+Section 2.2 covers Markov chains based on involutions as well as those based
+on discretizations of Langevin diffusions. We also discuss geometric ergodicity,
+mixture transition kernels, and the equivalence between single-step HMC and
+MALA in the Euclidean regime.
+
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+4
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+2.1 Numerical Integrators
+Hamiltonian and Lagrangian mechanics are equivalent, with the momentum
+and velocity being related by the Legendre transform [Marsden and Ratiu,
+2010]. In this section, we will consider numerical integrators of these equations
+of motion. Throughout our discussion, we will consider a Hamiltonian function
+of the following form.
+Definition 1 Let π : Rm → R+ be a smooth probability density and let G : Rm →
+PD(m). The Riemannian Hamiltonian is the function H : Rm × Rm → R defined by
+H(q, p) = − log π(q) + 1
+2 log det(G(q)) + 1
+2p⊤G−1(q)p.
+(1)
+Moreover, assuming
+�
+Rm
+�
+Rm exp(−H(q, p)) dq dp < +∞, the Riemannian density
+is the probability density π(q, p) ∝ exp(−H(q, p)).
+Remark 1 The term 1
+2 log det(G(q)) + 1
+2p⊤G−1(q)p appearing in the definition of
+the function H in eq. (1) is chosen such that the conditional density π(q, p) satisfies
+π(p|q) = Normal(0, G(q)).
+(2)
+It is easily seen that the q-marginal density of π(q, p) obtained by marginalizing out
+p is π(q).
+Hamiltonian functions H : Rm ×Rm → R produce Hamilton’s equations of
+motion which, by definition, are solutions to the coupled differential equations
+˙qt = ∇pH(qt, pt)
+(3)
+˙pt = −∇qH(qt, pt).
+(4)
+Except in special cases, a closed-form solution for the map t �→ (qt, pt) ∈
+Rm × Rm does not exist; this necessitates the use of numerical integrators in
+order to approximate the equations of motion obeying eqs. (3) and (4). One
+such example is the generalized leapfrog method, which we now define.
+Definition 2 The generalized leapfrog integrator with step-size ϵ ∈ R applied to the
+Riemannian Hamiltonian in eq. (1) is the map (q, p) �→ ˆΦϵ(q, p) defined by,
+˘p = p − ϵ
+2∇qH(q, ˘p)
+(5)
+˜q = q + ϵ
+2 (∇pH(q, ˘p) + ∇pH(˜q, ˘p))
+(6)
+= q + ϵ
+2
+�
+G−1(q) + G−1(˜q)
+�
+˘p
+(7)
+˜p = ˘p − ϵ
+2∇qH(˜q, ˘p)
+(8)
+= p − ϵ
+2∇qH(q, ˘p) − ϵ
+2∇qH(˜q, ˘p),
+(9)
+where ˆΦϵ(q, p) = (˜q, ˜p).
+
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+Modified Riemannian Manifold and Lagrangian Monte Carlo
+5
+Remark 2 One can prove [Leimkuhler and Reich, 2005, Hairer et al., 2006] that
+the generalized leapfrog integrator is a symplectic transformation and that therefore
+|det(∇Φ(q, p))| = 1.
+Remark 3 We must now give a remark about notation. Numerical integration plays
+a central role in our analysis, and we give particular attention to multiple steps of
+integration. At the same time, we will also discuss Markov chains, which also consist
+of multiple steps. As an attempt to differentiate these two notions of step, we will
+use a lower index to refer to steps of numerical integration, whereas we will use
+upper indices to denote Markov chain steps. Given initial data (q0, p0) ∈ Rm × Rm,
+we denote k-steps of the generalized leapfrog integrator by (qk, pk) = ˆΦkϵ (q0, p0) =
+ˆΦϵ(qk−1, pk−1). Similarly, we let ˘pk = pk−1 − ϵ
+2∇qH(qk−1, ˘pk), which we call the
+intermediate momentum at the k-th step.
+A Hamiltonian of the form in eq. (1) may be transformed into a Lagrangian
+as
+L(q, ˙q) = − log π(q) + 1
+2 log det(G(q)) − 1
+2 ˙q⊤G(q) ˙q.
+(10)
+As noted at the beginning of this subsection, Lagrangian and Hamiltonian
+mechanics are formally equivalent, with the momentum p ∈ Rm being related
+to the velocity ˙q ∈ Rm according to the Legendre transformation p = G(q) ˙q.
+The Lagrangian produces equivalent equations of motion called the Euler-
+Lagrange equations as
+∇qL(qt, ˙qt) = d
+dt∇ ˙qL(qt, ˙qt).
+(11)
+One advantage possessed by the Lagrangian formalism over the Hamiltonian
+approach is that one may identify explicit numerical integrators [Lan et al.,
+2015] of the equations of motion, such as the Lagrangian leapfrog.
+Definition 3 The Lagrangian leapfrog integrator with step-size ϵ ∈ R applied to the
+Riemannian Hamiltonian in eq. (1) is the map (q, p) �→ ˜Φϵ(q, p) defined by,
+v = G−1(q)p
+(12)
+˘v = v − ϵ
+2G−1(q)∇qU(q) − ϵ
+2Ω(q, v)˘v
+(13)
+˜q = q + ϵ˘v
+(14)
+˜v = ˘v − ϵ
+2G−1(˜q)∇qU(˜q) − ϵ
+2Ω(˜q, ˘v)˜v
+(15)
+= v − ϵ
+2G−1(q)∇qU(q) − ϵ
+2Ω(q, v)˘v − ϵ
+2G−1(˜q)∇qU(˜q) − ϵ
+2Ω(˜q, ˘v)˜v
+(16)
+˜p = G−1(˜q)˜v
+(17)
+where (˜q, ˜p) = ˜Φϵ(q, p) and
+U(q) = − log π(q) + 1
+2 log det(G(q))
+(18)
+
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+6
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+Ωij(q, v) =
+m
+�
+k=1
+Γi
+kj(q)vk
+(19)
+Γi
+kj(q) = 1
+2
+m
+�
+l=1
+G−1
+kl (q)
+� ∂
+∂qi
+Glj(q) +
+∂
+∂qj
+Gil(q) − ∂
+∂ql
+Gij(q)
+�
+.
+(20)
+The Jacobian determinant of the transformation ˜Φϵ (defined in definition 3)
+is,
+|det(∇˜Φϵ(q, p))| = det(G−1(q))det(G(˜q))|det(Id + ϵΩ(q, v)/2)det(Id − ϵΩ(˜q, ˜v)/2)|
+|det(Id + ϵΩ(˜q, ¯v)/2)det(Id − ϵΩ(q, ¯v)/2)|.
+(21)
+Remark 4 Given initial data (q0, p0) ∈ Rm×Rm, we denote k-steps of the generalized
+(or Lagrangian) leapfrog integrator by (qk, pk) = ˜Φkϵ (q0, p0) = ˜Φϵ(qk−1, pk−1).
+In the case where G is a constant function of q, the generalized leapfrog
+integrator reduces to the standard leapfrog integrator, which is defined as
+follows.
+Definition 4 Let π : Rm → R+ be a smooth probability density. A Euclidean
+Hamiltonian is a smooth map H : Rm × Rm → R of the form,
+H(q, p) = − log π(q) + 1
+2p⊤G−1p,
+(22)
+where G ∈ PD(m).
+Definition 5 Consider a Euclidean Hamiltonian as defined in definition 4. The
+Euclidean leapfrog integrator is the map (q, p) �→ ˇΦϵ(q, p) defined by
+˘p = p + ϵ
+2∇ log π(q)
+(23)
+˜q = q + ϵG−1˘p
+(24)
+˜p = ˘p + ϵ
+2∇ log π(˜q),
+(25)
+where ˇΦϵ(q, p) = (˜q, ˜p).
+The Euclidean leapfrog integrator is the de-facto standard numerical inte-
+grator employed in EHMC. This is because of its accuracy and computational
+efficiency; in contrast to the generalized leapfrog integrator in definition 2, the
+Euclidean leapfrog is fully explicit. Nevertheless, the generalized leapfrog and
+Euclidean leapfrog are exactly equivalent when applied to a Hamiltonian in
+the form of definition 4; this is made precise in the following result.
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+7
+Proposition 1 When the generalized leapfrog (definition 2) or Lagrangian leapfrog
+(definition 3) is applied to a Euclidean Hamiltonian in the form of definition 4 (in
+the sense that we take G(q) ≡ G for every q ∈ Rm), the resulting map is equivalent
+to the Euclidean leapfrog method in definition 5.
+Proof In the case of the generalized leapfrog integrator, this is immediate from the
+fact that ∇qH(q, p) = −∇ log π(q) which does not depend on p and ∇pH(q, p) = Gp
+which does not depend on q.
+In the case of the Lagrangian leapfrog, the form of the Euclidean Hamiltonian
+means that Ω(q, v) is uniformly zero (because all of the derivatives of the metric
+vanish). The Lagrangian leapfrog integrator may be re-expressed as
+v = G−1p
+(26)
+˘v = v + ϵ
+2G−1∇ log π(q)
+(27)
+= G−1˘p
+(28)
+˜q = q + ϵ˘v
+(29)
+= q + ϵG−1˘p
+(30)
+˜v = ˘v + ϵ
+2G−1∇ log π(˜q)
+(31)
+= G−1˜p,
+(32)
+where ˘p and ˜p are as in eqs. (23) and (25).
+□
+2.2 Markov Chains
+The purpose of this section is to review fundamentals of Markov chains, includ-
+ing their construction, convergence properties, and central limit theorems.
+Markov chains are stochastic processes that may be inductively defined by
+their transition kernel. Since the transition kernel yields a probability measure
+depending only on the current state, one sees by inspection that the Markov
+chain satisfies the Markov property.
+Definition 6 A Markov chain transition kernel on Rm is a function K : Rm ×
+B(Rm) → [0, 1] such that (i) for q ∈ Rm, K(q, ·) is a probability measure and (ii)
+for fixed A ∈ B(Rm) the function q �→ K(q, A) is measurable.
+Within the context of simulation-based inference and mainly Bayesian
+inference, one is interested in constructing a Markov chain which converges
+to a specified target probability distribution; as noted in the introduction,
+this target distribution is typically known by its density up to a normalizing
+constant. We now define two notions of convergence.
+Definition 7 Given a Markov chain transition kernel K, a Markov chain is a
+sequence of random variables defined inductively by qn+1|qn ∼ K(qn, ·).
+
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+8
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+Definition 8 The total variation distance between two probability measures Π :
+B(Rm) → [0, 1] and Ξ : B(Rm) → [0, 1] is defined by
+∥Π − Ξ∥TV = 2
+sup
+A∈B(Rm)
+|Π(A) − Ξ(A)|.
+(33)
+Definition 9 A Markov chain with transition kernel K is said to be ergodic if
+lim
+n→∞ ∥Kn(q, ·) − Π(·)∥TV = 0,
+(34)
+where for A ∈ B(Rm), Kn(q, A) = Pr(qn ∈ A|q0 = q) is the n-step Markov transition
+probability.
+It will turn out that one can establish ergodicity of a Markov chain if
+one can establish three separate properties: irreducibility, aperiodicity, and
+stationarity. We now define each of these concepts.
+Definition 10 A Markov chain transition kernel K is said to be Π-irreducible if for
+every set A ∈ B(Rm) with Π(A) > 0 there exists n ∈ N for which Kn(q, A) > 0 for
+all q ∈ Rm.
+Definition 11 Given a Markov chain transition kernel K : Rm × B(Rm) → [0, 1],
+a set C ⊆ Rm is called small if there exists a n ∈ N, a δ > 0, and a probability
+measure ν : B(Rm) → [0, 1] such that for any q ∈ C and A ∈ B(Rm) we have
+Kn(q, A) ≥ δ · ν(A).
+We may therefore say that the set C is (n, δ, ν)-small.
+Definition 12 A Markov chain transition kernel K is called aperiodic if there exists
+a small set C for which the greatest common divisor of the set
+�
+n′ ∈ N : ∃ δ > 0 and probability measure ν such that C is (n′, δ, ν)-small
+�
+(35)
+is one.
+Definition 13 A Markov chain with transition kernel K is said to be stationary for
+the probability measure Π if for any Borel set A ∈ B(Rm) we have
+E
+q∼ΠK(q, A) =
+Π(A).
+Definition 14 A Markov chain with transition kernel K is called geometrically
+ergodic if there exists ρ ∈ (0, 1) and function V : Rm → R+ such that
+∥Kn(q, ·) − Π(·)∥TV ≤ V (q)ρn.
+(36)
+It is clear from the definitions that geometric ergodicity is a stronger form of
+convergence than mere ergodicity. In the latter case, however, simple conditions
+under which a Markov chain is ergodic can be provided.
+
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+Modified Riemannian Manifold and Lagrangian Monte Carlo
+9
+Theorem 2 (Tierney [1994]) Suppose that K is a Markov chain transition kernel
+that is Π-irreducible, aperiodic, and for which Π is the stationary probability measure.
+Then K produces an ergodic Markov chain.
+However, in addition to giving an upper bound on the mixing of the Markov
+chain into the target probability measure, geometric ergodicity is also useful
+for establishing a central limit theorem for expectations computed from the
+sequence (q1, q2, . . .). We make this precise as follows.
+Theorem 3 (Meyn and Tweedie [1993]) Let h : Rm → R be a function for which
+E
+q∼Π|h(q)|2+γ < ∞ for some γ > 0. Let K be a Markov chain transition kernel that
+is aperiodic, Π-irreducible, and for which Π is the unique stationary distribution.
+Assume further that K converges geometrically to Π. Define SK[h] = n−1 �n
+i=1 h(qi)
+where qi+1|qi ∼ K(qi, ·). Then, as n → ∞,
+√n
+�
+SK[h] − E
+q∼Π [h(q)]
+�
+d→ Normal(0, τ2
+h)
+(37)
+where τ2
+h is a constant, less than infinity, that depends on h (and K). The quantity
+τ2
+h has a closed-form given by
+τ2
+h = Var
+q∼Π (h(q)) + 2
+∞
+�
+k=1
+Cov
+q∼Π
+�
+h(q), h(qk)
+�
+.
+(38)
+Establishing that a Markov chain has Π as a stationary distribution is often
+easily achieved by showing that the chain satisfies detailed balance with respect
+to Π. As we require detailed balance when discussing the marginal transition
+kernels of RMHMC and LMC in section 3, we define this notion now.
+Definition 15 A Markov chain transition kernel is said to satisfy detailed balance
+with respect to the probability distribution Π (equivalently, the Markov chain is
+called reversible with respect to Π) if for any sets Q, Q′ ∈ B(Rm),
+�
+Q
+K(q, Q′) Π(dq) =
+�
+Q′ K(q, Q) Π(dq).
+(39)
+Remark 5 When detailed balance holds for a Markov chain transition kernel K, it
+follows that Π is the stationary probability measure (definition 13) of the Markov
+transition kernel K.
+2.2.1 Metropolis-Hastings Kernels and Generalized Langevin
+Algorithms
+In this section we review Metropolis-Hastings Markov chains. We begin
+by formally defining these objects before proceeding to a special case of
+Metropolis-Hastings method based on Langevin diffusion, which recalls the
+constructions considered in Roy and Zhang [2023].
+
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+10
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+Definition 16 (Metropolis-Hastings Algorithm) Let π : Rm → R+ be a probabil-
+ity density and, for each q ∈ Rm, let ˜π(·|q) : Rm → R+ be a probability density
+depending on q. The Metropolis-Hastings transition kernel with proposal density
+˜π(·|q) is,
+K(q, A) =
+�
+A
+min
+�
+1, π(˜q)˜π(q|˜q)
+π(q)˜π(˜q|q)
+�
+˜π(˜q|q) d˜q
++
+�
+1 −
+�
+Rm min
+�
+1, π(˜q)˜π(q|˜q)
+π(q)˜π(˜q|q)
+�
+˜π(˜q|q) d˜q
+�
+1 {q ∈ A} ,
+(40)
+for q ∈ Rm and A ∈ B(Rm).
+Definition 17 (Generalized Metropolis-Adjusted Langevin Algorithm) Fix ϵ ∈ R \
+{0}. Let cϵ : Rm → Rm and let A : Rm → PD(m). The generalized Metropolis-
+adjusted Langevin algorithm is an instance of the Metropolis-Hastings algorithm
+with proposal density
+˜π(˜q|q) = Normal(˜q; cϵ(q), ϵ2A(q)).
+(41)
+We denote the Markov chain transition kernel of the generalized Metropolis-adjusted
+Langevin algorithm with step-size ϵ by Jϵ : Rm × B(Rm) → [0, 1].
+Definition 18 (Riemannian Manifold Metropolis-Adjusted Langevin Algorithm) In
+the special case where
+cϵ(q) = ϵ2
+2 A(q)∇ log π(q) + ϵ2
+2 Γ(q)
+(42)
+Γi(q) =
+m
+�
+j=1
+∂
+∂qj
+Aij(q)
+(43)
+we call the resulting method the Riemannian manifold Metropolis-adjusted Langevin
+algorithm (MMALA).
+Remark 6 The form of the proposal in eq. (42) is based on an Euler-Maruyama
+discretization of a Langevin diffusion on a manifold with Riemannian metric G(q) =
+A−1(q); see Xifara et al. [2014] for details. In summary, eq. (42) is the drift component
+of the Euler-Maruyama discretization (with step-size ϵ2) applied to the following
+stochastic differential equation:
+dXt = 1
+2G−1(Xt)∇ log π(Xt) dt + Γ(Xt) dt + G−1/2(Xt) dBt,
+(44)
+where Bt is Euclidean Brownian motion at time t ∈ R. When G(q) is the Fisher
+information matrix, the term G−1(Xt)∇ log π(Xt) can be identified as the natural
+gradient of the function log π under the geometry generated by the Fisher metric
+[Amari and Nagaoka, 2000] at position Xt. On the other hand, the term Γ(Xt) dt +
+G−1/2(Xt) dBt corresponds to manifold Brownian motion, since its infinitesimal
+generator is the Laplace-Beltrami operator on the manifold [Hsu, 2002]. Therefore,
+as the stochastic differential equation in eq. (44) consists of a term comprising the
+manifold gradient of a log-density and another term comprising manifold Brownian
+motion, it is called the Riemannian Langevin equation in correspondence with the
+Euclidean case.
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+11
+Definition 19 (Simplified Riemannian Manifold Metropolis-Adjusted Langevin
+Algorithm) Computing the function Γ in definition 18 may be inconvenient to eval-
+uate. It can be ignored while still yielding a Markov chain transition kernel, giving
+the special case,
+cϵ(q) = ϵ2
+2 A(q)∇ log π(q).
+(45)
+The resulting method is called the simplified Riemannian manifold Metropolis-
+adjusted Langevin algorithm, which was considered by Girolami and Calderhead
+[2011].
+We adopt the abbreviation SMALA to refer to the simplified Metropolis-
+adjusted Langevin algorithm.
+Remark 7 There are three common choices of the function A. The first is that A
+is a constant function, in which case we simply write A ∈ PD(m). A second option
+is that A is chosen as the inverse of the sum of the Fisher information matrix and
+the negative Hessian of the log-prior, which can capture second-order geometry of
+both the likelihood and the prior; the use of the inverse of the sum of the Fisher
+information and the negative Hessian of the log-prior as a preconditioner is the
+approach advocated by Girolami and Calderhead [2011]. A third option is that A is
+the inverse of the SoftAbs metric, which is a smooth transformation of the Hessian
+of the log-density of the target distribution; for details see Betancourt [2012].
+2.2.2 Geometric Methods of Bayesian Inference
+Definition 20 (Involutive Monte Carlo [Neklyudov et al., 2020]) Let Φ : Rm → Rm
+be a smooth involution and let π : Rm → R+ be a probability density on Rm. The
+Markov chain transition kernel of involutive Monte Carlo with target density π is
+K(q, A) = min
+�
+1, π(Φ(q))
+π(q)
+|det(∇Φ(q))|
+�
+1 {Φ(q) ∈ A}
++
+�
+1 − min
+�
+1, π(Φ(q))
+π(q)
+|det(∇Φ(q))|
+��
+1 {q ∈ A} ,
+(46)
+for q ∈ Rm and A ∈ B(Rm).
+It is easily verified that the transition kernel of involutive Monte Carlo
+satisfies detailed balance with respect to the density π. Involutive Monte Carlo
+gives rise to two special transition kernels corresponding to RMHMC and LMC.
+Definition 21 Let ˆΦϵ be as in definition 2 and let H be as in definition 1. The
+involution of Riemannian manifold Hamiltonian Monte Carlo with step-size ϵ and k
+integration steps is,
+Φ = F ◦ ˆΦϵ ◦ · · · ◦ ˆΦϵ
+�
+��
+�
+k times
+,
+(47)
+where F(q, p) = (q, −p). The target density of Riemannian manifold Hamiltonian
+Monte Carlo is π(q, p) ∝ exp(−H(q, p)). The transition kernel of involutive Monte
+
+Springer Nature 2021 LATEX template
+12
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+Carlo (definition 20) with involution Φ given in eq. (47) is called the transition kernel
+of Riemannian manifold Hamiltonian Monte Carlo with step-size ϵ and k integration
+steps.
+Remark 8 It can be shown that if Φ is an invertible function and if F ◦ Φ is an
+involution that F ◦ Φ ◦ · · · · Φ is also an involution. This explains why eq. (47) is also
+an involution. For further details, see proposition 15.
+In the special case where G is a constant function of q, the resulting Markov
+chain is called Euclidean Hamiltonian Monte Carlo.
+Definition 22 Let ˇΦϵ be as in definition 5 and let H be as in definition 4. The
+involution of Euclidean Hamiltonian Monte Carlo (EHMC) with step-size ϵ and k
+integration steps is,
+Φ = F ◦ ˇΦϵ ◦ · · · ◦ ˇΦϵ
+�
+��
+�
+k times
+,
+(48)
+where F(q, p) = (q, −p). The target density of Euclidean Hamiltonian Monte Carlo
+is π(q, p) ∝ exp(−H(q, p)). The transition kernel of involutive Monte Carlo (defini-
+tion 20) with involution Φ given in eq. (48) is called the transition kernel of Euclidean
+Hamiltonian Monte Carlo with step-size ϵ and k integration steps.
+Definition 23 Let ˜Φϵ be as in definition 3 and let H be as in definition 1. The
+involution of Lagrangian Monte Carlo with step-size ϵ and k integration steps is,
+Φ = F ◦ ˜Φϵ ◦ · · · ◦ ˜Φϵ
+�
+��
+�
+k times
+,
+(49)
+with F(q, p) = (q, −p). The target density of Lagrangian Hamiltonian Monte Carlo
+is π(q, p) ∝ exp(−H(q, p)). We call this the transition kernel of Lagrangian Monte
+Carlo with step-size ϵ and k integration steps.
+Remark 9 Unlike the transition kernel employed in Riemannian manifold Hamil-
+tonian Monte Carlo, the Jacobian determinant of the involution appearing in
+Lagrangian Monte Carlo must be computed through k applications of eq. (21), where,
+as before, k is the number of integration steps.
+In the special case where G is a constant function of q, Riemannian
+manifold Hamiltonian Monte Carlo and Lagrangian Monte Carlo produces a
+transition kernel that is equivalent to Euclidean Hamiltonian Monte Carlo with
+constant mass matrix G.
+Proposition 4 The transition kernels of RMHMC (or LMC) with step-size ϵ and k
+integration steps when applied to the Euclidean Hamiltonian in definition 4 (in the
+sense that we take G(q) ≡ G for every q ∈ Rm) are both equivalent to the EHMC
+transition kernel.
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+13
+Proof It follows from proposition 1 that the proposals produced by RMHMC and
+LMC are identical to EHMC. It remains to be verified that the acceptance decisions
+are identical, too. This can be verified by observing that the Riemannian Hamiltonian
+in eq. (1) is the Euclidean Hamiltonian in eq. (22) up to an additive constant (since
+G does not depend on position). Hence, the Euclidean and Riemannian densities are
+proportional to one another, and the acceptance probabilities will be identical.
+□
+We now elaborate on the connection between EHMC with a single-step and
+the Metropolis-adjusted Langevin algorithm: in particular, these two meth-
+ods can be constructed to be exactly equivalent in q-space, producing equal
+proposals and identical acceptance probabilities.
+Proposition 5 (Neal [2010a]) Let A ∈ Rm×m be a fixed positive definite matrix.
+Consider EHMC with the Hamiltonian H(q, p) = − log π(q) + 1
+2p⊤A−1p and
+MALA with proposal distribution ˜q|q ∼ Normal
+�
+q + ϵ2
+2 A−1∇ log π(q), ϵ2A−1�
+. The
+marginal transition kernel of EHMC with step-size ϵ and a single integration step can
+be constructed to be exactly equivalent to the transition kernel of MALA.
+A proof is provided in section A.1. The analysis of mixture transition ker-
+nels will be central to our analysis. In standard implementations of Riemannian
+manifold Hamiltonian and Lagrangian Monte Carlo, it is typical to random-
+ize the number of integration steps. This is done to avert any Markov chain
+pathologies (such as irreducibility failures) that may result from using a fixed
+number of integration steps. We therefore consider Markov chain transition
+kernels which are mixtures.
+Definition 24 Let Kϵ,k : (Rm × Rm) × B(Rm × Rm) → R+ be the Markov chain
+transition kernel of RMHMC (or LMC) with step-size ϵ and k integration steps.
+Let ˜Kϵ,k be the marginal transition kernel (defined in lemma 6) of Kϵ,k. Define the
+mixture transition kernel of RMHMC (or LMC) to be
+˜Kϵ =
+∞
+�
+k=1
+αk ˜Kϵ,k,
+(50)
+where (α1, α2, . . .) is a probability vector.
+The fact that EHMC, RMHMC, and LMC all satisfy detailed balance in
+(q, p)-space causes us to examine the progression of q-states alone in between
+Gibbs resampling steps of the momentum. This leads to marginal Markov chain
+transition kernels of the following form:
+Lemma 6 Let K : (Rm × Rm) × B(Rm × Rm) → R be a Markov chain tran-
+sition kernel. Suppose that K satisfies detailed balance with respect to the density
+π(q, p) with q-marginal distribution π(q) and conditional density π(p|q); i.e. π(q, p) =
+
+Springer Nature 2021 LATEX template
+14
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+π(q)π(p|q). Consider the marginal chain constructed as follows. Given qn = q, sam-
+ple pn ∼ π(p|q), sample (qn+1, pn+1) ∼ K((qn, pn), ·) and discard both momenta.
+The transition kernel of the marginal chain satisfies
+˜K(q, Q) =
+�
+Rm K((q, p), (Q, Rm)) π(p|q) dp
+(51)
+where Q ∈ B(Rm).
+A proof is given in section A.2
+Proposition 7 Let ˜K be the marginal transition kernel described in lemma 6. The
+marginal chain satisfies detailed balance with respect to the distribution whose density
+is π(q).
+A proof is provided in section A.3.
+2.2.3 Ergodicity Theorems
+The geometric ergodicity of MALA, MMALA, and SMALA have been exam-
+ined in the Markov chain literature. In section 3 we will see how the geometric
+ergodicity of a single component of a mixture Markov chain transition kernel
+implies the geometric ergodicity of the mixture itself. With this destination in
+mind, we now recall two key results in this direction.
+Theorem 8 (Roberts and Tweedie [1996]) Consider the transition kernel K of the
+Metropolis-adjusted Langevin algorithm with A = Id and cϵ(q) = x + ϵ2
+2 ∇ log π(q).
+Then K is geometrically ergodic under the following conditions:
+1. We have
+lim inf
+∥q∥→∞
+�
+∥q∥ − ∥q + ϵ2
+2 ∇ log π(q)∥
+�
+> 0.
+(52)
+2. We have
+lim
+∥q∥→∞
+�
+(A(q)∪I(q))∩(A(q)∩I(q))
+˜π(˜q|q) d˜q = 0,
+(53)
+where
+A(q) = {˜q ∈ X : π(q)˜π(˜q|q) ≤ π(y)˜π(q|˜q)}
+(54)
+I(q) = {˜q ∈ X : ∥˜q∥ ≤ ∥q∥} .
+(55)
+Theorem 9 (Roy and Zhang [2023]) Consider the transition kernel K of the gener-
+alized Metropolis-adjusted Langevin algorithm. Then K is geometrically ergodic under
+the following conditions:
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+15
+1. There exist matrices Al ∈ PD(m) and Au ∈ PD(m) such that Al ≤ A(q) ≤
+Au.
+2. When A ⊂ Rm is bounded, the function cϵ : Rm → Rm is bounded on A.
+3. There is a quantity
+C = lim sup
+∥q∥→∞
+�
+R(q)
+�
+1 − ˜π(˜q|q) min
+�
+1, π(˜q)˜π(q|˜q)
+π(q)˜π(˜q|q)
+��
+dy
+(56)
+which is strictly less than one, where
+R(q) = {˜q ∈ Rm : π(q)˜π(˜q|x) > π(˜q)˜π(q|˜q)} .
+(57)
+4. There exists s > 0 such that,
+lim inf
+∥q∥→∞
+�
+∥A−1/2
+u
+(q)x∥ − ∥A−1/2
+u
+(q)cϵ(q)∥
+�
+> log(D(s)) − log(1 − C)
+s
+,
+(58)
+where,
+D(s) = ϵ−m/2 �π
+2
+�m/2−1 �det(Au)
+det(Al)
+�1/2
+exp(ϵs2/2)
+� ∞
+0
+exp
+�
+−(r − ϵs)2/2ϵ
+�
+rm−1 dr
+(59)
+In section 3 we will examine mixture transition kernels and their geometric
+ergodicity. In the case that one transition kernel is geometrically ergodic and all
+transition kernels are reversible, the mixture transition kernel is geometrically
+ergodic, too, from the following result.
+Theorem 10 (Lee and �Latuszy´nski [2014]) Let (α1, α2, . . .) be a sequence satisfying
+αi ≥ 0 and �∞
+i=1 αi = 1. Let ˜K = �∞
+i=1 αiKi be a mixture of reversible transition
+kernels with invariant distribution Π. Suppose that K1 satisfies the properties: (i) Π
+is the unique invariant distribution, and (ii) K1 is geometrically ergodic. Then ˜K is
+geometrically ergodic.
+We term the derived geometric ergodicity of ˜K from K1 as “inherited”
+geometric ergodicity.
+3 Inherited Geometric Ergodicity
+In this section we discuss the notion of “inherited” geometric ergodicity,
+wherein we modify RMHMC and LMC to be geometrically ergodic when
+MMALA is. In the Euclidean case, by proposition 5, in cases where the
+Metropolis-adjusted Langevin algorithm is geometrically ergodic, so is HMC
+
+Springer Nature 2021 LATEX template
+16
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+by invoking theorem 10. The situation in the geometric setting is more com-
+plicated, since there is no analogue of proposition 5 to show that the marginal
+transition kernel of a single-step of RMHMC or LMC is exactly equivalent
+to the Markov chain transition kernel of the generalized Metropolis-adjusted
+Langevin algorithm for a suitable choice of mean function. This leads us to
+propose the following Markov chain transition kernel.
+Definition 25 Let Kϵ,k : (Rm × Rm) × B(Rm × Rm) → R+ be the Markov chain
+transition kernel of RMHMC (or LMC) with step-size ϵ and k integration steps.
+Let ˜Kϵ,k be the marginal transition kernel (defined in lemma 6). Let (α1, α2, . . .)
+be a sequence satisfying αi ≥ 0 and �∞
+i=1 αi = 1. Let Jϵ be the Markov chain
+transition kernel of the generalized Metropolis-adjusted Langevin algorithm (defined
+in definition 17). The Langevin mixture transition kernel of RMHMC (or LMC),
+which we abbreviate by LMRMHMC (or LMLMC), is defined by
+˜Kϵ = α1Jϵ +
+∞
+�
+k=2
+αk ˜Kϵ,k.
+(60)
+We call α1 the MMALA mixture weight.
+The modified transition kernel simply replaces a single-step of RMHMC (or
+LMC) by the transition kernel of the generalized Metropolis-adjusted Langevin
+algorithm. In order to apply theorem 10, it is necessary to verify that the
+marginal transition kernels of RMHMC and LMC are reversible. Fortunately,
+this is readily shown as follows.
+Corollary 11 For fixed ϵ ∈ R and k ∈ N, the RMHMC and LMC transition kernels
+with step-size ϵ and k integration steps satisfy detailed balance in (q, p)-space, it
+follows that their marginal chains are reversible with respect to the distribution Π by
+proposition 7.
+Proposition 12 Suppose α1 > 0. Under the conditions of theorem 9, the Markov
+chain transition kernel of LMRMHMC (or LMLMC) in eq. (60) is geometrically
+ergodic.
+Proof This follows as an immediate consequence of theorem 10 using the fact that
+the marginal transition kernels or RMHMC (or LMC) are reversible by corollary 11.
+□
+Remark 10 As a practical matter, we choose the mixture probabilities (α1, α2, . . .)
+in the following way. For a particular target distribution, we will choose a maximal
+number of integration steps kmax for which αk = 0 for k > kmax. Then, given a
+particular selection of α1 ∈ (0, 1], we split the remaining probability mass equally
+for each k ∈ {2, . . . , kmax}; that is, the fraction can be expressed as αk =
+1−α1
+kmax−1
+for k = 2, . . . , kmax. In our experiments, we will consider variable choices for the
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+17
+mixing parameter α1. The special case α1 = 0, will correspond by convention to the
+unmodified RMHMC and LMC transition kernels with a single-step computed using
+the prescribed involution, as described in definition 24.
+Mixing with the MMALA transition kernel also immediately establishes
+irreducibility, aperiodicity, and the smallness of all compact sets, as the
+following result reveals.
+Lemma 13 Let π(q) ∝ exp(L(q)) be continuous and bounded on compact sets and
+denote by Π the probability measure with density π. Suppose α1 > 0. The marginal
+transition kernel of LMRMHMC (or LMLMC) is Π-irreducible, aperiodic, and all
+non-negligible compact sets are small.
+Proof With probability α1, the Markov chain transitions according to a Metropolis-
+Hastings accept-reject decision with a normal proposal distribution. Hence, for any
+set A ⊂ B(Rm) for which
+�
+A π(q) dq > 0, we have,
+˜Kϵ(q, A) ≥ α1Jϵ(q, A)
+(61)
+> 0
+(62)
+since a normal proposal distribution is non-vanishing everywhere on Rm. The fact
+that ˜Kϵ is aperiodic and that all non-negligible compact sets are small follows as an
+immediate consequence of Lemma 1.2 from Mengersen and Tweedie [1996].
+□
+Corollary 14 From lemma 13, we have that the modified Markov chain of RMHMC
+(or LMC) is Π-irreducible, aperiodic, and from proposition 7 Π is the stationary
+distribution. Hence, it follows from theorem 2 that the modified Markov chain of
+RMHMC (or LMC) produces an ergodic Markov chain.
+4 Experimentation
+We turn now to the investigation of the proposed modified variations of
+RMHMC and LMC. These examples are chosen to represent a wide class of
+posterior distributions. In computing the convergence of the Markov chain
+under the maximum mean discrepancy metric, we measure similarity between
+10,000 i.i.d. samples of the target distribution, and an equal number of inde-
+pendent Markov chains; in this case, we compute the unbiased estimator of the
+maximum mean discrepancy over the course of the first one-hundred sampling
+steps. For the expected squared jump distance (ESJD) and the effective sample
+size (ESS) metrics, we consider a ten replicates of a long Markov chain consist-
+ing of 1,000,000 samples, except in the case of the Fitzhugh-Nagumo posterior
+where we sample only 100,000 times. In each experiment, we consider mix-
+ing LMC and RMHMC with the transition kernel described in definition 18,
+except in the case of Neal’s funnel distribution, wherein we consider a mixture
+with definition 19. We begin in section 4.1 to describe measures and metrics
+
+Springer Nature 2021 LATEX template
+18
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+by which we may assess the performance of the Markov chains corresponding
+to LMRHMC and LMLMC.
+Code for reproducing these experimental results may be found at https:
+//tinyurl.com/29kz7krz.
+4.1 Measures and Metrics
+We now give details of the evaluation metrics by which we compare Markov
+chains. We first recall the method of maximum mean discrepancy due to
+Gretton et al. [2012].
+Definition 26 Let k : Rm×Rm → R be a positive definite function that is symmetric
+in its arguments. Let Π and Π′ be two probability distributions on Rm. The squared
+maximum mean discrepancy between Π and Π′ is defined by,
+MMD2(k, Π, Π′) =
+E
+q,q′∼Πk(q, q′) +
+E
+q,q′∼Π′k(q, q′) − 2
+E
+q∼Π,q′∼Π′k(q, q′).
+(63)
+Let (q1∗, . . . , qr∗) ∼ Π and (q1, . . . , qs) ∼ Π′. An unbiased estimator of the squared
+maximum mean discrepancy is,
+MMD2
+u(k,
+�
+qi
+∗
+�r
+i=1 ,
+�
+qi�s
+i=1) =
+1
+r(r − 1)
+r
+�
+i=1
+r
+�
+j̸=i
+k(qi
+∗, qj
+∗) +
+1
+s(s − 1)
+s
+�
+i=1
+s
+�
+j̸=i
+k(qi, qj)
+−
+2
+rs
+r
+�
+i=1
+s
+�
+j=1
+k(qi
+∗, qj).
+(64)
+In our evaluations we adopt a squared exponential positive definite kernel
+k(q, q′) = exp(−∥q − q′∥2/2h), where h ∈ R+ is a parameter called the kernel
+bandwidth. In our experiments we set h to be the median distance between i.i.d.
+samples from the target distribution. Recall the definition of geometric ergod-
+icity given in definition 14. The measure of convergence is the total variation
+norm, which we can discuss theoretically but cannot evaluate in a computa-
+tional setting. Instead, we can measure convergence to the target distribution
+Π as a function of n, the number of steps, by means of eq. (63), for which we
+can obtain an unbiased estimate via eq. (64) if we have samples from Kn(q0, ·)
+and samples from Π. Indeed, as in definition 26, let (q1
+∗, . . . , qr
+∗) ∼ Π and, for a
+fixed initial position q0 and number of steps n let (q1, . . . , qs) ∼ Kn(q0, ·) and
+we can compute an unbiased estimate of the squared maximum mean discrep-
+ancy. In the latter case, (q1, . . . , qs) can be obtained by running s independent
+Markov chains for n steps; independent samples from the target distribution
+Π may be available in certain benchmark cases. In our experiments, i.i.d. sam-
+ples from the target distribution may be generated from the banana-shaped
+posterior, Neal’s funnel distribution, the Fitzhugh-Nagumo posterior, and the
+multi-scale Student-t distribution.
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+19
+Remark 11 Recall the definition of geometric ergodicity given in definition 14. Taking
+logarithms reveals
+log ∥Kn(q, ·) − Π(·)∥TV ≤ n log ρ + log V (q).
+(65)
+Therefore, one may claim to see evidence of geometric ergodicity if, as a function of
+n, there is a linear decrease in the total variation distance on a logarithmic scale. Of
+course, we cannot directly compute the total variation distance, but we may look for
+a similar negative linear trend when |MMD2u| is plotted on a logarithmic scale.
+As an additional measure of ergodicity, we consider comparing Markov
+chain samples against i.i.d. samples via random projection onto one-
+dimensional sub-spaces. Let (q1
+∗, . . . , qr
+∗) ∼ Π and let qn ∼ Kn(q0, ·) for n =
+1, . . . , s. Let u be a random unit vector. We compute the Kolmogorov-Smirnov
+(KS) statistic for the projections (u⊤q1
+∗, . . . , u⊤qr
+∗) and (u⊤q1, . . . , u⊤qs).
+Repeating this process for one-hundred randomly generated unit vectors yields
+a distribution over Kolmogorov-Smirnov statistics. The more tightly concen-
+trated this distribution is near zero, the closer the distribution of Markov chain
+iterates (q1, . . . , qs) is to the collection of i.i.d. samples (q1
+∗, . . . , qr
+∗) from the
+target distribution.
+We also consider the expected squared jump distance (ESJD) [Gelman and
+Pasarica, 2007], which measures the dissimilarity between subsequent states
+of the Markov chain. Intuitively, a Markov chain that moves more efficiently
+through the sample space (higher ESJD) will exhibit smaller sample auto-
+correlation. Formally, the ESJD is
+E
+q∼Π∥q′ − q∥2 where q′|q ∼ K(q, ·). Note
+that the choice of norm ∥ · ∥ is left to the practitioner and may be selected
+to capture geometric properties of the target distribution. This expectation is
+typically approximated as follows: let qk be the state of the Markov chain at
+step k and at step k + 1 a candidate state is generated, denoted qk+1
+prop; the
+proposal state is accepted with probability αk; the following empirical mean
+is then taken as our approximation to the ESJD
+ESJD(
+�
+(qk, qk+1
+prop, αk)
+�n
+k=1) = 1
+n
+n
+�
+k=1
+αk∥qk+1
+prop − qk∥2.
+(66)
+In our experiments in section 4.8, we observe that the ESJD can be misleading
+as a measure. Therefore, we also introduce the median squared jump distance
+(MSJD) which we define as
+MSJD(
+�
+(qk, qk+1
+prop, αk)
+�n
+k=1) = Median
+��
+αk∥qk+1
+prop − qk∥2�n
+k=1
+�
+.
+(67)
+As the median, we expect the MSJD to exhibit less sensitivity to outliers than
+the ESJD.
+We additionally consider the effective sample size (ESS) as a metric for
+our Markov chain procedures. We compute ESS using the technique of Kumar
+et al. [2019], who describe the method succinctly as follows. For i = 1, . . . , p,
+
+Springer Nature 2021 LATEX template
+20
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+(a) EHMC
+(b) LMRMHMC
+(c) LMLMC
+Fig. 1: We examine the ergodicity of the Markov chains as a function of the
+sampling step in the banana-shaped posterior distribution. We observe that
+the use of a smaller step-size degrades the ergodicity of the RMHMC, par-
+ticularly when compared against the LMC algorithm. In both RMHMC and
+LMC, we observe that aggressively mixing with MMALA causes the Markov
+chain to mix more slowly; on the other hand, a relatively small mixing proba-
+bility leads to ergodicity that is nearly indistinguishable from the unmodified
+implementations of RMHMC and LMC.
+let (qi,1, qi,2, . . . , qi,n) be a sequence of Rm-valued parameters. The integer p
+is called the number of chains. The effective sample size of the j-th parameter
+is computed according to,
+ESS
+��
+qi,1
+j , . . . , qi,n
+j
+�p
+i=1
+�
+= pn
+ˆτ
+(68)
+ˆτ =
+�
+2
+r
+�
+k=1
+ˆρ2k + ˆρ2k+1
+�
+− 1,
+(69)
+where ˆρk is an estimate, based on all of the p sequences, of the autocorrelation
+of the j-th parameter with a k-step lag (for details see Vehtari et al. [2021])
+and r is the smallest integer for which ˆρ2(r+1) + ˆρ2(r+1)+1 ≤ 0. In our exper-
+iments, we set p = 2 by taking a single long Markov chain and splitting it in
+half at the middle. As a practical matter, we will also report the minimum,
+over all parameters of the posterior, ESS per second in order to represent the
+computational efficiency of the method.
+4.2 Banana-Shaped Distribution
+Our first example considers the following generative model, which represents
+an example of non-identifiable parameters.
+θ1, θ2
+i.i.d.
+∼ Normal(0, σ2
+θ)
+(70)
+yi|θ1, θ2
+i.i.d.
+∼ Normal(θ1 + θ2
+2, σ2
+y) for i = 1, . . . , N.
+(71)
+Given observations {yi}N
+i=1, we wish to sample the posterior distribution of
+(θ1, θ2), a two-dimensional posterior. In our experiments, we consider n =
+
+-1.0
+0.9
+10-1
+0.8
+0.7
+2u
+0.6
+MD
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+10-
+0
+50
+100
+Sampling Step100
+10-1
+D
+≥ 10-3.
+10-4
+10-5
+0
+20
+40
+60
+80
+100
+Sampling Step100
+10-1
+≥ 10-3
+0
+20
+40
+60
+80
+100
+Sampling StepSpringer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+21
+(a) ESJD
+(b) Min. ESS
+(c) Min. ESS / Sec.
+Fig. 2: We show the ESJD, the minimum ESS, and the time-normalized
+minimum ESS for inference in the banana-shaped posterior distribution. We
+observe that for both RMHMC and LMC, aggressively mixing with MMALA
+causes the ESJD distance to decrease; this correspondingly produces a decrease
+in the minimum ESS. For LMC, the time-normalized minimum ESS is a
+decreasing function of the mixing probability.
+100. We employ Euclidean HMC with a step-size of ϵ = 0.1; RMHMC is
+implemented with a step-size of 0.04; in LMC we use an integration step-size
+of ϵ = 0.1. In each case, we set kmax = 10. For the geometric methods, the sum
+of the Fisher information and the negative Hessian of the log-prior is used as a
+Riemannian metric. The reason that a smaller integration step-size is employed
+in RMHMC is that eq. (5) in the defining involution will fail to have a solution
+for ˘p for large step-sizes; this necessitates the use of a smaller integration step.
+In fig. 1, we visualize the statistic |MMD2
+u| over one-hundred steps of the
+Markov chain for euclidean HMC (EHMC), RMHMC and LMC; we color the
+RMHMC and LMC Markov chains according to how aggressively they mix with
+MMALA. Neither EHMC nor RMHMC exhibit clear evidence of geometric
+ergodicity in this target distribution; on the other hand, there exist a broad
+range of mixing probabilities for which LMC exhibits a linear decrease, as a
+function of n, in |MMD2
+u| on a logarithmic scale. In fig. 2 we visualize the ESJD,
+the minimum ESS, and the minimum ESS per second for the three MCMC
+algorithms. We observe that LMC exhibits by far the strongest performance
+on these metrics, while RMHMC languishes due to its small step-size and the
+pathologies of the generalized leapfrog integrator applied to this posterior.
+4.3 Hierarchical Bayesian Logistic Regression
+We consider sampling from the hierarchical Bayesian logistic regression model
+α ∼ Gamma(ω, θ)
+(72)
+βi|α
+i.i.d.
+∼ Normal(0, α−1) for i = 1, . . . , m
+(73)
+yi|xi, β
+indep.
+∼
+Bernoulli
+�
+1
+1 + exp(−x⊤
+i β)
+�
+for i = 1, . . . , N.
+(74)
+
+0.6
+LMRMHMC
+0.5
+LMLMO
+0.4
+EHMC
+0.2
+0.1
+0.0
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability60000
+50000
+ ESS
+40000
+Min.
+30000
+20000
+10000
+0
+MMALA Probability40
+Sec.
+30
+ESS
+20
+Min.
+10
+0
+0.00.10.20.30.40.50.6 0.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+22
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+(a) ESJD
+(b) Min. ESS
+(c) Min. ESS / Sec.
+Fig. 3: We show the ESJD, the minimum ESS, and the time-normalized mini-
+mum ESS for inference in the hierarchical Bayesian logistic regression posterior
+distribution. We observe that both of the geometric methods do not perform
+as well as standard HMC in this inference task. For each mixing weight with
+MMALA, we observe that RMHMC enjoys a greater ESJD and minimum ESS
+when compared against LMC; however, as LMC is a more computationally
+expedient procedure, LMC has a greater time-normalized ESS. In all of these
+metrics, however, EHMC dominates the geometric inference algorithms.
+We consider a logistic regression dataset with N = 270 observations and
+m = 14 covariates. We set ω = 10 and θ = 2 in our experiments. We
+employ a Metropolis-within-Gibbs sampling procedure wherein we alternate
+between sampling the posterior distributions α|β, k, θ and β|α, {(xi, yi)}n
+i=1;
+sampling the former can be performed analytically, whereas we employ EHMC,
+RMHMC, and LMC to sample the latter. In Euclidean HMC, we set ϵ = 0.1; in
+RMHMC and LMC we set ϵ = 0.8. We set kmax = 10 as the upper bound on the
+number of integration steps in each case. For implementing both LMRMHMC
+and LMLMC, we use the sum of the Fisher information and the negative
+Hessian of the log-prior as a metric.
+In fig. 3 we show the ESJD, the minimum ESS, and the minimum ESS
+per second for the hierarchical Bayesian logistic regression posterior. Neither
+of the geometric methods perform well in this posterior, consistently under-
+performing EHMC. A criticism of LMC is that its performance degrades
+significantly in higher dimensions [Betancourt et al., 2014]; one sees evidence
+of this phenomenon in the smaller ESS generated by LMC; however, the com-
+putational savings due to eliminating the fixed point iterations still allow LMC
+to edge out a stronger time-normalized ESS compared to RMHMC.
+4.4 Neal’s Funnel Distribution
+Neal’s funnel distribution [Neal, 2003] is a density defined in the following
+hierarchical manner.
+v ∼ Normal(0, 9)
+(75)
+xi|v
+i.i.d.
+∼ Normal(0, exp(−v)) for i = 1, . . . , N.
+(76)
+
+11.50
+LMRMHMC
+11.25
+LMLMC
+11.00
+10.75
+EHMC
+D
+10.50
+S
+10.25
+10.00
+9.75
+9.50
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability450000
+400000
+S
+2 350000
+I
+300000
+in.
+250000
+200000
+150000
+MMALA Probability800
+600
+ESS
+400
+Min. 1
+200
+二二二二二二
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+23
+(a) EHMC
+(b) LMRMHMC
+Fig. 4: We examine the ergodicity of the Markov chains as a function of the
+sampling step in Neal’s funnel distribution. We observe that EHMC struggles
+in this distribution due to the multiple spatial scales inherent in the funnel-
+shaped distribution. Moreover, we observe that aggressively mixing RMHMC
+with SMALA causes convergence in MMD to become increasingly slow, with
+convergence of the pure SMALA algorithm being slower than EHMC. On the
+other hand, a modest amount of mixing with SMALA produces ergodicity that
+is nearly indistinguishable from the unmodified implementation of RMHMC.
+(a) ESJD
+(b) KS
+(c) Min. ESS / Sec.
+Fig. 5: We show the ESJD, the distribution of Kolmogorov-Smirnov statistics,
+and the time-normalized minimum ESS for inference in Neal’s funnel distribu-
+tion. Although RMHMC and EHMC have comparable time-normalized ESS, it
+is clear from the distribution of KS statistics that RMHMC produces samples
+that are closer to the target distribution in EHMC.
+This distribution is shaped like a funnel, in which the thickness of the “neck”
+is being controlled by the random variable v. This model is reflective of pos-
+teriors encountered in hierarchical models with sparse data. The objective in
+this task is to jointly sample (v, x1, . . . , xN), producing a (N + 1)-dimensional
+target distribution. In Euclidean HMC we employ an integration step-size of
+ϵ = 0.1 and kmax = 10. Our implementation of RMHMC uses kmax = 20
+integration steps with a step-size of ϵ = 0.1 with the SoftAbs metric. We do
+not consider LMC in this task since we found it non-obvious how the SoftAbs
+structure could be extended into the LMC framework while preserving the
+
+-1.0
+0.9
+10-1
+0.8
+0.7
+2u
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+10-5
+0
+50
+100
+Sampling Step100
+10-1
+≥ 10-3
+0
+20
+40
+60
+80
+100
+Sampling Step17.5
+LMRMHMC
+15.0
+EHMC
+12.5
+10.0
+7.5
+5.0
+2.5
+0.0
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA ProbabilitySmirnov
+0.030
+0.025
+0.020
+0.015
+0.010
+0.005
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability1.8
+ 1.6
+1.4
+1.2
+ESS
+1.0
+0.8
+Tin.
+0.6
+0.4
+0.2
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+24
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+(a) ESJD
+(b) Min. ESS
+(c) Min. ESS / Sec.
+Fig. 6: We show the ESJD, the minimum ESS, and the time-normalized mini-
+mum ESS for inference in the posterior distribution of the stochastic volatility
+model. We observe that both of the geometric methods with a modest mixing
+probability with MMALA produce time-normalized ESS that are competitive
+with EHMC. For each mixing weight with MMALA, we observe that LMC
+enjoys a greater ESJD when compared against RMHMC.
+cubic computational cost at each step. In this experiment, we use the SoftAbs
+metric.
+In fig. 4, we visualize |MMD2
+u| for both EHMC and RMHMC. As expected,
+EHMC struggles to sample from Neal’s funnel distribution due to the mul-
+tiscale phenomena. On the other hand, RMHMC exhibits much stronger
+convergence properties, having a linear decrease over several possible mixing
+probabilities with SMALA. We find that aggressively mixing with SMALA
+can be counter-productive, however, due to the less efficient traversal of the
+target distribution by single-step methods. In fig. 5 we show the ESJD, the
+Kolmogorov-Smirnov statistics, and the minimum ESS per second. On all of
+these metrics, RMHMC clearly outperforms EHMC.
+4.5 Stochastic Volatility Model
+We consider a stochastic volatility model with the following generative model.
+φ + 1
+2
+∼ Beta(20, 3/2)
+(77)
+σ2 ∼ InverseChiSquared(10, 1/20)
+(78)
+x1|φ, σ2 ∼ Normal(0, σ2/(1 − φ2))
+(79)
+xt+1|xt, φ, σ2 ∼ Normal(φxt, σ2) for t = 2, . . . , T − 1
+(80)
+yt|β, xt ∼ Normal(0, β2 exp(xt)) for t = 1, . . . , T.
+(81)
+Additionally, the parameter β is equipped with an improper prior pro-
+portional to β−1. In this example, we seek to generate samples from the
+joint distribution x1, . . . , xT , β, φ, σ2 given observations y1, . . . , yT . We employ
+Metropolis-within-Gibbs-like strategy wherein we alternate between sampling
+(x1, . . . , xT )|(y1, . . . , yT )φ, σ2, β and (φ, σ2, β)| {(xi, yi)}T
+i=1; in the former case
+we employ Euclidean HMC whereas in the latter case we compare Euclidean
+
+0.0006
+0.0005
+D
+0.0004
+S
+ 0.0003
+LMRMHMO
+0.0002
+LMLMC
+EHMC
+0.0001
+MMALA Probability4500
+4000
+ESS
+3500
+3000
+in.
+2500
+2000
+1500
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability0.40
+0.35
+0.30
+Min. ESS
+0.25
+0.20
+0.15
+0.10
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+25
+(a) ESJD
+(b) Min. ESS
+(c) Min. ESS / Sec.
+Fig. 7: We show the ESJD, the minimum ESS, and the time-normalized
+minimum ESS for inference in the posterior distribution of the log-Gaussian
+Cox-Poisson model. For any mixing probability, the geometric methods enjoy
+larger time-normalized effective sample sizes than EHMC, with LMC outper-
+forming RMHMC on these metrics. For nearly every mixing probability, we
+additionally observe that the ESJD is greater for LMC and RMHMC than for
+EHMC.
+HMC, RMHMC, and LMC. In sampling either distribution, the Riemannian
+metric is chosen as the sum of the Fisher information and the negative Hessian
+of the log-prior. In our experiments we set T = 1, 000. When using Euclidean
+HMC to sample (φ, σ2, β) we use kmax = 50 integration steps and a step-size
+of ϵ = 0.01; in the case of RMHMC and LMC we use kmax = 6 integration
+steps and a step-size of ϵ = 0.5.
+In fig. 6 we show the ESJD, the minimum ESS, and the minimum ESS
+per second. We observe that employing a modest mixture probability with
+MMALA produces a Markov chain that is marginally better than EHMC.
+4.6 Log-Gaussian Cox-Poisson Process
+We consider inference in a log-Gaussian Cox-Poisson model with the following
+generative model:
+Σ(i,j),(i′,j′)|σ2, β = σ2 exp
+�
+−
+�
+(i − i′)2 + (j − j′)2/(Nβ)
+�
+(82)
+vec(x)|Σ ∼ MultivariateNormal(µ1, Σ)
+(83)
+yij|xij ∼ Poisson(exp(xij)/N 2),
+(84)
+with priors β ∼ Gamma(2, 1/2) and σ2 ∼ Gamma(2, 1/2). In this exam-
+ple, the objective is to sample the joint distribution (β, σ2, {xij}N
+i,j=1) given
+observations {yij}N
+i,j=1. As in the case of stochastic volatility model, we alter-
+natively sample between {xij}N
+i,j=1 given {yij}N
+i,j=1, σ2, and β, and (β, σ2)
+given {xij}N
+i,j=1. In each case, the metric is given by the sum of the Fisher
+information and the negative Hessian of the log-prior. In our experiments we
+set N = 16. In the former case we employ Euclidean HMC, whereas in the
+
+0.6
+0.5
+LMRMHMG
+0.3
+LMLMC
+EHMC
+0.2
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability50000
+40000
+S
+S
+上 30000
+Min.
+20000
+10000
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability2.0
+.5
+Min. ESS
+1.0
+0.5
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+26
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+(a) EHMC
+(b) LMRMHMC
+(c) LMLMC
+Fig. 8: We examine the ergodicity of the Markov chains as a function of
+the sampling step in the Fitzhugh-Nagumo posterior distribution. We observe
+that when a modest mixing probability is employed, the modified RMHMC
+and LMC transition kernels mix more efficiently than the EHMC transition
+kernel, being nearly indistinguishable from the unmodified RMHMC and LMC
+transition kernels. When one uses a large mixing probability, we see that the
+mixing rate is decreased.
+latter case we compare Euclidean HMC, RMHMC and LMC. When imple-
+menting Euclidean HMC we employ kmax = 50 integration steps and a step-size
+of ϵ = 0.01; in RMHMC and LMC we use kmax = 6 integration steps and a
+step-size of ϵ = 0.5.
+In fig. 7 we show the ESJD, the minimum ESS, and the minimum ESS per
+second. We observe that there is a range of mixture probabilities for which
+mixing with the MMALA in LMRMHMC and LMLMC exhibit superior perfor-
+mance compared to EHMC. Both RMHMC and LMC exhibit similar minimum
+ESS metrics, but due to its computational advantage, LMC produces a larger
+minimum ESS per second.
+4.7 Fitzhugh-Nagumo Model
+Given R-valued parameters a, b, and c, the Fitzhugh-Nagumo differential
+equations are defined by,
+˙vt = c
+�
+vt − v3
+t
+3 + rt
+�
+(85)
+˙rt = −
+�vt − a + brt
+c
+�
+.
+(86)
+Given initial conditions v0 and r0, we consider the following generative model:
+(a, b, c)
+i.i.d.
+∼ Normal(0, 1)
+(87)
+ˆrtk|a, b, c, rtk, σ2 indep.
+∼
+Normal(rtk, σ2) for k = 1, . . . , n
+(88)
+ˆvtk|a, b, c, vtk, σ2 indep.
+∼
+Normal(vtk, σ2) for k = 1, . . . , n,
+(89)
+
+1.0
+0.9
+10-1
+0.8
+0.7
+2u
+0.6
+MD
+0.5
+10-3
+0.4
+0.3
+0.2
+0.1
+0.0
+10-
+0
+50
+100
+Sampling Step100
+≥ 10-3
+10-
+0
+20
+40
+60
+80
+100
+Sampling Step100
+≥ 10-3
+10-
+0
+20
+40
+60
+80
+100
+Sampling StepSpringer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+27
+(a) ESJD
+(b) Min. ESS
+(c) Min. ESS / Sec.
+Fig. 9: We show the ESJD, the minimum ESS, and the time-normalized min-
+imum ESS for inference in the Fitzhugh-Nagumo posterior distribution. LMC
+dominates both RMHMC and EHMC in this inference task under the time-
+normalized ESS metrics with its greater computational expediency. When
+timing is not accounted for, we observe that RMHMC and LMC enjoy simi-
+lar ESJD and minimum ESS metrics. We observe that for both RMHMC and
+LMC, aggressively mixing with MMALA causes the ESJD distance to decrease;
+this correspondingly produces a decrease in the minimum ESS.
+where t1, . . . , tn are equally spaced points between [0, T]. In our experiments,
+we set v0 = 1, r0 = −1, σ2 = 1/4, T = 10, and n = 200. Our metric is given
+by the sum of the Fisher information and negative Hessian of the log-prior. In
+our implementations, we use Euclidean HMC with a step-size of ϵ = 0.01 and
+kmax = 10 integration steps. In RMHMC and LMC, we employ an integration
+step-size of ϵ = 0.5 and kmax = 6 integration steps.
+Figure 8 shows the ergodicity measures for EHMC, RMHMC, and LMC.
+We observe that each of these MCMC algorithms exhibit a linear decrease in
+|MMD2
+u| on a logarithmic scale. In fig. 9 we visualize the ESJD, the minimum
+ESS, and the minimum ESS per second for the three MCMC algorithms. We
+observe that LMC exhibits the strongest performance in terms of the mini-
+mum ESS per second, whereas RMHMC only approaches the time-normalized
+performance of EHMC due to its complexity.
+4.8 Multi-Scale Student Distribution
+Fix m ∈ N and let Σ ∈ PD(m) and ν > 2. The density function of the
+multivariate Student-t distribution is,
+π(q) ∝
+�
+1 + 1
+ν x⊤Σ−1x
+�−(m+ν)/2
+.
+(90)
+In our experiments we consider m = 20, ν = 5, and Σ = diag(1, . . . , 1, 104) ∈
+Rm×m. The presence of severely differing spatial scales in the multivariate
+Student-t distribution will cause Euclidean HMC with identity mass matrix
+to exhibit highly oscillatory behavior [Pourzanjani and Petzold, 2019], which
+will limit the efficiency of the method both in terms of ergodicity and the
+effective sample size. We employ Euclidean HMC with a step-size of ϵ = 0.8,
+
+6
+5
+ESJD
+LMRMHMC
+3
+LMLMC
+2
+EHMC
+0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
+MMALA Probability160000
+140000
+120000
+in. ESS
+100000
+80000
+60000
+40000
+20000
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA Probability80
+Sec.
+70
+60
+ESS
+50
+40
+30
+20
+10
+0.00.10.20.30.40.50.6 0.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+28
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+(a) EHMC
+(b) LMRMHMC
+(c) LMLMC
+Fig. 10: We examine the ergodicity of the Markov chains as a function of the
+sampling step in the multiscale Student-t distribution. Here we see that, in the
+case of LMC, it is beneficial from an ergodicity perspective to mix heavily with
+MMALA, whereas for RMHMC, only modest mixing probabilities can produce
+a modified Markov chain that is competitive with the unmodified version. The
+Euclidean HMC struggles in this posterior distribution due to the multiple
+spatial scales.
+(a) ESJD
+(b) MSJD
+Fig. 11: We show the ESJD and the MSJD for inference in the multiscale
+Student-t distribution. Curiously, EHMC enjoys the largest ESJD, but this
+does not correspond to more efficient sampling, as shown in the inferior distri-
+bution of KS statistics. To examine this property further, we also measured the
+MSJD, which reveals that EHMC is less effective in traversing the parameter
+space compared to the RMHMC geometric method.
+and RMHMC and LMC with a step-size of ϵ = 0.7; in each case we employ
+kmax = 20 integration steps. In the case of RMHMC and LMC, we consider a
+metric given by the positive definite term in the Hessian of the log-density of
+the multivariate Student-t distribution.
+In fig. 10 we show |MMD2
+u| as a function of the number of Markov chain
+steps. All of the MCMC algorithms produce a linear decrease in the estimate
+of the maximum mean discrepancy on a logarithmic scale; however, the meth-
+ods differ drastically in terms of the slope of this linear relationship. We see
+that RMHMC exhibits by far the fastest convergence. Notably, LMC exhibits
+
+-1.0
+0.9
+10-1
+0.8
+0.7
+2u
+0.6
+MD
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+10-5
+0
+50
+100
+Sampling Step100
+10-1
+≥ 10-3
+0
+20
+40
+60
+80
+100
+Sampling Step100
+10-1
+≥ 10-3
+10-4
+10-5
+0
+20
+40
+60
+80
+100
+Sampling StepLMRMHMC
+30
+LMLMC
+25
+EHMC
+D
+20
+15
+10
+0.00.1 0.2 0.30.4 0.50.6 0.7 0.80.9 1.0
+MMALA Probability25
+20
+10
+5
+0.00.1 0.2 0.30.4 0.50.6 0.7 0.80.9 1.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+29
+(a) KS
+(b) Min. ESS / Sec.
+Fig. 12: We show the distribution of Kolmogorov-Smirnov statistics and the
+time-normalized minimum ESS for inference in the multiscale Student-t distri-
+bution. For the geometric methods, we observe that RMHMC has much larger
+ESJD than LMC; however, in terms of time-normalized performance, it is best
+to fully mix either method with MMALA. Indeed, we find that MMALA on
+its own produces a distribution of KS statistics that is competitive with any
+of the mixture kernels.
+slower convergence on this target distribution than MMALA; this is due to
+the dimensionality of the posterior, in which LMC struggles to maintain the
+Hamiltonian energy required to accept proposals. In fig. 12 we show the ESJD,
+KS, and the minimum ESS per second. EHMC produces the largest ESJD but
+this does not translate into a large ESS due to the oscillatory behavior of the
+EHMC proposal mechanism, with RMHMC producing the largest minimum
+ESS per second despite its computational complexity.
+5 Conclusion
+This work has considered methods by which to equip RMHMC and LMC
+with a geometric ergodicity theory. The fundamental technique we adopt is to
+replace the RMHMC (or LMC) transition kernel consisting of a single inte-
+gration step with the MMALA transition kernel. This modification is inspired
+by the Euclidean case, in which single-step HMC and MALA can be con-
+structed to be exactly equivalent. By establishing reversibility of the marginal
+transition kernels, geometric ergodicity can be inherited from MMALA. We
+evaluated the modified variations of RMHMC and LMC, called LMRMHMC
+and LMLMC, respectively, on a suite of Bayesian inference tasks. We found
+that aggressively mixing with MMALA transition kernel can be detrimental
+for the performance of the Markov chain on a variety of metrics, but that
+more modest mixing can produce behaviors competitive with, or exceeding,
+the original RMHMC or LMC methods while still imbuing the methods with
+a supporting theory of geometric ergodicity.
+
+rnov
+0.012
+LMRMHMC
+LMLMC
+0.010
+EHMC
+0.008
+0.006
+0.004
+0.002
+MMALA Probability160
+140
+120
+ESS
+100
+80
+lin.
+60
+40
+20
+0.00.10.20.30.40.50.60.70.80.91.0
+MMALA ProbabilitySpringer Nature 2021 LATEX template
+30
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+Acknowledgments.
+We thank the Yale Center for Research Computing
+for use of the research computing infrastructure. This material is based upon
+work supported by the National Science Foundation Graduate Research Fel-
+lowship under Grant No. 1752134. Any opinion, findings, and conclusions or
+recommendations expressed in this material are those of the authors(s) and
+do not necessarily reflect the views of the National Science Foundation. The
+work is also supported in part by NIH/NIGMS R01GM136780 and AFOSR
+FA9550-21-1-0317.
+Statements and Declarations
+No competing interests to declare.
+Appendix A
+Proofs
+A.1
+Proof of Proposition 5
+Proof The proposal distribution of MALA is
+˜q|q ∼ Normal
+�
+q + ϵ2
+2 A−1∇ log π(q), ϵ2A−1
+�
+(A1)
+˜π(˜q|q) ∝ exp
+�
+− 1
+2ϵ2
+�
+˜q − q − ϵ2
+2 A−1∇ log π(q)
+�⊤
+A
+�
+˜q − q − ϵ2
+2 A−1∇ log π(q)
+��
+(A2)
+= exp
+�
+− 1
+2ϵ2
+�
+A˜q − Aq − ϵ2
+2 ∇ log π(q)
+�⊤
+A−1
+�
+A˜q − Aq − ϵ2
+2 ∇ log π(q)
+��
+.
+(A3)
+Therefore, the acceptance probability of MALA is,
+min
+�
+1, π(˜q)˜π(q|˜q)
+π(q)˜π(˜q|q)
+�
+= min
+�
+�
+�
+�
+�
+�
+�
+1, π(˜q)
+π(q) ·
+exp
+�
+− 1
+2ϵ2
+�
+Aq − A˜q − ϵ2
+2 ∇ log π(˜q)
+�⊤
+A−1 �
+Aq − A˜q − ϵ2
+2 ∇ log π(˜q)
+��
+exp
+�
+− 1
+2ϵ2
+�
+A˜q − Aq − ϵ2
+2 ∇ log π(q)
+�⊤
+A−1
+�
+A˜q − Aq − ϵ2
+2 ∇ log π(q)
+��
+�
+�
+�
+�
+�
+�
+�
+.
+(A4)
+The proposal ˜q given q can be sampled by generating z ∼ Normal(0, Id) and setting
+˜q = q + ϵ
+2A−1∇ log π(q) + ϵ
+√
+A−1z.
+(A5)
+In HMC, the leapfrog integrator is applied to the Hamiltonian H(q, p) = − log π(q)+
+1
+2p⊤A−1p. The sequence of updates is,
+¯p = p + ϵ
+2∇ log π(q)
+(A6)
+˜q′ = q + ϵA−1¯p
+(A7)
+= q + ϵ2
+2 A−1∇ log π(q) + ϵA−1p
+(A8)
+
+Springer Nature 2021 LATEX template
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+31
+˜p′ = ¯p + ϵ
+2∇ log π(˜q′)
+(A9)
+= p + ϵ
+2∇ log π(q) + ϵ
+2∇ log π(˜q′).
+(A10)
+When sampling from the distribution with density proportional to exp(−H(q, p)), one
+sees by inspection that p is independent of q and that p ∼ Normal(0, A). Therefore,
+using the same z as in eq. (A5), we can sample p by setting p =
+√
+Az so that
+ϵA−1p = ϵ
+√
+A−1z; we therefore see that, in this case, HMC and MALA produce
+exactly the same proposal (i.e. ˜q′ = ˜q). Recall that the HMC acceptance probability
+is,
+min
+�
+�
+�1, π(˜q)
+π(q) ·
+exp
+�
+− 1
+2(˜p′)⊤A−1(˜p′)
+�
+exp
+�
+− 1
+2p⊤A−1p
+�
+�
+�
+� .
+(A11)
+By rearranging eq. (A7) we find,
+p = 1
+ϵ
+�
+A˜q − Aq − ϵ2
+2 ∇ log π(q)
+�
+(A12)
+˜p′ = 1
+ϵ
+�
+A˜q − Aq + ϵ2
+2 ∇ log π(˜q)
+�
+(A13)
+= −1
+ϵ
+�
+Aq − A˜q − ϵ2
+2 ∇ log π(˜q)
+�
+.
+(A14)
+Substituting these into eq. (A11) and comparing to eq. (A4) shows that not only are
+˜q and ˜q′ identical but that the acceptance probabilities are also identical. Therefore,
+the marginal chain of single-step HMC is exactly equivalent to the MALA chain.
+□
+A.2
+Proof of Lemma 6
+Proof
+Pr(qn+1 ∈ Q|qn = q) = Pr(qn+1 ∈ Q and pn+1 ∈ Rm|qn = q)
+(A15)
+=
+�
+Rm Pr(qn+1 ∈ Q and pn+1 ∈ Rm|qn = q, pn = p) π(p|q) dp
+(A16)
+=
+�
+Rm K((q, p), (Q, Rm)) π(p|q) dp.
+(A17)
+□
+A.3
+Proof of Proposition 7
+Proof Given Q, Q′ ∈ B(Rm) we have
+�
+Q
+˜K(q, Q′) π(q) dq =
+�
+Q
+�
+Rm K((q, p), (Q′, Rm)) π(p|q) dp π(q) dq
+(A18)
+=
+�
+Q
+�
+Rm K((q, p), (Q′, Rm)) π(q, p) dp dq
+(A19)
+=
+�
+Q′
+�
+Rm K((q, p), (Q, Rm)) π(q, p) dp dq
+(A20)
+
+Springer Nature 2021 LATEX template
+32
+Modified Riemannian Manifold and Lagrangian Monte Carlo
+=
+�
+Q′
+�
+Rm K((q, p), (Q, Rm)) π(p|q) dp π(q) dq
+(A21)
+=
+�
+Q′
+˜K(q, Q) π(q) dq,
+(A22)
+where in eq. (A20) we have used the fact that the phase space chain satisfies detailed
+balance with respect to π(q, p). This verifies that the marginal chain satisfies detailed
+balance.
+□
+A.4
+Proof of Involution Composition
+Proposition 15 Suppose that Φ : Rm × Rm → Rm × Rm is an invertible function
+and let F be the momentum flip function given in definition 21. Suppose that F ◦ Φ
+is an involution, then F ◦ Φk is also an involution.
+Proof This will be proved by induction with the base case established by assumption.
+As the inductive hypothesis, assume that F ◦ Φk is an involution. Using the fact
+that F ◦ Φ is an involution, we immediately obtain that Φ−1 ◦ F = F ◦ Φ. Using the
+inductive hypothesis, one also has Φ−k ◦ F = F ◦ Φk. Therefore, we obtain,
+Φ−1 ◦ F = F ◦ Φ
+(A23)
+=⇒ Φ−1 ◦ F ◦ Φk = F ◦ Φk+1
+(A24)
+=⇒ Φ−1 ◦ Φ−k ◦ F = F ◦ Φk+1
+(A25)
+=⇒ Φ−k−1 ◦ F = F ◦ Φk+1
+(A26)
+=⇒ F ◦ Φk+1 ◦ F ◦ Φk+1 = Id.
+(A27)
+This verifies that F ◦ Φk+1 is also an involution.
+□
+Data Availability Statement
+The datasets generated during and/or analysed during the current study are
+available in the GitHub repository, https://tinyurl.com/29kz7krz.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf,len=1300
+page_content='Springer Nature 2021 LATEX template  Geometric Ergodicity in Modified Variations  of Riemannian Manifold and Lagrangian  Monte Carlo  James A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Brofos1*, Vivekananda Roy2 and Roy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Lederman1  1Department of Statistics and Data Science, Yale University, 24  Hillhouse Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=', New Haven, 06510, Connecticut, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2Department of Statistics, Iowa State University, 2438 Osborn  Drive, Ames, 50011, Iowa, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' E-mail(s): james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='brofos@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Contributing authors: vroy@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='lederman@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Abstract  Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte  Carlo (LMC) have emerged as powerful methods of Bayesian infer-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Unlike Euclidean Hamiltonian Monte Carlo (EHMC) and the  Metropolis-adjusted Langevin algorithm (MALA), the geometric ergod-  icity of these Riemannian algorithms has not been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  On the other hand, the manifold Metropolis-adjusted Langevin algo-  rithm (MMALA) has recently been shown to exhibit geometric ergodicity  under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This work investigates the mixture of the LMC  and RMHMC transition kernels with MMALA in order to equip the  resulting method with an “inherited” geometric ergodicity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We  motivate this mixture kernel based on an analogy between single-step  HMC and MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We then proceed to evaluate the original and mod-  ified transition kernels on several benchmark Bayesian inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Keywords: Markov chain Monte Carlo, Riemannian Manifold Hamiltonian  Monte Carlo, Statistical Computing, Lagrangian Monte Carlo  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='01409v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='ME]  4 Jan 2023  Springer Nature 2021 LATEX template  2  Modified Riemannian Manifold and Lagrangian Monte Carlo  1 Introduction  Bayesian inference seeks to combine subjective sources of information with  observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' By specifying one’s prior beliefs and correctly capturing  sources of uncertainty in a stochastic system, one may employ the Bayesian  approach in order to capture and reason about uncertainty under the posterior  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Formally, the posterior distribution is a probability distribution  Π : B(Rm) → [0, 1] with density π : Rm → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' As a practical matter in  Bayesian inference, one is concerned with the computation of expectations with  respect to Π of integrable functions h : Rm → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' that is, we wish to compute  E  q∼Π[h(q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Since π is generally intractable, these expectations are not available  in closed form and Monte Carlo methods based on samples from π are often  used to approximate these means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For instance, one may compute the mean  and variance of the posterior by employing appropriate choices of expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Therefore, a central problem in Bayesian inference is the generation of samples  from the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  One popular Monte Carlo method is based on approximate samples from  the posterior distribution generated via the technique of Markov chain Monte  Carlo (MCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In MCMC, given some initial state of the chain, q0, a sequence  of random variables (q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=') is generated inductively: the conditional dis-  tribution qn|qn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , q0 obeys the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Under certain desirable  regularity conditions, the random variables qn will have a distribution which,  asymptotically, converges to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Establishing the rate of convergence, or at  least an upper bound on that rate, serves to quantify how quickly the chain  (q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=') mixes toward the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the case where the rate of  convergence is geometrically fast, the Markov chain enjoys a strong stability  property in that the estimator n−1 �n  i=1 h(qi) of  E  q∼Π[h(q)] can be equipped  with a central limit theorem under additional technical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This cen-  tral limit theorem has important practical implications in that it allows the use  of asymptotically valid standard errors of MCMC estimates, which, in turn,  can be used to decide how long to run the Markov chains [Roy, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  One of the most popular MCMC methods is Euclidean Hamiltonian Monte  Carlo (EHMC) [Neal, 2010b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In EHMC, one computes approximate solutions  to Hamilton’s equations of motion using numerical integrators, taking care to  ensure that the resulting Markov chain satisfies detailed balance in a phase-  space consisting of the position variables q ∈ Rm and auxiliary momentum  variables p ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The rate of convergence of HMC was established by Liv-  ingstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2016] in the presence of conditions on the log-density [see  also Durmus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Traditional EHMC, however, struggles in distri-  butions that exhibit multiple spatial scales [Pourzanjani and Petzold, 2019,  Betancourt, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This observation led to the development of geometric meth-  ods of HMC, specifically the Riemannian manifold Hamiltonian Monte Carlo  (RMHMC) method of Girolami and Calderhead [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Unlike EHMC, RMHMC adapts to the second-order structure of the poste-  rior, which allows it to align its proposals in the direction of the posterior that  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  3  exhibits the greatest local variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' However, the sophisticated form of the  Hamiltonian employed in RMHMC necessitates the use of complex numerical  integrators that are significantly more expensive than the numerical integrator  employed in EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This concern was partially alleviated in the introduction  of Lagrangian Monte Carlo (LMC) in Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2015], which was able to con-  struct a MCMC method with a more efficient numerical integration procedure,  while simultaneously continuing to take advantage of second-order geometric  knowledge in order to efficiently traverse the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Both RMHMC and  LMC can be viewed as generalizations of the Euclidean Hamiltonian Monte  Carlo algorithm, which incorporate second-order geometric information about  the posterior into the Markov chain transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Incorporating second-  order information allows these geometric methods of MCMC to explore the  typical region of the target distribution more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  However, neither RMHMC nor LMC have, as of yet, been equipped with  a geometric ergodicity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The focus in this work is in establishing the  geometric ergodicity of modified versions of RMHMC and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In imple-  mentations of RMHMC and LMC it is common to randomize the number of  integration steps, and to take one-step with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the case of  Euclidean HMC, the equivalence between single-step HMC and the Metropolis-  adjusted Langevin algorithm (MALA) is critical for establishing geometric  ergodicity of HMC in Livingstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Our approach is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' First,  we propose a simple modification to both RMHMC and LMC which is moti-  vated by a unique correspondence between single-step Euclidean HMC and  MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In particular, we propose that instead of applying the transition kernel  corresponding to RMHMC or LMC with a single integration step, one instead  applies the transition kernel of the manifold Metropolis-adjusted Langevin  algorithm (MMALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Recent work in Roy and Zhang [2023] gave conditions  under which MMALA is geometrically ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We may imbue these modi-  fied variations of RMHMC and LMC with a geometric ergodicity theory via  inheritance once one establishes that the RMHMC and LMC transition kernels  (with a fixed, non-random number of integration steps) are reversible in the  required sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This construction is described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In section 4 we pro-  ceed to a numerical evaluation of the proposed modification of RMHMC and  LMC, with special attention given to the probability of applying the MMALA  transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We begin, however, in section 2 with an overview of required  mathematical concepts from numerical integration and Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2 Preliminaries  In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 we review the generalized leapfrog and Lagrangian leapfrog  employed in geometric MCMC methods for performing Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 covers Markov chains based on involutions as well as those based  on discretizations of Langevin diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We also discuss geometric ergodicity,  mixture transition kernels, and the equivalence between single-step HMC and  MALA in the Euclidean regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  4  Modified Riemannian Manifold and Lagrangian Monte Carlo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 Numerical Integrators  Hamiltonian and Lagrangian mechanics are equivalent, with the momentum  and velocity being related by the Legendre transform [Marsden and Ratiu,  2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In this section, we will consider numerical integrators of these equations  of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Throughout our discussion, we will consider a Hamiltonian function  of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 1 Let π : Rm → R+ be a smooth probability density and let G : Rm →  PD(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The Riemannian Hamiltonian is the function H : Rm × Rm → R defined by  H(q, p) = − log π(q) + 1  2 log det(G(q)) + 1  2p⊤G−1(q)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (1)  Moreover, assuming  �  Rm  �  Rm exp(−H(q, p)) dq dp < +∞, the Riemannian density  is the probability density π(q, p) ∝ exp(−H(q, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Remark 1 The term 1  2 log det(G(q)) + 1  2p⊤G−1(q)p appearing in the definition of  the function H in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (1) is chosen such that the conditional density π(q, p) satisfies  π(p|q) = Normal(0, G(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (2)  It is easily seen that the q-marginal density of π(q, p) obtained by marginalizing out  p is π(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Hamiltonian functions H : Rm ×Rm → R produce Hamilton’s equations of  motion which, by definition, are solutions to the coupled differential equations  ˙qt = ∇pH(qt, pt)  (3)  ˙pt = −∇qH(qt, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (4)  Except in special cases, a closed-form solution for the map t �→ (qt, pt) ∈  Rm × Rm does not exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' this necessitates the use of numerical integrators in  order to approximate the equations of motion obeying eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' One  such example is the generalized leapfrog method, which we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 2 The generalized leapfrog integrator with step-size ϵ ∈ R applied to the  Riemannian Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (1) is the map (q, p) �→ ˆΦϵ(q, p) defined by,  ˘p = p − ϵ  2∇qH(q, ˘p)  (5)  ˜q = q + ϵ  2 (∇pH(q, ˘p) + ∇pH(˜q, ˘p))  (6)  = q + ϵ  2  �  G−1(q) + G−1(˜q)  �  ˘p  (7)  ˜p = ˘p − ϵ  2∇qH(˜q, ˘p)  (8)  = p − ϵ  2∇qH(q, ˘p) − ϵ  2∇qH(˜q, ˘p),  (9)  where ˆΦϵ(q, p) = (˜q, ˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  5  Remark 2 One can prove [Leimkuhler and Reich, 2005, Hairer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=', 2006] that  the generalized leapfrog integrator is a symplectic transformation and that therefore  |det(∇Φ(q, p))| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Remark 3 We must now give a remark about notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Numerical integration plays  a central role in our analysis, and we give particular attention to multiple steps of  integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' At the same time, we will also discuss Markov chains, which also consist  of multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' As an attempt to differentiate these two notions of step, we will  use a lower index to refer to steps of numerical integration, whereas we will use  upper indices to denote Markov chain steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Given initial data (q0, p0) ∈ Rm × Rm,  we denote k-steps of the generalized leapfrog integrator by (qk, pk) = ˆΦkϵ (q0, p0) =  ˆΦϵ(qk−1, pk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Similarly, we let ˘pk = pk−1 − ϵ  2∇qH(qk−1, ˘pk), which we call the  intermediate momentum at the k-th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  A Hamiltonian of the form in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (1) may be transformed into a Lagrangian  as  L(q, ˙q) = − log π(q) + 1  2 log det(G(q)) − 1  2 ˙q⊤G(q) ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (10)  As noted at the beginning of this subsection, Lagrangian and Hamiltonian  mechanics are formally equivalent, with the momentum p ∈ Rm being related  to the velocity ˙q ∈ Rm according to the Legendre transformation p = G(q) ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  The Lagrangian produces equivalent equations of motion called the Euler-  Lagrange equations as  ∇qL(qt, ˙qt) = d  dt∇ ˙qL(qt, ˙qt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (11)  One advantage possessed by the Lagrangian formalism over the Hamiltonian  approach is that one may identify explicit numerical integrators [Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=',  2015] of the equations of motion, such as the Lagrangian leapfrog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 3 The Lagrangian leapfrog integrator with step-size ϵ ∈ R applied to the  Riemannian Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (1) is the map (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' p) �→ ˜Φϵ(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' p) defined by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  v = G−1(q)p  (12)  ˘v = v − ϵ  2G−1(q)∇qU(q) − ϵ  2Ω(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' v)˘v  (13)  ˜q = q + ϵ˘v  (14)  ˜v = ˘v − ϵ  2G−1(˜q)∇qU(˜q) − ϵ  2Ω(˜q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ˘v)˜v  (15)  = v − ϵ  2G−1(q)∇qU(q) − ϵ  2Ω(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' v)˘v − ϵ  2G−1(˜q)∇qU(˜q) − ϵ  2Ω(˜q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ˘v)˜v  (16)  ˜p = G−1(˜q)˜v  (17)  where (˜q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ˜p) = ˜Φϵ(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' p) and  U(q) = − log π(q) + 1  2 log det(G(q))  (18)  Springer Nature 2021 LATEX template  6  Modified Riemannian Manifold and Lagrangian Monte Carlo  Ωij(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' v) =  m  �  k=1  Γi  kj(q)vk  (19)  Γi  kj(q) = 1  2  m  �  l=1  G−1  kl (q)  � ∂  ∂qi  Glj(q) +  ∂  ∂qj  Gil(q) − ∂  ∂ql  Gij(q)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (20)  The Jacobian determinant of the transformation ˜Φϵ (defined in definition 3)  is,  |det(∇˜Φϵ(q, p))| = det(G−1(q))det(G(˜q))|det(Id + ϵΩ(q, v)/2)det(Id − ϵΩ(˜q, ˜v)/2)|  |det(Id + ϵΩ(˜q, ¯v)/2)det(Id − ϵΩ(q, ¯v)/2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (21)  Remark 4 Given initial data (q0, p0) ∈ Rm×Rm, we denote k-steps of the generalized  (or Lagrangian) leapfrog integrator by (qk, pk) = ˜Φkϵ (q0, p0) = ˜Φϵ(qk−1, pk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In the case where G is a constant function of q, the generalized leapfrog  integrator reduces to the standard leapfrog integrator, which is defined as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 4 Let π : Rm → R+ be a smooth probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' A Euclidean  Hamiltonian is a smooth map H : Rm × Rm → R of the form,  H(q, p) = − log π(q) + 1  2p⊤G−1p,  (22)  where G ∈ PD(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 5 Consider a Euclidean Hamiltonian as defined in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  Euclidean leapfrog integrator is the map (q, p) �→ ˇΦϵ(q, p) defined by  ˘p = p + ϵ  2∇ log π(q)  (23)  ˜q = q + ϵG−1˘p  (24)  ˜p = ˘p + ϵ  2∇ log π(˜q),  (25)  where ˇΦϵ(q, p) = (˜q, ˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  The Euclidean leapfrog integrator is the de-facto standard numerical inte-  grator employed in EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This is because of its accuracy and computational  efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in contrast to the generalized leapfrog integrator in definition 2, the  Euclidean leapfrog is fully explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Nevertheless, the generalized leapfrog and  Euclidean leapfrog are exactly equivalent when applied to a Hamiltonian in  the form of definition 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' this is made precise in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  7  Proposition 1 When the generalized leapfrog (definition 2) or Lagrangian leapfrog  (definition 3) is applied to a Euclidean Hamiltonian in the form of definition 4 (in  the sense that we take G(q) ≡ G for every q ∈ Rm), the resulting map is equivalent  to the Euclidean leapfrog method in definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proof In the case of the generalized leapfrog integrator, this is immediate from the  fact that ∇qH(q, p) = −∇ log π(q) which does not depend on p and ∇pH(q, p) = Gp  which does not depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In the case of the Lagrangian leapfrog, the form of the Euclidean Hamiltonian  means that Ω(q, v) is uniformly zero (because all of the derivatives of the metric  vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The Lagrangian leapfrog integrator may be re-expressed as  v = G−1p  (26)  ˘v = v + ϵ  2G−1∇ log π(q)  (27)  = G−1˘p  (28)  ˜q = q + ϵ˘v  (29)  = q + ϵG−1˘p  (30)  ˜v = ˘v + ϵ  2G−1∇ log π(˜q)  (31)  = G−1˜p,  (32)  where ˘p and ˜p are as in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (23) and (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 Markov Chains  The purpose of this section is to review fundamentals of Markov chains, includ-  ing their construction, convergence properties, and central limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Markov chains are stochastic processes that may be inductively defined by  their transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Since the transition kernel yields a probability measure  depending only on the current state, one sees by inspection that the Markov  chain satisfies the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 6 A Markov chain transition kernel on Rm is a function K : Rm ×  B(Rm) → [0, 1] such that (i) for q ∈ Rm, K(q, ·) is a probability measure and (ii)  for fixed A ∈ B(Rm) the function q �→ K(q, A) is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Within the context of simulation-based inference and mainly Bayesian  inference, one is interested in constructing a Markov chain which converges  to a specified target probability distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' as noted in the introduction,  this target distribution is typically known by its density up to a normalizing  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We now define two notions of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 7 Given a Markov chain transition kernel K, a Markov chain is a  sequence of random variables defined inductively by qn+1|qn ∼ K(qn, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  8  Modified Riemannian Manifold and Lagrangian Monte Carlo  Definition 8 The total variation distance between two probability measures Π :  B(Rm) → [0, 1] and Ξ : B(Rm) → [0, 1] is defined by  ∥Π − Ξ∥TV = 2  sup  A∈B(Rm)  |Π(A) − Ξ(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (33)  Definition 9 A Markov chain with transition kernel K is said to be ergodic if  lim  n→∞ ∥Kn(q, ·) − Π(·)∥TV = 0,  (34)  where for A ∈ B(Rm), Kn(q, A) = Pr(qn ∈ A|q0 = q) is the n-step Markov transition  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  It will turn out that one can establish ergodicity of a Markov chain if  one can establish three separate properties: irreducibility, aperiodicity, and  stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We now define each of these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 10 A Markov chain transition kernel K is said to be Π-irreducible if for  every set A ∈ B(Rm) with Π(A) > 0 there exists n ∈ N for which Kn(q, A) > 0 for  all q ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 11 Given a Markov chain transition kernel K : Rm × B(Rm) → [0, 1],  a set C ⊆ Rm is called small if there exists a n ∈ N, a δ > 0, and a probability  measure ν : B(Rm) → [0, 1] such that for any q ∈ C and A ∈ B(Rm) we have  Kn(q, A) ≥ δ · ν(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We may therefore say that the set C is (n, δ, ν)-small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 12 A Markov chain transition kernel K is called aperiodic if there exists  a small set C for which the greatest common divisor of the set  �  n′ ∈ N : ∃ δ > 0 and probability measure ν such that C is (n′, δ, ν)-small  �  (35)  is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 13 A Markov chain with transition kernel K is said to be stationary for  the probability measure Π if for any Borel set A ∈ B(Rm) we have  E  q∼ΠK(q, A) =  Π(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 14 A Markov chain with transition kernel K is called geometrically  ergodic if there exists ρ ∈ (0, 1) and function V : Rm → R+ such that  ∥Kn(q, ·) − Π(·)∥TV ≤ V (q)ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (36)  It is clear from the definitions that geometric ergodicity is a stronger form of  convergence than mere ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the latter case, however, simple conditions  under which a Markov chain is ergodic can be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  9  Theorem 2 (Tierney [1994]) Suppose that K is a Markov chain transition kernel  that is Π-irreducible, aperiodic, and for which Π is the stationary probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Then K produces an ergodic Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  However, in addition to giving an upper bound on the mixing of the Markov  chain into the target probability measure, geometric ergodicity is also useful  for establishing a central limit theorem for expectations computed from the  sequence (q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We make this precise as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Theorem 3 (Meyn and Tweedie [1993]) Let h : Rm → R be a function for which  E  q∼Π|h(q)|2+γ < ∞ for some γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let K be a Markov chain transition kernel that  is aperiodic, Π-irreducible, and for which Π is the unique stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Assume further that K converges geometrically to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Define SK[h] = n−1 �n  i=1 h(qi)  where qi+1|qi ∼ K(qi, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Then, as n → ∞,  √n  �  SK[h] − E  q∼Π [h(q)]  �  d→ Normal(0, τ2  h)  (37)  where τ2  h is a constant, less than infinity, that depends on h (and K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The quantity  τ2  h has a closed-form given by  τ2  h = Var  q∼Π (h(q)) + 2  ∞  �  k=1  Cov  q∼Π  �  h(q), h(qk)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (38)  Establishing that a Markov chain has Π as a stationary distribution is often  easily achieved by showing that the chain satisfies detailed balance with respect  to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' As we require detailed balance when discussing the marginal transition  kernels of RMHMC and LMC in section 3, we define this notion now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 15 A Markov chain transition kernel is said to satisfy detailed balance  with respect to the probability distribution Π (equivalently, the Markov chain is  called reversible with respect to Π) if for any sets Q, Q′ ∈ B(Rm),  �  Q  K(q, Q′) Π(dq) =  �  Q′ K(q, Q) Π(dq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (39)  Remark 5 When detailed balance holds for a Markov chain transition kernel K, it  follows that Π is the stationary probability measure (definition 13) of the Markov  transition kernel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 Metropolis-Hastings Kernels and Generalized Langevin  Algorithms  In this section we review Metropolis-Hastings Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We begin  by formally defining these objects before proceeding to a special case of  Metropolis-Hastings method based on Langevin diffusion, which recalls the  constructions considered in Roy and Zhang [2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  10  Modified Riemannian Manifold and Lagrangian Monte Carlo  Definition 16 (Metropolis-Hastings Algorithm) Let π : Rm → R+ be a probabil-  ity density and, for each q ∈ Rm, let ˜π(·|q) : Rm → R+ be a probability density  depending on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The Metropolis-Hastings transition kernel with proposal density  ˜π(·|q) is,  K(q, A) =  �  A  min  �  1, π(˜q)˜π(q|˜q)  π(q)˜π(˜q|q)  �  ˜π(˜q|q) d˜q  +  �  1 −  �  Rm min  �  1, π(˜q)˜π(q|˜q)  π(q)˜π(˜q|q)  �  ˜π(˜q|q) d˜q  �  1 {q ∈ A} ,  (40)  for q ∈ Rm and A ∈ B(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 17 (Generalized Metropolis-Adjusted Langevin Algorithm) Fix ϵ ∈ R \\  {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let cϵ : Rm → Rm and let A : Rm → PD(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The generalized Metropolis-  adjusted Langevin algorithm is an instance of the Metropolis-Hastings algorithm  with proposal density  ˜π(˜q|q) = Normal(˜q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' cϵ(q), ϵ2A(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (41)  We denote the Markov chain transition kernel of the generalized Metropolis-adjusted  Langevin algorithm with step-size ϵ by Jϵ : Rm × B(Rm) → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 18 (Riemannian Manifold Metropolis-Adjusted Langevin Algorithm) In  the special case where  cϵ(q) = ϵ2  2 A(q)∇ log π(q) + ϵ2  2 Γ(q)  (42)  Γi(q) =  m  �  j=1  ∂  ∂qj  Aij(q)  (43)  we call the resulting method the Riemannian manifold Metropolis-adjusted Langevin  algorithm (MMALA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Remark 6 The form of the proposal in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (42) is based on an Euler-Maruyama  discretization of a Langevin diffusion on a manifold with Riemannian metric G(q) =  A−1(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' see Xifara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2014] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In summary, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (42) is the drift component  of the Euler-Maruyama discretization (with step-size ϵ2) applied to the following  stochastic differential equation:  dXt = 1  2G−1(Xt)∇ log π(Xt) dt + Γ(Xt) dt + G−1/2(Xt) dBt,  (44)  where Bt is Euclidean Brownian motion at time t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' When G(q) is the Fisher  information matrix, the term G−1(Xt)∇ log π(Xt) can be identified as the natural  gradient of the function log π under the geometry generated by the Fisher metric  [Amari and Nagaoka, 2000] at position Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' On the other hand, the term Γ(Xt) dt +  G−1/2(Xt) dBt corresponds to manifold Brownian motion, since its infinitesimal  generator is the Laplace-Beltrami operator on the manifold [Hsu, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Therefore,  as the stochastic differential equation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (44) consists of a term comprising the  manifold gradient of a log-density and another term comprising manifold Brownian  motion, it is called the Riemannian Langevin equation in correspondence with the  Euclidean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  11  Definition 19 (Simplified Riemannian Manifold Metropolis-Adjusted Langevin  Algorithm) Computing the function Γ in definition 18 may be inconvenient to eval-  uate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' It can be ignored while still yielding a Markov chain transition kernel, giving  the special case,  cϵ(q) = ϵ2  2 A(q)∇ log π(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (45)  The resulting method is called the simplified Riemannian manifold Metropolis-  adjusted Langevin algorithm, which was considered by Girolami and Calderhead  [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We adopt the abbreviation SMALA to refer to the simplified Metropolis-  adjusted Langevin algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Remark 7 There are three common choices of the function A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The first is that A  is a constant function, in which case we simply write A ∈ PD(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' A second option  is that A is chosen as the inverse of the sum of the Fisher information matrix and  the negative Hessian of the log-prior, which can capture second-order geometry of  both the likelihood and the prior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' the use of the inverse of the sum of the Fisher  information and the negative Hessian of the log-prior as a preconditioner is the  approach advocated by Girolami and Calderhead [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' A third option is that A is  the inverse of the SoftAbs metric, which is a smooth transformation of the Hessian  of the log-density of the target distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' for details see Betancourt [2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 Geometric Methods of Bayesian Inference  Definition 20 (Involutive Monte Carlo [Neklyudov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=', 2020]) Let Φ : Rm → Rm  be a smooth involution and let π : Rm → R+ be a probability density on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  Markov chain transition kernel of involutive Monte Carlo with target density π is  K(q, A) = min  �  1, π(Φ(q))  π(q)  |det(∇Φ(q))|  �  1 {Φ(q) ∈ A}  +  �  1 − min  �  1, π(Φ(q))  π(q)  |det(∇Φ(q))|  ��  1 {q ∈ A} ,  (46)  for q ∈ Rm and A ∈ B(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  It is easily verified that the transition kernel of involutive Monte Carlo  satisfies detailed balance with respect to the density π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Involutive Monte Carlo  gives rise to two special transition kernels corresponding to RMHMC and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 21 Let ˆΦϵ be as in definition 2 and let H be as in definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  involution of Riemannian manifold Hamiltonian Monte Carlo with step-size ϵ and k  integration steps is,  Φ = F ◦ ˆΦϵ ◦ · · · ◦ ˆΦϵ  �  ��  �  k times  ,  (47)  where F(q, p) = (q, −p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The target density of Riemannian manifold Hamiltonian  Monte Carlo is π(q, p) ∝ exp(−H(q, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The transition kernel of involutive Monte  Springer Nature 2021 LATEX template  12  Modified Riemannian Manifold and Lagrangian Monte Carlo  Carlo (definition 20) with involution Φ given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (47) is called the transition kernel  of Riemannian manifold Hamiltonian Monte Carlo with step-size ϵ and k integration  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Remark 8 It can be shown that if Φ is an invertible function and if F ◦ Φ is an  involution that F ◦ Φ ◦ · · · · Φ is also an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This explains why eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (47) is also  an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For further details, see proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In the special case where G is a constant function of q, the resulting Markov  chain is called Euclidean Hamiltonian Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 22 Let ˇΦϵ be as in definition 5 and let H be as in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  involution of Euclidean Hamiltonian Monte Carlo (EHMC) with step-size ϵ and k  integration steps is,  Φ = F ◦ ˇΦϵ ◦ · · · ◦ ˇΦϵ  �  ��  �  k times  ,  (48)  where F(q, p) = (q, −p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The target density of Euclidean Hamiltonian Monte Carlo  is π(q, p) ∝ exp(−H(q, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The transition kernel of involutive Monte Carlo (defini-  tion 20) with involution Φ given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (48) is called the transition kernel of Euclidean  Hamiltonian Monte Carlo with step-size ϵ and k integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 23 Let ˜Φϵ be as in definition 3 and let H be as in definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  involution of Lagrangian Monte Carlo with step-size ϵ and k integration steps is,  Φ = F ◦ ˜Φϵ ◦ · · · ◦ ˜Φϵ  �  ��  �  k times  ,  (49)  with F(q, p) = (q, −p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The target density of Lagrangian Hamiltonian Monte Carlo  is π(q, p) ∝ exp(−H(q, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We call this the transition kernel of Lagrangian Monte  Carlo with step-size ϵ and k integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Remark 9 Unlike the transition kernel employed in Riemannian manifold Hamil-  tonian Monte Carlo, the Jacobian determinant of the involution appearing in  Lagrangian Monte Carlo must be computed through k applications of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (21), where,  as before, k is the number of integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In the special case where G is a constant function of q, Riemannian  manifold Hamiltonian Monte Carlo and Lagrangian Monte Carlo produces a  transition kernel that is equivalent to Euclidean Hamiltonian Monte Carlo with  constant mass matrix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proposition 4 The transition kernels of RMHMC (or LMC) with step-size ϵ and k  integration steps when applied to the Euclidean Hamiltonian in definition 4 (in the  sense that we take G(q) ≡ G for every q ∈ Rm) are both equivalent to the EHMC  transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  13  Proof It follows from proposition 1 that the proposals produced by RMHMC and  LMC are identical to EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' It remains to be verified that the acceptance decisions  are identical, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This can be verified by observing that the Riemannian Hamiltonian  in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (1) is the Euclidean Hamiltonian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (22) up to an additive constant (since  G does not depend on position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Hence, the Euclidean and Riemannian densities are  proportional to one another, and the acceptance probabilities will be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  We now elaborate on the connection between EHMC with a single-step and  the Metropolis-adjusted Langevin algorithm: in particular, these two meth-  ods can be constructed to be exactly equivalent in q-space, producing equal  proposals and identical acceptance probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proposition 5 (Neal [2010a]) Let A ∈ Rm×m be a fixed positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Consider EHMC with the Hamiltonian H(q, p) = − log π(q) + 1  2p⊤A−1p and  MALA with proposal distribution ˜q|q ∼ Normal  �  q + ϵ2  2 A−1∇ log π(q), ϵ2A−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  marginal transition kernel of EHMC with step-size ϵ and a single integration step can  be constructed to be exactly equivalent to the transition kernel of MALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  A proof is provided in section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The analysis of mixture transition ker-  nels will be central to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In standard implementations of Riemannian  manifold Hamiltonian and Lagrangian Monte Carlo, it is typical to random-  ize the number of integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This is done to avert any Markov chain  pathologies (such as irreducibility failures) that may result from using a fixed  number of integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We therefore consider Markov chain transition  kernels which are mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 24 Let Kϵ,k : (Rm × Rm) × B(Rm × Rm) → R+ be the Markov chain  transition kernel of RMHMC (or LMC) with step-size ϵ and k integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Let ˜Kϵ,k be the marginal transition kernel (defined in lemma 6) of Kϵ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Define the  mixture transition kernel of RMHMC (or LMC) to be  ˜Kϵ =  ∞  �  k=1  αk ˜Kϵ,k,  (50)  where (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=') is a probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  The fact that EHMC, RMHMC, and LMC all satisfy detailed balance in  (q, p)-space causes us to examine the progression of q-states alone in between  Gibbs resampling steps of the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This leads to marginal Markov chain  transition kernels of the following form:  Lemma 6 Let K : (Rm × Rm) × B(Rm × Rm) → R be a Markov chain tran-  sition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Suppose that K satisfies detailed balance with respect to the density  π(q, p) with q-marginal distribution π(q) and conditional density π(p|q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' π(q, p) =  Springer Nature 2021 LATEX template  14  Modified Riemannian Manifold and Lagrangian Monte Carlo  π(q)π(p|q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Consider the marginal chain constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Given qn = q, sam-  ple pn ∼ π(p|q), sample (qn+1, pn+1) ∼ K((qn, pn), ·) and discard both momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  The transition kernel of the marginal chain satisfies  ˜K(q, Q) =  �  Rm K((q, p), (Q, Rm)) π(p|q) dp  (51)  where Q ∈ B(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  A proof is given in section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  Proposition 7 Let ˜K be the marginal transition kernel described in lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  marginal chain satisfies detailed balance with respect to the distribution whose density  is π(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  A proof is provided in section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3 Ergodicity Theorems  The geometric ergodicity of MALA, MMALA, and SMALA have been exam-  ined in the Markov chain literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In section 3 we will see how the geometric  ergodicity of a single component of a mixture Markov chain transition kernel  implies the geometric ergodicity of the mixture itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' With this destination in  mind, we now recall two key results in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Theorem 8 (Roberts and Tweedie [1996]) Consider the transition kernel K of the  Metropolis-adjusted Langevin algorithm with A = Id and cϵ(q) = x + ϵ2  2 ∇ log π(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Then K is geometrically ergodic under the following conditions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We have  lim inf  ∥q∥→∞  �  ∥q∥ − ∥q + ϵ2  2 ∇ log π(q)∥  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (52)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We have  lim  ∥q∥→∞  �  (A(q)∪I(q))∩(A(q)∩I(q))  ˜π(˜q|q) d˜q = 0,  (53)  where  A(q) = {˜q ∈ X : π(q)˜π(˜q|q) ≤ π(y)˜π(q|˜q)}  (54)  I(q) = {˜q ∈ X : ∥˜q∥ ≤ ∥q∥} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (55)  Theorem 9 (Roy and Zhang [2023]) Consider the transition kernel K of the gener-  alized Metropolis-adjusted Langevin algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Then K is geometrically ergodic under  the following conditions:  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' There exist matrices Al ∈ PD(m) and Au ∈ PD(m) such that Al ≤ A(q) ≤  Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' When A ⊂ Rm is bounded, the function cϵ : Rm → Rm is bounded on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' There is a quantity  C = lim sup  ∥q∥→∞  �  R(q)  �  1 − ˜π(˜q|q) min  �  1, π(˜q)˜π(q|˜q)  π(q)˜π(˜q|q)  ��  dy  (56)  which is strictly less than one, where  R(q) = {˜q ∈ Rm : π(q)˜π(˜q|x) > π(˜q)˜π(q|˜q)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (57)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' There exists s > 0 such that,  lim inf  ∥q∥→∞  �  ∥A−1/2  u  (q)x∥ − ∥A−1/2  u  (q)cϵ(q)∥  �  > log(D(s)) − log(1 − C)  s  ,  (58)  where,  D(s) = ϵ−m/2 �π  2  �m/2−1 �det(Au)  det(Al)  �1/2  exp(ϵs2/2)  � ∞  0  exp  �  −(r − ϵs)2/2ϵ  �  rm−1 dr  (59)  In section 3 we will examine mixture transition kernels and their geometric  ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the case that one transition kernel is geometrically ergodic and all  transition kernels are reversible, the mixture transition kernel is geometrically  ergodic, too, from the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Theorem 10 (Lee and �Latuszy´nski [2014]) Let (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=') be a sequence satisfying  αi ≥ 0 and �∞  i=1 αi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let ˜K = �∞  i=1 αiKi be a mixture of reversible transition  kernels with invariant distribution Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Suppose that K1 satisfies the properties: (i) Π  is the unique invariant distribution, and (ii) K1 is geometrically ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Then ˜K is  geometrically ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We term the derived geometric ergodicity of ˜K from K1 as “inherited”  geometric ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  3 Inherited Geometric Ergodicity  In this section we discuss the notion of “inherited” geometric ergodicity,  wherein we modify RMHMC and LMC to be geometrically ergodic when  MMALA is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the Euclidean case, by proposition 5, in cases where the  Metropolis-adjusted Langevin algorithm is geometrically ergodic, so is HMC  Springer Nature 2021 LATEX template  16  Modified Riemannian Manifold and Lagrangian Monte Carlo  by invoking theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The situation in the geometric setting is more com-  plicated, since there is no analogue of proposition 5 to show that the marginal  transition kernel of a single-step of RMHMC or LMC is exactly equivalent  to the Markov chain transition kernel of the generalized Metropolis-adjusted  Langevin algorithm for a suitable choice of mean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This leads us to  propose the following Markov chain transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 25 Let Kϵ,k : (Rm × Rm) × B(Rm × Rm) → R+ be the Markov chain  transition kernel of RMHMC (or LMC) with step-size ϵ and k integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Let ˜Kϵ,k be the marginal transition kernel (defined in lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=')  be a sequence satisfying αi ≥ 0 and �∞  i=1 αi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let Jϵ be the Markov chain  transition kernel of the generalized Metropolis-adjusted Langevin algorithm (defined  in definition 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The Langevin mixture transition kernel of RMHMC (or LMC),  which we abbreviate by LMRMHMC (or LMLMC), is defined by  ˜Kϵ = α1Jϵ +  ∞  �  k=2  αk ˜Kϵ,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (60)  We call α1 the MMALA mixture weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  The modified transition kernel simply replaces a single-step of RMHMC (or  LMC) by the transition kernel of the generalized Metropolis-adjusted Langevin  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In order to apply theorem 10, it is necessary to verify that the  marginal transition kernels of RMHMC and LMC are reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Fortunately,  this is readily shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Corollary 11 For fixed ϵ ∈ R and k ∈ N, the RMHMC and LMC transition kernels  with step-size ϵ and k integration steps satisfy detailed balance in (q, p)-space, it  follows that their marginal chains are reversible with respect to the distribution Π by  proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proposition 12 Suppose α1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Under the conditions of theorem 9, the Markov  chain transition kernel of LMRMHMC (or LMLMC) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (60) is geometrically  ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proof This follows as an immediate consequence of theorem 10 using the fact that  the marginal transition kernels or RMHMC (or LMC) are reversible by corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  Remark 10 As a practical matter, we choose the mixture probabilities (α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=')  in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For a particular target distribution, we will choose a maximal  number of integration steps kmax for which αk = 0 for k > kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Then, given a  particular selection of α1 ∈ (0, 1], we split the remaining probability mass equally  for each k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , kmax};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' that is, the fraction can be expressed as αk =  1−α1  kmax−1  for k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments, we will consider variable choices for the  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  17  mixing parameter α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The special case α1 = 0, will correspond by convention to the  unmodified RMHMC and LMC transition kernels with a single-step computed using  the prescribed involution, as described in definition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Mixing with the MMALA transition kernel also immediately establishes  irreducibility, aperiodicity, and the smallness of all compact sets, as the  following result reveals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Lemma 13 Let π(q) ∝ exp(L(q)) be continuous and bounded on compact sets and  denote by Π the probability measure with density π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Suppose α1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The marginal  transition kernel of LMRMHMC (or LMLMC) is Π-irreducible, aperiodic, and all  non-negligible compact sets are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proof With probability α1, the Markov chain transitions according to a Metropolis-  Hastings accept-reject decision with a normal proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Hence, for any  set A ⊂ B(Rm) for which  �  A π(q) dq > 0, we have,  ˜Kϵ(q, A) ≥ α1Jϵ(q, A)  (61)  > 0  (62)  since a normal proposal distribution is non-vanishing everywhere on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The fact  that ˜Kϵ is aperiodic and that all non-negligible compact sets are small follows as an  immediate consequence of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 from Mengersen and Tweedie [1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  Corollary 14 From lemma 13, we have that the modified Markov chain of RMHMC  (or LMC) is Π-irreducible, aperiodic, and from proposition 7 Π is the stationary  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Hence, it follows from theorem 2 that the modified Markov chain of  RMHMC (or LMC) produces an ergodic Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4 Experimentation  We turn now to the investigation of the proposed modified variations of  RMHMC and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' These examples are chosen to represent a wide class of  posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In computing the convergence of the Markov chain  under the maximum mean discrepancy metric, we measure similarity between  10,000 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' samples of the target distribution, and an equal number of inde-  pendent Markov chains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in this case, we compute the unbiased estimator of the  maximum mean discrepancy over the course of the first one-hundred sampling  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For the expected squared jump distance (ESJD) and the effective sample  size (ESS) metrics, we consider a ten replicates of a long Markov chain consist-  ing of 1,000,000 samples, except in the case of the Fitzhugh-Nagumo posterior  where we sample only 100,000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In each experiment, we consider mix-  ing LMC and RMHMC with the transition kernel described in definition 18,  except in the case of Neal’s funnel distribution, wherein we consider a mixture  with definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We begin in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 to describe measures and metrics  Springer Nature 2021 LATEX template  18  Modified Riemannian Manifold and Lagrangian Monte Carlo  by which we may assess the performance of the Markov chains corresponding  to LMRHMC and LMLMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Code for reproducing these experimental results may be found at https:  //tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='com/29kz7krz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 Measures and Metrics  We now give details of the evaluation metrics by which we compare Markov  chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We first recall the method of maximum mean discrepancy due to  Gretton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Definition 26 Let k : Rm×Rm → R be a positive definite function that is symmetric  in its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let Π and Π′ be two probability distributions on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The squared  maximum mean discrepancy between Π and Π′ is defined by,  MMD2(k, Π, Π′) =  E  q,q′∼Πk(q, q′) +  E  q,q′∼Π′k(q, q′) − 2  E  q∼Π,q′∼Π′k(q, q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (63)  Let (q1∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qr∗) ∼ Π and (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qs) ∼ Π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' An unbiased estimator of the squared  maximum mean discrepancy is,  MMD2  u(k,  �  qi  ∗  �r  i=1 ,  �  qi�s  i=1) =  1  r(r − 1)  r  �  i=1  r  �  j̸=i  k(qi  ∗, qj  ∗) +  1  s(s − 1)  s  �  i=1  s  �  j̸=i  k(qi, qj)  −  2  rs  r  �  i=1  s  �  j=1  k(qi  ∗, qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (64)  In our evaluations we adopt a squared exponential positive definite kernel  k(q, q′) = exp(−∥q − q′∥2/2h), where h ∈ R+ is a parameter called the kernel  bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments we set h to be the median distance between i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  samples from the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Recall the definition of geometric ergod-  icity given in definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The measure of convergence is the total variation  norm, which we can discuss theoretically but cannot evaluate in a computa-  tional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Instead, we can measure convergence to the target distribution  Π as a function of n, the number of steps, by means of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (63), for which we  can obtain an unbiased estimate via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (64) if we have samples from Kn(q0, ·)  and samples from Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Indeed, as in definition 26, let (q1  ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qr  ∗) ∼ Π and, for a  fixed initial position q0 and number of steps n let (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qs) ∼ Kn(q0, ·) and  we can compute an unbiased estimate of the squared maximum mean discrep-  ancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the latter case, (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qs) can be obtained by running s independent  Markov chains for n steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' independent samples from the target distribution  Π may be available in certain benchmark cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' sam-  ples from the target distribution may be generated from the banana-shaped  posterior, Neal’s funnel distribution, the Fitzhugh-Nagumo posterior, and the  multi-scale Student-t distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  19  Remark 11 Recall the definition of geometric ergodicity given in definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Taking  logarithms reveals  log ∥Kn(q, ·) − Π(·)∥TV ≤ n log ρ + log V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (65)  Therefore, one may claim to see evidence of geometric ergodicity if, as a function of  n, there is a linear decrease in the total variation distance on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Of  course, we cannot directly compute the total variation distance, but we may look for  a similar negative linear trend when |MMD2u| is plotted on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  As an additional measure of ergodicity, we consider comparing Markov  chain samples against i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' samples via random projection onto one-  dimensional sub-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let (q1  ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qr  ∗) ∼ Π and let qn ∼ Kn(q0, ·) for n =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Let u be a random unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We compute the Kolmogorov-Smirnov  (KS) statistic for the projections (u⊤q1  ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , u⊤qr  ∗) and (u⊤q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , u⊤qs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Repeating this process for one-hundred randomly generated unit vectors yields  a distribution over Kolmogorov-Smirnov statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The more tightly concen-  trated this distribution is near zero, the closer the distribution of Markov chain  iterates (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qs) is to the collection of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' samples (q1  ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qr  ∗) from the  target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We also consider the expected squared jump distance (ESJD) [Gelman and  Pasarica, 2007], which measures the dissimilarity between subsequent states  of the Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Intuitively, a Markov chain that moves more efficiently  through the sample space (higher ESJD) will exhibit smaller sample auto-  correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Formally, the ESJD is  E  q∼Π∥q′ − q∥2 where q′|q ∼ K(q, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Note  that the choice of norm ∥ · ∥ is left to the practitioner and may be selected  to capture geometric properties of the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This expectation is  typically approximated as follows: let qk be the state of the Markov chain at  step k and at step k + 1 a candidate state is generated, denoted qk+1  prop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' the  proposal state is accepted with probability αk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' the following empirical mean  is then taken as our approximation to the ESJD  ESJD(  �  (qk, qk+1  prop, αk)  �n  k=1) = 1  n  n  �  k=1  αk∥qk+1  prop − qk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (66)  In our experiments in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8, we observe that the ESJD can be misleading  as a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Therefore, we also introduce the median squared jump distance  (MSJD) which we define as  MSJD(  �  (qk, qk+1  prop, αk)  �n  k=1) = Median  ��  αk∥qk+1  prop − qk∥2�n  k=1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (67)  As the median, we expect the MSJD to exhibit less sensitivity to outliers than  the ESJD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We additionally consider the effective sample size (ESS) as a metric for  our Markov chain procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We compute ESS using the technique of Kumar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2019], who describe the method succinctly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , p,  Springer Nature 2021 LATEX template  20  Modified Riemannian Manifold and Lagrangian Monte Carlo  (a) EHMC  (b) LMRMHMC  (c) LMLMC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 1: We examine the ergodicity of the Markov chains as a function of the  sampling step in the banana-shaped posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that  the use of a smaller step-size degrades the ergodicity of the RMHMC, par-  ticularly when compared against the LMC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In both RMHMC and  LMC, we observe that aggressively mixing with MMALA causes the Markov  chain to mix more slowly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' on the other hand, a relatively small mixing proba-  bility leads to ergodicity that is nearly indistinguishable from the unmodified  implementations of RMHMC and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  let (qi,1, qi,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qi,n) be a sequence of Rm-valued parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The integer p  is called the number of chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The effective sample size of the j-th parameter  is computed according to,  ESS  ��  qi,1  j , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , qi,n  j  �p  i=1  �  = pn  ˆτ  (68)  ˆτ =  �  2  r  �  k=1  ˆρ2k + ˆρ2k+1  �  − 1,  (69)  where ˆρk is an estimate, based on all of the p sequences, of the autocorrelation  of the j-th parameter with a k-step lag (for details see Vehtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' [2021])  and r is the smallest integer for which ˆρ2(r+1) + ˆρ2(r+1)+1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our exper-  iments, we set p = 2 by taking a single long Markov chain and splitting it in  half at the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' As a practical matter, we will also report the minimum,  over all parameters of the posterior, ESS per second in order to represent the  computational efficiency of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 Banana-Shaped Distribution  Our first example considers the following generative model, which represents  an example of non-identifiable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  θ1, θ2  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼ Normal(0, σ2  θ)  (70)  yi|θ1, θ2  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼ Normal(θ1 + θ2  2, σ2  y) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (71)  Given observations {yi}N  i=1, we wish to sample the posterior distribution of  (θ1, θ2), a two-dimensional posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments, we consider n =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9  10-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7  2u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  MD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  10-  0  50  100  Sampling Step100  10-1  D  ≥ 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  10-4  10-5  0  20  40  60  80  100  Sampling Step100  10-1  ≥ 10-3  0  20  40  60  80  100  Sampling StepSpringer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  21  (a) ESJD  (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  (c) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 2: We show the ESJD, the minimum ESS, and the time-normalized  minimum ESS for inference in the banana-shaped posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We  observe that for both RMHMC and LMC, aggressively mixing with MMALA  causes the ESJD distance to decrease;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' this correspondingly produces a decrease  in the minimum ESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For LMC, the time-normalized minimum ESS is a  decreasing function of the mixing probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We employ Euclidean HMC with a step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' RMHMC is  implemented with a step-size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='04;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in LMC we use an integration step-size  of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In each case, we set kmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For the geometric methods, the sum  of the Fisher information and the negative Hessian of the log-prior is used as a  Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The reason that a smaller integration step-size is employed  in RMHMC is that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (5) in the defining involution will fail to have a solution  for ˘p for large step-sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' this necessitates the use of a smaller integration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 1, we visualize the statistic |MMD2  u| over one-hundred steps of the  Markov chain for euclidean HMC (EHMC), RMHMC and LMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' we color the  RMHMC and LMC Markov chains according to how aggressively they mix with  MMALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Neither EHMC nor RMHMC exhibit clear evidence of geometric  ergodicity in this target distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' on the other hand, there exist a broad  range of mixing probabilities for which LMC exhibits a linear decrease, as a  function of n, in |MMD2  u| on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 2 we visualize the ESJD,  the minimum ESS, and the minimum ESS per second for the three MCMC  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that LMC exhibits by far the strongest performance  on these metrics, while RMHMC languishes due to its small step-size and the  pathologies of the generalized leapfrog integrator applied to this posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3 Hierarchical Bayesian Logistic Regression  We consider sampling from the hierarchical Bayesian logistic regression model  α ∼ Gamma(ω, θ)  (72)  βi|α  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼ Normal(0, α−1) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , m  (73)  yi|xi, β  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼  Bernoulli  �  1  1 + exp(−x⊤  i β)  �  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (74)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  LMRMHMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  LMLMO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  EHMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability60000  50000  ESS  40000  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  30000  20000  10000  0  MMALA Probability40  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  30  ESS  20  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  22  Modified Riemannian Manifold and Lagrangian Monte Carlo  (a) ESJD  (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  (c) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 3: We show the ESJD, the minimum ESS, and the time-normalized mini-  mum ESS for inference in the hierarchical Bayesian logistic regression posterior  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that both of the geometric methods do not perform  as well as standard HMC in this inference task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For each mixing weight with  MMALA, we observe that RMHMC enjoys a greater ESJD and minimum ESS  when compared against LMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' however, as LMC is a more computationally  expedient procedure, LMC has a greater time-normalized ESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In all of these  metrics, however, EHMC dominates the geometric inference algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We consider a logistic regression dataset with N = 270 observations and  m = 14 covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We set ω = 10 and θ = 2 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We  employ a Metropolis-within-Gibbs sampling procedure wherein we alternate  between sampling the posterior distributions α|β, k, θ and β|α, {(xi, yi)}n  i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  sampling the former can be performed analytically, whereas we employ EHMC,  RMHMC, and LMC to sample the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In Euclidean HMC, we set ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in  RMHMC and LMC we set ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We set kmax = 10 as the upper bound on the  number of integration steps in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For implementing both LMRMHMC  and LMLMC, we use the sum of the Fisher information and the negative  Hessian of the log-prior as a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 3 we show the ESJD, the minimum ESS, and the minimum ESS  per second for the hierarchical Bayesian logistic regression posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Neither  of the geometric methods perform well in this posterior, consistently under-  performing EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' A criticism of LMC is that its performance degrades  significantly in higher dimensions [Betancourt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=', 2014];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' one sees evidence  of this phenomenon in the smaller ESS generated by LMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' however, the com-  putational savings due to eliminating the fixed point iterations still allow LMC  to edge out a stronger time-normalized ESS compared to RMHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4 Neal’s Funnel Distribution  Neal’s funnel distribution [Neal, 2003] is a density defined in the following  hierarchical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  v ∼ Normal(0, 9)  (75)  xi|v  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼ Normal(0, exp(−v)) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (76)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50  LMRMHMC  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='25  LMLMC  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='75  EHMC  D  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50  S  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='25  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='75  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability450000  400000  S  2 350000  I  300000  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  250000  200000  150000  MMALA Probability800  600  ESS  400  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 1  200  二二二二二二  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  23  (a) EHMC  (b) LMRMHMC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 4: We examine the ergodicity of the Markov chains as a function of the  sampling step in Neal’s funnel distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that EHMC struggles  in this distribution due to the multiple spatial scales inherent in the funnel-  shaped distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Moreover, we observe that aggressively mixing RMHMC  with SMALA causes convergence in MMD to become increasingly slow, with  convergence of the pure SMALA algorithm being slower than EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' On the  other hand, a modest amount of mixing with SMALA produces ergodicity that  is nearly indistinguishable from the unmodified implementation of RMHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (a) ESJD  (b) KS  (c) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 5: We show the ESJD, the distribution of Kolmogorov-Smirnov statistics,  and the time-normalized minimum ESS for inference in Neal’s funnel distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Although RMHMC and EHMC have comparable time-normalized ESS, it  is clear from the distribution of KS statistics that RMHMC produces samples  that are closer to the target distribution in EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  This distribution is shaped like a funnel, in which the thickness of the “neck”  is being controlled by the random variable v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This model is reflective of pos-  teriors encountered in hierarchical models with sparse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The objective in  this task is to jointly sample (v, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , xN), producing a (N + 1)-dimensional  target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In Euclidean HMC we employ an integration step-size of  ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 and kmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Our implementation of RMHMC uses kmax = 20  integration steps with a step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 with the SoftAbs metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We do  not consider LMC in this task since we found it non-obvious how the SoftAbs  structure could be extended into the LMC framework while preserving the  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9  10-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7  2u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  10-5  0  50  100  Sampling Step100  10-1  ≥ 10-3  0  20  40  60  80  100  Sampling Step17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  LMRMHMC  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  EHMC  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySmirnov  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  ESS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8  Tin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  24  Modified Riemannian Manifold and Lagrangian Monte Carlo  (a) ESJD  (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  (c) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 6: We show the ESJD, the minimum ESS, and the time-normalized mini-  mum ESS for inference in the posterior distribution of the stochastic volatility  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that both of the geometric methods with a modest mixing  probability with MMALA produce time-normalized ESS that are competitive  with EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For each mixing weight with MMALA, we observe that LMC  enjoys a greater ESJD when compared against RMHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  cubic computational cost at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In this experiment, we use the SoftAbs  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 4, we visualize |MMD2  u| for both EHMC and RMHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' As expected,  EHMC struggles to sample from Neal’s funnel distribution due to the mul-  tiscale phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' On the other hand, RMHMC exhibits much stronger  convergence properties, having a linear decrease over several possible mixing  probabilities with SMALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We find that aggressively mixing with SMALA  can be counter-productive, however, due to the less efficient traversal of the  target distribution by single-step methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 5 we show the ESJD, the  Kolmogorov-Smirnov statistics, and the minimum ESS per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' On all of  these metrics, RMHMC clearly outperforms EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5 Stochastic Volatility Model  We consider a stochastic volatility model with the following generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  φ + 1  2  ∼ Beta(20, 3/2)  (77)  σ2 ∼ InverseChiSquared(10, 1/20)  (78)  x1|φ, σ2 ∼ Normal(0, σ2/(1 − φ2))  (79)  xt+1|xt, φ, σ2 ∼ Normal(φxt, σ2) for t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , T − 1  (80)  yt|β, xt ∼ Normal(0, β2 exp(xt)) for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (81)  Additionally, the parameter β is equipped with an improper prior pro-  portional to β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In this example, we seek to generate samples from the  joint distribution x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , xT , β, φ, σ2 given observations y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , yT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We employ  Metropolis-within-Gibbs-like strategy wherein we alternate between sampling  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , xT )|(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , yT )φ, σ2, β and (φ, σ2, β)| {(xi, yi)}T  i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in the former case  we employ Euclidean HMC whereas in the latter case we compare Euclidean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0005  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0004  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0003  LMRMHMO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0002  LMLMC  EHMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0001  MMALA Probability4500  4000  ESS  3500  3000  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  2500  2000  1500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  25  (a) ESJD  (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  (c) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 7: We show the ESJD, the minimum ESS, and the time-normalized  minimum ESS for inference in the posterior distribution of the log-Gaussian  Cox-Poisson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For any mixing probability, the geometric methods enjoy  larger time-normalized effective sample sizes than EHMC, with LMC outper-  forming RMHMC on these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For nearly every mixing probability, we  additionally observe that the ESJD is greater for LMC and RMHMC than for  EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  HMC, RMHMC, and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In sampling either distribution, the Riemannian  metric is chosen as the sum of the Fisher information and the negative Hessian  of the log-prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments we set T = 1, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' When using Euclidean  HMC to sample (φ, σ2, β) we use kmax = 50 integration steps and a step-size  of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in the case of RMHMC and LMC we use kmax = 6 integration  steps and a step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 6 we show the ESJD, the minimum ESS, and the minimum ESS  per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that employing a modest mixture probability with  MMALA produces a Markov chain that is marginally better than EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6 Log-Gaussian Cox-Poisson Process  We consider inference in a log-Gaussian Cox-Poisson model with the following  generative model:  Σ(i,j),(i′,j′)|σ2, β = σ2 exp  �  −  �  (i − i′)2 + (j − j′)2/(Nβ)  �  (82)  vec(x)|Σ ∼ MultivariateNormal(µ1, Σ)  (83)  yij|xij ∼ Poisson(exp(xij)/N 2),  (84)  with priors β ∼ Gamma(2, 1/2) and σ2 ∼ Gamma(2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In this exam-  ple, the objective is to sample the joint distribution (β, σ2, {xij}N  i,j=1) given  observations {yij}N  i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' As in the case of stochastic volatility model, we alter-  natively sample between {xij}N  i,j=1 given {yij}N  i,j=1, σ2, and β, and (β, σ2)  given {xij}N  i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In each case, the metric is given by the sum of the Fisher  information and the negative Hessian of the log-prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments we  set N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the former case we employ Euclidean HMC, whereas in the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  LMRMHMG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3  LMLMC  EHMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability50000  40000  S  S  上 30000  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  20000  10000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  26  Modified Riemannian Manifold and Lagrangian Monte Carlo  (a) EHMC  (b) LMRMHMC  (c) LMLMC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 8: We examine the ergodicity of the Markov chains as a function of  the sampling step in the Fitzhugh-Nagumo posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe  that when a modest mixing probability is employed, the modified RMHMC  and LMC transition kernels mix more efficiently than the EHMC transition  kernel, being nearly indistinguishable from the unmodified RMHMC and LMC  transition kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' When one uses a large mixing probability, we see that the  mixing rate is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  latter case we compare Euclidean HMC, RMHMC and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' When imple-  menting Euclidean HMC we employ kmax = 50 integration steps and a step-size  of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in RMHMC and LMC we use kmax = 6 integration steps and a  step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 7 we show the ESJD, the minimum ESS, and the minimum ESS per  second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that there is a range of mixture probabilities for which  mixing with the MMALA in LMRMHMC and LMLMC exhibit superior perfor-  mance compared to EHMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Both RMHMC and LMC exhibit similar minimum  ESS metrics, but due to its computational advantage, LMC produces a larger  minimum ESS per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7 Fitzhugh-Nagumo Model  Given R-valued parameters a, b, and c, the Fitzhugh-Nagumo differential  equations are defined by,  ˙vt = c  �  vt − v3  t  3 + rt  �  (85)  ˙rt = −  �vt − a + brt  c  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (86)  Given initial conditions v0 and r0, we consider the following generative model:  (a, b, c)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼ Normal(0, 1)  (87)  ˆrtk|a, b, c, rtk, σ2 indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼  Normal(rtk, σ2) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , n  (88)  ˆvtk|a, b, c, vtk, σ2 indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  ∼  Normal(vtk, σ2) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , n,  (89)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9  10-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7  2u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  MD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  10-3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  10-  0  50  100  Sampling Step100  ≥ 10-3  10-  0  20  40  60  80  100  Sampling Step100  ≥ 10-3  10-  0  20  40  60  80  100  Sampling StepSpringer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  27  (a) ESJD  (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  (c) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 9: We show the ESJD, the minimum ESS, and the time-normalized min-  imum ESS for inference in the Fitzhugh-Nagumo posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' LMC  dominates both RMHMC and EHMC in this inference task under the time-  normalized ESS metrics with its greater computational expediency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' When  timing is not accounted for, we observe that RMHMC and LMC enjoy simi-  lar ESJD and minimum ESS metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We observe that for both RMHMC and  LMC, aggressively mixing with MMALA causes the ESJD distance to decrease;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  this correspondingly produces a decrease in the minimum ESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  where t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , tn are equally spaced points between [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In our experiments,  we set v0 = 1, r0 = −1, σ2 = 1/4, T = 10, and n = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Our metric is given  by the sum of the Fisher information and negative Hessian of the log-prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In  our implementations, we use Euclidean HMC with a step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='01 and  kmax = 10 integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In RMHMC and LMC, we employ an integration  step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5 and kmax = 6 integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Figure 8 shows the ergodicity measures for EHMC, RMHMC, and LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We observe that each of these MCMC algorithms exhibit a linear decrease in  |MMD2  u| on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 9 we visualize the ESJD, the minimum  ESS, and the minimum ESS per second for the three MCMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We  observe that LMC exhibits the strongest performance in terms of the mini-  mum ESS per second, whereas RMHMC only approaches the time-normalized  performance of EHMC due to its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8 Multi-Scale Student Distribution  Fix m ∈ N and let Σ ∈ PD(m) and ν > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The density function of the  multivariate Student-t distribution is,  π(q) ∝  �  1 + 1  ν x⊤Σ−1x  �−(m+ν)/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (90)  In our experiments we consider m = 20, ν = 5, and Σ = diag(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' , 1, 104) ∈  Rm×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The presence of severely differing spatial scales in the multivariate  Student-t distribution will cause Euclidean HMC with identity mass matrix  to exhibit highly oscillatory behavior [Pourzanjani and Petzold, 2019], which  will limit the efficiency of the method both in terms of ergodicity and the  effective sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We employ Euclidean HMC with a step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8,  6  5  ESJD  LMRMHMC  3  LMLMC  2  EHMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability160000  140000  120000  in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS  100000  80000  60000  40000  20000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability80  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  70  60  ESS  50  40  30  20  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  28  Modified Riemannian Manifold and Lagrangian Monte Carlo  (a) EHMC  (b) LMRMHMC  (c) LMLMC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 10: We examine the ergodicity of the Markov chains as a function of the  sampling step in the multiscale Student-t distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Here we see that, in the  case of LMC, it is beneficial from an ergodicity perspective to mix heavily with  MMALA, whereas for RMHMC, only modest mixing probabilities can produce  a modified Markov chain that is competitive with the unmodified version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  Euclidean HMC struggles in this posterior distribution due to the multiple  spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (a) ESJD  (b) MSJD  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 11: We show the ESJD and the MSJD for inference in the multiscale  Student-t distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Curiously, EHMC enjoys the largest ESJD, but this  does not correspond to more efficient sampling, as shown in the inferior distri-  bution of KS statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' To examine this property further, we also measured the  MSJD, which reveals that EHMC is less effective in traversing the parameter  space compared to the RMHMC geometric method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  and RMHMC and LMC with a step-size of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' in each case we employ  kmax = 20 integration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In the case of RMHMC and LMC, we consider a  metric given by the positive definite term in the Hessian of the log-density of  the multivariate Student-t distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 10 we show |MMD2  u| as a function of the number of Markov chain  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' All of the MCMC algorithms produce a linear decrease in the estimate  of the maximum mean discrepancy on a logarithmic scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' however, the meth-  ods differ drastically in terms of the slope of this linear relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We see  that RMHMC exhibits by far the fastest convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Notably, LMC exhibits  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9  10-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7  2u  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6  MD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  10-5  0  50  100  Sampling Step100  10-1  ≥ 10-3  0  20  40  60  80  100  Sampling Step100  10-1  ≥ 10-3  10-4  10-5  0  20  40  60  80  100  Sampling StepLMRMHMC  30  LMLMC  25  EHMC  D  20  15  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA Probability25  20  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  29  (a) KS  (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ESS / Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 12: We show the distribution of Kolmogorov-Smirnov statistics and the  time-normalized minimum ESS for inference in the multiscale Student-t distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' For the geometric methods, we observe that RMHMC has much larger  ESJD than LMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' however, in terms of time-normalized performance, it is best  to fully mix either method with MMALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Indeed, we find that MMALA on  its own produces a distribution of KS statistics that is competitive with any  of the mixture kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  slower convergence on this target distribution than MMALA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' this is due to  the dimensionality of the posterior, in which LMC struggles to maintain the  Hamiltonian energy required to accept proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 12 we show the ESJD,  KS, and the minimum ESS per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' EHMC produces the largest ESJD but  this does not translate into a large ESS due to the oscillatory behavior of the  EHMC proposal mechanism, with RMHMC producing the largest minimum  ESS per second despite its computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  5 Conclusion  This work has considered methods by which to equip RMHMC and LMC  with a geometric ergodicity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The fundamental technique we adopt is to  replace the RMHMC (or LMC) transition kernel consisting of a single inte-  gration step with the MMALA transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This modification is inspired  by the Euclidean case, in which single-step HMC and MALA can be con-  structed to be exactly equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' By establishing reversibility of the marginal  transition kernels, geometric ergodicity can be inherited from MMALA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We  evaluated the modified variations of RMHMC and LMC, called LMRMHMC  and LMLMC, respectively, on a suite of Bayesian inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' We found  that aggressively mixing with MMALA transition kernel can be detrimental  for the performance of the Markov chain on a variety of metrics, but that  more modest mixing can produce behaviors competitive with, or exceeding,  the original RMHMC or LMC methods while still imbuing the methods with  a supporting theory of geometric ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  rnov  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='012  LMRMHMC  LMLMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='010  EHMC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='002  MMALA Probability160  140  120  ESS  100  80  lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  60  40  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='0  MMALA ProbabilitySpringer Nature 2021 LATEX template  30  Modified Riemannian Manifold and Lagrangian Monte Carlo  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  We thank the Yale Center for Research Computing  for use of the research computing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This material is based upon  work supported by the National Science Foundation Graduate Research Fel-  lowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' 1752134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Any opinion, findings, and conclusions or  recommendations expressed in this material are those of the authors(s) and  do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  work is also supported in part by NIH/NIGMS R01GM136780 and AFOSR  FA9550-21-1-0317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Statements and Declarations  No competing interests to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Appendix A  Proofs  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1  Proof of Proposition 5  Proof The proposal distribution of MALA is  ˜q|q ∼ Normal  �  q + ϵ2  2 A−1∇ log π(q), ϵ2A−1  �  (A1)  ˜π(˜q|q) ∝ exp  �  − 1  2ϵ2  �  ˜q − q − ϵ2  2 A−1∇ log π(q)  �⊤  A  �  ˜q − q − ϵ2  2 A−1∇ log π(q)  ��  (A2)  = exp  �  − 1  2ϵ2  �  A˜q − Aq − ϵ2  2 ∇ log π(q)  �⊤  A−1  �  A˜q − Aq − ϵ2  2 ∇ log π(q)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A3)  Therefore, the acceptance probability of MALA is,  min  �  1, π(˜q)˜π(q|˜q)  π(q)˜π(˜q|q)  �  = min  �  �  �  �  �  �  �  1, π(˜q)  π(q) ·  exp  �  − 1  2ϵ2  �  Aq − A˜q − ϵ2  2 ∇ log π(˜q)  �⊤  A−1 �  Aq − A˜q − ϵ2  2 ∇ log π(˜q)  ��  exp  �  − 1  2ϵ2  �  A˜q − Aq − ϵ2  2 ∇ log π(q)  �⊤  A−1  �  A˜q − Aq − ϵ2  2 ∇ log π(q)  ��  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A4)  The proposal ˜q given q can be sampled by generating z ∼ Normal(0, Id) and setting  ˜q = q + ϵ  2A−1∇ log π(q) + ϵ  √  A−1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A5)  In HMC, the leapfrog integrator is applied to the Hamiltonian H(q, p) = − log π(q)+  1  2p⊤A−1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The sequence of updates is,  ¯p = p + ϵ  2∇ log π(q)  (A6)  ˜q′ = q + ϵA−1¯p  (A7)  = q + ϵ2  2 A−1∇ log π(q) + ϵA−1p  (A8)  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  31  ˜p′ = ¯p + ϵ  2∇ log π(˜q′)  (A9)  = p + ϵ  2∇ log π(q) + ϵ  2∇ log π(˜q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A10)  When sampling from the distribution with density proportional to exp(−H(q, p)), one  sees by inspection that p is independent of q and that p ∼ Normal(0, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Therefore,  using the same z as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (A5), we can sample p by setting p =  √  Az so that  ϵA−1p = ϵ  √  A−1z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' we therefore see that, in this case, HMC and MALA produce  exactly the same proposal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' ˜q′ = ˜q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Recall that the HMC acceptance probability  is,  min  �  �  �1, π(˜q)  π(q) ·  exp  �  − 1  2(˜p′)⊤A−1(˜p′)  �  exp  �  − 1  2p⊤A−1p  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A11)  By rearranging eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (A7) we find,  p = 1  ϵ  �  A˜q − Aq − ϵ2  2 ∇ log π(q)  �  (A12)  ˜p′ = 1  ϵ  �  A˜q − Aq + ϵ2  2 ∇ log π(˜q)  �  (A13)  = −1  ϵ  �  Aq − A˜q − ϵ2  2 ∇ log π(˜q)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A14)  Substituting these into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (A11) and comparing to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (A4) shows that not only are  ˜q and ˜q′ identical but that the acceptance probabilities are also identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Therefore,  the marginal chain of single-step HMC is exactly equivalent to the MALA chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2  Proof of Lemma 6  Proof  Pr(qn+1 ∈ Q|qn = q) = Pr(qn+1 ∈ Q and pn+1 ∈ Rm|qn = q)  (A15)  =  �  Rm Pr(qn+1 ∈ Q and pn+1 ∈ Rm|qn = q, pn = p) π(p|q) dp  (A16)  =  �  Rm K((q, p), (Q, Rm)) π(p|q) dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A17)  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='3  Proof of Proposition 7  Proof Given Q, Q′ ∈ B(Rm) we have  �  Q  ˜K(q, Q′) π(q) dq =  �  Q  �  Rm K((q, p), (Q′, Rm)) π(p|q) dp π(q) dq  (A18)  =  �  Q  �  Rm K((q, p), (Q′, Rm)) π(q, p) dp dq  (A19)  =  �  Q′  �  Rm K((q, p), (Q, Rm)) π(q, p) dp dq  (A20)  Springer Nature 2021 LATEX template  32  Modified Riemannian Manifold and Lagrangian Monte Carlo  =  �  Q′  �  Rm K((q, p), (Q, Rm)) π(p|q) dp π(q) dq  (A21)  =  �  Q′  ˜K(q, Q) π(q) dq,  (A22)  where in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' (A20) we have used the fact that the phase space chain satisfies detailed  balance with respect to π(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' This verifies that the marginal chain satisfies detailed  balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='4  Proof of Involution Composition  Proposition 15 Suppose that Φ : Rm × Rm → Rm × Rm is an invertible function  and let F be the momentum flip function given in definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Suppose that F ◦ Φ  is an involution, then F ◦ Φk is also an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Proof This will be proved by induction with the base case established by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  As the inductive hypothesis, assume that F ◦ Φk is an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Using the fact  that F ◦ Φ is an involution, we immediately obtain that Φ−1 ◦ F = F ◦ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Using the  inductive hypothesis, one also has Φ−k ◦ F = F ◦ Φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Therefore, we obtain,  Φ−1 ◦ F = F ◦ Φ  (A23)  =⇒ Φ−1 ◦ F ◦ Φk = F ◦ Φk+1  (A24)  =⇒ Φ−1 ◦ Φ−k ◦ F = F ◦ Φk+1  (A25)  =⇒ Φ−k−1 ◦ F = F ◦ Φk+1  (A26)  =⇒ F ◦ Φk+1 ◦ F ◦ Φk+1 = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  (A27)  This verifies that F ◦ Φk+1 is also an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  □  Data Availability Statement  The datasets generated during and/or analysed during the current study are  available in the GitHub repository, https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='com/29kz7krz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content='  ISBN 9780821843024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  URL https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='com/books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='id=  vc2FWSo7wLUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Betancourt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' A general metric for Riemannian manifold Hamiltonian Monte  Carlo, 12 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Betancourt, Simon Byrne, Samuel Livingstone, and Mark Girolami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' The  geometric foundations of Hamiltonian Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content='  Springer Nature 2021 LATEX template  Modified Riemannian Manifold and Lagrangian Monte Carlo  33  Alain Durmus, ´Eric Moulines, and Eero Saksman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Irreducibility and geometric  ergodicity of Hamiltonian Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  The Annals of Statistics, 48(6):  3545–3564, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Andrew Gelman and Cristian Pasarica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Adaptively scaling the Metropolis  algorithm using expected squared jumped distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Statistica Sinica, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  URL http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='2139/ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='1010403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Mark Girolami and Ben Calderhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content='  Markov Chains and Stochastic Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content=' Implicit Hamiltonian Monte Carlo  for sampling multiscale distributions, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content=' Tweedie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Exponential convergence of  Langevin distributions and their discrete approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Bernoulli, 2(4):  341 – 363, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content='  Vivekananda Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Convergence diagnostics for Markov chain Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Annual Review of Statistics and Its Application, 7:387–412, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content=' Bayesian  Analysis, 16(2), Jun 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Xifara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Sherlock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Livingstone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Byrne, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Girolami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content=' Langevin  diffusions and the Metropolis-adjusted langevin algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
+page_content='  Statistics &  Probability Letters, 91:14 – 19, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQfcvy7/content/2301.01409v1.pdf'}
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+Draft version January 12, 2023
+Typeset using LATEX preprint style in AASTeX631
+A Model for Gradual Phase Heating Driven by MHD Turbulence in Solar Flares
+William Ashfield IV1, 2 and Dana Longcope3
+1Bay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA 94035, USA
+2Lockheed Martin Solar & Astrophysics Laboratory, Organization A021S, Building 252, 3251 Hanover Street, Palo
+Alto, CA 94304, USA
+3Dept. of Physics, Montana State University, Bozeman, MT 59717
+(Accepted January 8th, 2023)
+Submitted to ApJ
+ABSTRACT
+Coronal flare emission is commonly observed to decay on timescales longer than those
+predicted by impulsively-driven, one-dimensional flare loop models.
+This discrepancy
+is most apparent during the gradual phase, where emission from these models decays
+over minutes, in contrast to the hour or more often observed. Magnetic reconnection
+is invoked as the energy source of a flare, but should deposit energy into a given loop
+within a matter of seconds. Models which supplement this impulsive energization with a
+long, persistent ad hoc heating have successfully reproduced long-duration emission, but
+without providing a clear physical justification. Here we propose a model for extended
+flare heating by the slow dissipation of turbulent Alfv´en waves initiated during the re-
+traction of newly-reconnected flux tubes through a current sheet. Using one-dimensional
+simulations, we track the production and evolution of MHD wave turbulence trapped
+by reflection from high-density gradients in the transition region. Turbulent energy
+dissipates through non-linear interaction between counter-propagating waves, modeled
+here using a phenomenological one-point closure model. AIA EUV light curves synthe-
+sized from the simulation were able to reproduce emission decay on the order of tens
+of minutes. We find this simple model offers a possible mechanism for generating the
+extended heating demanded by observed coronal flare emissions self-consistently from
+reconnection-powered flare energy release.
+1. INTRODUCTION
+Solar flares are typically characterized by the increase in radiation observed during their onset.
+Accepted as the result of energy released through magnetic reconnection, the sudden rise in emission
+across different wavelengths is considered to be an indication of impulsive plasma heating. Following
+the conclusion that flare loops convert magnetic energy on the order of the Alfv´enic transit time
+across a given loop (Kopp & Pneuman 1976; Priest & Forbes 2002), numerical flare loop models
+aiming to study the impulsive phase have commonly used energy source terms that reflect these
+short timescales. Many of these models have also been constrained by observations to infer energy
+deposition rates — either through hard X-rays (Kowalski et al. 2017; Graham et al. 2020) or UV
+arXiv:2301.04592v1  [astro-ph.SR]  11 Jan 2023
+
+2
+Ashfield & Longcope
+emission (Longcope & Bradshaw 2010; Qiu et al. 2012; Ashfield et al. 2022) — where the heating
+duration was found to last up to tens of seconds.
+Although impulsively-driven models have been a successful tool for explaining a number of flare
+phenomena, their failure to reproduce gradual phase emission is widely recognized. Observations
+in soft X-rays and EUV have shown the gradual decay of hot coronal plasma to last between tens
+of minutes to several hours following the rise phase.
+The cooling of coronal plasma, marked by
+the successive peaks of emission light curves from increasingly cooler ion species, happens through
+a combination of heat conduction (Culhane et al. 1970, 1994) and radiative losses (Aschwanden &
+Alexander 2001; Vrˇsnak et al. 2006). Compared to the characteristic cooling timescales modeled by
+individual flare loops (e.g. minutes, Kerr et al. 2020), the sustained duration of observed coronal
+emissions is considerably longer than it would be if regulated by cooling alone.
+The discrepancy between modeled and observed decay rates during the gradual phase has led
+many to believe that additional heating is required beyond the impulsive phase to offset the effects
+of radiative losses (Withbroe 1978; Svestka 1989; Ryan et al. 2013; Sun et al. 2013). A popular
+interpretation of this heating has been the successive energization of multiple individual flux tubes
+within a flare arcade — so-called multi-loop models (Dere & Cook 1979; Hori et al. 1997; Warren
+2006; Reep et al. 2022) — with each loop emulating the prescribed impulsive energy release. While
+these models are consistent with observation, several investigations were unable to rectify the lack
+of sustained coronal emissions (Reeves & Warren 2002; Qiu et al. 2012; Liu et al. 2013; Kerr et al.
+2020).
+In response to
+this persistent discrepancy, Qiu & Longcope (2016) recently developed a model
+where individual loops were driven with a two-part heating profile consisting of an impulsive energy
+release followed by a prolonged, low-rate heating. This slow-tail heating, lasting on the order of
+20 minutes, was able to successfully forward-model key characteristics of EUV lightcurves measured
+using the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012), including long cooling timescales.
+While motivated by flare UV footpoint emissions that display similar, two-phase behavior (see also
+Cheng et al. 2012; Qiu et al. 2013; Zhu et al. 2018), the heating profiles introduced in this work were
+only ad hoc. Physical justifications behind extended loop heating were not provided outside of a few,
+potentially viable scenarios.
+Several mechanisms for producing such extended heating rates alluded to in Qiu & Longcope (2016)
+and other investigations are contingent upon post-reconnection flare loops retracting under magnetic
+tension and releasing energy (Forbes & Acton 1996; Linton & Longcope 2006). One such speculation
+is the continual heating from slow-shocks generated by the resistance loops receive from earlier formed
+loops that exist lower in the flare arcade (Cargill & Priest 1982, 1983). Another related mechanism
+is the excitation of MHD waves by reconnection outflows (Aschwanden 2006) that then decay and
+heat plasma within a post-reconnection loop over timescales comparable to slow-tail heating (Wang
+2011).
+Central to both of the scenarios described above is the interaction between retracting flare loops
+and their surroundings. Previous investigations of outflows driven by magnetic tension have modeled
+flux tubes moving through an ideal current sheet unobstructed, and have therefore retracted at the
+local Alfv´en speed (Forbes & Priest 1983; Longcope & Klimchuk 2015). Observed outflows — inferred
+from the motion of supra-arcade downflows (SADs; McKenzie & Hudson 1999; Savage & McKenzie
+2011; Reeves et al. 2017) interpreted to be the wakes behind retracting flux tubes (Savage et al. 2012)
+
+3
+— however, move at fractions of the Alfv´en speed. In the case of a non-ideal current sheet, such
+as that with a high-β plasma (Scott et al. 2016), a retracting loop would likely undergo a loss of
+momentum and speed as it imparted work on the surrounding medium. This interaction, modeled as
+a simple aerodynamic drag force, was recently found to slow outflow speeds to within their observed
+ranges (Unverferth & Longcope 2021).
+The interpretation of SADs as the wakes of retracting flux tubes experiencing a drag force would
+also suggest the generation of turbulence, given the aerodynamic analogy.
+In fact, evidence for
+turbulence has been reported in current sheets containing SADs, inferred through measurements of
+non-thermal line broadening (Ciaravella & Raymond 2008; Warren et al. 2018; Cheng et al. 2018)
+and local correlation tracking (McKenzie 2013). Separate observations using AIA and the Extreme-
+ultraviolet Imaging Spectrometer (EIS; Culhane et al. 2007) have also shown turbulence to coexist
+in regions with hot, ∼ 30 MK downward moving loops, but did not make reference to SADs (Imada
+et al. 2013; Doschek et al. 2014). Using observations of several strong flares, Larosa & Moore (1993)
+found that the interactions between individual flux tubes and interactions with a diffusive current
+sheet are likely to develop outflows with a turbulent structure. Moreover, they also found MHD
+turbulence to be capable of rapidly thermalizing bulk kinetic energy via a turbulent cascade, thus
+becoming a mechanism for impulsive phase energy release. It was further suggested by Jiang et al.
+(2006) that plasma wave turbulence could produce heating into the decay phase, indicated by the
+long cooling times of loop-top HXR sources. Although the interconnected behavior of retracting flux
+loops, turbulence, and extended flare heating has been well established by observation, it has yet to
+be modeled in detail.
+The present work aims to develop a self-consistent mechanism for gradual-phase energy release.
+Motivated by the observations of SADs, we create a relatively simple model for the excitation of
+MHD turbulence along a thin flux tube (TFT) as it retracts through a current sheet. The retracting
+tube experiences a drag force, which converts energy from the tube into an internal population of
+turbulent Alfv´en waves that then propagate along the tube and reflect off high-density gradients
+in the transition region. Turbulent energy is dissipated through the non-linear interaction between
+counter-propagating waves, thus creating a local heat source within the tube. A model for drag with
+turbulent transport is outlined in Section 2. The behavior of our model is then demonstrated using an
+initial reference simulation in Section 3. Using the simulation results, synthetic AIA EUV emissions
+are created by which our model can be qualitatively compared to observation. The parameters critical
+to the duration of turbulent energy release — the fraction of drag power converted to turbulent Alfv´en
+waves and the energy correlation length of the propagating waves — are explored in Section 4. A
+final simulation run with a set of optimized parameters was found to extend the duration of EUV
+emission past 40 minutes. The duration of the heating produced from turbulent dissipation was also
+found to persist long after the retraction had ended. The results presented in this paper point to a
+physical mechanism that can produce late-phase heating required to sustain coronal flare emission
+for the first time.
+2. FLARE MODEL WITH ALFV´EN WAVE TURBULENCE
+In order to model the energy released by reconnection and its conversion to MHD turbulence,
+we employ the TFT equations described by Longcope et al. (2009). The model assumes localized
+reconnection has already occurred within a small diffusion region, within the current sheet, to create
+a closed magnetic flux tube. The moment following reconnection is illustrated in Figure 1. Two
+
+4
+Ashfield & Longcope
+(a)
+(b)
+Figure 1. Schematic of a reconnected flux tube embedded in a current sheet. (a) Face-on view
+of the
+current sheet. External magnetic fields on either side of the current sheet — the blue and red-dashed lines —
+are skewed according to the sheer angle ∆θ. The grey tube shows the geometry of the newly formed flare loop
+immediately following reconnection. (b) Perspective orientation illustrating the deflection of the external
+magnetic fields around the flux tube, with the tube sliced through its apex. The grey area corresponds to
+the current sheet shown in (a). Only the ˆz-component of the external magnetic fields is shown in the latter.
+layers of equilibrium magnetic field are separated by a current sheet and have directions differing by
+a shear angle ∆θ. The newly reconnected loop, shown in grey, is embedded within the current sheet
+that lies in the x–z plane. It is at this point the TFT model begins. Mechanisms of the reconnection
+process itself are not considered here.
+Once the bent tube is initialized, it retracts through the current sheet under a magnetic tension
+force. The internal magnetic pressure of the tube is balanced by the magnetic pressure of the external
+magnetic fields separated by the current sheet, setting the magnetic field strength inside the tube.
+In the present case, we assume the external magnetic fields to be uniform, such that the magnetic
+field B is also uniform along the tube.
+The TFT equations govern the evolution of the axis of a flux tube, r(ℓ, t), where ℓ is the param-
+eterized length coordinate of the curved tube. The fluid velocity of the plasma confined within the
+tube, v = dr/dt, is advanced according to the momentum equation (Longcope & Klimchuk 2015)
+ρdv
+dt =
+�
+B2
+4π − p
+�
+∂ˆl
+∂ℓ −ˆl∂p
+∂ℓ + ρg + 4
+3µ ∂
+∂ℓ
+�
+ˆlˆl · ∂v
+∂ℓ
+�
+,
+(1)
+where p = ρkBT/ ¯m is the plasma pressure and ¯m = 0.593 mp is the mean particle mass of the
+plasma, assumed to be fully ionized with a coronal abundance. For a fluid element along the tube
+with differential length dℓ and mass per unit flux dm, the mass density is given by ρ = B(dm/ dℓ).
+Outside of gravitational force ρg, which acts in the downward direction, forces acting on a fluid
+element in Equation (1) are dictated by the tangent vector, ˆl = ∂r/∂ℓ. The first term on the rhs
+describes the magnetic tension force along the tube’s curvature vector, ∂ˆl/∂ℓ, perpendicular to the
+tube’s axis. The second is the gas pressure directed along the tube. Viscous interactions between
+
+ZZ5
+fluid elements are described by the final term, with µ = 0.012 κsp(T)/cv being the temperature-
+dependent dynamical viscosity coefficient and the classical Spitzer-H¨arm thermal conductivity being
+κsp = 10−6 T 5/2, in conventional cgs units.
+The energy of the tube is advanced according to the temperature of a fluid element
+cvρdT
+dt = −p
+�
+ˆl · ∂v
+∂ℓ
+�
++ 4
+3µ
+�
+ˆl · ∂v
+∂ℓ
+�2
+− n2
+eΛ(T) + ∂
+∂ℓ
+�
+κ∂T
+∂ℓ
+�
+,
+(2)
+where cv = 3kB/2 ¯m is the specific heat and ne = 0.874 (ρ/mp) is the electron number density. The
+first term on the rhs constitutes the adiabatic compression done on the gas. This term is followed
+by viscous heating, which arises from the loss of kinetic energy in Equation (1).
+Optically thin
+radiative losses are given by the third term, where the radiative loss function Λ(T) is taken from
+the output of CHIANTI 7.1 (Landi et al. 2012). The final term describes the field-aligned thermal
+conduction. A flux limiter is included in the model, such that the thermal conductivity is restricted
+to the theoretical electron free-streaming limit at sufficiently large temperature gradients (Longcope
+& Klimchuk 2015).
+2.1. Drag Force and Alfv´en Wave Turbulence
+The TFT model describes the evolution of flare loops retracting under a tension force brought
+about by a change in magnetic topology. Previous studies assumed the retraction occurs without any
+interaction with the background current sheet, and therefore saw typical outflow velocities reaching
+the local Alfv´en speed on the order of 3 Mm s−1 (Longcope et al. 2018; Unverferth & Longcope 2020).
+Alfv´enic outflow speeds are common to all idealized models of magnetic reconnection, but are not
+supported by much observational evidence. Supra-arcade downflows (SADs McKenzie & Hudson
+1999) are often taken to be signatures of post-reconnection retraction, but always seem to move well
+below the local Alfv´en speed (Savage & McKenzie 2011; Savage et al. 2012).
+Based on observations of sub-Alfv´enic outflow speeds it has been proposed that the retracting
+flux must somehow interact with the surrounding plasma, possibly by deforming the flux outside
+the sheet or entraining plasma (see Figure 1b and discussions in Linton & Longcope 2006; Scott
+et al. 2013). Most forms of interaction would leave the surrounding plasma with greater magnetic or
+kinetic energy, at the expense of the retracting tube. This would appear as some kind of drag force
+on the retracting flux, removing some of its energy and reducing the retraction speed. Unverferth &
+Longcope (2021) investigated this possibility, introducing a drag force modeled using a high Reynolds
+number aerodynamic drag (Choudhuri & Gilman 1987)
+fd = −D ρ |v⊥| v⊥.
+(3)
+Here v⊥ = v − ˆl(ˆl · v) is the component of fluid velocity perpendicular to the flux tube and D
+is a constant proportional to the traditional drag coefficient. Because the interaction between the
+flux tube and the current sheet is likely more intricate than the conventional drag force exerted by a
+neutral fluid on a rigid body, D is taken to be a free parameter to capture these unknown complexities
+(Unverferth & Longcope 2020). This force, per unit area, is added to the momentum equation (1).
+The force exerted by the surrounding plasma ultimately opposes the acceleration of the loop by
+transferring momentum and energy away. Energy is therefore removed from the retracting loop at a
+rate of
+Pd = ρ−1 v · fd ≤ 0,
+(4)
+
+6
+Ashfield & Longcope
+which is always negative.
+The energy is not really lost, but must appear in some form in the
+surrounding plasma. We assume here that it takes the form of MHD turbulence, of which some
+fraction remains on the tube, but at length scales heretofore unresolved. We therefore assume a
+fraction of Pd takes the form of unresolved Alfv´en wave turbulence occurring on the tube itself, even
+as it retracts.
+2.2. Evolution of Alfv´en Wave Turbulence
+Gaining inspiration from turbulent transport models of the solar wind (i.e. Marsch & Tu 1989;
+Matthaeus et al. 1999; Dmitruk et al. 2001; Verdini & Velli 2007; Lionello et al. 2014), the equations
+used for the evolution of turbulent wave energy in this work are developed by first decomposing
+the velocity and magnetic fields into large-scale contributions, denoted by capitals, and small-scale
+perturbations, denoted by lower case:
+v = U + u
+(5)
+B = B0 + b = Bˆl + b.
+(6)
+The perturbations can then be collected in terms of the Els¨asser variables, z± ≡ u ± b/√4πρ.
+The evolution of the fluctuations is expressed by the scale-separated, linearized MHD equations in
+their Els¨asser representation (Zhou & Matthaeus 1990a,b; Zank et al. 2011)
+Dz±
+Dt = ±(VA · ∇)z± + 1
+2(z∓ − z±)∇ ·
+�U
+2 ± VA
+�
+.
+(7)
+Here, VA = B0/√4πρ is the local Alfv´en speed along the loop. We assume the large-scale fields of
+U and B vary slowly across the flux tube and have therefore ignored the gradient terms in Equation
+(7). The source and sink terms for the Els¨asser variables are also temporarily neglected here and are
+instead discussed in detail below.
+The aggregate energy densities of the turbulent Alfv´en waves are then found by spatially averaging
+over the small-scale fluctuations. The energy per unit mass of the two turbulent waves is given by:
+w± = 1
+4⟨z± · z±⟩
+(8)
+with the total energy density of the perturbations being wtot = w+ + w− = ⟨u2/2⟩ + ⟨b2/8πρ⟩. By
+averaging over the perturbations, we remove their explicit dependence from the system and instead
+model their collective evolution implicitly.
+Taking the inner product of z± with Equation (7) for z±, respectively, gives the expression for each
+Els¨asser energy w±:
+Dw±
+Dt
+= ±(VA · ∇)w± − w±∇ ·
+�U
+2 ± VA
+�
++ R±.
+(9)
+Here, R± is a term proportional to ⟨z+ · z−⟩ that accounts for wave reflection. Analogous to the
+‘mixing’ effects described in Zhou & Matthaeus (1990b), the interaction between opposite Els¨asser
+variables allows for energy to be redistributed between the two populations, such that w± will generate
+counter-propagating w∓. This process is linked to large-scale gradients in the system, and is likely to
+be most effective where ∂lnρ/∂ℓ is large (Hollweg 1981; Velli 1993). Computing R± self-consistently
+
+7
+would require additional equations beginning with one for the evolution of ⟨z+ · z−⟩. In the interest
+of simplicity, we forego this approach and instead set R± = 0 and account for wave reflection with a
+boundary condition described below.
+Following the TFT model, we parameterize the wave energy equations according to length coordi-
+nate ℓ. As the large-scale fields are taken to vary only in the direction parallel to ˆl, divergences in U
+and VA reduce to spatial derivatives in ℓ. The full turbulent transport equations in their conservative
+forms are then
+dw±
+dt
+= ±vA
+∂w±
+∂ℓ − 1
+2w±
+∂v∥
+∂ℓ ∓ w±
+∂vA
+∂ℓ + S± + NL±,
+(10)
+for v∥ = ˆl · v. Because the terms in the above expression are dependent only on system variables
+aligned with the mean magnetic field, Equation (10) describes the energies of small-scale Alfv´en
+waves propagating along the field in the ±ˆl direction. Propagation of the wave energies is described
+by the first three terms on the rhs. The first is the advective term, showing how the wave energies
+propagate at the Alfv´en speed. The second and third terms describe work done on the waves by
+compression of the plasma and magnetic pressures against that of the wave, respectively.
+2.3. Sources and Sinks of Alfv´en Wave Turbulence
+The last two terms of Equation (10), absent from Equation (9), are added to describe the source,
+S±, and sink, NL±, of the turbulent energies, respectively. The source, as described above, is a
+fraction, fturb, of the power the tube loses through drag as it retracts through the current sheet
+described by Equation (4):
+S± = − 1
+2 fturb Pd.
+(11)
+Here the 1
+2 prefactor assumes the input energy is divided equally between the counter-propagating
+wave species.
+The loss of turbulent energy in the system is modeled according to the simple phenomenological
+decay rate used in numerous MHD turbulence investigations (e.g. Hossain et al. 1995; Matthaeus et al.
+1999; Dmitruk et al. 2001; Zank et al. 2011). Following one-point closure models for hydrodynamic
+turbulence first described by de Karman & Howarth (1938), the energy loss arises from the non-
+linear interaction between the counter-propagating waves. This interaction is assumed to produce a
+cascading spectrum of turbulent Alfv´en waves — defined by a set of wave numbers that correspond
+to the cascade of energy-containing eddies into increasingly smaller scales — that ultimately results
+in the dissipation of turbulent energy.
+The dissipation rate for the Els¨asser energies is expressed as
+NL± = −w±√w∓
+λ⊥
+,
+(12)
+where λ⊥ is the single similarity length scale that characterizes the transverse dimensions of the
+energy-containing eddies for both the rightward and leftward propagating waves. As such, λ⊥ is
+the characteristic length scale that couples the non-linear spectral transfer between the counter-
+propagating modes, and can thus be thought to correspond to the correlation length between the
+two energy populations (Batchelor 1953). Although turbulence is typically described by a range of
+length scales (i.e. wave numbers), this simple one-point model assumes the decay can be represented
+by a single non-linear term.
+
+8
+Ashfield & Longcope
+We assume that all energy lost from the turbulence is ultimately thermalized and appears as a heat
+source for the plasma. This is achieved simply by adding the term
+Hturb = ρ
+�
+NL+ + NL−
+�
+,
+(13)
+to the rhs of Equation (2). With this additional term, our model establishes a connection, albeit
+indirect, between the energy lost to drag and a heating rate that can be used to explain long-duration
+EUV emission seen during the gradual phase of flares.
+The TFT equations are further modified by considering the effects of turbulence on the tube’s
+thermal conductivity. As magnetic field perturbations will lengthen the field lines along which ther-
+mal electrons carry heat, the thermal conductivity will consequently decrease. We account for this
+suppression by modifying the classical Spitzer-H¨arm conductivity
+κsp → κ(turb)
+sp
+=
+κsp
+1 + ρ(w+ + w−)/B2,
+(14)
+such that the turbulent energy will directly impede the heat flux in the tube.
+Although this is
+a relatively simple addition to the model, it attempts to address the notion of heat conduction
+suppression via turbulence in a self-consistent manner.
+Finally, we achieve turbulent wave reflection from the chromosphere using the boundary conditions
+w−(ℓ0) = η w+(ℓ0)
+,
+w+(ℓ1) = η w−(ℓ1),
+(15)
+where ℓ0 and ℓ1 are the left and right boundaries, respectively. Because reflections are likely to occur
+in the transition region where temperature and density gradients are large, these boundaries are set
+by a characteristic temperature TTR such that ℓ0(1) = min(max) {T > TTR}. In this case, waves
+incident on the transition region at either end of the loop will be transformed into their respective
+counter-propagating waves, effectively trapping the wave energy in the corona. Furthermore, the
+boundary condition assumes a reflection coefficient of η ≤ 1 to account for energy lost from wave
+transmission. The value of this coefficient is taken as a free parameter and is discussed, along with
+the other free parameters of our model, in the following section.
+3. SIMULATION WITH ALFV´EN WAVE TURBULENCE
+To illustrate the behavior of our model, we construct and run a single reference simulation. Rather
+than attempt to model a particular observation, we aim at properties typical of a long-duration event,
+including peak temperatures above 20 MK and flux retraction speeds of order 500 km/s. We attempt
+to extend the cooling time to values comparable to those typically found. For concreteness, we use
+values reported in Qiu & Longcope (2016), since they also infer input energy. To make a meaningful
+comparison, we choose parameters to give the simulation the same energy flux as they report. The
+run and its parameters are discussed alongside their motivations below.
+3.1. Initial Conditions
+Prior to the simulation run, a flux tube is initialized in a configuration analogous to a bent field
+line created from reconnection. The tube is confined to the x–z plane, corresponding to the current
+sheet that separates uniform layers of magnetic flux differing by shear angle ∆θ = 120◦, as illustrated
+by Figure 1a. The initial tube is therefore bent at its apex by 180◦ − ∆θ = 50◦, and is set to have
+
+9
+a uniform magnetic field of magnitude B = 100 G. Because this process forms two straight segments
+joined at a single point, the apex of the loop is rounded to be a semi-circle composed of eight grid
+points to prevent issues arising from under-resolution.
+Once bent, the flux tube is initialized into three components: a coronal loop and two footpoints
+attached to a rudimentary chromosphere. The pre-flare chromosphere is stratified under gravity with
+pressure increasing exponentially according to scale height H = 500 km. Taken to be isothermal with
+temperature Tmin=0.01 MK, the chromosphere primarily serves as a mass reservoir of cool, dense
+plasma for evaporated material. The coronal region of our flux tube is structured according to the
+relations given in Rosner et al. (1978). Defined by an apex temperature set to Tco,0=1.3 MK, the
+initial corona is configured in isobaric equilibrium maintained by an ad hoc volumetric heating input.
+The heating required to maintain equilibrium, however, is only used during the initialization process.
+Subsequent evolution of the loop does not contain this heating term in order to more directly observe
+the consequences of turbulent heating induced along the tube.
+The initial length of the tube is calculated from the amount of flare energy we want to be released
+by the system. In the TFT model, the source of this energy comes from contracting magnetic field
+lines, releasing magnetic free energy into other forms as the flux tube retracts. The magnetic energy
+per unit of magnetic flux of the tube is (Longcope & Klimchuk 2015)
+WM = 1
+4π
+�
+B[r(ℓ)]dℓ.
+(16)
+Because the strength of the magnetic field is fixed, a change in magnetic energy is therefore powered
+by a reduction in the tube’s length. A net length decrease ∆L, releases flare energy, per magnetic
+flux,
+Efl = B∆L
+4π
+sin2
+�∆θ
+4
+�
+.
+(17)
+Here, the sine-squared factor represents the fraction of magnetic energy converted to kinetic en-
+ergy parallel to the axis of the tube through rotational discontinuities that form during retraction
+(Longcope & Klimchuk 2015). This mechanism constitutes the primary mode of magnetic energy
+conversion in previous studies using the TFT model to investigate reconnection dynamics, with and
+without drag (Guidoni & Longcope 2010; Unverferth & Longcope 2021).
+To make contact with the work done in Qiu & Longcope (2016), we incorporate the same flare
+energy deposited by the impulsive part of their heating profile, H(t), into the energy released in
+our model1. By integrating over the heating rate, the total energy per unit area is calculated to be
+�
+H(t)dt = ζEflB = ζ2.9 × 1011 erg cm−2. The prefactor ζ is introduced to account for imperfect
+conversion between magnetic energy and other forms. Assuming the primary energy driving flare
+phenomena arises from MHD turbulence, as we argue below, the conversion is ultimately related to
+the fraction of power lost through drag, such that ζ = 1/fturb. We initially take this fraction to be
+reasonably conservative at fturb=0.2, giving a total energy release of 1.45 × 1012 erg cm−2.
+Equation(17) is then solved for ∆L, giving ∆L = 73 Mm.
+The loop is initialized with length
+L0 = Lf + ∆L, with Lf being the final length of the loop once retraction is complete. We choose Lf
+to be 93 Mm, which gives an initial loop length of L0=166 Mm.
+1 The heating rate, given by Equation (6) in Section 4 of Qiu & Longcope (2016), is a piecewise expression composed of
+three Gaussians. The first two describe the impulsive flare energy release as inferred from AIA 1600 ˚A ribbon emission
+observations, while the third is appended to model gradual-phase heating. Here we integrate the first two, along with
+the same parameter values described in the same section, to calculate a total energy per unit area.
+
+10
+Ashfield & Longcope
+3.2. Parameters of the Drag Model
+Although the total energy released by the loop through retraction is set by Equation (17), the
+inclusion of a drag force in our model changes the nature of energy release in comparison to previous
+TFT models. In these earlier works, parallel kinetic energy is responsible for shocks that compress
+and thermalize the plasma via viscous heating, while the energy found in perpendicular flows is found
+to have little effect on the overall magnetic energy conversion. Unverferth & Longcope (2021) showed
+that kinetic energy will be effectively removed from the tube by including drag. While this reduction
+was shown to affect the perpendicular kinetic energy primarily, rotational discontinuities were also
+shown to be less effective at driving parallel flows as well, thus limiting the available energy to be
+converted to heat. In the case of drag-induced Alfv´en wave turbulence, perpendicular kinetic energy
+is no longer lost. Instead, a fraction is reintroduced to the tube, as dictated by Equations (4) and
+(11). We expect the subsequent transfer of turbulent energy to heat from dissipation will capture
+a portion of the magnetic energy that would otherwise have gone into perpendicular flows and be a
+source of heat comparable to, if not greater than, heat driven by parallel plasma flows.
+The drag coefficient, D, in Equation (3) controls both the amount of turbulent energy siphoned
+from perpendicular kinetic energy and the rate of retraction in the −ˆz direction. As described in the
+introduction, drag was first motivated by observations of reconnection outflows (e.g. SADs Savage
+et al. 2012), and serves as the mechanism by which sub-Alfv´enic speeds can become consistent with
+Petschek-like reconnection models. Here we set D = 50 Mm−1 in order for our model to agree with
+measured outflow speeds. An initial investigation using this coefficient shows the loop retracting
+at speeds ∼500 km s−1, which is comparable to the average velocity of SADs measured in Longcope
+et al. (2018).
+The turbulence model includes several new parameters, used here for the first time. These will
+be varied in order to understand their effects, but we first describe their significance and plausible
+ranges. We will then return in the discussion to consider what their variation has revealed about the
+physics of turbulence.
+The turbulent energy dissipation rate is set by the energy correlation length, λ⊥. Investigations
+into the effects of turbulence on coronal heating have assumed λ⊥ to be on the order of inter-network
+spacing of 30 Mm (Matthaeus et al. 1999; Downs et al. 2016). Abramenko et al. (2013) measured
+the characteristic length of the energy-containing structures using local correlation tracking in the
+photosphere, finding λ⊥ to lie in the range of 0.6–2 km. Although these studies deal with the issue
+of coronal heating driven by convective motions in the photosphere, it is not unreasonable to take
+the correlation length in the coronal to be of similar values. If we instead take λ⊥ to be on the order
+of the diameter of a given flux tube, then λ⊥ ∼ 0.1 − 2 Mm, given the distribution of loop widths
+measured using images taken of coronal loops with AIA and the Hi-C imager (Aschwanden & Peter
+2017).
+There is an alternative approach through the observed decay time of Alfv´en wave energy. If that
+decay is due only to a non-linear turbulent cascade, then the decay time will be given by the eddy
+turnover time
+τnl =
+λ⊥
+⟨vnth⟩ = λ⊥
+√w.
+(18)
+Investigations into the non-thermal broadening of EIS Fe xxiv 255 ˚A lines during flares have seen
+spatially averaged non-thermal velocities ⟨vnth⟩ on the order of 60-100 km s−1 decay to pre-flare values
+
+11
+in roughly 20 minutes (Kontar et al. 2017; Stores et al. 2021). Setting Equation (18) to this time,
+and using this turbulent velocity, suggests a correlation length of λ⊥ = 30 − 150 Mm.
+There is at least one other aspect to include in our interpretation of λ⊥. Traditional derivations
+of the non-linear dissipation captured by Equation (12) assume the counter-propagating waves are
+uncorrelated with one another. Our waves, however, are trapped between reflecting ends and are
+likely to be correlated. This fact could be accounted for by one more multiplicative factor, 0 < ξ < 1,
+in Equation (12). Rather than including one more free parameter, we combine the two into a single
+free parameter λ′
+⊥ = λ⊥/ξ, which will thus be larger than the turbulent length scale. We hereafter
+interpret λ⊥ as λ′
+⊥.
+Altogether, the above analysis produces a very rough range of λ⊥ = 0.1–150 Mm. Nonetheless,
+given the simplicity of this model, these values are only meant to demonstrate that a variety of
+length scales can be used to characterize turbulent energy dissipation, while also being grounded
+in observation. For the purpose of a single reference model, λ⊥ is initially set to 2 Mm — a value
+consistent with the diameter of observed coronal loops. This choice thereby assumes the energy
+correlation length to be on the order of the small-scale transverse fluctuations along the flux tube.
+The effects of different length scales on the duration of turbulent heating are explored further in the
+following section.
+Lastly, the initial value of the reflection coefficient for the propagating energy waves is set to
+be η = 0.95. This value, albeit close to its upper threshold of 1, is chosen so that energy losses
+arising from turbulent dissipation can be distinguished from losses due to energy escape through the
+transition region. The characteristic temperature TTR defining the reflection boundaries is chosen to
+correspond to regions with a high-density gradient. For our tube initialized with Tco,0=1.3 MK, we
+therefore set TTR= 0.5 MK.
+3.3. The Reference Simulation
+With the flux tube initialized as described above, Equations (1) (2), and (10) are solved using
+the PREFT numerical code as documented in Longcope & Klimchuk (2015). Fluid elements along
+the tube are represented by cells on a 1D Lagrangian grid, with the density and differential length
+of each calculated to ensure each has a constant mass per unit flux. Aside from the evolution of
+temperature, which is advanced semi-implicitly, all expressions are advanced explicitly.
+In order
+to ensure stability in the system, each time step is chosen such that the Courant–Friedrichs–Lewy
+conditions are satisfied. Initial parameters and the resulting properties of the reference simulation
+are summarized in Table 1.
+The evolution of the tube during its retraction is illustrated in Figure 2. Starting with an apex
+at height z0=71 Mm, the tube retracts until its total length has decreased to Lf = 93 Mm at time
+t = 452 s (7.5 min), yielding a final apex height of zf = 18 Mm.
+The change in apex height,
+∆z = z0 − zf = 53 Mm, is analogous to a post-flare loop moving through a current sheet of the same
+length (Longcope et al. 2018, for example). Once the retraction is complete, the tube is artificially
+straightened to lie on the ˆx-axis, mimicking the effect of the tube reaching the post-flare arcade.
+Velocity flows after this point are set to equal the parallel velocity v∥ immediately prior to the
+straightening, and the perpendicular flow is set to zero. The simulation is then run for an additional
+32 minutes in the straightened configuration. With no perpendicular flow, the source of Alfv´en waves
+is S± = 0, so the wave energies are left to decay freely.
+
+12
+Ashfield & Longcope
+0
+20
+40
+60
+80
+0
+10
+20
+30
+40
+50
+60
+70
+z [Mm]
+0 s
+20 s
+40 s
+75 s
+200 s
+300 s
+452 s
+0
+20
+40
+60
+80
+1010
+1012
+1014
+ne [cm−3]
+0
+20
+40
+60
+80
+−1000
+−500
+0
+500
+1000
+v∥ [km s−1]
+0
+20
+40
+60
+80
+x [Mm]
+0
+10
+20
+30
+T [MK]
+Figure 2. Dynamics of the reference simulation over the course of retraction. A face-on view of the tube
+axis r(ℓ) at seven times is shown in the top panel. The following three panels
+show the corresponding
+electron number density ne, velocity parallel to the axis of the tube v∥, and temperature T. Each panel is
+plotted against the x-coordinate, scaled such that x=0 Mm corresponds to the
+leftmost boundary of the
+tube.
+
+13
+Table 1.
+Simulation properties of the reference and optimized runs
+Reference
+Optimized
+Drag Parameter
+fturb
+...
+0.2
+0.6
+λ⊥
+Mm
+2
+100
+D
+Mm−1
+50
+50
+η
+...
+0.95
+0.95
+Model Result
+Tpeak
+MK
+41
+32
+Peak ˙Q
+1010 erg cm−2 s−1
+2.15
+1.02
+Retraction time
+min.
+7.5
+8.1
+˙Q duration
+min.
+13
+41
+EUV duration
+min.
+32
+40
+During the retraction, the corona undergoes characteristic flare dynamics. Temperature quickly
+increases from Tco,0 = 1.3 MK to values well over 30 MK before decaying down to ∼ 10 MK. The
+parallel velocity along the tube shows that evaporation starts by t = 20 s, reaching speeds up to
+nearly 1 Mm s−1. By t = 5 minutes, most of v∥ has dropped out completely. Once evaporation has
+begun, the tube’s density increases by several orders of magnitude in the corona, and remains elevated
+after the temperature and velocity in the tube have subsided.
+The explosive initial evolution appears more reminiscent of drag-free solutions (e.g. Longcope &
+Klimchuk 2015), than the gentle dynamics seen in Unverferth & Longcope (2021). The effect of drag,
+however, can be seen from the apparent deceleration of the tube in Figure 2. To illustrate the change
+in downward velocity, Figure 3 plots the apex height of the tube throughout its retraction. Here the
+retraction appears to occur in two nearly constant-velocity phases; a fast downflow, lasting a little
+less than one minute, followed by a dramatic reduction of speed that remains constant until the tube
+is straightened. A straightforward explanation for the deceleration is provided by evaporation. The
+upward driving of chromospheric material increases the density in the corona, subsequently lowering
+the local Alfv´en speed and slowing the retraction.
+Linear fits to each phase show the tube initially moving downward at a speed of 540 km s−1 before
+slowing down to 57 km s−1. The initial phase agrees with downflows measured in Longcope et al.
+(2018), where two-thirds of the 35 downward moving features had velocities lower than 600 km s−1.
+While the features analyzed in their work saw only straight streaks in a height-time map of a current
+sheet viewed edge-on, observations taken using TRACE 195 ˚A filtergrams saw downflows decelerating
+in a distinct two-step fashion (for example, see Figure 2 in Sheeley et al. 2004). This deceleration
+of roughly an order of magnitude occurred prior to the feature reaching the post-flare arcade. Our
+model thus offers a novel insight into the behavior of these observations.
+The initial phase of the flux tube’s evolution is shown in Figure 4, where we see the dynamics of
+the turbulent Alfv´en wave energies w±. The total energy of the waves wtot peaks at t = 0.3 s (cyan)
+and subsequently decreases as the waves propagate towards the footpoints of the loop. At t = 4.3 s,
+the waves reach the boundary set by the characteristic temperature TTR=0.5 MK, corresponding to
+
+14
+Ashfield & Longcope
+Figure 3. Apex position of the flux tube over the retraction period. Linear fits to the fast (red dashed)
+and slow (blue dashed) retraction phases are over-plotted.
+position ±x = 2.6 Mm from the edge of the tube. Upon reaching the boundary, the waves then
+begin to reflect, the effects of which can be seen at t = 5.4 s.
+The solid lines in the upper left
+panels correspond to the leftward propagating Els¨asser energy and the dashed lines correspond to
+the rightward, counter-propagating energy created from the reflection. Over times t > 4.3 s, the point
+of reflection moves progressively downward as the
+position where T = 0.5 MK moves by thermal
+conduction.
+Counter-propagating waves appear first at the loop top, where drag creates both waves together.
+Interaction between waves results in significant loop-top heating according to Equation (13).
+Tur-
+bulent dissipation drives the temperature of the loop to a peak temperature of Tpeak = 41 MK that
+rapidly drives thermal conduction fronts outward towards the footpoints. The steep temperature
+gradients shown in the lower left panels of Figure 4 are evidence of the flux limiter taking effect.
+Once the reflection begins (t > 4.3 s), interactions between the incident and reflected waves create a
+second locus of heating near the footpoint. A novel result of this simulation is the interaction between
+the thermal conduction fronts and the increase in temperature driven by the turbulent heating at
+the reflection boundary. As the conduction fronts move leftward in Figure 4, they are impeded by a
+localized temperature increase in the transition region that grows over time. This growth, initially
+
+15
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+(c)
+0.0
+0.1
+0.2
+0.3
+Hturb[1016 erg g−1 s−1 ]
+0.0
+0.5
+1.0
+1.5
+2.0
+(a)
+0.0
+0.2
+0.4
+0.6
+w± [1016 erg g−1]
+0
+20
+40
+x [Mm]
+10−2
+10−1
+100
+101
+102
+(e)
+1.5
+2.0
+x [Mm]
+10−2
+10−1
+100
+101
+T [MK]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+(b)
+0.0
+0.2
+0.4
+0.6
+0.8
+wtot [1016 erg g−1]
+−600
+−400
+−200
+0
+200
+(d)
+−100
+0
+100
+200
+v∥ [km s−1]
+0
+20
+40
+x [Mm]
+10−1
+100
+101
+(f)
+1.5
+2.0
+x [Mm]
+10−1
+100
+101
+p [erg cm−3]
+0.3 s
+1.0 s
+3.0 s
+5.4 s
+5.7 s
+6.0 s
+6.3 s
+6.5 s
+7.0 s
+Figure 4. Propagation of six flux tube variables over the first seven seconds of the reference simulation.
+Middle panels show each variable over the entire left half of the loop, while the corresponding leftmost and
+rightmost panels show enlargements of the chromosphere and upper transition region in detail. Working top
+to bottom, the variables plotted are (a) the leftward w+ —solid — and rightward w− — dashed — propa-
+gating Els¨asser energies (b) total Els¨asser energy wtot (c) Heat produced from turbulent energy dissipation
+Hturb (d) parallel velocity v∥ (e) temperature T and (f) pressure p.
+due to Hturb, is likely compounded by the conduction front, as the initial temperature bump seen
+at t = 5.4 s gradually smooths out with increasing temperature. Nevertheless, the localized heating
+from reflection creates a separate conduction front , as well as a pressure increase above the transition
+region. Remarkably, it is this conduction front that appears to drive evaporation upon reaching the
+chromosphere, the beginnings of which are seen at t = 6.3 s in both parallel velocity and pressure.
+It should be noted that the leftward movement of the reflection boundary, as shown by the leftward
+propagation of w±, is also due to the localized conduction front, which heats the plasma and moves
+TTR in that direction.
+
+16
+Ashfield & Longcope
+0
+10
+20
+30
+40
+x [Mm]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+wtot [1016 erg g−1]
+0
+10
+20
+30
+40
+x [Mm]
+10−3
+10−2
+10−1
+100
+101
+S [1016 erg g−1 s−1 ]
+0.1 s
+0.3 s
+1.0 s
+3.0 s
+7.0 s
+20.0 s
+40.0 s
+60.0 s
+Figure 5. Total Els¨asser energy wtot (left) and the corresponding source from drag power S (right) over
+the first 60 s of the reference simulation. An instance of the wave energy envelope per unit mass without
+sources or reflection (Equation (19)) is shown as the dashed-dotted curve at t=7.0 s.
+Figure 5 plots the evolution of the total turbulent Alfv´en wave energies alongside their corresponding
+sources S = S+ + S− for the first minute of the simulation. Here, the amplitude of the source peaks
+immediately after the retraction begins and is centered around the loop’s apex. As the tube continues
+to retract, the source amplitude decreases by two orders of magnitude and spreads along the extent
+of the tube. This effect illustrates how the sharp initial bend of the tube decomposes into a more
+rounded shape via the drag force, subsequently allowing for additional segments of the tube to be
+deflected downward — an effect that compounds throughout the retraction. Even so, most of the
+power in drag remains concentrated at the loop center, leading to a continual creation of turbulent
+Alfv´en waves there.
+After the Els¨asser energies reach their peak, they then decrease along with the source. Of particular
+note is the shape of the energies themselves, where the amplitude sharply decreases before undergoing
+reflection at the boundary. To investigate this behavior, we calculate an envelope representing the
+energy per unit mass in the waves that would otherwise be stationary without sources or reflection.
+To do this we solve the static version of equation (10), with dw±/dt = v∥ ∂w±/∂ℓ, and the left two
+terms omitted. The result
+w(env)
+±
+(ℓ) = A exp
+�
+�
+ℓ
+�
+1
+(vA ∓ v∥)
+∂(vA ± v∥/2)
+∂ℓ′
+dℓ′
+�
+� ,
+(19)
+is an envelope profile plotted as a dot-dashed line alongside the normal wave at t = 7 s in Figure 5,
+with arbitrary scaling A adjusted for clarity. The envelope shows a drop in wave energy due mainly
+to a decrease in the Alfv´en speed in the transition region. Excess energy density around x = 48 Mm
+
+17
+10−1
+100
+101
+102
+103
+0
+10
+20
+30
+40
+50
+60
+Wave
+Kinetic
+∥ Kinetic
+Thermal
+10−1
+100
+101
+102
+103
+Time [sec]
+−600
+−500
+−400
+−300
+−200
+−100
+0
+Total
+Magnetic
+Net Drag Loss
+Net Rad Loss
+Energy [108 erg Mx−1]
+Figure 6. Evolution of energies within the tube in units of 108 erg Mx−1, plotted against a logarithmic time
+axis. The bottom panel shows free magnetic energy (dark orange), net drag loss (purple), and radiative losses
+(magenta). The top panel shows turbulent wave energy (blue), kinetic (orange), parallel kinetic (dashed
+green), and thermal (red) on an enlarged scale. The total energy (teal) is the sum of the wave, kinetic,
+thermal, and magnetic energies.
+Note the kinetic and parallel kinetic energies are virtually indistinguishable.
+is then due to reflection and the production of counter-propagating waves. Similarly, although the
+source is still relatively high, turbulent dissipation brings the energy density below the envelope for
+x ≳58 Mm.
+The long-term evolution of the tube is shown by the energies plotted in Figure 6. The magnetic
+energy (dark orange) lost from its initial value of WM,0 = 1.317×1011 erg Mx−1 is shown in the bottom
+panel. Over its retraction, the tube releases a total of ∆WM = 5.76 × 1010 erg Mx−1 — 44% of the
+initial amount. Drag (purple) is responsible for removing nearly all free magnetic energy released.
+The total work done by drag by the end of the retraction is 5.61 × 1010 erg Mx−1, constituting over
+97% of the available magnetic energy. The slight deflection seen at t ∼ 60 s results from the change
+in retraction speed, while constant values starting at t = 452 s indicate the tube’s straightening.
+
+18
+Ashfield & Longcope
+The top panel of Figure 6 shows the remaining energies present in the tube. The total energy found
+in turbulent Alfv´en waves (blue) mirrors the drag power, reaching a maximum of 12.8×108 erg Mx−1
+before non-linear interactions between counter-propagating waves dominate. The inflection seen in
+the wave energy at t =7.8 s therefore corresponds to an uptick in thermal energy, shown in red,
+as the heating from turbulent dissipation increases. The source of wave energy vanishes once the
+retraction ends (t = 452 s), and the wave energy decays rapidly thereafter. It seems that the non-
+linear dissipation term works on relatively short time scales provided there is enough reflected power
+to facilitate non-linear interaction.
+The kinetic energy (orange) increases notably at t = 10 s, corresponding to the onset of evaporation
+flows. The parallel kinetic energy (green dashed) is effectively identical to the kinetic energy and
+makes up 99% of the total. This is a direct effect of the drag limiting the perpendicular velocity;
+in drag-free solutions, the parallel flow contributes only a small fraction to the total kinetic energy
+(Longcope & Klimchuk 2015). Furthermore, thermal energy decreases when the kinetic energy has
+subsided at t = 206 s. This moment, also corresponding to an increase in the radiative losses, can be
+used to indicate the cooling phase of the loop. By the end of the run, 1.25 × 1010 erg Mx−1 has been
+lost to radiation.
+The importance of wave energy in driving the initial flare evolution is readily seen when compared
+to the parallel kinetic energy. As alluded to in Section 3.1, the energy contained in turbulent Alfv´en
+waves is found to be comparable to that contained in parallel plasma flows. Moreover, wave energy
+in the tube increases immediately upon the start of retraction, whereas kinetic energy increases 10 s
+later, after much of the initial dynamics have already taken place. The kinetic energy here is due
+to flows driven by evaporation — shown above to be a result of heating via turbulent dissipation
+at the reflection boundary in the transition region — and not from plasma deflected by rotational
+discontinuities traveling at supersonic speeds (Longcope et al. 2009; Guidoni & Longcope 2010). That
+is to say, the evolution of energies seen here supports the notion of turbulence acting as the primary
+mode of heating during impulsive flare energy release.
+The close relation between explosive flare behavior and turbulent wave energy is also illustrated
+in the temperature evolution shown in Figure 7. The apex temperature of the reference simulation,
+shown in blue, is plotted over the entire evolution of the tube. The evolution is characteristic of those
+observed during flares, reaching a peak of T = 41 MK before slowly decaying down to its pre-flare
+value. In this case, the decay time for the flare is on the order of 35 minutes. We note that the
+additional peak at t ∼ 2 minutes is an effect of evaporation. For comparison, a simulation was run
+using the same drag coefficient, D=50 Mm−1, as the reference, but without including turbulent Alfv´en
+waves (i.e. fturb = 0). The result, shown in green, only reaches a peak temperature of T = 5.5 MK.
+While the temperature decay is on the same order of time as the reference run, the initial flare
+evolution is considerably weaker, suggesting kinetic energy alone is insufficient to power strong flare
+dynamics.
+It seems that the interaction of waves at the loop apex explains much of the peak temperature
+evolution. To demonstrate this notion, we lowered the reflection coefficient to η=0.10, while leaving
+the drag and source term the same. The results, plotted in red on Figure 7, shows the temperature
+evolution closely following that of the η=0.95 run, albeit at slightly lower values, reaching a peak of
+T = 32 MK and decaying to the pre-flare value in virtually the same time. The lower temperatures
+seen in this case suggest that reflected waves play some role in heating, albeit relatively minor.
+
+19
+0
+10
+20
+30
+time [min.]
+0
+10
+20
+30
+40
+Apex Temp. [MK]
+η=0.95
+η=0.10
+no waves
+Figure 7. Evolution for apex temperature from the results of the reference simulation (blue). Plotted
+for comparison are apex temperatures for a simulation with a reflection coefficient η = 0.10 (red) and a
+simulation with no turbulent Alfv´en waves (green).
+The small role played by reflection is consistent with the observation above that non-linear wave
+dissipation is strong enough to eliminate the wave energy rapidly. It is evidently also capable
+of
+dissipating the energy over length scales short enough that most does not reach the feet.
+3.4. Effect of Suppressed Thermal Conduction
+We briefly note that the effects of thermal conduction suppression, given by the modification in
+Equation (14), were found to be negligible. At its largest deviation, conductivity remained within less
+than 1% of its classical value. This is a natural consequence of the low energy density in Alfv´en waves
+(√w ≃ 0.5 Mm s−1) relative to the loop’s Alfv´en speed vA ≃ 10 Mm s−1. The magnetic perturbations
+will deflect the field lines by a random angle ∼ √w/vA ≃ 0.05. This deflection results in only a minor
+increase in field line path lengths, resulting in only a minor decrease in diffusivity.
+3.5. Synthetic EUV Emission
+As mentioned in the introduction, a major motivation for including turbulent Alfv´en waves in the
+TFT model was to reproduce observations of sustained coronal flare emission. In order to create
+a benchmark by which our model can be qualitatively compared to such observations, light curves
+corresponding to the six coronal EUV channels measured by AIA were synthesized using the results
+of our reference simulation.
+
+20
+Ashfield & Longcope
+For a given wavelength channel i, the pixel value pi for the total emission integrated along the flux
+tube is calculated by
+pi =
+� ∞
+0
+n2
+e(ℓ)Ki[T(ℓ)] dℓ,
+(20)
+given the temperature-response function K(T) (Boerner et al. 2012). These functions were accessed
+using the aia get response function in the SolarSoft IDL library (Freeland & Handy 1998).
+Because we are only interested in the time evolution, and not the integrated intensity, the light
+curves are normalized according to their maximum value, making it unnecessary to assign a cross-
+sectional area to the 1D tube.
+Synthesized emissions for each AIA EUV channel — 131, 94, 335, 211, 193, and 171 ˚A— are shown
+in Figure 8. The lightcurves evolve in a typical fashion and peak successively according to their
+characteristic temperatures, decaying from 20 to 0.4 MK. The higher temperature passbands of 131,
+94, and 335 ˚A , also have noticeably wider emission profiles than the lower temperature passbands of
+211, 193, and 171 ˚A. This difference in duration indicates the temperature evolution through these
+passbands, representing a temperature range of 0.4-2 MK, occurs more quickly. To be expected, the
+overall evolution of the light curves agrees well with the temperature decay in Figure 7. If we quantify
+the decay of the EUV emission by the time it takes the lowest channel in 171 ˚A to peak, then the
+duration of the flare emission in this case is approximately 32 minutes long.
+To illustrate the effect of turbulent Alfv´en waves on the duration of the EUV emission, the total heat
+from turbulent dissipation integrated along the tube, ˙Q =
+�
+Hturbdℓ, is plotted alongside the light
+curves, converted here into a heating rate per unit area. The heating rate roughly tracks the evolution
+of the total wave energy in Figure 6, climbing to a peak of 2.15×1010 erg cm−2 s−1 in the initial seconds
+of the simulation before sharply decreasing. Because the loop starts to cool at t = 206 s, turbulent
+heating after this time can be considered the gradual-phase heating supplemental to the impulsive
+energy release — analogous to the so-called slow-tail heating used in (Qiu & Longcope 2016). Here,
+the gradual-phase heating component lasts much less than the duration of the synthesized emission,
+falling to zero
+13 minutes after the simulation start time. Seeking an explanation for extended
+gradual-phase heating, and longer duration of EUV emission, we revisit the initial parameters of our
+turbulent transport model in the following section.
+4. EXPLORING PARAMETERS
+The preceding analysis found the dissipation of turbulent Alfv´en waves to effectively drive char-
+acteristic flare behavior and produce long-duration EUV emissions on the order of 30 minutes. We
+explore what is required to extend long-term coronal emission by adjusting the parameters of our
+turbulent transport model. Having already visited the reflection coefficient and the modification to
+the Spitzer-H¨arm conductivity, we now focus on the two remaining parameters: the percentage of
+drag power converted to turbulent wave energy, fturb, and the energy correlation length, λ⊥.
+4.1. Converted Drag Power
+The amount of energy converted into MHD turbulence from drag losses is dictated by the fraction
+fturb in Equation (11). Because the generation of Alfv´en wave perturbations is not explicitly modeled
+from the interaction between the flux tube and the current sheet, it is unclear what value of fturb
+would serve as a good approximation for this interaction.
+We therefore explore the response in
+turbulent energy to a change in drag loss conversion by varying the degree of interaction through
+
+21
+0
+5
+10
+15
+20
+25
+30
+35
+0
+20
+40
+Tmax [MK]
+131
+94
+335
+211
+193
+171
+0
+5
+10
+15
+20
+25
+30
+35
+time [min.]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+norm. counts
+131
+94
+335
+211
+193
+171
+0
+50
+100
+150
+200
+˙Q [108 erg cm−2 s−1]
+Figure 8. Bottom panel: Synthetic light curves of AIA EUV channels 131, 94, 335, 211, 193, and 171 ˚A
+produced using results from the reference simulation. Heating due to the dissipation of turbulent Alfv´en
+waves is shown in purple, read against the axis on the right. Top panel: Apex temperature evolution as
+shown in Figure 7. Symbols denote the times when Tapex crosses the characteristic formation temperature(s)
+of the respective AIA channel.
+fturb. To this end, six simulations are run with fturb ranging from 0.05–0.8. The remaining loop
+parameters match the reference simulation outlined in Section 3.1.
+Results from the six simulations are shown in the top two panels of Figure 9. The apex temperature
+evolution — used here as a proxy for the effectiveness of a set of parameters to produce gradual-phase
+heating — is plotted on the left. The overall evolution in each run is similar to that of the reference
+simulation, shown in green (fturb=0.2), with temperatures quickly a peak before decaying to ground.
+In particular, the peak temperature is seen to scale with fturb. As more drag loss becomes available
+to be converted into turbulent energy, the tube reaches higher internal temperatures from increased
+dissipation and subsequent heating. This conclusion is supported by the integrated heating rates ˙Q,
+plotted on the right, where higher degrees of drag conversion correspond to larger heating rates.
+Greater degrees of drag conversion were also found to produce more prolonged heating rates. Com-
+pared to the fturb=0.05 case, in which ˙Q decayed to zero within 8 minutes, a conversion of fturb=0.8
+resulted in an additional 7 minutes of heating. Regardless, the time taken for the apex temperature
+to decay to its pre-flare level remained
+nearly the same for each of the six simulations, despite
+increasing heating duration. The similarity between these times is ultimately due to the different
+evaporation strengths. Stronger evaporation, arising from increased heating following a higher fturb,
+leads to a higher density in the coronal segment of the loop, making the losses from radiative cooling
+
+22
+Ashfield & Longcope
+0
+5
+10
+15
+20
+25
+30
+35
+100
+101
+102
+Apex Temp. [MK]
+fturb = 0.8
+fturb = 0.6
+fturb = 0.4
+fturb = 0.2
+fturb = 0.1
+fturb = 0.05
+0
+10
+20
+30
+40
+0
+100
+200
+300
+400
+500
+600
+˙Q [108 erg cm−2 s−1]
+0
+10
+20
+30
+40
+50
+time [min.]
+100
+101
+102
+Apex Temp. [MK]
+λ⊥ [Mm] = 0.1
+λ⊥ [Mm] = 2
+λ⊥ [Mm] = 10
+λ⊥ [Mm] = 100
+λ⊥ [Mm] = 1000
+0
+10
+20
+30
+40
+50
+time [min.]
+0
+50
+100
+150
+200
+˙Q [108 erg cm−2 s−1]
+4
+6
+8
+10
+12
+14
+0
+10
+20
+30
+5
+10
+15
+20
+25
+30
+0
+1
+2
+3
+4
+5
+Figure 9. Apex temperature evolution (left) and integrated heating rate ˙Q (right) for 11 simulations run
+with different values of drag power conversion fturb (top) and correlation length λ⊥ (bottom). The cases of
+fturb=0.2 and λ⊥=2 Mm in the top and bottom panels , respectively, correspond to the reference simulation
+analyzed in Section 3. Vertical dashed lines in the upper right panel indicate the time at which retraction
+ends.
+more effective and the loop cools faster. As a result, the duration of the temperature evolution
+appears to be insensitive to the amount of energy put into turbulent Alfv´en waves.
+4.2. Energy Correlation Length
+
+23
+The heat generated via turbulence occurs at a rate set by the non-linear interaction between
+the counter-propagating waves.
+Because we adopt a one-point closure model, this interaction is
+characterized by a single parameter, λ⊥, indicative of the degree of correlation between the two
+energy populations. In Section 3.1, a range of possible values for λ⊥ was motivated from observation.
+Here, λ⊥ is taken to be the parameter controlling the efficiency of energy loss from the waves, adjusted
+from a phenomenological perspective to produce decay timescales that best agree with observation.
+The bottom two panels of Figure 9 show the apex temperature evolution (left) and integrated
+heating rate (right) for
+five simulations run with a range of λ⊥=0.1–1000 Mm. As λ⊥ scales in-
+versely with ˙Q, smaller correlation lengths produce larger heating rates and subsequently larger apex
+temperatures. Apex temperature evolution is virtually indistinguishable, except during the earliest
+phase, for all values λ⊥ ≤ 10 Mm. Heating rates for these runs are also similar following the initial
+phase.
+This similarity appears to be a consequence of energy dissipation within a single transit. The net
+energy input into every case is roughly the same. For cases with λ⊥ ≤ 10 Mm, that energy is mostly
+dissipated before reaching the feet, and thus does not remain on the loop beyond the end of retraction
+at t = 7.5 min. This end was noted for the illustration case (λ⊥ = 2 Mm), and is repeated in the
+olive curve. Those curves with still smaller values of λ⊥ drop even more rapidly after the termination
+of retraction (see λ⊥ = 0.1 Mm in red).
+The two simulations with λ⊥ > 10 Mm produced markedly different behavior. Only in these cases
+does significant wave energy remain beyond the end of
+retraction at t = 7.5 min (see inset in
+the lower right of Figure 9). It seems that the non-linear dissipation is small enough for waves to
+reflect several times. This persistent wave energy is partly responsible for apex temperatures which
+decayed to pre-flare levels in 42 and 50 minutes (λ⊥ = 100 and 1000 Mm, respectively) rather than
+the 30-minute decay observed in all other cases. Larger correlation lengths are accompanied by less
+dissipation in the early phases, resulting in more gentle flare behavior, as evidenced by peak apex
+temperatures under 20 MK. The conductive flux to the feet is therefore smaller, leading to weaker
+evaporation flows, lower loop density, and thereby diminished radiative cooling at late times. This
+is a second factor extending the cooling time of the loop. We therefore see that larger correlation
+lengths, and the weaker non-linear damping they produce, are critical for our model to extend the
+duration of coronal emissions.
+4.3. Optimized Simulation
+To capitalize on our better understanding of the role wave energy can play in extending cooling
+times we perform one final simulation. This uses a large correlation length, λ⊥=100 Mm, and high
+conversion efficiency of fturb=0.6.
+Although the parameter exploration of fturb determined that
+temperature evolution was not sensitive to the degree of drag conversion, a higher value of fturb is
+used to produce higher apex temperatures to agree with flare observations.
+AIA EUV light curves synthesized from this tuned run are shown in Figure 10. Compared to
+the emissions from the reference simulation in Figure 8, the profiles here for the hotter channels
+are broader and peak at later times, illustrating how the temperature remains elevated during this
+period. The cooler channels, however, are still relatively narrow. At this point in the simulation,
+the heating rate has decreased from its peak of 1.0 × 1010 to 0.2 × 1010 erg cm−2 s−1, indicating that
+energy dissipated from turbulence is not enough to counteract energy lost from radiative cooling.
+
+24
+Ashfield & Longcope
+0
+10
+20
+30
+40
+time [min.]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+norm. counts
+131
+94
+335
+211
+193
+171
+0
+20
+40
+60
+80
+100
+˙Q [108 erg cm−2 s−1]
+Figure 10. Synthetic EUV emission produced from the optimized simulation, plotted in the same format
+as Figure 8.
+In this case, the duration of the heating rate lasted 41 minutes and produced coronal emissions
+lasting for approximately 40 minutes, given the peak in 171 ˚A. Even though the correlation length and
+degree of drag loss conversion were 50 and 3 times larger than the reference simulation, respectively,
+the duration was only 8 minutes longer.
+Notable differences between the optimized and the reference
+simulations are summarized in Table 1. Additional simulations run with increased fturb and λ⊥, while
+not shown here, also failed to extend the duration of the emission past 40 minutes, further signifying
+the balance between strong flare dynamics and gradual turbulent dissipation. As such, the emission
+duration shown in Figure 10 represents the upper limit of our turbulent transport model.
+Alfv´en
+wave turbulence should be evident in
+an excess broadening of hot spectral lines. The
+turbulent energy per mass gives the turbulent velocity squared averaged over all fluctuation time
+scales:
+wtot = w+ + w− = 1
+2⟨|v|2⟩ + 1
+2
+�|b|2
+4π
+�
+= ⟨|v|2⟩,
+(21)
+after assuming turbulence is Alfv´enic. The net effect on an unresolved observation of spectral line λ
+is characterized by a spatial average weighted by the local intensity of the line, Iλ(ℓ),
+¯wλ =
+��
+Iλ(ℓ) dℓ
+�−1 �
+wtot(ℓ) Iλ(ℓ) dℓ.
+(22)
+If the axis of the tube is viewed perpendicularly, then the averaged broadening of the spectral line
+λ along the line of sight, say ˆz, is
+σλ =
+�
+⟨v2
+z⟩ =
+�
+¯wλ
+2 .
+(23)
+Figure 11 shows the broadening of three lines typically associated with flares, both versus time (right)
+and normalized line intensity (left). Cooler lines show lower broadening, and all decrease with time
+as the turbulence decays.
+
+25
+Figure 11. Turbulent line broadening, defined in Equation (23) for spectral lines of Fe xxiv 192˚A (red), Fe
+xxi 129 ˚A (violet), and Fe xviii 94 ˚A (blue). The left panel plots these against the normalized intensity of
+the line, and the right against time when the line is bright.
+Time flows respectively from top to bottom for
+each curve in the left panel. The time of loop straightening (t = 8.13 min) is indicated by a vertical dashed
+line and triangles on each curve. Diamonds along the right axis are the average values for each line.
+5. DISCUSSION
+We have extended a post-reconnection flare model to include the production and dissipation of
+turbulent Alfv´en waves. This extension involved a system of turbulent transport equations inspired
+by existing models of solar wind turbulence. One-dimensional simulations tracked the evolution of
+MHD turbulence through their aggregate energy densities, which remained trapped in the corona by
+reflection from high-density gradients in the transition region. Turbulent energy dissipated through
+the non-linear interaction between counter-propagating waves and produced heating longer than
+previous versions of the PREFT flare code. Synthetic light curves from AIA EUV bands showed
+increased similarity with typical flare observations.
+Ours is the first model of its kind to generate Alfv´en wave turbulence from reconnection outflow
+and model its effect on long-term flare signatures.
+In the spirit of a first attempt, we chose to
+represent the physical processes involved using simple, fixed parameters. Energy is removed from the
+retracting flux by a simple aerodynamic drag force, parameterized by the parameter D. We chose the
+value D = 50 Mm−1 in order to slow the retraction to a level consistent with typical observations. A
+fraction, fturb, of the energy lost to drag appears as unresolved Alfv´en wave turbulence. While this
+process is likely to be very complicated, with a fraction varying in time as well as space, we chose
+to use a fixed value of fturb. We performed a series of runs and different values of fturb and found its
+most important effect was in setting the peak flare temperature.
+The physics of Alfv´en wave propagation, reflection, and dissipation is also captured through a small
+number of free parameters. Reflection is modeled through a reflection coefficient, η, applied at the
+
+26
+Ashfield & Longcope
+point where T = TTR = 0.5 MK. Wave dissipation occurs through non-linear dissipation involving
+waves of both species. This process is parameterized through an effective correlation length, λ⊥,
+which was varied to explore its effects on the flare evolution.
+The duration of turbulent heating was found to be most dependent on the correlation length λ⊥.
+Correlation lengths λ⊥ ≤ 10 Mm lead to very rapid dissipation and wave energy vanishes rapidly
+after retraction ends. Only in those with λ⊥ ≳ 100 Mm does wave energy persist beyond the end of
+retraction, thereby prolonging the cooling of the flare. If Alfv´en waves are actually responsible for
+the long flare cooling times observed, non-linear wave dissipation would need to be characterized by
+such large correlation lengths.
+The finding described above requires reconsideration of the physical significance behind the pa-
+rameter λ⊥. Our turbulence model assumes the small-scale fluctuations responsible for the MHD
+turbulence to be on scales smaller than the radius of the flux tube, so λ⊥ cannot be simply taken
+as the correlation length of the turbulence. It was mentioned above that our parameter incorpo-
+rates the correlation between the counter-propagating waves, such that the true correlation length is
+λ′
+⊥ = λ⊥/ξ, where ξ is the fraction of the counter-propagating turbulence which is actually uncorre-
+lated. For an effective correlation length of λ⊥ = 100 Mm and a tube with radius R = λ⊥ =2 Mm,
+this gives ξ=0.02, meaning that only 2% of the counter-propagating waves are uncorrelated and able
+to produce a turbulent cascade. While an imperfect correlation should be expected between the two
+energy populations, it is worth noting that the length scale for the small-scale fluctuations (e.g. R)
+and the length scale for the energy correlation do not necessarily need to be related (Zank et al.
+2011). Should this perspective be taken into account, it would also substantiate the range of length
+scales derived in Section 3.1 using the decay rates of non-thermal broadenings observed in Fe xxiv ˚A
+spectral lines, where λ⊥ ∼30-150 Mm.
+The heating rates produced in this work also fit the description of a two-step heating profile as
+described in Qiu & Longcope (2016). The initial phase of turbulent dissipation was highly impulsive
+in all cases, reaching a peak within 20 s before quickly decaying. Following the impulsive energy
+release, defined here by the time it takes a loop to enter a regime of increased radiative losses,
+heating from dissipation then became more gradual and lasted anywhere from 5 to 50 minutes. In
+particular, the heating rate produced in Figure 10 was found to match the general heating profile
+described in Zhu et al. (2018), where energy deposited during the impulsive and gradual phases
+was required to be 60% and 40% of the total energy released in order to reproduce sustained EUV
+emission. Here, after an impulsive phase that lasted 240 s, the gradual phase released 46% of the
+total turbulent wave energy over 36 minutes. To the best of our knowledge, this mechanism was the
+first to reproduce these two-step heating rates in a self-consistent manner.
+A byproduct of using turbulent Alfv´en waves for gradual-phase heating was their capacity to be
+equally effective at driving characteristic flare behavior during the initial phase of the simulation. By
+inducing turbulence from drag, we were able to reintroduce energy back into the tube that would have
+otherwise been lost to the background of our flux tube. Hence, our model successfully reproduced
+impulsive flare behavior while still keeping the speed of reconnection outflows within their observed
+ranges.
+Investigations into the earlier phases of turbulent dynamics presented here, such as the
+forward-modeling of Fe xxi ˚A lines, may explain the large broadenings of such lines seen during flares
+(Polito et al. 2019) and could serve to constrain our model.
+
+27
+While our model produced long-duration heating in conjunction with impulsive flare dynamics, it
+was not able to reproduce sustained EUV emission on the order of hours, as seen in observation.
+One possible explanation for this discrepancy is the value of the drag coefficient D=50 Mm−1 used
+in this paper. Increasing D would lead to lower retraction rates, likely resulting in more prolonged
+heating and more energy available to convert to turbulent Alfv´en waves. Because the first stage of
+our reconnection downflow was observed to be 540 km s−1, the drag coefficient could be increased to
+the point of generating retraction rates on the order of 50 km s−1 — the lower speed limit of SADs
+observed in Savage et al. (2012).
+In this new model, the chromospheric response to reconnection is driven in two different ways.
+The first is thermal conduction, which was part of even the earliest flare loop models (Nagai 1980;
+Pallavicini et al. 1983). Augmenting this, Alfv´en waves are created by retraction at the loop apex
+and propagate to the chromosphere where they are dissipated by interacting with their own reflection.
+This second transport channel drives chromospheric evaporation beyond that attributable to conduc-
+tion alone. It is worth noting that a fraction 1 − fturb of the drag work will appear as turbulence on
+field lines neighboring that just reconnected. Some of the neighboring flux will still be unreconnected,
+and will now experience some chromospheric evaporation before reconnecting. There are several lines
+of observational evidence suggesting that flare reconnection occurs on field lines whose density is
+higher than one would expect in the ambient, pre-flare corona (Veronig & Brown 2004; Veronig et al.
+2005; Guo et al. 2012). Our model offers a possible explanation for this previously puzzling fact.
+Another mechanism proposed to extend the cooling time of a flare loop is through suppression of
+thermal conduction. Previous investigations into the evolution of flare loops using zero-dimensional
+models have shown that alterations to the mean-free path of thermal electrons extended cooling times
+considerably (Zhu et al. 2018; Bian et al. 2018). Our model accounts, in Equation (14), for a single
+MHD mechanism for decreasing κ: decreasing the mean axial step size ℓz by the lengthening of the
+actual path, ℓ, due to field line meandering on sub-resolution scales, i.e. MHD turbulence. When the
+perturbation velocity is significantly lower than the local Alfv´en speed, as implied by observations
+and explicitly found in our model, then field lines are lengthened only slightly. The result is the
+very small reduction in κ we have observed. Effective reduction in the mean-free path must occur
+not through MHD mechanisms but through the kinds of particle kinetics effects suggested by others
+(Bian et al. 2016).
+The level of turbulence predicted here can be compared to observation through turbulent line
+broadening of high-temperature spectral lines. The level in our optimized run produces σ between
+100–400 km/s, a factor of 2-3 higher than typically observed (Warren et al. 2018; Tian et al. 2014;
+Polito et al. 2019). There are, however, many reasons to expect our estimate to be too large. The
+estimate in Equation (23) assumes the field lines are viewed exactly perpendicularly and that the
+observation integrates over all turbulent frequencies. The turbulence consists of Alfv´en waves on the
+loop, whose frequencies will extend down to the fundamental with a period of two Alfv´en transit
+times — perhaps up to 20 seconds. Integration times shorter than this will omit the lower frequency
+component of the spectrum, which contains the most energy. This will reduce the value of σ actually
+observed. It should be noted that the turbulent broadening in our model naturally decreases on
+an approximately 15-minute scale. This is consistent with observations (Kontar et al. 2017; Warren
+et al. 2018).
+
+28
+Ashfield & Longcope
+The model presented here is self-consistent to the extent that turbulence is ultimately generated
+by the retraction of a flux tube through a current sheet. The interaction between the current sheet
+and the tube responsible for both the drag force and the excitement of turbulent Alfv´en waves,
+however, remains an assumption of our model and is not directly investigated here. Moreover, the
+resolution of PREFT required MHD turbulence to be modeled according to its aggregate energy
+density. Small-scale perturbations in the velocity and magnetic fields that constitute the turbulence
+were not considered. While the simplicity of the turbulent transport model in this work warrants
+further study, any improvements would likely require techniques beyond the formalism of 1D flare
+loop simulations. Despite these limitations, the results presented here illustrate the efficacy of MHD
+turbulence as a mechanism for flare energy transport and heating, and provides an additional tool
+by which other flare phenomena can be studied.
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+doi: 10.3847/1538-4357/aaad10
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf,len=1331
+page_content='Draft version January 12, 2023  Typeset using LATEX preprint style in AASTeX631  A Model for Gradual Phase Heating Driven by MHD Turbulence in Solar Flares  William Ashfield IV1, 2 and Dana Longcope3  1Bay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA 94035, USA  2Lockheed Martin Solar & Astrophysics Laboratory, Organization A021S, Building 252, 3251 Hanover Street, Palo  Alto, CA 94304, USA  3Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' of Physics, Montana State University, Bozeman, MT 59717  (Accepted January 8th, 2023)  Submitted to ApJ  ABSTRACT  Coronal flare emission is commonly observed to decay on timescales longer than those  predicted by impulsively-driven, one-dimensional flare loop models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  This discrepancy  is most apparent during the gradual phase, where emission from these models decays  over minutes, in contrast to the hour or more often observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Magnetic reconnection  is invoked as the energy source of a flare, but should deposit energy into a given loop  within a matter of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Models which supplement this impulsive energization with a  long, persistent ad hoc heating have successfully reproduced long-duration emission, but  without providing a clear physical justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here we propose a model for extended  flare heating by the slow dissipation of turbulent Alfv´en waves initiated during the re-  traction of newly-reconnected flux tubes through a current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Using one-dimensional  simulations, we track the production and evolution of MHD wave turbulence trapped  by reflection from high-density gradients in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Turbulent energy  dissipates through non-linear interaction between counter-propagating waves, modeled  here using a phenomenological one-point closure model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' AIA EUV light curves synthe-  sized from the simulation were able to reproduce emission decay on the order of tens  of minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We find this simple model offers a possible mechanism for generating the  extended heating demanded by observed coronal flare emissions self-consistently from  reconnection-powered flare energy release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' INTRODUCTION  Solar flares are typically characterized by the increase in radiation observed during their onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Accepted as the result of energy released through magnetic reconnection, the sudden rise in emission  across different wavelengths is considered to be an indication of impulsive plasma heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Following  the conclusion that flare loops convert magnetic energy on the order of the Alfv´enic transit time  across a given loop (Kopp & Pneuman 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Priest & Forbes 2002), numerical flare loop models  aiming to study the impulsive phase have commonly used energy source terms that reflect these  short timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Many of these models have also been constrained by observations to infer energy  deposition rates — either through hard X-rays (Kowalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2020) or UV  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='04592v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='SR]  11 Jan 2023  2  Ashfield & Longcope  emission (Longcope & Bradshaw 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Ashfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2022) — where the heating  duration was found to last up to tens of seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Although impulsively-driven models have been a successful tool for explaining a number of flare  phenomena, their failure to reproduce gradual phase emission is widely recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Observations  in soft X-rays and EUV have shown the gradual decay of hot coronal plasma to last between tens  of minutes to several hours following the rise phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The cooling of coronal plasma, marked by  the successive peaks of emission light curves from increasingly cooler ion species, happens through  a combination of heat conduction (Culhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 1970, 1994) and radiative losses (Aschwanden &  Alexander 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Vrˇsnak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Compared to the characteristic cooling timescales modeled by  individual flare loops (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' minutes, Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2020), the sustained duration of observed coronal  emissions is considerably longer than it would be if regulated by cooling alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The discrepancy between modeled and observed decay rates during the gradual phase has led  many to believe that additional heating is required beyond the impulsive phase to offset the effects  of radiative losses (Withbroe 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Svestka 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Ryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A popular  interpretation of this heating has been the successive energization of multiple individual flux tubes  within a flare arcade — so-called multi-loop models (Dere & Cook 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Hori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Warren  2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Reep et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2022) — with each loop emulating the prescribed impulsive energy release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' While  these models are consistent with observation, several investigations were unable to rectify the lack  of sustained coronal emissions (Reeves & Warren 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In response to  this persistent discrepancy, Qiu & Longcope (2016) recently developed a model  where individual loops were driven with a two-part heating profile consisting of an impulsive energy  release followed by a prolonged, low-rate heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This slow-tail heating, lasting on the order of  20 minutes, was able to successfully forward-model key characteristics of EUV lightcurves measured  using the Atmospheric Imaging Assembly (AIA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012), including long cooling timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  While motivated by flare UV footpoint emissions that display similar, two-phase behavior (see also  Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018), the heating profiles introduced in this work were  only ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Physical justifications behind extended loop heating were not provided outside of a few,  potentially viable scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Several mechanisms for producing such extended heating rates alluded to in Qiu & Longcope (2016)  and other investigations are contingent upon post-reconnection flare loops retracting under magnetic  tension and releasing energy (Forbes & Acton 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Linton & Longcope 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' One such speculation  is the continual heating from slow-shocks generated by the resistance loops receive from earlier formed  loops that exist lower in the flare arcade (Cargill & Priest 1982, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Another related mechanism  is the excitation of MHD waves by reconnection outflows (Aschwanden 2006) that then decay and  heat plasma within a post-reconnection loop over timescales comparable to slow-tail heating (Wang  2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Central to both of the scenarios described above is the interaction between retracting flare loops  and their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Previous investigations of outflows driven by magnetic tension have modeled  flux tubes moving through an ideal current sheet unobstructed, and have therefore retracted at the  local Alfv´en speed (Forbes & Priest 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Longcope & Klimchuk 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Observed outflows — inferred  from the motion of supra-arcade downflows (SADs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' McKenzie & Hudson 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Savage & McKenzie  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Reeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2017) interpreted to be the wakes behind retracting flux tubes (Savage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012)  3  — however, move at fractions of the Alfv´en speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In the case of a non-ideal current sheet, such  as that with a high-β plasma (Scott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2016), a retracting loop would likely undergo a loss of  momentum and speed as it imparted work on the surrounding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This interaction, modeled as  a simple aerodynamic drag force, was recently found to slow outflow speeds to within their observed  ranges (Unverferth & Longcope 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The interpretation of SADs as the wakes of retracting flux tubes experiencing a drag force would  also suggest the generation of turbulence, given the aerodynamic analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In fact, evidence for  turbulence has been reported in current sheets containing SADs, inferred through measurements of  non-thermal line broadening (Ciaravella & Raymond 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Warren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018)  and local correlation tracking (McKenzie 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Separate observations using AIA and the Extreme-  ultraviolet Imaging Spectrometer (EIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Culhane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2007) have also shown turbulence to coexist  in regions with hot, ∼ 30 MK downward moving loops, but did not make reference to SADs (Imada  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Doschek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Using observations of several strong flares, Larosa & Moore (1993)  found that the interactions between individual flux tubes and interactions with a diffusive current  sheet are likely to develop outflows with a turbulent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Moreover, they also found MHD  turbulence to be capable of rapidly thermalizing bulk kinetic energy via a turbulent cascade, thus  becoming a mechanism for impulsive phase energy release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It was further suggested by Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (2006) that plasma wave turbulence could produce heating into the decay phase, indicated by the  long cooling times of loop-top HXR sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Although the interconnected behavior of retracting flux  loops, turbulence, and extended flare heating has been well established by observation, it has yet to  be modeled in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The present work aims to develop a self-consistent mechanism for gradual-phase energy release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Motivated by the observations of SADs, we create a relatively simple model for the excitation of  MHD turbulence along a thin flux tube (TFT) as it retracts through a current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The retracting  tube experiences a drag force, which converts energy from the tube into an internal population of  turbulent Alfv´en waves that then propagate along the tube and reflect off high-density gradients  in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Turbulent energy is dissipated through the non-linear interaction between  counter-propagating waves, thus creating a local heat source within the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A model for drag with  turbulent transport is outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The behavior of our model is then demonstrated using an  initial reference simulation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Using the simulation results, synthetic AIA EUV emissions  are created by which our model can be qualitatively compared to observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The parameters critical  to the duration of turbulent energy release — the fraction of drag power converted to turbulent Alfv´en  waves and the energy correlation length of the propagating waves — are explored in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A  final simulation run with a set of optimized parameters was found to extend the duration of EUV  emission past 40 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The duration of the heating produced from turbulent dissipation was also  found to persist long after the retraction had ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The results presented in this paper point to a  physical mechanism that can produce late-phase heating required to sustain coronal flare emission  for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' FLARE MODEL WITH ALFV´EN WAVE TURBULENCE  In order to model the energy released by reconnection and its conversion to MHD turbulence,  we employ the TFT equations described by Longcope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The model assumes localized  reconnection has already occurred within a small diffusion region, within the current sheet, to create  a closed magnetic flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The moment following reconnection is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Two  4  Ashfield & Longcope  (a)  (b)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Schematic of a reconnected flux tube embedded in a current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (a) Face-on view  of the  current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' External magnetic fields on either side of the current sheet — the blue and red-dashed lines —  are skewed according to the sheer angle ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The grey tube shows the geometry of the newly formed flare loop  immediately following reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (b) Perspective orientation illustrating the deflection of the external  magnetic fields around the flux tube, with the tube sliced through its apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The grey area corresponds to  the current sheet shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Only the ˆz-component of the external magnetic fields is shown in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  layers of equilibrium magnetic field are separated by a current sheet and have directions differing by  a shear angle ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The newly reconnected loop, shown in grey, is embedded within the current sheet  that lies in the x–z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It is at this point the TFT model begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Mechanisms of the reconnection  process itself are not considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Once the bent tube is initialized, it retracts through the current sheet under a magnetic tension  force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The internal magnetic pressure of the tube is balanced by the magnetic pressure of the external  magnetic fields separated by the current sheet, setting the magnetic field strength inside the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In the present case, we assume the external magnetic fields to be uniform, such that the magnetic  field B is also uniform along the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The TFT equations govern the evolution of the axis of a flux tube, r(ℓ, t), where ℓ is the param-  eterized length coordinate of the curved tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The fluid velocity of the plasma confined within the  tube, v = dr/dt, is advanced according to the momentum equation (Longcope & Klimchuk 2015)  ρdv  dt =  �  B2  4π − p  �  ∂ˆl  ∂ℓ −ˆl∂p  ∂ℓ + ρg + 4  3µ ∂  ∂ℓ  �  ˆlˆl · ∂v  ∂ℓ  �  ,  (1)  where p = ρkBT/ ¯m is the plasma pressure and ¯m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='593 mp is the mean particle mass of the  plasma, assumed to be fully ionized with a coronal abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For a fluid element along the tube  with differential length dℓ and mass per unit flux dm, the mass density is given by ρ = B(dm/ dℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Outside of gravitational force ρg, which acts in the downward direction, forces acting on a fluid  element in Equation (1) are dictated by the tangent vector, ˆl = ∂r/∂ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The first term on the rhs  describes the magnetic tension force along the tube’s curvature vector, ∂ˆl/∂ℓ, perpendicular to the  tube’s axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The second is the gas pressure directed along the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Viscous interactions between  ZZ5  fluid elements are described by the final term, with µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='012 κsp(T)/cv being the temperature-  dependent dynamical viscosity coefficient and the classical Spitzer-H¨arm thermal conductivity being  κsp = 10−6 T 5/2, in conventional cgs units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The energy of the tube is advanced according to the temperature of a fluid element  cvρdT  dt = −p  �  ˆl · ∂v  ∂ℓ  �  + 4  3µ  �  ˆl · ∂v  ∂ℓ  �2  − n2  eΛ(T) + ∂  ∂ℓ  �  κ∂T  ∂ℓ  �  ,  (2)  where cv = 3kB/2 ¯m is the specific heat and ne = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='874 (ρ/mp) is the electron number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The  first term on the rhs constitutes the adiabatic compression done on the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This term is followed  by viscous heating, which arises from the loss of kinetic energy in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Optically thin  radiative losses are given by the third term, where the radiative loss function Λ(T) is taken from  the output of CHIANTI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1 (Landi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The final term describes the field-aligned thermal  conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A flux limiter is included in the model, such that the thermal conductivity is restricted  to the theoretical electron free-streaming limit at sufficiently large temperature gradients (Longcope  & Klimchuk 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Drag Force and Alfv´en Wave Turbulence  The TFT model describes the evolution of flare loops retracting under a tension force brought  about by a change in magnetic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Previous studies assumed the retraction occurs without any  interaction with the background current sheet, and therefore saw typical outflow velocities reaching  the local Alfv´en speed on the order of 3 Mm s−1 (Longcope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Unverferth & Longcope 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Alfv´enic outflow speeds are common to all idealized models of magnetic reconnection, but are not  supported by much observational evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Supra-arcade downflows (SADs McKenzie & Hudson  1999) are often taken to be signatures of post-reconnection retraction, but always seem to move well  below the local Alfv´en speed (Savage & McKenzie 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Savage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Based on observations of sub-Alfv´enic outflow speeds it has been proposed that the retracting  flux must somehow interact with the surrounding plasma, possibly by deforming the flux outside  the sheet or entraining plasma (see Figure 1b and discussions in Linton & Longcope 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Scott  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Most forms of interaction would leave the surrounding plasma with greater magnetic or  kinetic energy, at the expense of the retracting tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This would appear as some kind of drag force  on the retracting flux, removing some of its energy and reducing the retraction speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Unverferth &  Longcope (2021) investigated this possibility, introducing a drag force modeled using a high Reynolds  number aerodynamic drag (Choudhuri & Gilman 1987)  fd = −D ρ |v⊥| v⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (3)  Here v⊥ = v − ˆl(ˆl · v) is the component of fluid velocity perpendicular to the flux tube and D  is a constant proportional to the traditional drag coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because the interaction between the  flux tube and the current sheet is likely more intricate than the conventional drag force exerted by a  neutral fluid on a rigid body, D is taken to be a free parameter to capture these unknown complexities  (Unverferth & Longcope 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This force, per unit area, is added to the momentum equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The force exerted by the surrounding plasma ultimately opposes the acceleration of the loop by  transferring momentum and energy away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Energy is therefore removed from the retracting loop at a  rate of  Pd = ρ−1 v · fd ≤ 0,  (4)  6  Ashfield & Longcope  which is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The energy is not really lost, but must appear in some form in the  surrounding plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We assume here that it takes the form of MHD turbulence, of which some  fraction remains on the tube, but at length scales heretofore unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We therefore assume a  fraction of Pd takes the form of unresolved Alfv´en wave turbulence occurring on the tube itself, even  as it retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Evolution of Alfv´en Wave Turbulence  Gaining inspiration from turbulent transport models of the solar wind (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Marsch & Tu 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Matthaeus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Dmitruk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Verdini & Velli 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Lionello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2014), the equations  used for the evolution of turbulent wave energy in this work are developed by first decomposing  the velocity and magnetic fields into large-scale contributions, denoted by capitals, and small-scale  perturbations, denoted by lower case:  v = U + u  (5)  B = B0 + b = Bˆl + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (6)  The perturbations can then be collected in terms of the Els¨asser variables, z± ≡ u ± b/√4πρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The evolution of the fluctuations is expressed by the scale-separated, linearized MHD equations in  their Els¨asser representation (Zhou & Matthaeus 1990a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Zank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2011)  Dz±  Dt = ±(VA · ∇)z± + 1  2(z∓ − z±)∇ ·  �U  2 ± VA  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (7)  Here, VA = B0/√4πρ is the local Alfv´en speed along the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We assume the large-scale fields of  U and B vary slowly across the flux tube and have therefore ignored the gradient terms in Equation  (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The source and sink terms for the Els¨asser variables are also temporarily neglected here and are  instead discussed in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The aggregate energy densities of the turbulent Alfv´en waves are then found by spatially averaging  over the small-scale fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The energy per unit mass of the two turbulent waves is given by:  w± = 1  4⟨z± · z±⟩  (8)  with the total energy density of the perturbations being wtot = w+ + w− = ⟨u2/2⟩ + ⟨b2/8πρ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' By  averaging over the perturbations, we remove their explicit dependence from the system and instead  model their collective evolution implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Taking the inner product of z± with Equation (7) for z±, respectively, gives the expression for each  Els¨asser energy w±:  Dw±  Dt  = ±(VA · ∇)w± − w±∇ ·  �U  2 ± VA  �  + R±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (9)  Here, R± is a term proportional to ⟨z+ · z−⟩ that accounts for wave reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Analogous to the  ‘mixing’ effects described in Zhou & Matthaeus (1990b), the interaction between opposite Els¨asser  variables allows for energy to be redistributed between the two populations, such that w± will generate  counter-propagating w∓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This process is linked to large-scale gradients in the system, and is likely to  be most effective where ∂lnρ/∂ℓ is large (Hollweg 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Velli 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Computing R± self-consistently  7  would require additional equations beginning with one for the evolution of ⟨z+ · z−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In the interest  of simplicity, we forego this approach and instead set R± = 0 and account for wave reflection with a  boundary condition described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Following the TFT model, we parameterize the wave energy equations according to length coordi-  nate ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As the large-scale fields are taken to vary only in the direction parallel to ˆl, divergences in U  and VA reduce to spatial derivatives in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The full turbulent transport equations in their conservative  forms are then  dw±  dt  = ±vA  ∂w±  ∂ℓ − 1  2w±  ∂v∥  ∂ℓ ∓ w±  ∂vA  ∂ℓ + S± + NL±,  (10)  for v∥ = ˆl · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because the terms in the above expression are dependent only on system variables  aligned with the mean magnetic field, Equation (10) describes the energies of small-scale Alfv´en  waves propagating along the field in the ±ˆl direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Propagation of the wave energies is described  by the first three terms on the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The first is the advective term, showing how the wave energies  propagate at the Alfv´en speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The second and third terms describe work done on the waves by  compression of the plasma and magnetic pressures against that of the wave, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Sources and Sinks of Alfv´en Wave Turbulence  The last two terms of Equation (10), absent from Equation (9), are added to describe the source,  S±, and sink, NL±, of the turbulent energies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The source, as described above, is a  fraction, fturb, of the power the tube loses through drag as it retracts through the current sheet  described by Equation (4):  S± = − 1  2 fturb Pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (11)  Here the 1  2 prefactor assumes the input energy is divided equally between the counter-propagating  wave species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The loss of turbulent energy in the system is modeled according to the simple phenomenological  decay rate used in numerous MHD turbulence investigations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Hossain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Matthaeus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Dmitruk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Zank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Following one-point closure models for hydrodynamic  turbulence first described by de Karman & Howarth (1938), the energy loss arises from the non-  linear interaction between the counter-propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This interaction is assumed to produce a  cascading spectrum of turbulent Alfv´en waves — defined by a set of wave numbers that correspond  to the cascade of energy-containing eddies into increasingly smaller scales — that ultimately results  in the dissipation of turbulent energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The dissipation rate for the Els¨asser energies is expressed as  NL± = −w±√w∓  λ⊥  ,  (12)  where λ⊥ is the single similarity length scale that characterizes the transverse dimensions of the  energy-containing eddies for both the rightward and leftward propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As such, λ⊥ is  the characteristic length scale that couples the non-linear spectral transfer between the counter-  propagating modes, and can thus be thought to correspond to the correlation length between the  two energy populations (Batchelor 1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Although turbulence is typically described by a range of  length scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' wave numbers), this simple one-point model assumes the decay can be represented  by a single non-linear term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  8  Ashfield & Longcope  We assume that all energy lost from the turbulence is ultimately thermalized and appears as a heat  source for the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This is achieved simply by adding the term  Hturb = ρ  �  NL+ + NL−  �  ,  (13)  to the rhs of Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' With this additional term, our model establishes a connection, albeit  indirect, between the energy lost to drag and a heating rate that can be used to explain long-duration  EUV emission seen during the gradual phase of flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The TFT equations are further modified by considering the effects of turbulence on the tube’s  thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As magnetic field perturbations will lengthen the field lines along which ther-  mal electrons carry heat, the thermal conductivity will consequently decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We account for this  suppression by modifying the classical Spitzer-H¨arm conductivity  κsp → κ(turb)  sp  =  κsp  1 + ρ(w+ + w−)/B2,  (14)  such that the turbulent energy will directly impede the heat flux in the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Although this is  a relatively simple addition to the model, it attempts to address the notion of heat conduction  suppression via turbulence in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Finally, we achieve turbulent wave reflection from the chromosphere using the boundary conditions  w−(ℓ0) = η w+(ℓ0)  ,  w+(ℓ1) = η w−(ℓ1),  (15)  where ℓ0 and ℓ1 are the left and right boundaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because reflections are likely to occur  in the transition region where temperature and density gradients are large, these boundaries are set  by a characteristic temperature TTR such that ℓ0(1) = min(max) {T > TTR}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In this case, waves  incident on the transition region at either end of the loop will be transformed into their respective  counter-propagating waves, effectively trapping the wave energy in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Furthermore, the  boundary condition assumes a reflection coefficient of η ≤ 1 to account for energy lost from wave  transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The value of this coefficient is taken as a free parameter and is discussed, along with  the other free parameters of our model, in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' SIMULATION WITH ALFV´EN WAVE TURBULENCE  To illustrate the behavior of our model, we construct and run a single reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Rather  than attempt to model a particular observation, we aim at properties typical of a long-duration event,  including peak temperatures above 20 MK and flux retraction speeds of order 500 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We attempt  to extend the cooling time to values comparable to those typically found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For concreteness, we use  values reported in Qiu & Longcope (2016), since they also infer input energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To make a meaningful  comparison, we choose parameters to give the simulation the same energy flux as they report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The  run and its parameters are discussed alongside their motivations below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Initial Conditions  Prior to the simulation run, a flux tube is initialized in a configuration analogous to a bent field  line created from reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The tube is confined to the x–z plane, corresponding to the current  sheet that separates uniform layers of magnetic flux differing by shear angle ∆θ = 120◦, as illustrated  by Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The initial tube is therefore bent at its apex by 180◦ − ∆θ = 50◦, and is set to have  9  a uniform magnetic field of magnitude B = 100 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because this process forms two straight segments  joined at a single point, the apex of the loop is rounded to be a semi-circle composed of eight grid  points to prevent issues arising from under-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Once bent, the flux tube is initialized into three components: a coronal loop and two footpoints  attached to a rudimentary chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The pre-flare chromosphere is stratified under gravity with  pressure increasing exponentially according to scale height H = 500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Taken to be isothermal with  temperature Tmin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='01 MK, the chromosphere primarily serves as a mass reservoir of cool, dense  plasma for evaporated material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The coronal region of our flux tube is structured according to the  relations given in Rosner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Defined by an apex temperature set to Tco,0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 MK, the  initial corona is configured in isobaric equilibrium maintained by an ad hoc volumetric heating input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The heating required to maintain equilibrium, however, is only used during the initialization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Subsequent evolution of the loop does not contain this heating term in order to more directly observe  the consequences of turbulent heating induced along the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The initial length of the tube is calculated from the amount of flare energy we want to be released  by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In the TFT model, the source of this energy comes from contracting magnetic field  lines, releasing magnetic free energy into other forms as the flux tube retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The magnetic energy  per unit of magnetic flux of the tube is (Longcope & Klimchuk 2015)  WM = 1  4π  �  B[r(ℓ)]dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (16)  Because the strength of the magnetic field is fixed, a change in magnetic energy is therefore powered  by a reduction in the tube’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A net length decrease ∆L, releases flare energy, per magnetic  flux,  Efl = B∆L  4π  sin2  �∆θ  4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (17)  Here, the sine-squared factor represents the fraction of magnetic energy converted to kinetic en-  ergy parallel to the axis of the tube through rotational discontinuities that form during retraction  (Longcope & Klimchuk 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This mechanism constitutes the primary mode of magnetic energy  conversion in previous studies using the TFT model to investigate reconnection dynamics, with and  without drag (Guidoni & Longcope 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Unverferth & Longcope 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  To make contact with the work done in Qiu & Longcope (2016), we incorporate the same flare  energy deposited by the impulsive part of their heating profile, H(t), into the energy released in  our model1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' By integrating over the heating rate, the total energy per unit area is calculated to be  �  H(t)dt = ζEflB = ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='9 × 1011 erg cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The prefactor ζ is introduced to account for imperfect  conversion between magnetic energy and other forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Assuming the primary energy driving flare  phenomena arises from MHD turbulence, as we argue below, the conversion is ultimately related to  the fraction of power lost through drag, such that ζ = 1/fturb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We initially take this fraction to be  reasonably conservative at fturb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2, giving a total energy release of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='45 × 1012 erg cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Equation(17) is then solved for ∆L, giving ∆L = 73 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The loop is initialized with length  L0 = Lf + ∆L, with Lf being the final length of the loop once retraction is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We choose Lf  to be 93 Mm, which gives an initial loop length of L0=166 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  1 The heating rate, given by Equation (6) in Section 4 of Qiu & Longcope (2016), is a piecewise expression composed of  three Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The first two describe the impulsive flare energy release as inferred from AIA 1600 ˚A ribbon emission  observations, while the third is appended to model gradual-phase heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here we integrate the first two, along with  the same parameter values described in the same section, to calculate a total energy per unit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  10  Ashfield & Longcope  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Parameters of the Drag Model  Although the total energy released by the loop through retraction is set by Equation (17), the  inclusion of a drag force in our model changes the nature of energy release in comparison to previous  TFT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In these earlier works, parallel kinetic energy is responsible for shocks that compress  and thermalize the plasma via viscous heating, while the energy found in perpendicular flows is found  to have little effect on the overall magnetic energy conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Unverferth & Longcope (2021) showed  that kinetic energy will be effectively removed from the tube by including drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' While this reduction  was shown to affect the perpendicular kinetic energy primarily, rotational discontinuities were also  shown to be less effective at driving parallel flows as well, thus limiting the available energy to be  converted to heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In the case of drag-induced Alfv´en wave turbulence, perpendicular kinetic energy  is no longer lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Instead, a fraction is reintroduced to the tube, as dictated by Equations (4) and  (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We expect the subsequent transfer of turbulent energy to heat from dissipation will capture  a portion of the magnetic energy that would otherwise have gone into perpendicular flows and be a  source of heat comparable to, if not greater than, heat driven by parallel plasma flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The drag coefficient, D, in Equation (3) controls both the amount of turbulent energy siphoned  from perpendicular kinetic energy and the rate of retraction in the −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As described in the  introduction, drag was first motivated by observations of reconnection outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' SADs Savage  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012), and serves as the mechanism by which sub-Alfv´enic speeds can become consistent with  Petschek-like reconnection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here we set D = 50 Mm−1 in order for our model to agree with  measured outflow speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' An initial investigation using this coefficient shows the loop retracting  at speeds ∼500 km s−1, which is comparable to the average velocity of SADs measured in Longcope  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The turbulence model includes several new parameters, used here for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' These will  be varied in order to understand their effects, but we first describe their significance and plausible  ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We will then return in the discussion to consider what their variation has revealed about the  physics of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The turbulent energy dissipation rate is set by the energy correlation length, λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Investigations  into the effects of turbulence on coronal heating have assumed λ⊥ to be on the order of inter-network  spacing of 30 Mm (Matthaeus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Downs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Abramenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (2013) measured  the characteristic length of the energy-containing structures using local correlation tracking in the  photosphere, finding λ⊥ to lie in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6–2 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Although these studies deal with the issue  of coronal heating driven by convective motions in the photosphere, it is not unreasonable to take  the correlation length in the coronal to be of similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' If we instead take λ⊥ to be on the order  of the diameter of a given flux tube, then λ⊥ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1 − 2 Mm, given the distribution of loop widths  measured using images taken of coronal loops with AIA and the Hi-C imager (Aschwanden & Peter  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  There is an alternative approach through the observed decay time of Alfv´en wave energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' If that  decay is due only to a non-linear turbulent cascade, then the decay time will be given by the eddy  turnover time  τnl =  λ⊥  ⟨vnth⟩ = λ⊥  √w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (18)  Investigations into the non-thermal broadening of EIS Fe xxiv 255 ˚A lines during flares have seen  spatially averaged non-thermal velocities ⟨vnth⟩ on the order of 60-100 km s−1 decay to pre-flare values  11  in roughly 20 minutes (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Stores et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Setting Equation (18) to this time,  and using this turbulent velocity, suggests a correlation length of λ⊥ = 30 − 150 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  There is at least one other aspect to include in our interpretation of λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Traditional derivations  of the non-linear dissipation captured by Equation (12) assume the counter-propagating waves are  uncorrelated with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Our waves, however, are trapped between reflecting ends and are  likely to be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This fact could be accounted for by one more multiplicative factor, 0 < ξ < 1,  in Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Rather than including one more free parameter, we combine the two into a single  free parameter λ′  ⊥ = λ⊥/ξ, which will thus be larger than the turbulent length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We hereafter  interpret λ⊥ as λ′  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Altogether, the above analysis produces a very rough range of λ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1–150 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Nonetheless,  given the simplicity of this model, these values are only meant to demonstrate that a variety of  length scales can be used to characterize turbulent energy dissipation, while also being grounded  in observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For the purpose of a single reference model, λ⊥ is initially set to 2 Mm — a value  consistent with the diameter of observed coronal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This choice thereby assumes the energy  correlation length to be on the order of the small-scale transverse fluctuations along the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The effects of different length scales on the duration of turbulent heating are explored further in the  following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Lastly, the initial value of the reflection coefficient for the propagating energy waves is set to  be η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This value, albeit close to its upper threshold of 1, is chosen so that energy losses  arising from turbulent dissipation can be distinguished from losses due to energy escape through the  transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The characteristic temperature TTR defining the reflection boundaries is chosen to  correspond to regions with a high-density gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For our tube initialized with Tco,0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 MK, we  therefore set TTR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The Reference Simulation  With the flux tube initialized as described above, Equations (1) (2), and (10) are solved using  the PREFT numerical code as documented in Longcope & Klimchuk (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Fluid elements along  the tube are represented by cells on a 1D Lagrangian grid, with the density and differential length  of each calculated to ensure each has a constant mass per unit flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Aside from the evolution of  temperature, which is advanced semi-implicitly, all expressions are advanced explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In order  to ensure stability in the system, each time step is chosen such that the Courant–Friedrichs–Lewy  conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Initial parameters and the resulting properties of the reference simulation  are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The evolution of the tube during its retraction is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Starting with an apex  at height z0=71 Mm, the tube retracts until its total length has decreased to Lf = 93 Mm at time  t = 452 s (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 min), yielding a final apex height of zf = 18 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The change in apex height,  ∆z = z0 − zf = 53 Mm, is analogous to a post-flare loop moving through a current sheet of the same  length (Longcope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Once the retraction is complete, the tube is artificially  straightened to lie on the ˆx-axis, mimicking the effect of the tube reaching the post-flare arcade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Velocity flows after this point are set to equal the parallel velocity v∥ immediately prior to the  straightening, and the perpendicular flow is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The simulation is then run for an additional  32 minutes in the straightened configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' With no perpendicular flow, the source of Alfv´en waves  is S± = 0, so the wave energies are left to decay freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  12  Ashfield & Longcope  0  20  40  60  80  0  10  20  30  40  50  60  70  z [Mm]  0 s  20 s  40 s  75 s  200 s  300 s  452 s  0  20  40  60  80  1010  1012  1014  ne [cm−3]  0  20  40  60  80  −1000  −500  0  500  1000  v∥ [km s−1]  0  20  40  60  80  x [Mm]  0  10  20  30  T [MK]  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Dynamics of the reference simulation over the course of retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A face-on view of the tube  axis r(ℓ) at seven times is shown in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The following three panels  show the corresponding  electron number density ne, velocity parallel to the axis of the tube v∥, and temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Each panel is  plotted against the x-coordinate, scaled such that x=0 Mm corresponds to the  leftmost boundary of the  tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  13  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Simulation properties of the reference and optimized runs  Reference  Optimized  Drag Parameter  fturb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6  λ⊥  Mm  2  100  D  Mm−1  50  50  η  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='95  Model Result  Tpeak  MK  41  32  Peak ˙Q  1010 erg cm−2 s−1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='02  Retraction time  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1  ˙Q duration  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  13  41  EUV duration  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  32  40  During the retraction, the corona undergoes characteristic flare dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Temperature quickly  increases from Tco,0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 MK to values well over 30 MK before decaying down to ∼ 10 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The  parallel velocity along the tube shows that evaporation starts by t = 20 s, reaching speeds up to  nearly 1 Mm s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' By t = 5 minutes, most of v∥ has dropped out completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Once evaporation has  begun, the tube’s density increases by several orders of magnitude in the corona, and remains elevated  after the temperature and velocity in the tube have subsided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The explosive initial evolution appears more reminiscent of drag-free solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Longcope &  Klimchuk 2015), than the gentle dynamics seen in Unverferth & Longcope (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The effect of drag,  however, can be seen from the apparent deceleration of the tube in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To illustrate the change  in downward velocity, Figure 3 plots the apex height of the tube throughout its retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here the  retraction appears to occur in two nearly constant-velocity phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' a fast downflow, lasting a little  less than one minute, followed by a dramatic reduction of speed that remains constant until the tube  is straightened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A straightforward explanation for the deceleration is provided by evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The  upward driving of chromospheric material increases the density in the corona, subsequently lowering  the local Alfv´en speed and slowing the retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Linear fits to each phase show the tube initially moving downward at a speed of 540 km s−1 before  slowing down to 57 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The initial phase agrees with downflows measured in Longcope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (2018), where two-thirds of the 35 downward moving features had velocities lower than 600 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  While the features analyzed in their work saw only straight streaks in a height-time map of a current  sheet viewed edge-on, observations taken using TRACE 195 ˚A filtergrams saw downflows decelerating  in a distinct two-step fashion (for example, see Figure 2 in Sheeley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This deceleration  of roughly an order of magnitude occurred prior to the feature reaching the post-flare arcade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Our  model thus offers a novel insight into the behavior of these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The initial phase of the flux tube’s evolution is shown in Figure 4, where we see the dynamics of  the turbulent Alfv´en wave energies w±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The total energy of the waves wtot peaks at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s (cyan)  and subsequently decreases as the waves propagate towards the footpoints of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' At t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s,  the waves reach the boundary set by the characteristic temperature TTR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 MK, corresponding to  14  Ashfield & Longcope  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Apex position of the flux tube over the retraction period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Linear fits to the fast (red dashed)  and slow (blue dashed) retraction phases are over-plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  position ±x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6 Mm from the edge of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Upon reaching the boundary, the waves then  begin to reflect, the effects of which can be seen at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The solid lines in the upper left  panels correspond to the leftward propagating Els¨asser energy and the dashed lines correspond to  the rightward, counter-propagating energy created from the reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Over times t > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s, the point  of reflection moves progressively downward as the  position where T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 MK moves by thermal  conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Counter-propagating waves appear first at the loop top, where drag creates both waves together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Interaction between waves results in significant loop-top heating according to Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Tur-  bulent dissipation drives the temperature of the loop to a peak temperature of Tpeak = 41 MK that  rapidly drives thermal conduction fronts outward towards the footpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The steep temperature  gradients shown in the lower left panels of Figure 4 are evidence of the flux limiter taking effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Once the reflection begins (t > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s), interactions between the incident and reflected waves create a  second locus of heating near the footpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A novel result of this simulation is the interaction between  the thermal conduction fronts and the increase in temperature driven by the turbulent heating at  the reflection boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As the conduction fronts move leftward in Figure 4, they are impeded by a  localized temperature increase in the transition region that grows over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This growth, initially  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='25  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3  Hturb[1016 erg g−1 s−1 ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6  w± [1016 erg g−1]  0  20  40  x [Mm]  10−2  10−1  100  101  102  (e)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  x [Mm]  10−2  10−1  100  101  T [MK]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8  wtot [1016 erg g−1]  −600  −400  −200  0  200  (d)  −100  0  100  200  v∥ [km s−1]  0  20  40  x [Mm]  10−1  100  101  (f)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  x [Mm]  10−1  100  101  p [erg cm−3]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4 s  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='7 s  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 s  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Propagation of six flux tube variables over the first seven seconds of the reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Middle panels show each variable over the entire left half of the loop, while the corresponding leftmost and  rightmost panels show enlargements of the chromosphere and upper transition region in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Working top  to bottom, the variables plotted are (a) the leftward w+ —solid — and rightward w− — dashed — propa-  gating Els¨asser energies (b) total Els¨asser energy wtot (c) Heat produced from turbulent energy dissipation  Hturb (d) parallel velocity v∥ (e) temperature T and (f) pressure p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  due to Hturb, is likely compounded by the conduction front, as the initial temperature bump seen  at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4 s gradually smooths out with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Nevertheless, the localized heating  from reflection creates a separate conduction front , as well as a pressure increase above the transition  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Remarkably, it is this conduction front that appears to drive evaporation upon reaching the  chromosphere, the beginnings of which are seen at t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s in both parallel velocity and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  It should be noted that the leftward movement of the reflection boundary, as shown by the leftward  propagation of w±, is also due to the localized conduction front, which heats the plasma and moves  TTR in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  16  Ashfield & Longcope  0  10  20  30  40  x [Mm]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5  wtot [1016 erg g−1]  0  10  20  30  40  x [Mm]  10−3  10−2  10−1  100  101  S [1016 erg g−1 s−1 ]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1 s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3 s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Total Els¨asser energy wtot (left) and the corresponding source from drag power S (right) over  the first 60 s of the reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' An instance of the wave energy envelope per unit mass without  sources or reflection (Equation (19)) is shown as the dashed-dotted curve at t=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Figure 5 plots the evolution of the total turbulent Alfv´en wave energies alongside their corresponding  sources S = S+ + S− for the first minute of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here, the amplitude of the source peaks  immediately after the retraction begins and is centered around the loop’s apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As the tube continues  to retract, the source amplitude decreases by two orders of magnitude and spreads along the extent  of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This effect illustrates how the sharp initial bend of the tube decomposes into a more  rounded shape via the drag force, subsequently allowing for additional segments of the tube to be  deflected downward — an effect that compounds throughout the retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Even so, most of the  power in drag remains concentrated at the loop center, leading to a continual creation of turbulent  Alfv´en waves there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  After the Els¨asser energies reach their peak, they then decrease along with the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Of particular  note is the shape of the energies themselves, where the amplitude sharply decreases before undergoing  reflection at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To investigate this behavior, we calculate an envelope representing the  energy per unit mass in the waves that would otherwise be stationary without sources or reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  To do this we solve the static version of equation (10), with dw±/dt = v∥ ∂w±/∂ℓ, and the left two  terms omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The result  w(env)  ±  (ℓ) = A exp  �  �  ℓ  �  1  (vA ∓ v∥)  ∂(vA ± v∥/2)  ∂ℓ′  dℓ′  �  � ,  (19)  is an envelope profile plotted as a dot-dashed line alongside the normal wave at t = 7 s in Figure 5,  with arbitrary scaling A adjusted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The envelope shows a drop in wave energy due mainly  to a decrease in the Alfv´en speed in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Excess energy density around x = 48 Mm  17  10−1  100  101  102  103  0  10  20  30  40  50  60  Wave  Kinetic  ∥ Kinetic  Thermal  10−1  100  101  102  103  Time [sec]  −600  −500  −400  −300  −200  −100  0  Total  Magnetic  Net Drag Loss  Net Rad Loss  Energy [108 erg Mx−1]  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Evolution of energies within the tube in units of 108 erg Mx−1, plotted against a logarithmic time  axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The bottom panel shows free magnetic energy (dark orange), net drag loss (purple), and radiative losses  (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The top panel shows turbulent wave energy (blue), kinetic (orange), parallel kinetic (dashed  green), and thermal (red) on an enlarged scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The total energy (teal) is the sum of the wave, kinetic,  thermal, and magnetic energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Note the kinetic and parallel kinetic energies are virtually indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  is then due to reflection and the production of counter-propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Similarly, although the  source is still relatively high, turbulent dissipation brings the energy density below the envelope for  x ≳58 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The long-term evolution of the tube is shown by the energies plotted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The magnetic  energy (dark orange) lost from its initial value of WM,0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='317×1011 erg Mx−1 is shown in the bottom  panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Over its retraction, the tube releases a total of ∆WM = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='76 × 1010 erg Mx−1 — 44% of the  initial amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Drag (purple) is responsible for removing nearly all free magnetic energy released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The total work done by drag by the end of the retraction is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='61 × 1010 erg Mx−1, constituting over  97% of the available magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The slight deflection seen at t ∼ 60 s results from the change  in retraction speed, while constant values starting at t = 452 s indicate the tube’s straightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  18  Ashfield & Longcope  The top panel of Figure 6 shows the remaining energies present in the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The total energy found  in turbulent Alfv´en waves (blue) mirrors the drag power, reaching a maximum of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8×108 erg Mx−1  before non-linear interactions between counter-propagating waves dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The inflection seen in  the wave energy at t =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8 s therefore corresponds to an uptick in thermal energy, shown in red,  as the heating from turbulent dissipation increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The source of wave energy vanishes once the  retraction ends (t = 452 s), and the wave energy decays rapidly thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It seems that the non-  linear dissipation term works on relatively short time scales provided there is enough reflected power  to facilitate non-linear interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The kinetic energy (orange) increases notably at t = 10 s, corresponding to the onset of evaporation  flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The parallel kinetic energy (green dashed) is effectively identical to the kinetic energy and  makes up 99% of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This is a direct effect of the drag limiting the perpendicular velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  in drag-free solutions, the parallel flow contributes only a small fraction to the total kinetic energy  (Longcope & Klimchuk 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Furthermore, thermal energy decreases when the kinetic energy has  subsided at t = 206 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This moment, also corresponding to an increase in the radiative losses, can be  used to indicate the cooling phase of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' By the end of the run, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='25 × 1010 erg Mx−1 has been  lost to radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The importance of wave energy in driving the initial flare evolution is readily seen when compared  to the parallel kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As alluded to in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1, the energy contained in turbulent Alfv´en  waves is found to be comparable to that contained in parallel plasma flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Moreover, wave energy  in the tube increases immediately upon the start of retraction, whereas kinetic energy increases 10 s  later, after much of the initial dynamics have already taken place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The kinetic energy here is due  to flows driven by evaporation — shown above to be a result of heating via turbulent dissipation  at the reflection boundary in the transition region — and not from plasma deflected by rotational  discontinuities traveling at supersonic speeds (Longcope et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Guidoni & Longcope 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' That  is to say, the evolution of energies seen here supports the notion of turbulence acting as the primary  mode of heating during impulsive flare energy release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The close relation between explosive flare behavior and turbulent wave energy is also illustrated  in the temperature evolution shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The apex temperature of the reference simulation,  shown in blue, is plotted over the entire evolution of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The evolution is characteristic of those  observed during flares, reaching a peak of T = 41 MK before slowly decaying down to its pre-flare  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In this case, the decay time for the flare is on the order of 35 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We note that the  additional peak at t ∼ 2 minutes is an effect of evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For comparison, a simulation was run  using the same drag coefficient, D=50 Mm−1, as the reference, but without including turbulent Alfv´en  waves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' fturb = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The result, shown in green, only reaches a peak temperature of T = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  While the temperature decay is on the same order of time as the reference run, the initial flare  evolution is considerably weaker, suggesting kinetic energy alone is insufficient to power strong flare  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  It seems that the interaction of waves at the loop apex explains much of the peak temperature  evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To demonstrate this notion, we lowered the reflection coefficient to η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='10, while leaving  the drag and source term the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The results, plotted in red on Figure 7, shows the temperature  evolution closely following that of the η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='95 run, albeit at slightly lower values, reaching a peak of  T = 32 MK and decaying to the pre-flare value in virtually the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The lower temperatures  seen in this case suggest that reflected waves play some role in heating, albeit relatively minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  19  0  10  20  30  time [min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=']  0  10  20  30  40  Apex Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' [MK]  η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='95  η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='10  no waves  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Evolution for apex temperature from the results of the reference simulation (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Plotted  for comparison are apex temperatures for a simulation with a reflection coefficient η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='10 (red) and a  simulation with no turbulent Alfv´en waves (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The small role played by reflection is consistent with the observation above that non-linear wave  dissipation is strong enough to eliminate the wave energy rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It is evidently also capable  of  dissipating the energy over length scales short enough that most does not reach the feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Effect of Suppressed Thermal Conduction  We briefly note that the effects of thermal conduction suppression, given by the modification in  Equation (14), were found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' At its largest deviation, conductivity remained within less  than 1% of its classical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This is a natural consequence of the low energy density in Alfv´en waves  (√w ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 Mm s−1) relative to the loop’s Alfv´en speed vA ≃ 10 Mm s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The magnetic perturbations  will deflect the field lines by a random angle ∼ √w/vA ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This deflection results in only a minor  increase in field line path lengths, resulting in only a minor decrease in diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Synthetic EUV Emission  As mentioned in the introduction, a major motivation for including turbulent Alfv´en waves in the  TFT model was to reproduce observations of sustained coronal flare emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In order to create  a benchmark by which our model can be qualitatively compared to such observations, light curves  corresponding to the six coronal EUV channels measured by AIA were synthesized using the results  of our reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  20  Ashfield & Longcope  For a given wavelength channel i, the pixel value pi for the total emission integrated along the flux  tube is calculated by  pi =  � ∞  0  n2  e(ℓ)Ki[T(ℓ)] dℓ,  (20)  given the temperature-response function K(T) (Boerner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' These functions were accessed  using the aia get response function in the SolarSoft IDL library (Freeland & Handy 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Because we are only interested in the time evolution, and not the integrated intensity, the light  curves are normalized according to their maximum value, making it unnecessary to assign a cross-  sectional area to the 1D tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Synthesized emissions for each AIA EUV channel — 131, 94, 335, 211, 193, and 171 ˚A— are shown  in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The lightcurves evolve in a typical fashion and peak successively according to their  characteristic temperatures, decaying from 20 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The higher temperature passbands of 131,  94, and 335 ˚A , also have noticeably wider emission profiles than the lower temperature passbands of  211, 193, and 171 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This difference in duration indicates the temperature evolution through these  passbands, representing a temperature range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4-2 MK, occurs more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To be expected, the  overall evolution of the light curves agrees well with the temperature decay in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' If we quantify  the decay of the EUV emission by the time it takes the lowest channel in 171 ˚A to peak, then the  duration of the flare emission in this case is approximately 32 minutes long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  To illustrate the effect of turbulent Alfv´en waves on the duration of the EUV emission, the total heat  from turbulent dissipation integrated along the tube, ˙Q =  �  Hturbdℓ, is plotted alongside the light  curves, converted here into a heating rate per unit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The heating rate roughly tracks the evolution  of the total wave energy in Figure 6, climbing to a peak of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='15×1010 erg cm−2 s−1 in the initial seconds  of the simulation before sharply decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because the loop starts to cool at t = 206 s, turbulent  heating after this time can be considered the gradual-phase heating supplemental to the impulsive  energy release — analogous to the so-called slow-tail heating used in (Qiu & Longcope 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here,  the gradual-phase heating component lasts much less than the duration of the synthesized emission,  falling to zero  13 minutes after the simulation start time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Seeking an explanation for extended  gradual-phase heating, and longer duration of EUV emission, we revisit the initial parameters of our  turbulent transport model in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' EXPLORING PARAMETERS  The preceding analysis found the dissipation of turbulent Alfv´en waves to effectively drive char-  acteristic flare behavior and produce long-duration EUV emissions on the order of 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We  explore what is required to extend long-term coronal emission by adjusting the parameters of our  turbulent transport model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Having already visited the reflection coefficient and the modification to  the Spitzer-H¨arm conductivity, we now focus on the two remaining parameters: the percentage of  drag power converted to turbulent wave energy, fturb, and the energy correlation length, λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Converted Drag Power  The amount of energy converted into MHD turbulence from drag losses is dictated by the fraction  fturb in Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because the generation of Alfv´en wave perturbations is not explicitly modeled  from the interaction between the flux tube and the current sheet, it is unclear what value of fturb  would serve as a good approximation for this interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  We therefore explore the response in  turbulent energy to a change in drag loss conversion by varying the degree of interaction through  21  0  5  10  15  20  25  30  35  0  20  40  Tmax [MK]  131  94  335  211  193  171  0  5  10  15  20  25  30  35  time [min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=']  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' counts  131  94  335  211  193  171  0  50  100  150  200  ˙Q [108 erg cm−2 s−1]  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Bottom panel: Synthetic light curves of AIA EUV channels 131, 94, 335, 211, 193, and 171 ˚A  produced using results from the reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Heating due to the dissipation of turbulent Alfv´en  waves is shown in purple, read against the axis on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Top panel: Apex temperature evolution as  shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Symbols denote the times when Tapex crosses the characteristic formation temperature(s)  of the respective AIA channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  fturb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To this end, six simulations are run with fturb ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The remaining loop  parameters match the reference simulation outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Results from the six simulations are shown in the top two panels of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The apex temperature  evolution — used here as a proxy for the effectiveness of a set of parameters to produce gradual-phase  heating — is plotted on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The overall evolution in each run is similar to that of the reference  simulation, shown in green (fturb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2), with temperatures quickly a peak before decaying to ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In particular, the peak temperature is seen to scale with fturb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As more drag loss becomes available  to be converted into turbulent energy, the tube reaches higher internal temperatures from increased  dissipation and subsequent heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This conclusion is supported by the integrated heating rates ˙Q,  plotted on the right, where higher degrees of drag conversion correspond to larger heating rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Greater degrees of drag conversion were also found to produce more prolonged heating rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Com-  pared to the fturb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='05 case, in which ˙Q decayed to zero within 8 minutes, a conversion of fturb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8  resulted in an additional 7 minutes of heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Regardless, the time taken for the apex temperature  to decay to its pre-flare level remained  nearly the same for each of the six simulations, despite  increasing heating duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The similarity between these times is ultimately due to the different  evaporation strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Stronger evaporation, arising from increased heating following a higher fturb,  leads to a higher density in the coronal segment of the loop, making the losses from radiative cooling  22  Ashfield & Longcope  0  5  10  15  20  25  30  35  100  101  102  Apex Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' [MK]  fturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8  fturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6  fturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4  fturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  fturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1  fturb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='05  0  10  20  30  40  0  100  200  300  400  500  600  ˙Q [108 erg cm−2 s−1]  0  10  20  30  40  50  time [min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=']  100  101  102  Apex Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' [MK]  λ⊥ [Mm] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1  λ⊥ [Mm] = 2  λ⊥ [Mm] = 10  λ⊥ [Mm] = 100  λ⊥ [Mm] = 1000  0  10  20  30  40  50  time [min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=']  0  50  100  150  200  ˙Q [108 erg cm−2 s−1]  4  6  8  10  12  14  0  10  20  30  5  10  15  20  25  30  0  1  2  3  4  5  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Apex temperature evolution (left) and integrated heating rate ˙Q (right) for 11 simulations run  with different values of drag power conversion fturb (top) and correlation length λ⊥ (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The cases of  fturb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2 and λ⊥=2 Mm in the top and bottom panels , respectively, correspond to the reference simulation  analyzed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Vertical dashed lines in the upper right panel indicate the time at which retraction  ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  more effective and the loop cools faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As a result, the duration of the temperature evolution  appears to be insensitive to the amount of energy put into turbulent Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Energy Correlation Length  23  The heat generated via turbulence occurs at a rate set by the non-linear interaction between  the counter-propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Because we adopt a one-point closure model, this interaction is  characterized by a single parameter, λ⊥, indicative of the degree of correlation between the two  energy populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1, a range of possible values for λ⊥ was motivated from observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Here, λ⊥ is taken to be the parameter controlling the efficiency of energy loss from the waves, adjusted  from a phenomenological perspective to produce decay timescales that best agree with observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The bottom two panels of Figure 9 show the apex temperature evolution (left) and integrated  heating rate (right) for  five simulations run with a range of λ⊥=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1–1000 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As λ⊥ scales in-  versely with ˙Q, smaller correlation lengths produce larger heating rates and subsequently larger apex  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Apex temperature evolution is virtually indistinguishable, except during the earliest  phase, for all values λ⊥ ≤ 10 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Heating rates for these runs are also similar following the initial  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  This similarity appears to be a consequence of energy dissipation within a single transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The net  energy input into every case is roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For cases with λ⊥ ≤ 10 Mm, that energy is mostly  dissipated before reaching the feet, and thus does not remain on the loop beyond the end of retraction  at t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This end was noted for the illustration case (λ⊥ = 2 Mm), and is repeated in the  olive curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Those curves with still smaller values of λ⊥ drop even more rapidly after the termination  of retraction (see λ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1 Mm in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The two simulations with λ⊥ > 10 Mm produced markedly different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Only in these cases  does significant wave energy remain beyond the end of  retraction at t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 min (see inset in  the lower right of Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It seems that the non-linear dissipation is small enough for waves to  reflect several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This persistent wave energy is partly responsible for apex temperatures which  decayed to pre-flare levels in 42 and 50 minutes (λ⊥ = 100 and 1000 Mm, respectively) rather than  the 30-minute decay observed in all other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Larger correlation lengths are accompanied by less  dissipation in the early phases, resulting in more gentle flare behavior, as evidenced by peak apex  temperatures under 20 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The conductive flux to the feet is therefore smaller, leading to weaker  evaporation flows, lower loop density, and thereby diminished radiative cooling at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This  is a second factor extending the cooling time of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We therefore see that larger correlation  lengths, and the weaker non-linear damping they produce, are critical for our model to extend the  duration of coronal emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Optimized Simulation  To capitalize on our better understanding of the role wave energy can play in extending cooling  times we perform one final simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This uses a large correlation length, λ⊥=100 Mm, and high  conversion efficiency of fturb=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Although the parameter exploration of fturb determined that  temperature evolution was not sensitive to the degree of drag conversion, a higher value of fturb is  used to produce higher apex temperatures to agree with flare observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  AIA EUV light curves synthesized from this tuned run are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Compared to  the emissions from the reference simulation in Figure 8, the profiles here for the hotter channels  are broader and peak at later times, illustrating how the temperature remains elevated during this  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The cooler channels, however, are still relatively narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' At this point in the simulation,  the heating rate has decreased from its peak of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0 × 1010 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2 × 1010 erg cm−2 s−1, indicating that  energy dissipated from turbulence is not enough to counteract energy lost from radiative cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  24  Ashfield & Longcope  0  10  20  30  40  time [min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=']  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='0  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' counts  131  94  335  211  193  171  0  20  40  60  80  100  ˙Q [108 erg cm−2 s−1]  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Synthetic EUV emission produced from the optimized simulation, plotted in the same format  as Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In this case, the duration of the heating rate lasted 41 minutes and produced coronal emissions  lasting for approximately 40 minutes, given the peak in 171 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Even though the correlation length and  degree of drag loss conversion were 50 and 3 times larger than the reference simulation, respectively,  the duration was only 8 minutes longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Notable differences between the optimized and the reference  simulations are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Additional simulations run with increased fturb and λ⊥, while  not shown here, also failed to extend the duration of the emission past 40 minutes, further signifying  the balance between strong flare dynamics and gradual turbulent dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' As such, the emission  duration shown in Figure 10 represents the upper limit of our turbulent transport model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Alfv´en  wave turbulence should be evident in  an excess broadening of hot spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The  turbulent energy per mass gives the turbulent velocity squared averaged over all fluctuation time  scales:  wtot = w+ + w− = 1  2⟨|v|2⟩ + 1  2  �|b|2  4π  �  = ⟨|v|2⟩,  (21)  after assuming turbulence is Alfv´enic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The net effect on an unresolved observation of spectral line λ  is characterized by a spatial average weighted by the local intensity of the line, Iλ(ℓ),  ¯wλ =  ��  Iλ(ℓ) dℓ  �−1 �  wtot(ℓ) Iλ(ℓ) dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (22)  If the axis of the tube is viewed perpendicularly, then the averaged broadening of the spectral line  λ along the line of sight, say ˆz, is  σλ =  �  ⟨v2  z⟩ =  �  ¯wλ  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  (23)  Figure 11 shows the broadening of three lines typically associated with flares, both versus time (right)  and normalized line intensity (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Cooler lines show lower broadening, and all decrease with time  as the turbulence decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  25  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Turbulent line broadening, defined in Equation (23) for spectral lines of Fe xxiv 192˚A (red), Fe  xxi 129 ˚A (violet), and Fe xviii 94 ˚A (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The left panel plots these against the normalized intensity of  the line, and the right against time when the line is bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Time flows respectively from top to bottom for  each curve in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The time of loop straightening (t = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='13 min) is indicated by a vertical dashed  line and triangles on each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Diamonds along the right axis are the average values for each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' DISCUSSION  We have extended a post-reconnection flare model to include the production and dissipation of  turbulent Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This extension involved a system of turbulent transport equations inspired  by existing models of solar wind turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' One-dimensional simulations tracked the evolution of  MHD turbulence through their aggregate energy densities, which remained trapped in the corona by  reflection from high-density gradients in the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Turbulent energy dissipated through  the non-linear interaction between counter-propagating waves and produced heating longer than  previous versions of the PREFT flare code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Synthetic light curves from AIA EUV bands showed  increased similarity with typical flare observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Ours is the first model of its kind to generate Alfv´en wave turbulence from reconnection outflow  and model its effect on long-term flare signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In the spirit of a first attempt, we chose to  represent the physical processes involved using simple, fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Energy is removed from the  retracting flux by a simple aerodynamic drag force, parameterized by the parameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We chose the  value D = 50 Mm−1 in order to slow the retraction to a level consistent with typical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' A  fraction, fturb, of the energy lost to drag appears as unresolved Alfv´en wave turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' While this  process is likely to be very complicated, with a fraction varying in time as well as space, we chose  to use a fixed value of fturb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' We performed a series of runs and different values of fturb and found its  most important effect was in setting the peak flare temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The physics of Alfv´en wave propagation, reflection, and dissipation is also captured through a small  number of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Reflection is modeled through a reflection coefficient, η, applied at the  26  Ashfield & Longcope  point where T = TTR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='5 MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Wave dissipation occurs through non-linear dissipation involving  waves of both species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This process is parameterized through an effective correlation length, λ⊥,  which was varied to explore its effects on the flare evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The duration of turbulent heating was found to be most dependent on the correlation length λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Correlation lengths λ⊥ ≤ 10 Mm lead to very rapid dissipation and wave energy vanishes rapidly  after retraction ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Only in those with λ⊥ ≳ 100 Mm does wave energy persist beyond the end of  retraction, thereby prolonging the cooling of the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' If Alfv´en waves are actually responsible for  the long flare cooling times observed, non-linear wave dissipation would need to be characterized by  such large correlation lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The finding described above requires reconsideration of the physical significance behind the pa-  rameter λ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Our turbulence model assumes the small-scale fluctuations responsible for the MHD  turbulence to be on scales smaller than the radius of the flux tube, so λ⊥ cannot be simply taken  as the correlation length of the turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It was mentioned above that our parameter incorpo-  rates the correlation between the counter-propagating waves, such that the true correlation length is  λ′  ⊥ = λ⊥/ξ, where ξ is the fraction of the counter-propagating turbulence which is actually uncorre-  lated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' For an effective correlation length of λ⊥ = 100 Mm and a tube with radius R = λ⊥ =2 Mm,  this gives ξ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='02, meaning that only 2% of the counter-propagating waves are uncorrelated and able  to produce a turbulent cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' While an imperfect correlation should be expected between the two  energy populations, it is worth noting that the length scale for the small-scale fluctuations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' R)  and the length scale for the energy correlation do not necessarily need to be related (Zank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Should this perspective be taken into account, it would also substantiate the range of length  scales derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='1 using the decay rates of non-thermal broadenings observed in Fe xxiv ˚A  spectral lines, where λ⊥ ∼30-150 Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The heating rates produced in this work also fit the description of a two-step heating profile as  described in Qiu & Longcope (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The initial phase of turbulent dissipation was highly impulsive  in all cases, reaching a peak within 20 s before quickly decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Following the impulsive energy  release, defined here by the time it takes a loop to enter a regime of increased radiative losses,  heating from dissipation then became more gradual and lasted anywhere from 5 to 50 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' In  particular, the heating rate produced in Figure 10 was found to match the general heating profile  described in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (2018), where energy deposited during the impulsive and gradual phases  was required to be 60% and 40% of the total energy released in order to reproduce sustained EUV  emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Here, after an impulsive phase that lasted 240 s, the gradual phase released 46% of the  total turbulent wave energy over 36 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' To the best of our knowledge, this mechanism was the  first to reproduce these two-step heating rates in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  A byproduct of using turbulent Alfv´en waves for gradual-phase heating was their capacity to be  equally effective at driving characteristic flare behavior during the initial phase of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' By  inducing turbulence from drag, we were able to reintroduce energy back into the tube that would have  otherwise been lost to the background of our flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Hence, our model successfully reproduced  impulsive flare behavior while still keeping the speed of reconnection outflows within their observed  ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Investigations into the earlier phases of turbulent dynamics presented here, such as the  forward-modeling of Fe xxi ˚A lines, may explain the large broadenings of such lines seen during flares  (Polito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2019) and could serve to constrain our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  27  While our model produced long-duration heating in conjunction with impulsive flare dynamics, it  was not able to reproduce sustained EUV emission on the order of hours, as seen in observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  One possible explanation for this discrepancy is the value of the drag coefficient D=50 Mm−1 used  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Increasing D would lead to lower retraction rates, likely resulting in more prolonged  heating and more energy available to convert to turbulent Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Because the first stage of  our reconnection downflow was observed to be 540 km s−1, the drag coefficient could be increased to  the point of generating retraction rates on the order of 50 km s−1 — the lower speed limit of SADs  observed in Savage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  In this new model, the chromospheric response to reconnection is driven in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The first is thermal conduction, which was part of even the earliest flare loop models (Nagai 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Pallavicini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Augmenting this, Alfv´en waves are created by retraction at the loop apex  and propagate to the chromosphere where they are dissipated by interacting with their own reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  This second transport channel drives chromospheric evaporation beyond that attributable to conduc-  tion alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It is worth noting that a fraction 1 − fturb of the drag work will appear as turbulence on  field lines neighboring that just reconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Some of the neighboring flux will still be unreconnected,  and will now experience some chromospheric evaporation before reconnecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' There are several lines  of observational evidence suggesting that flare reconnection occurs on field lines whose density is  higher than one would expect in the ambient, pre-flare corona (Veronig & Brown 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Veronig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Our model offers a possible explanation for this previously puzzling fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Another mechanism proposed to extend the cooling time of a flare loop is through suppression of  thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Previous investigations into the evolution of flare loops using zero-dimensional  models have shown that alterations to the mean-free path of thermal electrons extended cooling times  considerably (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Our model accounts, in Equation (14), for a single  MHD mechanism for decreasing κ: decreasing the mean axial step size ℓz by the lengthening of the  actual path, ℓ, due to field line meandering on sub-resolution scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' MHD turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' When the  perturbation velocity is significantly lower than the local Alfv´en speed, as implied by observations  and explicitly found in our model, then field lines are lengthened only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The result is the  very small reduction in κ we have observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Effective reduction in the mean-free path must occur  not through MHD mechanisms but through the kinds of particle kinetics effects suggested by others  (Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  The level of turbulence predicted here can be compared to observation through turbulent line  broadening of high-temperature spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The level in our optimized run produces σ between  100–400 km/s, a factor of 2-3 higher than typically observed (Warren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  Polito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' There are, however, many reasons to expect our estimate to be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The  estimate in Equation (23) assumes the field lines are viewed exactly perpendicularly and that the  observation integrates over all turbulent frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The turbulence consists of Alfv´en waves on the  loop, whose frequencies will extend down to the fundamental with a period of two Alfv´en transit  times — perhaps up to 20 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Integration times shorter than this will omit the lower frequency  component of the spectrum, which contains the most energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This will reduce the value of σ actually  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' It should be noted that the turbulent broadening in our model naturally decreases on  an approximately 15-minute scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' This is consistent with observations (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Warren  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content='  28  Ashfield & Longcope  The model presented here is self-consistent to the extent that turbulence is ultimately generated  by the retraction of a flux tube through a current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' The interaction between the current sheet  and the tube responsible for both the drag force and the excitement of turbulent Alfv´en waves,  however, remains an assumption of our model and is not directly investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Moreover, the  resolution of PREFT required MHD turbulence to be modeled according to its aggregate energy  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Small-scale perturbations in the velocity and magnetic fields that constitute the turbulence  were not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' While the simplicity of the turbulent transport model in this work warrants  further study, any improvements would likely require techniques beyond the formalism of 1D flare  loop simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
+page_content=' Despite these limitations, the results presented here illustrate the efficacy of MHD  turbulence as a mechanism for flare energy transport and heating, and provides an additional tool  by which other flare phenomena can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE3T4oBgHgl3EQfjwrg/content/2301.04592v1.pdf'}
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+Under consideration for publication in J. Fluid Mech.
+1
+Banner appropriate to article type will appear here in typeset article
+Stratified inclined duct: two-layer hydraulics and
+instabilities
+Amir Atoufi1, Lu Zhu1†, Adrien Lefauve1, John R. Taylor1, Rich R. Kerswell1,
+Stuart B. Dalziel1, Gregory. A. Lawrence2, and P. F. Linden1
+1Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,
+Wilberforce Road, Cambridge CB3 0WA, UK
+2Department of Civil Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
+(Received xx; revised xx; accepted xx)
+The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct
+as a model for continuously-forced density-stratified flows such as those found in estuaries.
+Experiments have shown that the emergence of interfacial waves and their transition to
+turbulence as the tilt angle is increased appears linked to a threshold in the exchange flow
+rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms with (i)
+recent direct numerical simulations (DNS) providing full flow data in the key flow regimes
+(Zhu & Atoufi et al., arXiv:2301.09773, 2023), (ii) averaging these DNS into two layers,
+(iii) an inviscid two-layer shallow water and instability theory to diagnose interfacial wave
+behaviour and provide physical insight. The laminar flow is subcritical and stable throughout
+the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the
+flow becomes everywhere supercritical and unstable to long waves. An internal undular
+jump featuring stationary waves first appears near the centre of the duct, then leads to larger-
+amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence
+at the largest tilt angle. Long waves described by the (nonlinear) shallow water equation are
+locally interpreted as linear waves on a two-layer parallel base flow described by the Taylor-
+Goldstein equation. This link helps us interpret long-wave instability and contrast it to
+short-wave (e.g. Kelvin-Helmholtz) instability. Our results suggest a transition to turbulence
+in SID through long-wave instability relying on vertical confinement by the top and bottom
+walls.
+1. Introduction
+Buoyancy-driven exchange flows between water masses of different density are common in
+estuaries and straits, e.g. the straits of the Great Belt, Gibraltar, Bab el Mandab, and the
+Bosphorous (Gregg & Özsoy 2002). These essentially hydrostatic and two-layer flows are
+known to exhibit hydraulic jumps (Farmer & Armi 1988; Thorpe et al. 2018b), which are
+important discontinuities in the internal flow properties (layer thickness and speed) and often
+lead to instability (Lawrence & Armi 2022). A local jump can influence the large-scale
+properties of the flow such as the exchange flow rate (Lawrence 1993). These non-local
+hydraulic flow features are often studied in an idealised ‘shallow-water’ setting consisting of
+† Email address for correspondence: lz447@cam.ac.uk
+Abstract must not spill onto p.2
+arXiv:2301.13035v1  [physics.flu-dyn]  30 Jan 2023
+
+2
+A. Atoufi, L. Zhu, and others
+fluid organised in two counter-flowing, frictionless layers of specified thickness and speed
+with constant densities (Armi 1986; Lawrence 1990; Dalziel 1991).
+In this paper, we employ two-layer hydraulics as a diagnostic tool to derive insights from
+direct numerical simulation (DNS) data of the exchange flows in the stratified inclined duct
+(SID). SID is a canonical stratified shear flow through a long, square cross-section, tilted
+duct for which there is now ample data, both experimental (e.g. Partridge et al. 2019) and
+numerical (Zhu et al. 2022). The investigation of turbulence in two-layer shear flows through
+tubes dates back to the classic experiments of Reynolds (1883) and Thorpe (1968), who both
+used a closed tube. The opening of the tube into large reservoirs in SID is more recent and
+allows for interfacial waves and turbulence to be sustained for much longer time periods,
+and for various flow regimes to be distinguished. The successive transitions to increasingly
+turbulent regimes in SID, as the Reynolds number and tilt angle are increased, have been
+recognised since Macagno & Rouse (1961) and Kiel (1991), and have been much studied more
+recently (Meyer & Linden 2014; Lefauve et al. 2019; Lefauve & Linden 2020; Duran Matute
+et al. 2023). These transitions are underpinned by many fundamental features of stratified
+flows, including interfacial waves and turbulent intermittency.
+Our first aim with this new analysis is to uncover internal hydraulic effects in order to
+explain some of these leading-order dynamics that DNS and experimental data have revealed
+but not yet explained. In particular, we will provide the first direct evidence for the existence
+of internal hydraulic jumps, where the layer thickness expands in the direction of the flow
+of each layer. We also show how the development of jumps and large-amplitude interfacial
+waves coincides with a plateau in the exchange flow rate through the duct, after an initial
+increase with increasing Reynolds number and tilt in the laminar regime. This upper bound
+is a remarkably robust feature of SID, which has deep ramifications for the flow energetics
+and transition to turbulence (Lefauve et al. 2019). Though the emergence of waves and
+turbulence has long been linked to the notion of ‘hydraulic control’ in the experimental SID
+literature, this link has not yet been studied in detail, primarily because experiments lack the
+full velocity, density and pressure data along the length of the duct. The recent availability of
+DNS data (Zhu et al. 2022) overcoming these limitations finally makes a rigorous hydraulics
+analysis of SID possible.
+Our second aim is to link two-layer internal hydraulic effects to the growth of instabilities
+and to shorter (non-hydrostatic) waves, links which are rarely found explicitly in the
+literature. The streamwise variation of the base flow in the SID geometry, and generally
+in all exchange flows, distinguishes them from idealised parallel stratified shear flows. This
+variation, however, is essential to the formation of internal hydraulic jumps, to the ideas of
+hydraulic control and maximal exchange, and hence to the nature of SID turbulence. It is
+well known that under some flow conditions, the loss of hyperbolicity of the shallow-water
+equations renders long waves unstable. However much less is known about the consequences
+of this instability, and the relative importance of long-wave versus short-wave instability. We
+will clarify this by explaining how a certain range of unstable nonlinear shallow water waves
+associated with an internal jump can be interpreted as linear instabilities on a locally parallel
+base flow, and thus that the insights derived from stable hydraulic theory remain valid even
+under (moderately) unstable conditions.
+To tackle these aims, we introduce our DNS datasets in §2, and the two-layer averaging
+in §3, using the averaged datasets to show evidence of jumps and maximal exchange. In
+§4, we adapt the two-layer shallow water theory to SID, summarise important results from
+the literature, and connect them to linear stability theory. In §5, we use this hydraulics and
+stability framework to analyse our DNS. Then, in §6, we explore the transition between long
+(hydrostatic) waves and short (non-hydrostatic) waves, the influence of molecular diffusion
+(Prandtl number) and of smoother flow profiles. Finally, we draw our conclusions in §7.
+
+SID: two-layer hydraulics and instabilities
+3
+Figure 1: A schematic of the stratified inclined duct (SID) numerical setup. The duct is
+oriented at angle 𝜃 to the horizontal, which is equivalent to tilting the gravity vector.
+Densities are non-dimensionalised by the density differences between the reservoirs and
+lengths by the half-duct height. The duct’s volume is
+(𝑥, 𝑦, 𝑧) ∈ [−30, 30] × [−1, 1] × [−1, 1].
+2. Direct numerical simulations
+2.1. Methodology
+Our DNS solves the following non-dimensional Navier-Stokes equations under the Boussi-
+nesq approximation for the density-stratified flow in the SID setup sketched in figure 1:
+∇ · 𝒖 = 0,
+(2.1)
+𝜕𝒖
+𝜕𝑡 + (𝒖 · ∇) 𝒖 = −∇𝑝 + 1
+Re∇2𝒖 + ˆg Ri 𝜌 − 𝑭𝑢,
+(2.2)
+𝜕𝜌
+𝜕𝑡 + (𝒖 · ∇) 𝜌 =
+1
+Re Pr∇2𝜌 − 𝐹𝜌,
+(2.3)
+where Re = 𝐻∗𝑈∗/𝜈 is the input Reynolds number (𝑈∗ = √𝑔′𝐻∗ is the characteristic
+buoyancy velocity, 𝐻∗ is the dimensional half-height of the duct, 𝜈 is the kinematic viscosity,
+𝑔′ is the reduced gravity); Ri = 𝑔′𝐻∗/(2𝑈∗)2 is the input bulk Richardson number, giving a
+fixed Ri = 1/4; and Pr ≡ 𝜈/𝜅 is the Prandtl number (𝜅 is the thermal diffusivity). The unit
+gravity vector in the coordinate system (𝑥, 𝑦, 𝑧) aligned with the duct is ˆg ≡ [sin 𝜃, 0, − cos 𝜃].
+The DNS is performed with the open-source solver Xcompact3D (Bartholomew et al. 2020).
+The duct has a square cross-section of non-dimensional height and width 2 and of length
+60 (giving a long aspect ratio of 𝐴 = 30). No-slip boundary conditions for 𝒖 and no-flux
+boundary conditions for 𝜌 are applied on the duct walls in the spanwise (𝑦 = ±1) and vertical
+(𝑧 = ±1) directions with an immersed boundary method. The flow is driven by the density
+difference between the dense (𝜌 = 1) and light (𝜌 = −1) fluid in the left- and right-hand
+reservoirs, respectively, producing counter-flowing layers in the streamwise 𝑥-direction. The
+experimental reservoirs are modelled by ad hoc forcing terms 𝑭𝑢 and 𝐹𝜌, which damp flow
+in the reservoirs and restore the density field to 𝜌 = ±1, allowing us to maintain a quasi-
+steady exchange flow for arbitrarily long times at a minimal computational cost. Details
+about the numerical setup and validation against experiments and benchmark cases (with
+large reservoirs and 𝑭𝑢 = 𝐹𝜌 = 0) can be found in Zhu et al. (2022).
+The DNS is started at 𝑡 = 0 from lock-exchange initial conditions, after which two counter-
+flowing gravity currents develop from 𝑥 = 0, advance at absolute speed ≈ 1, and reach either
+end of the duct after ≈ 30 advective time unit (ATU). Shortly after, the statistically-steady
+exchange flow of interest in SID becomes established. Conservatively, we only retain 𝑡 ⩾ 80
+in the following analysis to discard any initial transients, and we run the simulation until
+𝑡 = 260.
+When the setup is tilted by an angle 𝜃 > 0◦, the streamwise component of gravity
+contributes Ri 𝜌 sin 𝜃 to the 𝑥-component of the momentum and adds extra kinetic energy
+
+4
+A. Atoufi, L. Zhu, and others
+Figure 2: A schematic of two-layer shallow water flow. Flow in a part of the reservoir is
+considered.
+to the flow. Increasing 𝜃 and/or 𝑅𝑒 leads to a variety of flow regimes from laminar to
+wavy to intermittently turbulent to fully turbulent, found both in DNS (Zhu et al. 2022) and
+experiments (Meyer & Linden 2014; Lefauve et al. 2019; Lefauve & Linden 2020).
+2.2. Database
+We use the DNS database recently acquired by Zhu et al. (2022), which shows good agreement
+with experiments when all five non-dimensional parameters Re, Ri, Pr, 𝜃, 𝐴 are matched. This
+database provides the complete set of flow variables all along the duct, which is a requirement
+to properly study hydraulic processes in SID.
+We consider flows at a fixed Reynolds number Re = 650 and fixed Prandtl number Pr = 7,
+corresponding to temperature-stratified water. Four cases are examined, corresponding to
+tilt angles 𝜃 = 2◦, 5◦, 6◦, 8◦, denoted by B2, B5, B6, B8, respectively in Zhu et al. (2022).
+Each of these four cases covers specific flow regimes: B2 is laminar (L), B5 has stationary
+waves (SW), B6 has travelling waves (TW), and B8 has intermittent turbulence (I). In the
+following, we refer to these datasets as L, SW, TW, and I respectively. We only briefly touch
+upon more turbulent cases (e.g. B10 having Re = 1000 and 𝜃 = 10◦, referred to here as T),
+as they deviate significantly from the assumptions of the model in this paper, that the flow
+is primarily composed of two layers with a hydrostatic pressure field (discussed in §A). For
+more details about the flow regimes, statistics and spatio-temporal dynamics, see Zhu et al.
+(2022).
+3. Two-layer model as a diagnostic tool
+3.1. Layer averaging procedure
+We seek to reduce the dimensionality of our DNS datasets to a set of two layers in order to
+interpret their dynamics using simple two-layer hydraulics, as sketched in figure 2. To do
+this, we will define the interface that separates the layers as the height 𝑧 = 𝜂(𝑥, 𝑡) where
+𝜌 = 0. The streamwise velocities, heights and densities of each layer are 𝑢𝑖, ℎ𝑖, and 𝜌𝑖 (where
+𝑖 = 1, 2 correspond to the upper and lower layer, respectively), are then obtained by averaging
+the DNS data over the 𝑦-direction and in 𝑧 over the height of each layer. Specifically, the flow
+properties of the layers are obtained by
+ℎ1(𝑥, 𝑡) = 1 − 𝜂(𝑥, 𝑡),
+𝑢1(𝑥, 𝑡) = ⟨⟨𝑢⟩𝑦⟩𝑧1,
+𝜌1(𝑥, 𝑡) = ⟨⟨𝜌⟩𝑦⟩𝑧1,
+(3.1)
+ℎ2(𝑥, 𝑡) = 1 + 𝜂(𝑥, 𝑡),
+𝑢2(𝑥, 𝑡) = ⟨⟨𝑢⟩𝑦⟩𝑧2,
+𝜌2(𝑥, 𝑡) = ⟨⟨𝜌⟩𝑦⟩𝑧2,
+(3.2)
+where the top-layer average is ⟨·⟩𝑧1 = (1/ℎ1)
+∫ 1
+𝜂 · 𝑑𝑧, the bottom-layer average is ⟨·⟩𝑧2 =
+(1/ℎ2)
+∫ 𝜂
+−1 · 𝑑𝑧 and and the spanwise average is ⟨·⟩𝑦 = (1/2)
+∫ 1
+−1 · 𝑑𝑦. Recall that 𝑧 = 1 and
+𝑧 = −1 are the non-dimensional height of the top and bottom walls, respectively. Figure 3
+shows a single snapshot at time 𝑡 = 110 of 𝑢 (colours) and 𝜌 (contours) at the 𝑦 = 0 midplane
+Focus on Fluids articles must not exceed this page length
+
+SID: two-layer hydraulics and instabilities
+5
+Figure 3: Snapshot at 𝑡 = 110 of streamwise velocity 𝑢 (blue to red shading) and density 𝜌
+(colour contours) in the midplane (𝑦 = 0) for the (𝑎) laminar (L) flow and (𝑏) travelling
+wave (TW) flow, the latter showing evidence of an jump (see the interface 𝜌 = 0
+emphasised by the thick green line).
+for the two datasets L and TW, highlighting the 𝜌 = 0 density interface with a thick green
+contour.
+3.2. Layer-averaged DNS data and evidence of jumps
+Figure 4 illustrates the results obtained after applying this layer averaging to the DNS
+database. We show 𝑥 − 𝑡 diagrams of the lower-layer height ℎ2 (top row) and lower-layer
+velocity 𝑢2 > 0 (bottom row) for the laminar (L), stationary wave (SW), travelling wave
+(TW), and intermittently turbulent (I) cases. The layer averaging (in 𝑦, 𝑧) was performed in
+the range |𝑥| ⩽ 32, which includes the duct |𝑥| ⩽ 30 and extends slightly into the reservoirs
+where the layers flow in and out.
+The L flow regime exhibits sudden changes in depth and speed only at the ends of the duct
+𝑥 = ±30, as is expected from the flow exiting into the deep reservoirs. Figure 4(a) and an
+instantaneous snapshot of the flow in figure 3(a) confirms that the interface 𝜂(𝑥) is steady
+and gently sloping down (𝜂′ < 0) throughout the duct, and is symmetric in the duct centre,
+i.e. 𝜂(𝑥, 𝑡) = 𝜂(𝑥) = −𝜂(−𝑥).
+In contrast, for the SW and TW flow regimes, the upper layer thickness ℎ2 suddenly
+increases along the direction of the flow (purple for 𝑥 < 0 to orange for 𝑥 > 0 in figure 4),
+which is accompanied by a sudden drop in 𝑢2 (dark to light red). This discontinuity indicates
+the presence of what is commonly called an ‘internal hydraulic jump’ (Baines 2016; Thorpe
+et al. 2018a), as can be seen in figure 3(b), where both layers experience a sudden expansion
+and deceleration in their respective flow direction. In the TW flow (figure 3(b)) the interface
+is sloping up (𝜂′ > 0) in the vicinity of the jump, and is negative (𝜂 < 0) throughout most of
+the left-hand side of the duct, and vice versa, which is the opposite of what is found in the L
+flow (figure 3(a)).
+Figure 5 shows the temporal evolution of the interface in all four flows, with 𝜂(𝑥, 𝑡)
+plotted at intervals separated by one ATU and vertically stacked. In the SW case, the jump
+remains in the narrow interval 𝑥 = ±1 whereas in the TW case, it oscillates over a much larger
+interval and sends off waves in either direction, hence the distinction between the ‘stationary’
+and ‘travelling’ wave regime. In the I case, moving jumps are observed in the quiet phase
+(150 ≲ 𝑡 ≲ 200), being initiated near both ends of the duct at 𝑡 ≈ 150 and progressively
+moving toward the centre of the duct. These jumps merge at around 𝑡 = 200 and then stay
+
+6
+A. Atoufi, L. Zhu, and others
+Figure 4: Spatio-temporal diagrams of (a-d) the lower-layer height ℎ2 and (e-h)
+lower-layer velocity 𝑢2 from the DNS data for the four flow regimes (a,e) L, (b,f) SW,
+(c,g) TW, (d,h) I. All data are for 𝑡 ∈ [80, 260].
+Figure 5: Temporal evolution of the density interface 𝜂(𝑥, 𝑡) in (a) L, (b) SW, (c) TW, (d)
+I. The interface curves are stacked in time at intervals of one ATU. Jumps are revealed in
+(b)-(d) by the discontinuity in 𝜂(𝑥).
+
+(a) L
+(b) SW
+(c) TW
+(d) I
+250
+h2
+1.5
+200
+1.0
+150
+100
+0.5
+(e) L
+() SW
+(g) TW
+(h) I
+250
+U2
+1
+200
+0
+150
+100
+-1
+-30-150
+1530-30-15 0
+1530-30-150
+1530-30-15 0
+15
+30
+x
+x
+x
+x(a) L
+(b) Sw
+(C)
+IW
+(d)
+260
+80
+-30
+0
+0
+0
+30 -30
+30 -30
+30 -30
+0
+30
+x
+x
+x
+xSID: two-layer hydraulics and instabilities
+7
+Figure 6: The mass flux 𝑄𝑚 from the 15 DNS of Zhu et al. (2022) increasing with 𝑅𝑒 and
+𝜃 until the upper bound of ≈ 0.5. The symbols, coded by the respective regime, show the
+time-averaged value while the error bars depict the extreme values
+at the middle of the duct 𝑥 = 0 for ≈ 20 ATU before the transition to turbulence occurs at
+𝑡 ≈ 220 (the active stage of the intermittent cycle).
+3.3. Flow rate and evidence of maximal exchange
+To a good approximation, the flow in SID has zero net (barotropic) volume flow rate
+⟨𝑢⟩𝑦,𝑧 ≡ 1
+4
+∫ 1
+−1
+∫ 1
+−1
+𝑢 𝑑𝑦 𝑑𝑧 = 1
+2 (𝑢1ℎ1 + 𝑢2ℎ2) ≈ 0,
+(3.3)
+but a non-zero exchange volume flow rate (or volume flux) 𝑄(𝑥, 𝑡) and mass flow rate (or
+mass flux) 𝑄𝑚(𝑥, 𝑡) defined as
+𝑄 ≡ ⟨|𝑢|⟩𝑦,𝑧 = 1
+2 (𝑢2ℎ2 − 𝑢1ℎ1) ≈ −𝑢1ℎ1 ≈ 𝑢2ℎ2
+using (3.3),
+(3.4)
+𝑄𝑚 ≡ ⟨𝜌𝑢⟩𝑦,𝑧 ≈ 𝜌1𝑢1ℎ1 ≈ 𝜌2𝑢2ℎ2.
+(3.5)
+The approximation in (3.5) comes from the fact that the layer averages of the product 𝜌𝑢 is not
+exactly equal to the product 𝜌𝑖𝑢𝑖 of the layer averages. Also, recall that the non-dimensional
+density is defined such that the mean density is 0 and the minimum and maximum are −1
+and 1 respectively.
+Hydraulic jumps in two-layer flows are often connected to the notion of maximal exchange,
+i.e. of an upper bound in the exchange volume flux 𝑄 and mass flux 𝑄𝑚. This means that
+flows lacking such jumps have a lower 𝑄, and that no realisable flow may have a higher
+𝑄. While hydraulic jumps have not yet been investigated in detail in the experimental SID
+literature, the mass flux 𝑄𝑚 has, due to the simplicity with which it can be measured.
+In figure 6 we show the dependence of the mass flux 𝑄𝑚 with RiRe sin 𝜃 ≈ (1/4)𝜃Re
+using its temporal mean (symbols) and extreme values (error bars) in 15 different DNS
+from Zhu et al. (2022), containing the four main datasets L, SW, TW and I, as well as 11
+others including one turbulent dataset (T). We find that 𝑄𝑚 increases approximately linearly
+with 𝜃Re in the L regime (where little mixing ensures that 𝑄 ≈ 𝑄𝑚), in agreement with
+the laminar analytical solution of Duran Matute et al. (2023). For higher values of 𝜃Re it
+reaches an upper bound 𝑄𝑚 ≈ 0.5 in the W regime, before dropping below 0.5 in the I and
+T regimes. These observations are consistent with the corresponding experimental data of
+Lefauve & Linden (2020) (their figures 5-6), if we exclude their data at small angles 𝜃 < 2◦
+(which behave slightly differently and we did not simulate). This apparent upper bound is an
+evidence of maximal exchange, since further increase in 𝜃 does not lead to an increase in the
+
+0.6
+T
+西1
+0.4
+中中中
+=
+0.2
+LWIT
+0.0
+0
+20
+40
+RiResin 08
+A. Atoufi, L. Zhu, and others
+velocity difference between the upper and lower layer. It is a significant departure from the
+laminar solution based on the balance of gravitational forcing and viscous drag, suggesting
+𝑄𝑚 ∝ 𝜃Re (Lefauve & Linden 2020). As it will be shown later (section 5.3) this 𝑄𝑚 ≈ 0.5
+is the threshold of the long-wave instabilities and onset of hydraulic transitions and a further
+increase in 𝑄𝑚 beyond this limit is taxed by turbulent dissipation.
+This large body of experimental and numerical evidence in favour of maximal exchange
+in SID and 𝑄𝑚 ≈ 0.5 at the transition between the L and W regime, combined with our
+observations in §3.2 of the existence of an internal jump, is strongly suggestive that hydraulic
+effects dominate the flow in the W regime onwards.
+In the next section, we develop the two-layer hydraulics framework to study these jumps
+and maximal exchange and their relation to the transition from laminar flow to waves and to
+turbulence in SID.
+4. Two-layer equations: characteristics and instabilities
+In this section, we aim to define and relate the characteristic velocity of the two-layer flows
+in the context of SID to the hydraulic regime of the flow. We also aim to provide a physical
+implication of when this characteristic velocity is not purely real. We start by simplifying
+the inviscid Navier-Stokes equations with shallow water (long wave) theory in §4.1-4.3
+before comparing it to the Taylor-Goldstein (linear wave) theory in §4.4, and interpreting our
+findings in §4.5.
+4.1. Shallow water equations: nonlinear long waves
+The validity of this hydrostatic assumption is verified in our DNS data in appendix A. The
+upper layer then obeys
+𝜕ℎ1
+𝜕𝑡 + 𝜕(ℎ1𝑢1)
+𝜕𝑥
+= 0,
+(4.1)
+𝜕(ℎ1𝑢1)
+𝜕𝑡
++ 𝜕
+𝜕𝑥
+�
+ℎ1𝑢2
+1 + Ri cos 𝜃𝜌1
+ℎ2
+1
+2
+�
+= Ri sin 𝜃𝜌1ℎ1 − Ri cos 𝜃ℎ1
+𝜕
+𝜕𝑥 (𝑝𝑤 + 𝜌1ℎ2 + 𝜌1𝑏) ,
+and the lower layer obeys
+𝜕ℎ2
+𝜕𝑡 + 𝜕(ℎ2𝑢2)
+𝜕𝑥
+= 0,
+(4.2)
+𝜕(ℎ2𝑢2)
+𝜕𝑡
++ 𝜕
+𝜕𝑥
+�
+ℎ2𝑢2
+2 + Ri cos 𝜃𝜌2
+ℎ2
+2
+2
+�
+= Ri sin 𝜃𝜌2ℎ2 − Ri cos 𝜃ℎ2
+𝜕
+𝜕𝑥 (𝑝𝑤 + 𝜌1ℎ1 + 𝜌2𝑏) .
+Here, 𝑝𝑤 (𝑥, 𝑡) is the pressure at the upper wall and 𝑏(𝑥) is the elevation of the bottom wall.
+Conveniently, the bottom wall in SID is fixed at 𝑏(𝑥) = −1. We subtract the momentum
+equations in (4.1) to (4.2) to remove 𝑝𝑤 and reduce the number of unknowns. The variation
+of density of the upper and lower layers in 𝑥 is also neglected compared to variations in
+height of the layers. The shallow water equations become
+𝜕ℎ2
+𝜕𝑡 − 𝜕ℎ1
+𝜕𝑡 + ℎ2
+𝜕𝑢2
+𝜕𝑥 − ℎ1
+𝜕𝑢1
+𝜕𝑥 + 𝑢2
+𝜕ℎ2
+𝜕𝑥 − 𝑢1
+𝜕ℎ1
+𝜕𝑥 = 0,
+(4.3)
+𝜕𝑢2
+𝜕𝑡 − 𝜕𝑢1
+𝜕𝑡 + 𝑢2
+𝜕𝑢2
+𝜕𝑥 − 𝑢1
+𝜕𝑢1
+𝜕𝑥 + Ri cos 𝜃(𝜌2 − 𝜌1) 𝜕ℎ2
+𝜕𝑥
+= Ri sin 𝜃(𝜌2 − 𝜌1)
+(4.4)
+−Ri cos 𝜃(𝜌2 − 𝜌1) 𝜕𝑏
+𝜕𝑥 ,
+
+SID: two-layer hydraulics and instabilities
+9
+with two auxiliary equations to satisfy the no-net (barotropic) flow condition and geometric
+constraint inside the duct (𝜕𝑏/𝜕𝑥 = 0)
+𝑢1ℎ1 + 𝑢2ℎ2 = 0,
+ℎ1 + ℎ2 = 2.
+(4.5)
+By taking the derivative of (4.5) with respect to 𝑥, these four equations can be written
+compactly as
+C 𝜕𝒒
+𝜕𝑡 + A(𝒒, Δ𝜌) 𝜕𝒒
+𝜕𝑥 = 𝒇,
+(4.6)
+where the state vector 𝒒 and coefficient matrices A, C are
+𝒒 =
+�������
+𝑢1
+𝑢2
+ℎ1
+ℎ2
+�������
+, A =
+����
+�
+−𝑢1
+𝑢2
+0
+Ri cos 𝜃 Δ𝜌
+−ℎ1
+ℎ2
+−𝑢1
+𝑢2
+0
+0
+1
+1
+ℎ1
+ℎ2
+𝑢1
+𝑢2
+����
+�
+, C =
+����
+�
+−1
+1
+0
+0
+0
+0
+−1
+1
+0
+0
+0
+0
+0
+0
+0
+0
+����
+�
+,
+(4.7)
+The shallow water equation (SWE) (4.6) is quasilinear, i.e. linear in the derivatives of 𝒒 but
+with the coefficient matrix A dependent on 𝒒. The quasi-constant local density difference
+between layers is defined as Δ𝜌(𝑥, 𝑡) ≡ 𝜌2−𝜌1 ∈ (0, 2) (specified by our layer-averaged DNS
+data). This equation does not assume that interfacial waves have infinitesimal amplitudes,
+but it does assume, through the hydrostatic assumption, that waves are long with respect to
+the layer height (i.e. that their non-dimensional wavenumber 𝑘 ≪ 1). In the following, we
+neglect the forcing
+𝒇 = [Ri sin 𝜃 Δ𝜌, 0, 0, 0]𝑇 ,
+(4.8)
+recalling that sin 𝜃 ≪ 1. We will study (4.6) in the homogeneous limit ( 𝒇 = 0), with a focus
+on the eigenvalues of A, in which 𝒇 plays no role. The role of forcing in shallow water theory
+is an interesting question left for future work. However, the forcing proportional to sin 𝜃 does
+influence the DNS, and is thus implicitly taken into account in the state vector 𝒒 obtained
+after layer averaging the DNS data.
+4.2. Characteristic curves and propagation of information
+Consider a left eigenvector 𝒗 and eigenvalue 𝜆 associated with the matrix pair (A,C) such
+that 𝒗𝐻A = 𝜆𝒗𝐻C, where 𝐻 denotes Hermitian transpose. Multiplying (4.6) by 𝒗𝐻 yields
+𝒗𝐻C
+� 𝜕𝒒
+𝜕𝑡 + 𝜆(𝑥, 𝑡) 𝜕𝒒
+𝜕𝑥
+�
+= 0.
+(4.9)
+The eigenvalues 𝜆 are called characteristic velocities, since they define characteristics curves
+𝑠 in the (𝑥, 𝑡) plane along which the partial differential equation (4.6) reduces to an ordinary
+differential equation (Whitham 2011). For this homogeneous equation, the combinations
+of flow variables 𝒗𝐻C 𝑑𝒒/𝑑𝑠 = 0 is conserved. These characteristic velocities, henceforth
+simply referred to as ‘characteristics’, can be complex 𝜆 = 𝜆𝑅 + 𝑖𝜆𝐼 ∈ C. Their real part
+𝜆𝑅 ≡ ℜ(𝜆) represents the phase speed of shallow water waves, describing the trajectories 𝑠.
+Their imaginary part 𝜆𝐼 ≡ ℑ(𝜆) represents any potential growth (𝜆𝐼 > 0) or decay (𝜆𝐼 < 0)
+of these waves. The direction of information propagation is set by the sign of 𝜆𝑅: when
+𝜆𝑅 > 0, information propagates rightward (towards increasing 𝑥), and vice versa, whereas
+𝜆𝑅 = 0, indicates stationary waves.
+
+10
+A. Atoufi, L. Zhu, and others
+Figure 7: Real and imaginary parts of the eigenvalues of the two-layer SWE in (4.11) at (a)
+positive (b) zero and (c) negative interface elevations 𝜂 as functions of the volumetric flow
+rate 𝑄. The eigenvalues are always complex for 𝑄 ⩾ 𝑄𝑐 = 0.5, and become complex for
+slightly lower critical values 𝑄𝑐 ≈ 0.4 in the asymmetric cases (a,c).
+The characteristics 𝜆 are given by the two distinct solutions of det(𝑨 − 𝜆𝑪) = 0:
+𝜆1,2(𝑥, 𝑡) =
+ℎ1𝑢2 + ℎ2𝑢1
+ℎ1 + ℎ2
+����������������������
+convective velocity ≡ ¯𝜆
+±
+√︂
+Δ𝜌Ri cos 𝜃 ℎ1ℎ2
+ℎ1 + ℎ2
+�1 − 𝐹2
+Δ
+� ,
+��������������������������������������������������������������������
+phase speed ≡ 𝛿𝜆
+(4.10)
+≈
+������
+−2𝜂𝑄
+1 − 𝜂2
+±
+������������������������������������������������������������������
+√︂
+Δ𝜌
+2 cos 𝜃(1 − 𝜂2)(1 − 𝐹2
+Δ)
+using (3.4),
+(4.11)
+where
+𝐹2
+Δ(𝑥, 𝑡) =
+(𝑢2 − 𝑢1)2
+Δ𝜌 Ri cos 𝜃 (ℎ1 + ℎ2) ,
+(4.12)
+≈
+2
+Δ𝜌 cos 𝜃
+� 2𝑄
+1 − 𝜂2
+�2
+using (3.4).
+(4.13)
+Here, (4.11) and (4.13) use the volume flux 𝑄 > 0 in (3.4) (which is an approximation
+relying on (3.3)) and the interface position instead of the layer heights and velocities, as well
+as the fact that Ri = 1/4 and ℎ1 + ℎ2 = 2 in SID. We see that the characteristics consist
+of a convective velocity ¯𝜆 and a phase speed 𝛿𝜆, which can be imaginary, depending on
+the value of the ‘stability Froude number’ 𝐹2
+Δ (Long 1956; Lawrence 1990; Dalziel 1991).
+Note that (4.7), (4.10) and (4.12) are slightly modified versions of those given in previous
+studies (Long 1956; Armi 1986; Lawrence 1993) adapted to SID flows.
+If 𝐹2
+Δ < 1 the two characteristics 𝜆1,2 = ¯𝜆 ± 𝛿𝜆 are real and information propagates in
+both directions relative to the convective velocity ¯𝜆. The absolute direction of propagation is
+given by the sign of 𝜆1,2, which we shall return to in §4.3.
+If 𝐹2
+Δ > 1 the characteristics become complex conjugates 𝜆1,2 = 𝜆𝑅 ± 𝑖𝜆𝐼 = ¯𝜆 ± 𝑖|𝛿𝜆|,
+indicating that the system is no longer hyperbolic and that waves grow temporally unstable.
+The condition 𝐹2
+Δ > 1 is sometimes known as Long’s instability criterion (Long 1956),
+although it is quoted in Lamb (1932) and likely dates back to Helmholtz. The real part is the
+convective velocity, i.e. information only propagates in one direction, and the growth rate is
+𝜆𝐼 = |𝛿𝜆| =
+√︃
+(Δ𝜌/2) cos 𝜃(1 − 𝜂2)(𝐹2
+Δ − 1).
+Figure 7(a-c) shows how𝜆1,2 vary with the volume flux𝑄 and interface position 𝜂 following
+(4.11), assuming no mixing (Δ𝜌 = 2) and a horizontal duct (cos 𝜃 = 1). We compare the case
+where the interface is locally symmetric (𝜂 = 0, panel b) to the cases where it is asymmetric,
+Rapids articles must not exceed this page length
+
+(h) n=0.5
+(b) n=0.0
+(b) n=-0.5
+2.5
+0.0
+R
+-2.5
+0.0
+0.5
+1.00.0
+0.5
+1.00.0
+0.5
+1.0
+Q
+Q
+QSID: two-layer hydraulics and instabilities
+11
+Figure 8: Summary of the scenarios revealed by the stability Froude number 𝐹2
+Δ (4.12),
+characteristics 𝜆1,2 (4.10), and composite Froude number 𝐺2 (4.16).
+i.e. above or below the mid-level (𝜂 = ±0.5 in panels a and c, respectively). Panel b shows that
+with a symmetric interface 𝜆1,2 = ±
+√︁
+1 − 4𝑄2 are real (red curves) for 𝑄 ⩽ 0.5 and become
+purely imaginary (blue curves) for 𝑄 > 0.5. However, when the interface is either below
+or above the midplane (𝜂 = ±0.5, panels a,c), this transition to instability occurs at a lower
+critical volume flux 𝑄 > 𝑄𝑐 ≈ 0.4. In summary, instability is caused, for a given volume flux
+𝑄, by an increasingly asymmetric interface |𝜂|, and vice versa, for a given interface position,
+by an increasing volume flux 𝑄. This offers a possible explanation for the transition from L
+to W flow observed in figure 6.
+4.3. Composite Froude number and hydraulic control
+To further stress the importance of the characteristics 𝜆1,2, we return to the original SWE
+(4.6), and note that a non-trivial steady solution 𝒒(𝑥) requires det A ≠ 0, i.e.
+det A = 2Δ𝜌ℎ1ℎ2Ri cos 𝜃(𝐺2 − 1) ≠ 0
+⇔
+𝐺2 ≠ 1
+(4.14)
+where 𝐺2 is the squared composite Froude number defined with the Froude numbers of the
+upper and lower layers, 𝐹1, and 𝐹2, respectively,
+𝐺2 = 𝐹2
+1 + 𝐹2
+2 ,
+𝐹𝑖 =
+𝑢𝑖
+√︁
+Δ𝜌 Ri cos 𝜃 ℎ𝑖
+𝑖 = 1, 2.
+(4.15)
+Points at which 𝐺2 = 1 are called control points (Armi 1986; Lawrence 1990; Dalziel 1991).
+At control points, A is non-invertible and a regularity condition must exist to recover a steady
+solution.
+The link between characteristics and the composite Froude number can be highlighted
+with the identity
+𝐺2 = 1 +
+ℎ1 + ℎ2
+(𝜌2 − 𝜌1)Ri cos 𝜃 ℎ1ℎ2
+𝜆1𝜆2 = 1 +
+2𝜆1𝜆2
+Δ𝜌 cos 𝜃(1 − 𝜂2)
+(4.16)
+≈ 1 +
+√︄
+2
+Δ𝜌 cos 𝜃
+𝜆1𝜆2𝐹Δ
+2𝑄
+using (4.13) (4.17)
+From this expression, we deduce the following, illustrated in figure 8.
+If the waves are stable (𝐹2
+Δ < 1), the characteristics 𝜆1,2 are real. The flow is called
+subcritical and information propagates in both directions (along positive and negative 𝑥) i.e.
+𝜆1𝜆2 < 0 ⇔ 𝐺2 < 1. In other words, the absolute phase speed |𝛿𝜆| is larger than the absolute
+
+G2<1
+-0>
+subcriticalα「>「i
+11.2 E R
+=
+stable
+>→G>1
+hyperbolic
+supercritical 」
+21.2ECIR
+>>1
+!=
+ supercritical
+unstable
+non-hyperbolic
+waves propagate at velocity aR - ^
+grow at rate a' = I8a12
+A. Atoufi, L. Zhu, and others
+convective velocity ¯𝜆 in (4.10). Vice versa, the flow is called supercritical when information
+propagates in only one direction i.e. 𝜆1𝜆2 > 0 ⇔ 𝐺2 > 1, and the absolute phase speed
+|𝛿𝜆| is smaller than the absolute convective velocity ¯𝜆. Note that for supercritical flow, the
+direction of propagation associated with 𝜆1 and 𝜆2 is the same as that given by the convective
+velocity ¯𝜆, i.e. the waves are swept downstream.
+If, on the other hand, the waves are unstable (𝐹2
+Δ > 1), the characteristics are complex
+conjugates 𝜆1,2 = 𝜆𝑅 ± 𝑖𝜆𝐼, i.e. information always propagates in a single direction given by
+the sign of 𝜆𝑅 = ¯𝜆 and the flow is always supercritical (𝜆1𝜆2 = (𝜆𝑅)2+(𝜆𝐼)2 > 0 ⇔ 𝐺2 > 1).
+Under unstable conditions, control points where 𝐺2 = 1 cannot exist, but points can exist
+where stationary (𝜆𝑅 = 0) waves grow (𝜆𝐼 > 0) and control the flow.
+In the next sections, we clarify the interpretation of unstable shallow water waves using
+linear stability analysis around a locally parallel base flow. In §4.4 we take the long-wave
+limit of solutions of the Taylor-Goldstein equations, and in §4.5 we linearise the shallow
+water equations assuming the waves are sufficiently (but not excessively) long.
+4.4. Taylor-Goldstein equations: linear short and long waves
+We relax the previous restriction that waves must be long (𝑘 ≪ 1) by studying waves of
+possibly larger 𝑘 but of infinitesimal amplitude developing on a parallel base flow described
+by a velocity profile U(𝑧) and a density profile R(𝑧). The perturbation streamfunction
+ˆ𝜓(𝑧) exp𝑖𝑘(𝑥 − 𝑐𝑡) describing the evolution of these two-dimensional linear waves is given
+by the inviscid Taylor-Goldstein equation (TGE)
+(U − 𝑐)
+� 𝑑2
+𝑑𝑧2 − 𝑘2
+�
+ˆ𝜓 − U′′ ˆ𝜓 − Ri cos 𝜃 R′
+U − 𝑐
+ˆ𝜓 = 0.
+(4.18)
+This can be analysed following standard methods (e.g. Drazin 2002; Smyth & Carpenter 2019)
+with details given in appendix B. Note that we assumed small tilt angles 0 < sin 𝜃 ≪ cos 𝜃,
+although the more general TGE in appendix B shows that sin 𝜃 has a destabilising effect
+(ignored here). In the ordinary differential equation above, ‘′’ denotes differentiation with
+respect to 𝑧, and 𝑐 ∈ C is the phase speed of the plane waves, akin to the characteristics 𝜆
+of the SWE. However, we use a different notation for the characteristics of shallow water
+nonlinear waves and the phase speed of TG linear waves to emphasise that while the former
+implicitly assumes 𝑘 ≪ 1, the latter implicitly assumes 𝑘 ≫ 𝐴−1. In other words, the TG
+linear waves we investigate should be shorter than the duct length 𝐴 for the local analysis on
+a parallel base flow to be sensible. In other words, 𝑘 ≫ 𝐴−1 ensures that the waves do not
+‘feel’ the streamwise variations of the base flow along the duct.
+To make analytical progress and obtain solutions for 𝑐, we assume a two-layer flow bounded
+by solid walls, with fixed layer heights ℎ1, ℎ2, velocities 𝑢1, 𝑢2, and an interface at 𝑧 = 0,
+consistent with the two-layer model adopted throughout this paper
+U(𝑧) =
+�
+𝑢1
+0 < 𝑧 ⩽ ℎ1
+𝑢2
+−ℎ2 ⩽ 𝑧 < 0 ,
+R(𝑧) =
+�
+𝜌1
+0 < 𝑧 ⩽ ℎ1
+𝜌2
+−ℎ2 ⩽ 𝑧 < 0 .
+(4.19)
+Note that ℎ1 + ℎ2 = 2 and, by a simple vertical shift, the above model is equivalent to having
+a domain restricted 𝑧 = ±1 and an interface at an arbitrary 𝑧 = 𝜂 (giving ℎ1,2 = 1 ± 𝜂).
+By enforcing the matching conditions for the stream function and pressure at the interface
+
+SID: two-layer hydraulics and instabilities
+13
+(see appendix B), we derive the dispersion relation for the complex phase speeds
+𝑐1,2 = 𝑢1 + 𝑢2
+2
++ (𝜎5 − 𝜎6)(𝑢1 − 𝑢2) ± 𝜎2
+2 (𝜎1 − 1)
+,
+(4.20)
+𝜎1 = cosh (4 𝑘 ) ,
+𝜎2 = sinh (2𝑘 ) Λ,
+𝜎3 = sinh (2 𝑘 ℎ2) ,
+𝜎4 = sinh (2 𝑘 ℎ1) ,
+𝜎5 = cosh (2 𝑘 ℎ2) ,
+𝜎6 = cosh (2 𝑘 ℎ1) ,
+Λ =
+√︂
+−4
+𝑘
+�𝑘𝜎4𝜎3[𝑢1 − 𝑢2]2 + Ri cos 𝜃[𝜎4(1 − 𝜎5) + 𝜎3(1 − 𝜎6)]Δ𝜌�.
+These waves are the well-known Kelvin-Helmholtz (KH) waves supported by a single vortex
+sheet (see e.g. Smyth & Carpenter (2019) § 4.6.1). They are modulated by stratification;
+increasing Ri cos 𝜃Δ𝜌 always stabilises them. Importantly, the dispersion relation (4.20)
+describes waves in a domain bounded by the top and bottom walls, which as we will see,
+strongly affect waves that have a wavelength comparable to or longer than the domain height.
+For ‘short’ waves, i.e. having a wavelength of the order of the duct height 𝑘 = 𝑂(1) or
+shorter 𝑘 > 1, the dispersion relation (4.20) cannot be simplified further, and these waves
+are dispersive. These will be referred to simply as KH waves, as in the short wave limit they
+are identical to those found in vertically unbounded domains.
+For long waves (which, for clarity, we do not call KH waves), i.e. having a wavelength
+much longer than the duct height, but still shorter than the duct length 𝐴−1 ≪ 𝑘 ≪ 1, the
+dispersion relation (4.20) simplifies as sinh 𝑘ℎ −→ 𝑘ℎ and cosh 𝑘ℎ −→ 1 and (4.20) reduces
+to (4.10), i.e.
+𝑐1,2(𝑘) −→ 𝜆1,2
+when 𝑘 ≪ 1,
+(4.21)
+and these waves are non-dispersive. In other words, the characteristics 𝜆1,2 of nonlinear
+shallow water waves can be identified to the phase speeds 𝑐1,2 of linear (infinitesimal) long
+waves calculated assuming the local two-layer flow (𝑢1, 𝑢2, ℎ1, ℎ2) to be parallel and steady
+as in (4.19).
+The dispersion relation (4.20) shows that there exists a smooth transition between short
+KH waves and long shallow water waves, as those waves differ by their wavelength but not the
+underlying physical mechanism. We return to this smooth transition and plot the dispersion
+relation in §6.
+Although the limit (4.21) can be inferred from the PhD thesis of Gu (2001) (§ 3.3) and is
+briefly alluded to in Boonkasame & Milewski (2012) (§ 3), it does not appear to be widely
+disseminated in the hydraulics literature.
+4.5. Link between complex characteristics and instability
+This natural link between 𝜆1,2 and 𝑐1,2 in the long-wave limit can be further understood
+by considering another limit. It is possible to directly linearise the SWE (4.6) to study the
+evolution of infinitesimal perturbations 𝜖 ˜𝒒(𝑥, 𝑡) (0 < 𝜖 ≪ 1) on a parallel, steady base
+flow 𝒒0 = (𝑢1, 𝑢2, ℎ1, ℎ2) akin to (4.19). We perform a first-order Taylor expansion of the
+coefficient matrix from (4.7) A(𝒒) = A(𝒒0 + 𝜖 ˜𝒒) and obtain
+C 𝜕(𝒒0 + 𝜖 ˜𝒒)
+𝜕𝑡
++
+�
+A(𝒒0) + 𝜖 ˜𝒒 𝜕A
+𝜕𝒒0
+� � 𝜕𝒒0
+𝜕𝑥 + 𝜖 𝜕 ˜𝒒
+𝜕𝑥
+�
+= 0,
+(4.22)
+At order 𝜖 we have the linear, local SWE
+C 𝜕 ˜𝒒
+𝜕𝑡 + A0
+𝜕 ˜𝒒
+𝜕𝑥 = −˜𝒒 𝜕A
+𝜕𝒒0
+𝜕𝒒0
+𝜕𝑥
+����
+= 0
+,
+(4.23)
+
+14
+A. Atoufi, L. Zhu, and others
+Figure 9: (a) Summary of the interpretation of characteristics in §4.2 using linear theory,
+either by first linearising the NSE, and then taking the long-wave limit (TGE, §4.4) or vice
+versa (SWE, §4.5). Both approaches yield the same result for waves much shorter than the
+duct length but much longer than the duct height (range sketched in b).
+where A0 ≡ A(𝒒0) is the local constant coefficient matrix. Importantly, the right-hand side
+coming from the Taylor expansion, and acting as a forcing term, vanishes when we assume
+the base flow to vary slowly. Substituting the plane wave ansatz ˜𝒒 = ˆ𝒒 exp𝑖𝑘(𝑥 − 𝜍𝑡) gives
+the phase speed 𝜍 as the eigenvalue of the matrix pair (A0, C). The two distinct solutions
+𝜍1,2 of det(A0 − 𝜍C) = 0 then become identical to the local characteristics 𝜆1,2 (at a fixed
+𝑥, 𝑡) derived in (4.10).
+In other words, the nonlinear shallow water wave (𝑘 ≪ 1) characteristics 𝜆1,2 can be
+interpreted as the linear shallow water waves phase speed 𝜍1,2 propagating on a locally
+parallel base flow (𝑘 ≫ 𝐴−1), which are themselves the long wave, non-dispersive limit case
+of the Taylor-Goldstein two-layer phase speeds 𝑐1,2. This result ultimately stems from the
+fact that, simply put, the linearisation and the long wave limit commute. In particular, the
+potential positive imaginary component of characteristics 𝜆𝐼 > 0 can be interpreted as the
+exponential growth rate of such unstable waves satisfying 𝐴−1 ≪ 𝑘 ≪ 1.
+This interpretation is summarised in figure 9. The long aspect ratio 𝐴 of SID (in this
+paper 𝐴 = 30), ensures that the range of waves 𝐴−1 ≪ 𝑘 ≪ 1 exists, and therefore that this
+interpretation is useful, unlike in shorter geometries having 𝐴 ≲ 10. We also note in passing
+that this interpretation can be generalised to any number of layers greater than two.
+5. Two-layer hydraulics applied to DNS
+In this section, we use the modelling results of §4 to understand the observations of hydraulic
+jumps and maximal exchange of §3. In §5.1 we study the stability Froude number flagging
+
+(a)
+Shallow water
+q(x, t) = (ui, u2, hi, h2)(x, t)
+Inviscid Navier-Stokes
+equations (SWE)
+equations (NSE)
+characteristics 21,2(x, t)
+two layers
+q(x, t) = (u,p)
+non-dispersive, global
+long waves k < 1
+o = (u1, U2, P1, P2)
+q(x, t) = qo + eqexp ik(x - st)
+q(x, t) = qo + eq exp ik(x - ct)
+parallel flow k > A-1
+two-layer parallel flow k > A-1
+linearise e < 1
+linearise e < 1
+plane waves with speed s
+plane waves with speed c
+21,2 → C1,2, S1,2 non-dispersive, local
+Taylor-Goldstein
+long waves k < 1
+-2nQ
+equations (TGE)
+-n2
+phase speeds C1,2(k)
+Unstable F2 > 1 → 1,2 = R ±ia!
+dispersive, local
+wave propagation
+wave growth
+sign AR= - sign n α (1 -n2)(F2 - 1)
+(b)
+duct length
+duct height
+waves of interest
+wavenumber
+1
+A-1
+long waves k < 1
+A-1<k1
+parallel flow k > A-1SID: two-layer hydraulics and instabilities
+15
+Figure 10: Spatio-temporal diagram (𝑥 − 𝑡) of the stability Froude number 𝐹2
+Δ for the (𝑎)
+L, (𝑏) SW, (𝑐) TW, and (𝑑) I cases. Long-wave instability is predicted for 𝐹2
+Δ > 1 (red
+colour).
+long-wave instability, in §5.2 we focus on the characteristics and composite Froude number
+to diagnose internal jumps and hydraulic control, and in §5.3 we explain the observed flow
+rate with notions of maximal exchange, viscous friction and mixing.
+5.1. Stability Froude number and instability
+In figure 10 we plot the spatio-temporal diagram of the stability Froude number 𝐹2
+Δ(𝑥, 𝑡)
+given by (4.12) in our four datasets L, SW, TW, and I (all at 𝑅𝑒 = 650) to diagnose any long
+wave instability.
+The laminar flow (L) has 𝐹2
+Δ < 1 everywhere (in blue in figure 10a), hence long waves
+are stable at 𝜃 = 2◦. In all other cases, 𝐹2
+Δ > 1 (in red in figure 10) in most of the duct,
+hence long waves are unstable at 𝑅𝑒 = 650 for 𝜃 > 𝜃𝑐 where 𝜃𝑐 ∈ [2◦, 5◦]. In the wave
+flows (SW and TW), the waves are most unstable (maximum 𝐹2
+Δ, deep red) near the centre
+of the duct. In the I flow, long waves are very unstable (𝐹2
+Δ ≫ 1) throughout most of the
+duct. These 𝐹2
+Δ(𝑥, 𝑡) diagrams correspond to the absence of a jump in the duct in the L flow
+(figure 4a) and the existence of a jump in the SW, TW, and I flows (figure 4b-d), although the
+reasons for the jump remain to be explained. These diagrams also show that as the tilt angle
+𝜃 is modestly increased between 5◦ and 8◦, further changes take place as the flow becomes
+increasingly unstable to long waves. Next, we delve deeper into the hydraulics analysis to
+understand jumps by focusing on the characteristics in each flow.
+5.2. Characteristics, composite Froude number and control
+In figure 11 we plot the characteristic velocities 𝜆1,2(𝑥) given by (4.10) (real parts in panel
+a and imaginary parts in panel b) and the composite Froude number 𝐺2(𝑥) given by (4.16)
+(panel c) at time 𝑡 = 110, to diagnose the propagation of information and criticality of the
+flow, respectively. In figure 12 we plot a set of discrete trajectories (white curves) by solving
+𝑑𝑋
+𝑑𝑡 = 𝜆𝑅(𝑋, 𝑡) for 𝑡 > 80,
+(5.1)
+and initialising 𝑋(𝑡 = 80) as 61 equidistant points along the duct 𝑥 ∈ [−30, 30]. Additionally,
+we plot the local growth rate 𝜆𝐼 (𝑥, 𝑡) in background colours (dark blue to yellow) for
+comparison.
+Figures 11(a) and 12(a) show that in the stable L flow, 𝜆 has two distinct real roots of
+
+0
+1
+2
+250
+200
+150
+100
+(b)
+(n)
+(c)
+-30
+0
+0
+0
+0
+30
+30-30
+30-30
+30-30
+x
+x
+x
+x16
+A. Atoufi, L. Zhu, and others
+Figure 11: Characteristics (a) real part 𝜆𝑅, (b) imaginary part 𝜆𝐼 (only the positive values
+are shown), and (c) composite Froude number 𝐺2 of the L, SW, TW, I flows at 𝑡 = 110.
+Note that we also show data immediately outside the duct, up to 𝑥 = ±32.
+Figure 12: Characteristic curves 𝑋(𝑡) (in white) obtained by (5.1) for (a) L, (b) SW, (c)
+TW, (d) I. The curves (in white) originate from 61 equally spaced positions between
+𝑥 = −30 and 30. The colour contours show the growth rate 𝜆𝐼 (𝑥, 𝑡) (which is zero in a).
+Note that we only show data inside the duct up to 𝑥 = ±30.
+opposite signs throughout most of the duct (𝜆𝑅
+1 𝜆𝑅
+2 < 0 ⇔ 𝐺2 < 1), allowing characteristics to
+cross. Information propagates in both directions and the flow is subcritical inside the duct. The
+speed of propagation is of order 0.2 − 0.4, i.e. significantly lower than the advective velocity
+1. However, at the ends of the duct, the characteristics vanish locally (𝜆𝑅
+1 𝜆𝑅
+2 = 0 ⇔ 𝐺2 = 1 at
+|𝑥| ≈ 30) signalling that long waves become stationary. Figure 11(a) shows that immediately
+outside the duct, information propagates only in one direction (𝐺2 > 1 at |𝑥| ≳ 30), in fact,
+leftward on the left-hand side (𝜆𝑅
+1 , 𝜆𝑅
+2 < 0) and rightward on the right-hand side (𝜆𝑅
+1 , 𝜆𝑅
+2 > 0),
+i.e. always away from the duct. The ends of the duct, therefore, act as control points, in the
+sense that no information from the reservoirs can propagate into the duct. The existence
+of two such hydraulic control points, with their respective characteristics pointed outwards,
+means that the interior of the duct is ‘fully controlled’ in the hydraulic sense, isolating the
+
+ 1, L --22,L ---SW
+.....TW
+R
+0
+1.0
+(q)
+～ 0.5
+0.0
+4
+2
+0
+-30
+0
+30
+x0
+0.3
+0.6
+(b) SW
+(c) TW
+(d) I
+(a) L
+入
+250
+200
+150
+100
+-30
+0
+30-30
+0
+30-30
+0
+30-30
+0
+30
+x
+x
+x
+xSID: two-layer hydraulics and instabilities
+17
+flow from hydrostatic disturbances within the reservoirs to either side. This is the first direct
+evidence that SID flows in the L regime are hydraulically controlled.
+In contrast, in the unstable SW, TW, and I cases, the roots are complex conjugates
+throughout most of the duct (figure 11(a,b)), a consequence of instability (𝐹2
+Δ > 1), causing
+supercriticality (𝐺2 > 1, figure 11(c)). These unstable waves always move at the local
+convective velocity of the flow 𝜆𝑅 = ¯𝜆(𝑥, 𝑡), which we recall from (4.11) is non-zero if the
+interface is not at mid-depth 𝜂(𝑥, 𝑡) ≠ 0.
+In the SW and TW cases, the unstable waves are initially carried rightward ( ¯𝜆 > 0)
+throughout most of the left-hand side of the duct, and leftward ( ¯𝜆 < 0) throughout most of
+the right-hand side of the duct, as seen in figure 12(b,c). Thus, all unstable waves are carried
+towards the centre (𝑥 = 0) where their characteristics converge, creating the hydraulic jump
+observed in figure 5. The largest values of 𝜆𝐼 = |𝛿𝜆| are found in the region where the 𝜆𝑅 = ¯𝜆
+convective components converge (see figure 11(b) and green-yellow shades in figure 12(b,c)).
+Their growth rate is fast (𝜆𝐼 ≈ 0.2−0.5) and slightly higher in TW than in SW. Such a jump,
+bounded by supercritical regions on either side, is distinguished from the standard hydraulic
+jumps which make the flow transition from a supercritical to a subcritical state. It can be
+viewed as the limit of the length of the subcritical region tending to zero. The jumps in SW
+and TW can be called ‘undular jumps’ because of their moderate ‘upstream’ Froude numbers
+𝐹2
+𝑖 ≈ 𝐺2/2 of each layer (between 1 and 2) and the small energy they dissipate, compared to
+direct, breaking hydraulic jumps. Some jumps in I are stronger, as evidenced by their locally
+higher 𝐺2 values and their visibly higher dissipation.
+This pattern of unstable characteristics converging toward the centre to form a jump can be
+summarised by sign 𝜆𝑅 = −sign 𝑥. This can be understood first by sign 𝜆𝑅 = sign ¯𝜆 = −sign 𝜂
+from (4.11), i.e. the waves are carried at the local convective velocity, and second, by
+sign 𝜂 = sign 𝑥, i.e. the interface does not slope down as in L but is instead lowered on the
+left-hand side of the duct, and lifted on the right-hand side (central jump).
+The main difference between SW (stationary wave regime) and TW (travelling wave
+regime) lies in the behaviour of 𝜆𝑅 near the jump around 𝑥 = 0 (figure 11(a)). While 𝜆𝑅(𝑥)
+goes smoothly through zero in SW, it oscillates more in TW, suggesting that the location of
+the jump is prone to oscillations. This is confirmed by comparing the 𝑥 − 𝑡 trajectory of the
+locus of the maximum 𝐹2
+Δ in figure 10(b,c) or the maximum 𝜆𝐼 in figure 12(b,c) as a proxy
+to the location of the jumps.
+Finally, we turn to the I flow, which exhibits more vigorous interfacial turbulence and
+wave instability (especially between 𝑡 = 150 − 250) and greater variability in 𝑥 and 𝑡 than
+SW and TW. The characteristics in figure 12(d) converge quickly to form a large number of
+local jumps around 𝑡 ≈ 100, which then organise into three main clusters: a central stationary
+cluster flanked by a left and a right cluster which themselves converge to form discrete jumps
+around 𝑡 ≈ 160 while being carried to the centre, eventually converging into a single jump
+𝑡 ≳ 200. The convergence of characteristics tends to coincide with the maximum instability
+(𝜆𝐼 ≈ 0.5). During the more stable (transitional) period at 𝑡 = 110 the multiple sign reversals
+of 𝜆𝑅(𝑥) (figure 11(a)) hinder a straightforward interpretation of wave propagation along 𝑥.
+This pattern by which unstable waves are carried in SW, TW and I flows differs so greatly
+from the classical picture of hydraulic control in L flow that it prompts the question: since
+information travels toward the duct centre, does it travel from the reservoirs into the duct,
+and is the flow still hydraulically controlled? Figure 11 shows that close to the duct exits
+(|𝑥| ≈ 28), the composite Froude number 𝐺2 (panel c) of all the cases becomes 1. Meanwhile,
+immediately outside the duct 30 < |𝑥| < 32 the waves are stable (𝜆𝐼 = 0, panel b), and both
+the curves of the real characteristics (panel a) and of the composite Froude number 𝐺2 (panel
+c) closely match those in the L flow. In other words, the SW and TW flows also have a
+
+18
+A. Atoufi, L. Zhu, and others
+control point (𝐺2 = 1), flanked by narrow regions of subcriticality (𝐺2 < 1). We conclude
+that the flow within the duct remains isolated from the reservoirs , and hence that it is also
+hydraulically controlled. This represents the first direct evidence that SID flows in the W and
+I regimes, in addition to having a (supercritical-to-supercritical) jump in the centre of the
+duct, are also hydraulically controlled at the ends of the duct.
+5.3. Maximal exchange and critical flow rate
+In the previous section, we showed that all four flows cases (L, SW, TW and I) were
+hydraulically controlled in the sense that control points at the ends of the duct isolated the
+flow within the duct from processes in the reservoirs and prevented the flow inside the duct
+from reaching velocities exceeding a maximal volume flux 𝑄. In this section, we seek to
+explain the differences in the value of this critical volume flux (and by extension mass flux)
+observed between the L, W, and I regimes in figure 6.
+In the stable L flow, we find a time-averaged 𝑄 ≈ 0.31 well below the absolute upper bound
+of 0.5 for instability given by (4.13). This is understood by the frictional hydraulic theory
+of Gu & Lawrence (2005), subsequently adapted to SID in Lefauve & Linden (2020) (their
+§ 5.2). In short, the relatively low values of the Reynolds number in these low-tilt L flows
+mean that viscous friction at the duct walls and at the interface must be parameterised in
+the shallow water equations. This parameterisation allows a correct prediction of the sloping
+interface 𝜂(𝑥) (a consequence of viscous friction, i.e. loss of momentum along the flow of
+each layer), which in turn allows prediction of 𝑄 by imposing the criticality condition at
+the ends of the duct 𝐺2(𝑥 = ±𝐴) = 1. Simply speaking, the lower 𝑅𝑒 and the longer the
+duct aspect ratio 𝐴, the more friction occurs along the duct, the more offset the interface
+|𝜂| becomes at the ends of the duct, and the lower the volume flux 𝑄 becomes to satisfy
+𝐺2(|𝜂|, 𝑄) = 1 (since 𝐺2 is an increasing function of both |𝜂| and 𝑄).
+In the unstable SW, TW and I flows, despite the existence of viscous friction, we find a
+remarkably consistent time-averaged 𝑄 ≈ 0.51 − 0.53, slightly above the critical 𝑄𝑐 = 0.5
+upper bound for frictionless two-layer hydraulics and a flat interface 𝜂 = 0 (figure 7). These
+values require a different explanation. Although the Reynolds number of SW, TW, and I are
+identical to L, their larger tilt angle 𝜃 pushes these flows beyond the instability threshold 𝐹2
+Δ =
+1, corresponding to the transition between ‘lazy’ and ‘forced’ flows (Lefauve et al. 2019).
+However, 𝑄 does not continue to increase with 𝜃. Rather, beyond the instability threshold,
+(4.13) suggests that, in the centre of the duct (where 𝜂 ≈ 0), 𝑄 = 0.5
+√︁
+(Δ𝜌)/2 cos 𝜃𝐹Δ, i.e.
+a linear increase with the stability Froude number. We deduce that, since 𝑄 never greatly
+exceeds 0.5 (the value reached the instability threshold), the subsequent increase in 𝐹Δ > 1
+(indirectly caused by the forcing ∝ 𝑠𝑖𝑛𝜃 in the DNS) must be compensated by a decrease
+in Δ𝜌/2 ∝ 1/𝐹2
+Δ, i.e. by increased mixing. The data show that the average Δ𝜌/2 indeed
+decreases from 0.79 in SW, to 0.76 in TW, to 0.68 in I. This mixing in turn explains why 𝑄𝑚
+(roughly ≈ (Δ𝜌/2)𝑄) stays robustly below 0.5 in SID, and indeed decreases from the W to
+the I regime (figure 6).
+6. Applicability of unstable hydraulics
+In this section, we study the applicability of the previous results to problems not usually
+considered in two-layer hydraulics. In §6.1 we study the transition between long waves (the
+propagation of which is identical to the local characteristics of the SWE) and shorter Kelvin-
+Helmholtz waves (only predicted by the TGE). In §6.2 we study the waves diagnosed from
+DNS run at different values of the Prandtl number to investigate their indirect dependence on
+scalar diffusion and the thickness of the density interface. In §6.3 we study how the growth
+
+SID: two-layer hydraulics and instabilities
+19
+Figure 13: Dispersion relation of all (long and short) inviscid two-layer waves: growth rate
+(colours) and phase speed (contours) solutions of the TGE varying with wavenumber 𝑘
+and volume flux 𝑄. (a) Symmetric interface 𝜂 = 0. (b) Asymmetric interface 𝜂 = −0.5. (c)
+Symmetric interface but without solid top and bottom boundaries at 𝑧 = ±1, in which case
+the long waves of SWE disappear.
+Figure 14: Dispersion relation (growth rate only) predicted by the TGE two-layer model
+applied to the DNS (a) L and (b) TW flows. The dashed line at 𝑘 = 1 represents the
+boundary between long and short waves. Short waves are predicted to be most unstable by
+this inviscid model but appear relatively stable in reality.
+of long waves is impacted by smooth, diffuse (i.e. not strictly two-layer) density and velocity
+profiles, which are expected in all real-world flows (having a finite Re and Pr).
+6.1. Long vs short waves
+Figure 13 shows the dispersion relation from the TGE (4.20) with the phase speed ℜ(𝑐1) = 𝑐𝑅
+1
+(blue to red contours) and growth rate ℑ(𝑐1) = 𝑐𝐼
+1 (colour map with deep blue being stable)
+as functions of the wavenumber 𝑘 (vertical axis) and the volume flux 𝑄 (horizontal axis). We
+compare a symmetric interface (𝜂 = 0, panel a), an asymmetric interface (𝜂 = −0.5, panel
+b) and a case with a symmetric interface but without solid top and bottom boundaries (panel
+c), whose dispersion relation (B 13) is derived analytically in §B.3.
+We recall that for 𝑘 ≪ 1 the phase speed 𝑐𝑅 and growth rate 𝑐𝐼 of TGE become identical
+to 𝜆𝑅 and 𝜆𝐼 of SWE, respectively. In this case the 𝑘 = 10−2 data of figure 13(a,b) become
+indistinguishable from those plotted in figure 7(b,c), respectively.
+From figure 13(a-b) we recover the results from previous sections. First, long waves are
+non-dispersive (the phase speed contours do not depend on 𝑘). Second, they become unstable
+(lighter shades of blue and green) above a critical volume flux 𝑄 > 𝑄𝑐, equal to 0.5 for
+
+0.0
+0.61.2
+一1
+0
+102
+K
+¥100
+(b)
+()
+(a)
+10-2.
+0.0
+0.5
+1.00.0
+0.5
+1.00.0
+0.5
+1.0
+Q
+Q
+Q0.0 0.3 0.6
+102.
+(b)
+100
+10-2.
+-30
+0
+30-30
+0
+30
+x
+x20
+A. Atoufi, L. Zhu, and others
+a symmetric interface (panel a) and lower than 0.5 for an asymmetric interface (panel b).
+Third, for a symmetric interface, all unstable waves (long and short) are stationary (absence
+of contours), but all stable waves are travelling (presence of contours). For an asymmetric
+interface, even unstable waves are travelling in the reference frame of the duct, as we explained
+in §5.2. Fourth, the transition between short and long waves is smooth, i.e. there is a continuity
+between the long shallow water waves controlling the hydraulics of two-layer flows and the
+short KH waves.
+Panels (a,b) of figure 13 also give new results. First, short waves (𝑘 � 1) become unstable
+at smaller values of 𝑄 compared to the long wave threshold 𝑄𝑐. This transition to short waves
+becomes noticeable from 𝑘 ≳ 10−0.5 ≈ 0.3 and is clear for 𝑘 > 1. The shortest waves shown
+here (𝑘 = 102) are predicted to become unstable above a very small volume flux 𝑄 ≳ 0.1.
+However, we note that this threshold would be closer to the long-wave 𝑄𝑐 if we included
+viscosity in the TGE, as viscosity would significantly damp the growth of short waves. These
+panels show that for a given value of 𝑄, the growth rate increases monotonically with 𝑘 (i.e.
+the shortest waves are the most unstable). As is often the case, this ‘ultraviolet catastrophe’
+would be regularised by viscosity, with the possible existence of a maximum growth rate at
+an intermediate 𝑘 for intermediate values of Re.
+In figure 13(c), the absence of solid walls does not affect short waves (the colours and
+contours at 𝑘 ≳ 3 are identical to panel a), because they do not ‘feel’ the presence of the
+walls. However, the absence of solid walls precludes the existence of long waves 𝑘 ≲ 0.3,
+because this setup (despite being bounded at 𝑧 = ±1) approximates layers of infinite depth,
+compared to which all waves are ‘short’. In other words, this analysis explicitly shows that the
+presence of solid walls in SID are crucial to explain the leading order dynamics in the DNS
+by allowing long-wave instability. Adding walls (panel a) creates the long waves on which
+hydraulic effects rely, an long waves transition smoothly into short waves as 𝑘 increases.
+Figure 14 shows the linear growth rate 𝑐𝐼 obtained by substituting into the TGE dispersion
+relation (4.20) (function of 𝑘) the two-layer properties (as functions of 𝑥) extracted from
+the DNS. Diagnostics are shown for the L flow (panel a) and the TW flow (panel b) at time
+𝑡 = 110. The TW flow is, unlike the L flow, unstable to long waves 𝑘 ≪ 1, with the maximal
+growth rate found near the centre of the duct, as previously seen in figure 11(b) as we know
+from (4.21) that 𝑐𝐼 (𝑘 ≪ 1) −→ 𝜆𝐼. However, we also find that the L flow appears mildly
+unstable to short waves (especially very short waves 𝑘 ≳ 10), and the TW flow appears even
+more strongly unstable to them. However, we know that in the DNS the L flow is visibly stable
+and does not have observable interfacial waves, while the TW flow is primarily unstable to
+long waves, whereas short waves play a more minor role. This suggests that, at least for the
+present values of Re = 650 and Pr = 7, the growth of short waves is sufficiently damped by
+viscosity, mass diffusion and/or other effects not taken into account in this inviscid two-layer
+model.
+6.2. Low vs high Prandtl numbers
+Although Pr does not appear explicitly in the SWE, the DNS dynamics from which the two-
+layer properties are extracted certainly depend on Pr. In applications, three typical values
+are of particular interest: Pr ≈ 1 (representative of temperature stratification in air), Pr ≈ 7
+(representative of temperature stratification in water, as studied in this paper) and Pr ≈ 700
+(representative of salt stratification in water).
+To study the impacts of diffusion, we carried out two additional DNS with parameters
+identical to the TW flow (with Pr = 7) at Pr = 1 and Pr = 28 (the latter requiring a much
+higher spatial resolution, hindering the study of higher Pr). In figure 15 we compare the
+characteristics curves 𝑋(𝑡) and growth rates 𝜆𝐼 at these three different values of Pr using the
+same visualisation as figure 12 (where the Pr = 7 data was already shown as TW). We find
+
+SID: two-layer hydraulics and instabilities
+21
+Figure 15: Characteristic curves 𝑋(𝑡) and growth rate 𝜆𝐼 (colours) from DNS data at
+Re = 650 𝜃 = 6◦ and (a) Pr = 1, (b) Pr = 7 (TW data), (c) Pr = 28. Note that for easier
+comparison, (b) is identical to figure 12(c), though we only show data for 𝑡 ∈ [80, 220].
+Also note the reduced colour map maximum 𝜆𝐼 (0.5 in this figure, compared to 0.6 in
+figure 12)
+Figure 16: Characteristics (a) 𝜆𝑅, (b) 𝜆𝐼 > 0, and (c) composite Froude number 𝐺2 of TW
+flows at 𝑡 = 110. We compare three different Prandtl numbers Pr = 1, 7, 28 as in figure 15.
+that curves from the Pr = 1 flow initially converge into a central jump and a small number
+of peripheral jumps, which eventually merge with the central jump. This pattern resembles
+that of the more stable SW flow from figure 12(b), except that it has a higher growth rate
+than TW. The curves from the Pr = 28 flow converge into a larger number of intermediate,
+travelling jumps before merging into a single jump. This pattern resembles that of the more
+unstable I flow from figure 12(d), except that it has a smaller growth rate. than TW.
+In figure 16 we compare the characteristics 𝜆𝑅(𝑥) (panel a) and 𝜆𝐼 (𝑥) (panel b) as well
+as the composite Froude number 𝐺2(𝑥) at 𝑡 = 110 (the Pr = 7 data was already shown in
+figure 11). All curves (solid red, dashed blue, and dotted green) have essentially the same
+
+0
+0.5
+(a) Pr = 1
+(b) Pr = 7 (TW)
+(c) Pr = 28
+200
+← 150
+100
+0
+30 -30
+0
+30
+-30
+0
+30 -30
+x
+x
+rPr= 1 --·Pr=7 ---Pr=28
+R
+0
+1.0
+(q)
+~ 0.5
+0.0
+4
+2
+0
+-30
+0
+30
+x22
+A. Atoufi, L. Zhu, and others
+qualitative features described earlier in figure 11. However, as noted in figure 15, the growth
+rate appears to decrease slightly with Pr.
+The synoptic features of the flow governed by long waves, therefore, appear relatively
+unaffected by Pr. This can be rationalised by the fact that Pr will primarily influence the
+thickness of the density interface separating the two layers, rather than its location 𝜂 (the
+locus of 𝜌 = 0) or the speed of the flow (𝑄), which are the two key model variables in the
+SWE.
+We expect short waves to be more strongly influenced by a decreasing thickness of the
+density interface with increasing Pr, and vice versa. However, the short waves observed in
+our DNS at Pr = 28 and in experiments at 𝑃𝑟 = 700 are Holmboe waves, not the KH waves
+supported by our two-layer model. Unlike the KH instability caused by a single vortex sheet,
+the Holmboe instability is caused by the resonance of vorticity waves (e.g. on the edges of a
+diffuse velocity interface) with a non-collocated gravity wave (on a sharper density interface)
+(Carpenter et al. 2011). Lefauve et al. (2018) performed a linear stability analysis on the
+experimentally measured mean flow (at Re = 440, Pr = 700), including viscosity and scalar
+diffusion. They found that intermediate 1 ≲ 𝑘 ≲ 2 Holmboe waves were most unstable (see
+their figure 6a). However, tackling Holmboe waves – and their presumed coexistence with
+the long waves governing hydraulic processes at high Pr SID – would require a three-layer
+model (for velocity) mixed with a two-layer model (for density).
+6.3. Sharp vs smooth two-layer flow profiles
+Finally, we study the influence of smooth density and velocity profiles U(𝑧) and R(𝑧) on
+the growth rate of long waves. To do so, we solve the eigenvalue problem from the Taylor-
+Goldstein equation (4.18) before the two-layer base flow ansatz (4.19). Numerical solutions
+for the growth rate 𝑐𝐼 are shown in figure 17 as functions of 𝑄. In panel a, we show the
+results for hyperbolic-tangent U(𝑧)/𝑄 = R(𝑧) = tanh 𝑧/𝛿 where the interface thickness is
+progressively decreased from 1/80 (almost exactly two layers, solid lines) to 1/8 (dashed
+lines) to 1/4 (thicker interface, dotted lines). We compare these ‘smooth tanh’ growth rates (in
+black) to the equivalent ‘sharp two-layer’ growth rates (in red) obtained from the analytical
+dispersion relation (4.20) by layer-averaging the tanh profiles (in which case 𝑐𝐼 = 𝜆𝐼). In
+panel b, we keep the same density profiles but use U(𝑧)/𝑄 = − sin 𝜋𝑧, which is a good
+approximation of the mean velocity at these relatively low values of Re = 𝑂(102 − 103).
+Both panels a and b show that the 𝑄𝑐 = 0.5 threshold for long-wave instability is virtually
+unchanged by smooth profiles, with only a slight increase of a few percent for the thickest
+interface 𝛿 = 1/4. This result supports the relevance of two-layer hydraulics even in ‘real-
+world’ flows which depart significantly from the two-layer model.
+However, the ‘smooth tanh’ growth rates are always lower than the corresponding ‘sharp
+two-layer’ growth rates. The thicker the interface, the lower the ‘smooth tanh’ growth rate.
+Comparing the vertical scale in panels a and b, we conclude that the sinusoidal velocity profile
+is more stable (by approximately a factor of 10) than the tanh profile. The combination of
+a sinusoidal velocity and a diffuse velocity interface (dotted blue line in panel b) yields the
+slowest growth as 𝑄 increases. As such profiles are a better approximation of the mean flow
+of the DNS at low Pr (e.g. Pr = 1) than sharp two-layer profiles, these results warn us not to
+interpret the large growth rates 𝜆𝐼 = 𝑂(0.1) found in this paper too literally.
+In other words, although the qualitative predictions of two-layer hydraulics are robust (in
+particular the long-wave instability threshold 𝑄𝑐) when the underlying data is not exactly
+two-layer, the quantitative growth rates predictions are over-estimated when the interface is
+diffuse, as in low-Pr flows.
+
+SID: two-layer hydraulics and instabilities
+23
+Figure 17: Growth rate 𝑐𝐼 of long waves on smooth velocity and density profiles, obtained
+by a numerical solution of the TGE (4.18) as the volume flux 𝑄 is increased (black
+curves). (a) Hyperbolic-tangent profiles for velocity U(𝑧) and density R(𝑧) profiles of
+increasing interface thickness. (b) Sinusoidal profiles for velocity U(𝑧) (and same R(𝑧) as
+in (a)). The growth is always slower than it would be using the sharp two-layer analytical
+solution (4.20) (red curves).
+7. Conclusions
+In this paper, we employed a two-layer averaging procedure to extract a reduced-order
+representation of four direct numerical simulations (DNS) datasets in the stratified inclined
+duct (SID) at Re = 650 and Pr = 7 (with two supplementary datasets at Pr = 1 and Pr = 28.
+This two-layer representation revealed in §3 that the flow is stable in the laminar regime
+(L, tilt angle 𝜃 = 2◦), but develops an internal hydraulic jump (discontinuity in the layer
+properties) in the centre of the duct in the stationary wave regime (SW, 𝜃 = 5◦). This jump
+moves around in the travelling wave (TW, 𝜃 = 6◦) regime, and causes further disorganised
+wave breaking in the intermittently turbulent (I, 𝜃 = 8◦) regime.
+7.1. Modelling results
+To understand these findings, in §4 we adapted to SID DNS the well-known inviscid
+Boussinesq shallow water equations (SWE) governing the nonlinear evolution of long waves
+(𝑘 ≪ 1) at a sharp density interface. The SWE predict that information propagates along
+a pair of trajectories, 𝜆1,2, that arise from the solution of a generalised eigenvalue problem
+and depend on the local (𝑥) and instantaneous (𝑡) state of the two-layer representation. The
+solutions can be written in the form 𝜆1,2 = ¯𝜆 ± 𝛿𝜆, where the convective velocity ¯𝜆 is
+always real but the phase speed 𝛿𝜆 may be either real or imaginary. When 𝜆1,2 are real
+(𝛿𝜆 ∈ R), they represent the propagation of two (neutrally stable) kinematic waves where
+¯𝜆 can be interpreted as a convective velocity and 𝛿𝜆 as the phase speed of waves relative
+to ¯𝜆. The respective signs of 𝜆1,2 determine the direction of information propagation and
+whether the flow is subcritical (composite Froude number 𝐺2 < 1; product 𝜆1𝜆2 < 0)
+with information propagating in both directions, or supercritical (𝐺2 > 1; 𝜆1𝜆2 > 0) with
+information propagating only in the direction given by the sign of ¯𝜆. When 𝜆1,2 are complex
+𝜆1,2 = 𝜆𝑅 ± 𝑖𝜆𝐼 = ¯𝜆 ± 𝑖|𝛿𝜆|, the real part still represents a convective velocity that carries
+information while the positive imaginary part indicates that the flow is unstable. Although in
+
+)0.14
+a
+R= -tanh(80 z)
+U
+ R = -tanh(80 z)
+-QU-
+-- R = -tanh(8 z)
+= R= -tanh(8 z)
+0.12
+... R = - tanh(4 z)
+QU
+0.8
+= R= -tanh(4 z)
+Q
+0.1
+0.6
+0.08
+C
+0.06
+0.4
+0.04
+0.2
+0.02
+0
+0
+0
+0.5
+0
+0.5
+Q
+Q24
+A. Atoufi, L. Zhu, and others
+this unstable case the SWE are no longer hyperbolic, the flow may be viewed as supercritical
+(𝐺2 > 1) in the sense that information is propagated only in the direction given by ¯𝜆.
+To interpret the unstable SWE, we compared the characteristics with the dispersion
+relation from the inviscid Taylor-Goldstein equation (TGE) governing the linear normal-
+mode stability of a two-layer base flow. We showed that the global nonlinear characteristics
+𝜆(𝑥, 𝑡) defined on the non-parallel base flow and sloping interface of the SWE could be
+interpreted locally as the phase speed and growth rate of linear waves in the long-wave limit
+(i.e. 𝑘 ≪ 1), propagating on a base flow that is locally assumed parallel. Importantly, this
+interpretation is only valid for waves that are much shorter than the duct length 2𝐴 (i.e.
+𝑘 ≫ 𝐴−1). It provides a local, linear stability interpretation for unstable two-layer wave
+characteristics satisfying 𝐴−1 ≪ 𝑘 ≪ 1, which is a relevant range in long ducts (𝐴−1 ≪ 1).
+The dispersion relation for the dispersive TGE waves 𝑐(𝑘) tend to the non-dispersive SWE
+characteristics 𝜆 as 𝑘 ≪ 1, but they also allow us to explore the smooth transition to shorter
+(𝑘 � 1), non-hydrostatic Kelvin-Helmholtz (KH) waves.
+7.2. Physical results
+Applying this two-layer hydraulics and instability framework to the two-layer-averaged DNS
+datasets yielded the main physical results of this paper in §5. We provided the first direct
+evidence that SID flows are, in all regimes (L, SW, TW and I), hydraulically controlled at the
+ends of the duct and thus in a state of maximal exchange. At these control points, the flow is
+locally supercritical; thus information from the reservoirs cannot enter the duct and influence
+the flow within it. In the SW, TW and I regime, the flow in the duct is always unstable to
+long waves (𝐹2
+Δ > 1) and thus supercritical (𝐺2 > 1), explaining the existence of an undular
+jump within the duct, as a consequence of characteristic trajectories converging to a single
+point. In the I regime, multiple local jumps gradually merge into clusters and eventually into
+a single, stronger jump, resulting in greater instability.
+The emergence of unstable, supercritical flow in SID beyond a certain tilt angle is
+rationalised by the fact that gravitational forcing continuously provides a surplus of kinetic
+energy which must be dissipated. From a hydraulics perspective, the required dissipation in a
+supercritical flow (i.e. having a surplus of kinetic energy compared to potential energy) must
+be associated with an decrease in kinetic energy and an increase in potential energy, hence a
+thickening of both layers downstream of the jump. The physical insight of energy surplus and
+dissipation dates back to Meyer & Linden (2014). It was later formalised by Lefauve et al.
+(2019) and Lefauve & Linden (2020), who showed using frictional two-layer hydraulics with
+a tilt 𝜃 that the mid-duct interfacial slope obeyed 𝜂′(0) ∝ 𝜃 − 𝐹, where 𝐹 represents viscous
+friction along the duct. The transition from subcritical to supercritical flow that we identified
+corresponds to the transition from ‘lazy’ to ‘forced’ flows, which they identified based on
+the relative importance of the tilt 𝜃 and the duct geometric angle 𝛼 = tan−1 𝐴−1 ≈ 2◦ in
+this paper. ‘Lazy’ flows are characterised by 𝜃 < 𝛼, and an interface gently sloping down.
+‘Forced’ flows are characterised by 𝜃 > 𝛼, and a relatively flat interface all along the duct.
+In forced flows, the tendency of 𝜃 to tilt up the density interface exceeds the tendency of
+frictional losses 𝐹 to tilt it down, thus 𝜂′(0) > 0, which causes central jumps.
+Next, we rationalised the values of the volume flux in all regimes. The value 𝑄 ≈ 0.3 in
+the L regime (lazy flow) is explained by the offset of the interface at the ends of the duct
+where control (𝐺2 = 1) takes place, recalling that the sloping interface is caused by viscous
+friction along the duct. The robust values 𝑄 ≈ 0.5 in the unstable SW, TW, and I regimes
+(forced flows) are all surprisingly close to the instability threshold 𝑄𝑐 = 0.5 for a symmetric
+interface. Increasing instability from SW to TW to I as the tilt angle 𝜃 is increased (which
+theory predicts should increase 𝑄 above 0.5) appears balanced by increasing mixing in the
+layers (which decreases 𝑄). This explains why the maximal exchange threshold 𝑄 = 0.5
+
+SID: two-layer hydraulics and instabilities
+25
+predicted by inviscid long wave theory remains a remarkably robust feature of SID flows,
+even under turbulence.
+Using the TGE analysis provided further physical insight into the applicability of unstable
+hydraulics in §6. We showed that short inviscid KH waves are always predicted to be more
+linearly unstable than long waves, despite the fact that long waves cause the internal hydraulic
+jumps observed in SW, TW and I and appear to dominate the dynamics of these SID flows.
+We explained this paradox by the neglect of viscosity in TGE, which would damp the shortest
+waves. We also showed that DNS at lower Pr = 1 or higher Pr = 28 showed qualitatively (but
+not quantitatively) similar two-layer long wave hydraulics to Pr = 7. We also highlighted that
+experimental observations at Pr = 700 of the simultaneous existence of unstable long waves
+(causing an internal jump and supercritical flow) with short finite-amplitude Holmboe waves
+could not be explained by the two-layer model, because it does not support Holmboe waves.
+Finally, we showed that the predictions of long wave instability (especially the threshold
+𝑄𝑐 = 0.5) were robust even in diffuse two-layer flows having a thick interface.
+7.3. Outlook
+These key ‘hydraulic’ features of SID flows, explaining the emergence of waves, increasingly
+supercritical jumps, and ultimately turbulence, result from long wave instability which
+(tautologically) relies on the existence of top and bottom solid boundaries confining the
+flow in a long, tilted duct. This ‘long-wave’ pathway to turbulence in SID appears a priori
+to differ from the classical ‘short-wave’ KH pathway in an unbounded stratified shear layer
+(see e.g. Caulfield & Peltier (2000); Mashayek & Peltier (2012)), often used as a paradigm
+for ocean mixing. Further work is needed to clarify the relative importance of long and
+short waves, and within short waves, of Kelvin-Helmholtz (two-layer) waves and Holmboe
+(three-layer) waves, and how they contribute to the transition to turbulence under varying
+𝜃, Re, Pr.
+The current formulation of the two-layer SWE does not account for the mixing layer that
+develops and appears to be important beyond the wave regime. The omission of a mixed
+layer may reduce the accuracy when investigating detailed spatial features of the flow, such
+as the localization of unstable wave regions in the centre of the duct in TW and SW. This
+model is able to predict the formation of shocks and provide clues to the long-length-scale
+dynamics and how they govern the synoptic features of the flow, but further work is needed
+to accurately represent the non-hydrostatic processes of turbulent mixing itself.
+Acknowledgments
+We acknowledge support from the European Research Council (ERC) under the European
+Union’s Horizon 2020 research and innovation Grant No 742480 ‘Stratified Turbulence And
+Mixing Processes’ (STAMP). Parts of the simulations were carried out with resources from
+Compute/Calcul Canada. A. L. is supported by a Leverhulme Trust Early Career Fellowship.
+For the purpose of open access, the authors have applied a Creative Commons Attribution
+(CC BY) licence to any Author Accepted Manuscript version arising from this submission.
+Declaration of interests
+The authors report no conflict of interest.
+Appendix A. Hydrostatic and non-hydrostatic pressure gradients
+The assumptions behind the SWE require flows to be dominantly hydrostatic. To validate
+this assumption, we explicitly decompose the non-dimensional pressure into a hydrostatic
+
+26
+A. Atoufi, L. Zhu, and others
+Figure 18: Cumulative Distribution Function (CDF) of the ratio between non-hydrostatic
+and hydrostatic streamwise pressure gradient.
+component defined by
+𝑝ℎ(𝑥, 𝑧, 𝑡) = −𝑅𝑖 cos 𝜃
+∫
+𝑧
+−1
+⟨𝜌⟩𝑦(𝑥, 𝜉, 𝑡) d𝜉,
+(A 1)
+and the remaining non-hydrostatic (but still spanwise averaged) component
+𝑝𝑛ℎ(𝑥, 𝑧, 𝑡) = ⟨𝑝⟩𝑦 − 𝑝ℎ.
+(A 2)
+Figure 18 shows the cumulative density function of the magnitude ratio between 𝜕𝑝ℎ/𝜕𝑥
+and 𝜕𝑝𝑛ℎ/𝜕𝑥 in all five datasets L, SW, TW, I and T. We find that the hydrostatic pressure
+gradient dominates over the non-hydrostatic gradient (i.e. |𝜕𝑝𝑛ℎ/𝜕𝑥|/|𝜕𝑝ℎ/𝜕𝑥| < 1 in
+⩾ 80% of the data in the L, SW, TW, and I cases, and ⩾ 60% of the data even in the
+T case.
+Non-hydrostatic effects, therefore, play a secondary role in all but the T case, and the
+two-layer SWE are expected to model adequately the primary two-layer dynamics of SID
+flows forced by relatively small tilt angles 𝜃. However, as the flow becomes turbulent (T
+case), non-hydrostatic effects become important and the applicability of the shallow water
+model breaks down. Turbulence also creates a third layer of intermediate density (see Zhu
+et al. (2022), figure 5(e,j)) and the two-layer model also breaks down. For these reasons, we
+exclude this T case from the analyses of this paper and focus on the L, SW, TW, and I cases.
+Appendix B. Derivation of the Taylor-Goldstein equation and solution
+In this section, we provide additional details regarding the derivation of the inviscid
+Boussinesq Taylor-Goldstein equation (TGE) (4.18) for a two-layer SID flow and the
+dispersion relation (4.20).
+B.1. Governing equation
+The linearised inviscid equations for two-dimensional wave-like perturbations [ ˜𝜓, ˜𝜌, ˜𝑝] =
+[ ˆ𝜓, ˆ𝜌, ˆ𝑝](𝑧) exp𝑖𝑘(𝑥 − 𝑐𝑡) around a parallel base flow [U, R](𝑧) are
+(U − 𝑐) ˆ𝜓′ − ˆ𝜓U′ = − ˆ𝑝 − 𝑖
+𝑘 Ri sin 𝜃 ˆ𝜌,
+(B 1)
+𝑘2 (U − 𝑐) ˆ𝜓 = − ˆ𝑝′ − Ri cos 𝜃 ˆ𝜌,
+(B 2)
+
+1.0
+0.8
+0.6
+CDF
+0.4
+ TW
+0.2
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0pnh/0x//l0ph/axlSID: two-layer hydraulics and instabilities
+27
+(U − 𝑐) ˆ𝜌 − ˆ𝜓R′ = 0,
+(B 3)
+By taking the 𝑧 derivative of (B 1) and using (B 2) and (B 3) we derive the general TGE as
+(U − 𝑐)
+� 𝑑2
+𝑑𝑧2 − 𝑘2
+�
+ˆ𝜓 − U′′ ˆ𝜓 − Ri cos 𝜃 R′
+U − 𝑐
+ˆ𝜓 = 𝐹,
+(B 4)
+with forcing 𝐹 = − 𝑖
+𝑘 Ri sin 𝜃
+�
+R′
+U − 𝑐
+ˆ𝜓′ +
+R′′
+U − 𝑐
+ˆ𝜓 −
+U′R′
+(U − 𝑐)2 ˆ𝜓
+�
+.
+The streamwise component of the gravitational force Ri sin 𝜃 appears multiplied by 𝑖 such
+that even if 𝑐 is real (the waves are stable) increasing the tilt angle will eventually lead to
+instability. We neglect this effect here.
+For small tilt angles, we assume for simplicity 𝐹 = 0. These unforced TG equations
+under small tilt angles will be the focus of the following stability analysis. The unforced TG
+equation then (i.e 𝐹 = 0) is given in (4.18).
+Taking the base flow as the two-layer piecewise constant profiles in (4.19) leads to U′ =
+(𝑢1−𝑢2) 𝛿(𝑧) and R′ = (𝜌1−𝜌2) 𝛿(𝑧), where 𝛿 is the Dirac delta function. Since U′′ = R′ = 0
+everywhere except at the interface, the TGE becomes trivial
+ˆ𝜓′′ − 𝑘2 ˆ𝜓 = 0.
+(B 5)
+B.2. Solution with solid top and bottom walls
+In this bounded duct configuration, we take a solution of the form
+ˆ𝜓 =
+�
+𝐶1 sinh 𝑘(ℎ1 − 𝑧)
+0 < 𝑧 ⩽ ℎ1,
+𝐶2 sinh 𝑘(ℎ2 + 𝑧)
+−ℎ2 ⩽ 𝑧 < 0,
+(B 6)
+which satisfies the no-penetration condition at 𝑧 = −ℎ2 and 𝑧 = ℎ1 ( ˆ𝑤 = 𝑖𝑘 ˆ𝜓 = 0) modelling
+the presence of solid walls.
+Following Drazin & Reid (2004), the matching conditions are derived by integrating (4.18)
+over the neighbouring region of the interface, and by using the integral property of the Dirac
+delta function, leading to
+⟦
+ˆ𝜓
+U − 𝜆⟧0 = 0,
+(B 7)
+⟦U ˆ𝜓′ − 𝑐 ˆ𝜓′⟧0 + Ri Δ𝜌 cos 𝜃
+�
+ˆ𝜓
+U − 𝑐
+�
+𝑧=0
+= 0,
+(B 8)
+where we recall that Δ𝜌 ≡ 𝜌2 − 𝜌1. The first condition (B 7) guarantees continuity of the
+streamfunction across the interface (and thus the wall-normal velocity). Together, the first
+and second term in the second condition (B 8) guarantees continuity of the pressure modes
+ˆ𝑝 based on (B 1). Note that ⟦U′ ˆ𝜓⟧0 vanishes as U′ = 0 on either side of the interface and
+U′ → ∞ at the interface. The last term in (B 8) guarantees continuity of the density modes
+ˆ𝜌 based on (B 3). Using these two conditions we can solve for 𝐶1 and 𝐶2 leading to the
+following algebraic system of equations:
+𝐶1 sinh 𝑘ℎ1(𝑢2 − 𝑐) − 𝐶2 sinh 𝑘ℎ2(𝑢1 − 𝑐) = 0,
+(B 9)
+�
+− (𝑢1 − 𝑐) 𝑘 cosh 𝑘ℎ1 + Ri Δ𝜌
+2
+cos 𝜃 sinh 𝑘ℎ1
+𝑢1 − 𝑐
+�
+𝐶1 +
+(B 10)
+�
+− (𝑢2 − 𝑐) 𝑘 cosh 𝑘ℎ2 + Ri Δ𝜌
+2
+cos 𝜃 sinh 𝑘ℎ2
+𝑢2 − 𝑐
+�
+𝐶2 = 0.
+
+28
+A. Atoufi, L. Zhu, and others
+To have a non-trivial solution, the determinant of the above 2 × 2 system must be zero,
+resulting in the dispersion relation (4.20).
+B.3. Solution without solid top and bottom walls
+In the unbounded configuration, we instead take a solution of the form
+ˆ𝜓 = 𝐶 exp (−𝑘|𝑧|),
+(B 11)
+satisfying continuous and finite ˆ𝜓 for all 𝑧 (Smyth & Carpenter 2019).
+Substituting (B 11) into (B 8), we obtain
+− (𝑢1 − 𝑐)2𝑘 − (𝑢2 − 𝑐)2𝑘 − Ri Δ𝜌 cos 𝜃 = 0,
+(B 12)
+and thus the dispersion relation for KH waves plotted in figure 13(c)
+𝑐 = 𝑢1 + 𝑢2
+2
+±
+√︄
+Ri Δ𝜌
+2 cos 𝜃
+𝑘
+− (𝑢1 − 𝑢2)2
+4
+.
+(B 13)
+The KH instability (𝑐𝐼 ≠ 0) is thus found for
+𝑘 > 2 Ri Δ𝜌 cos 𝜃
+(𝑢1 − 𝑢2)2 .
+(B 14)
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+Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
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+215, 457–480.
+Lawrence, G. A. 1993 The hydraulics of steady two-layer flow over a fixed obstacle. Journal of Fluid
+Mechanics 254, 605–633.
+
+SID: two-layer hydraulics and instabilities
+29
+Lawrence, G. A. & Armi, L. 2022 Stationary internal hydraulic jumps. Journal of Fluid Mechanics 936,
+A25.
+Lefauve, A. & Linden, P. F. 2020 Buoyancy-driven exchange flows in inclined ducts. J. Fluid Mech. 893.
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+and origin of confined holmboe waves. J. Fluid Mech. 848, 508–544.
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+Mechanics 708, 5–44.
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+Fluid Mechanics .
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf,len=1131
+page_content='Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  1  Banner appropriate to article type will appear here in typeset article  Stratified inclined duct: two-layer hydraulics and  instabilities  Amir Atoufi1, Lu Zhu1†, Adrien Lefauve1, John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Taylor1, Rich R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Kerswell1,  Stuart B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Dalziel1, Gregory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lawrence2, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Linden1  1Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,  Wilberforce Road, Cambridge CB3 0WA, UK  2Department of Civil Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada  (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' accepted xx)  The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct  as a model for continuously-forced density-stratified flows such as those found in estuaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Experiments have shown that the emergence of interfacial waves and their transition to  turbulence as the tilt angle is increased appears linked to a threshold in the exchange flow  rate given by inviscid two-layer hydraulics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We uncover these hydraulic mechanisms with (i)  recent direct numerical simulations (DNS) providing full flow data in the key flow regimes  (Zhu & Atoufi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='09773, 2023), (ii) averaging these DNS into two layers,  (iii) an inviscid two-layer shallow water and instability theory to diagnose interfacial wave  behaviour and provide physical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The laminar flow is subcritical and stable throughout  the duct and hydraulically controlled at the ends of the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' As the tilt is increased, the  flow becomes everywhere supercritical and unstable to long waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' An internal undular  jump featuring stationary waves first appears near the centre of the duct, then leads to larger-  amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence  at the largest tilt angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Long waves described by the (nonlinear) shallow water equation are  locally interpreted as linear waves on a two-layer parallel base flow described by the Taylor-  Goldstein equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This link helps us interpret long-wave instability and contrast it to  short-wave (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Kelvin-Helmholtz) instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Our results suggest a transition to turbulence  in SID through long-wave instability relying on vertical confinement by the top and bottom  walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Introduction  Buoyancy-driven exchange flows between water masses of different density are common in  estuaries and straits, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' the straits of the Great Belt, Gibraltar, Bab el Mandab, and the  Bosphorous (Gregg & Özsoy 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These essentially hydrostatic and two-layer flows are  known to exhibit hydraulic jumps (Farmer & Armi 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Thorpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2018b), which are  important discontinuities in the internal flow properties (layer thickness and speed) and often  lead to instability (Lawrence & Armi 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A local jump can influence the large-scale  properties of the flow such as the exchange flow rate (Lawrence 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These non-local  hydraulic flow features are often studied in an idealised ‘shallow-water’ setting consisting of  † Email address for correspondence: lz447@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='uk  Abstract must not spill onto p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='13035v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='flu-dyn]  30 Jan 2023  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  fluid organised in two counter-flowing, frictionless layers of specified thickness and speed  with constant densities (Armi 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lawrence 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Dalziel 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In this paper, we employ two-layer hydraulics as a diagnostic tool to derive insights from  direct numerical simulation (DNS) data of the exchange flows in the stratified inclined duct  (SID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' SID is a canonical stratified shear flow through a long, square cross-section, tilted  duct for which there is now ample data, both experimental (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Partridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2019) and  numerical (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The investigation of turbulence in two-layer shear flows through  tubes dates back to the classic experiments of Reynolds (1883) and Thorpe (1968), who both  used a closed tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The opening of the tube into large reservoirs in SID is more recent and  allows for interfacial waves and turbulence to be sustained for much longer time periods,  and for various flow regimes to be distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The successive transitions to increasingly  turbulent regimes in SID, as the Reynolds number and tilt angle are increased, have been  recognised since Macagno & Rouse (1961) and Kiel (1991), and have been much studied more  recently (Meyer & Linden 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lefauve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lefauve & Linden 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Duran Matute  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These transitions are underpinned by many fundamental features of stratified  flows, including interfacial waves and turbulent intermittency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Our first aim with this new analysis is to uncover internal hydraulic effects in order to  explain some of these leading-order dynamics that DNS and experimental data have revealed  but not yet explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In particular, we will provide the first direct evidence for the existence  of internal hydraulic jumps, where the layer thickness expands in the direction of the flow  of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We also show how the development of jumps and large-amplitude interfacial  waves coincides with a plateau in the exchange flow rate through the duct, after an initial  increase with increasing Reynolds number and tilt in the laminar regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This upper bound  is a remarkably robust feature of SID, which has deep ramifications for the flow energetics  and transition to turbulence (Lefauve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Though the emergence of waves and  turbulence has long been linked to the notion of ‘hydraulic control’ in the experimental SID  literature, this link has not yet been studied in detail, primarily because experiments lack the  full velocity, density and pressure data along the length of the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The recent availability of  DNS data (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2022) overcoming these limitations finally makes a rigorous hydraulics  analysis of SID possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Our second aim is to link two-layer internal hydraulic effects to the growth of instabilities  and to shorter (non-hydrostatic) waves, links which are rarely found explicitly in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The streamwise variation of the base flow in the SID geometry, and generally  in all exchange flows, distinguishes them from idealised parallel stratified shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This  variation, however, is essential to the formation of internal hydraulic jumps, to the ideas of  hydraulic control and maximal exchange, and hence to the nature of SID turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' It is  well known that under some flow conditions, the loss of hyperbolicity of the shallow-water  equations renders long waves unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However much less is known about the consequences  of this instability, and the relative importance of long-wave versus short-wave instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We  will clarify this by explaining how a certain range of unstable nonlinear shallow water waves  associated with an internal jump can be interpreted as linear instabilities on a locally parallel  base flow, and thus that the insights derived from stable hydraulic theory remain valid even  under (moderately) unstable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  To tackle these aims, we introduce our DNS datasets in §2, and the two-layer averaging  in §3, using the averaged datasets to show evidence of jumps and maximal exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In  §4, we adapt the two-layer shallow water theory to SID, summarise important results from  the literature, and connect them to linear stability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In §5, we use this hydraulics and  stability framework to analyse our DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Then, in §6, we explore the transition between long  (hydrostatic) waves and short (non-hydrostatic) waves, the influence of molecular diffusion  (Prandtl number) and of smoother flow profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Finally, we draw our conclusions in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  SID: two-layer hydraulics and instabilities  3  Figure 1: A schematic of the stratified inclined duct (SID) numerical setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The duct is  oriented at angle 𝜃 to the horizontal, which is equivalent to tilting the gravity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Densities are non-dimensionalised by the density differences between the reservoirs and  lengths by the half-duct height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The duct’s volume is  (𝑥, 𝑦, 𝑧) ∈ [−30, 30] × [−1, 1] × [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Direct numerical simulations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Methodology  Our DNS solves the following non-dimensional Navier-Stokes equations under the Boussi-  nesq approximation for the density-stratified flow in the SID setup sketched in figure 1:  ∇ · 𝒖 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1)  𝜕𝒖  𝜕𝑡 + (𝒖 · ∇) 𝒖 = −∇𝑝 + 1  Re∇2𝒖 + ˆg Ri 𝜌 − 𝑭𝑢,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2)  𝜕𝜌  𝜕𝑡 + (𝒖 · ∇) 𝜌 =  1  Re Pr∇2𝜌 − 𝐹𝜌,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3)  where Re = 𝐻∗𝑈∗/𝜈 is the input Reynolds number (𝑈∗ = √𝑔′𝐻∗ is the characteristic  buoyancy velocity, 𝐻∗ is the dimensional half-height of the duct, 𝜈 is the kinematic viscosity,  𝑔′ is the reduced gravity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Ri = 𝑔′𝐻∗/(2𝑈∗)2 is the input bulk Richardson number, giving a  fixed Ri = 1/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' and Pr ≡ 𝜈/𝜅 is the Prandtl number (𝜅 is the thermal diffusivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The unit  gravity vector in the coordinate system (𝑥, 𝑦, 𝑧) aligned with the duct is ˆg ≡ [sin 𝜃, 0, − cos 𝜃].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The DNS is performed with the open-source solver Xcompact3D (Bartholomew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The duct has a square cross-section of non-dimensional height and width 2 and of length  60 (giving a long aspect ratio of 𝐴 = 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' No-slip boundary conditions for 𝒖 and no-flux  boundary conditions for 𝜌 are applied on the duct walls in the spanwise (𝑦 = ±1) and vertical  (𝑧 = ±1) directions with an immersed boundary method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The flow is driven by the density  difference between the dense (𝜌 = 1) and light (𝜌 = −1) fluid in the left- and right-hand  reservoirs, respectively, producing counter-flowing layers in the streamwise 𝑥-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The  experimental reservoirs are modelled by ad hoc forcing terms 𝑭𝑢 and 𝐹𝜌, which damp flow  in the reservoirs and restore the density field to 𝜌 = ±1, allowing us to maintain a quasi-  steady exchange flow for arbitrarily long times at a minimal computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Details  about the numerical setup and validation against experiments and benchmark cases (with  large reservoirs and 𝑭𝑢 = 𝐹𝜌 = 0) can be found in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The DNS is started at 𝑡 = 0 from lock-exchange initial conditions, after which two counter-  flowing gravity currents develop from 𝑥 = 0, advance at absolute speed ≈ 1, and reach either  end of the duct after ≈ 30 advective time unit (ATU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Shortly after, the statistically-steady  exchange flow of interest in SID becomes established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Conservatively, we only retain 𝑡 ⩾ 80  in the following analysis to discard any initial transients, and we run the simulation until  𝑡 = 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  When the setup is tilted by an angle 𝜃 > 0◦, the streamwise component of gravity  contributes Ri 𝜌 sin 𝜃 to the 𝑥-component of the momentum and adds extra kinetic energy  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  Figure 2: A schematic of two-layer shallow water flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Flow in a part of the reservoir is  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Increasing 𝜃 and/or 𝑅𝑒 leads to a variety of flow regimes from laminar to  wavy to intermittently turbulent to fully turbulent, found both in DNS (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2022) and  experiments (Meyer & Linden 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lefauve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lefauve & Linden 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Database  We use the DNS database recently acquired by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2022), which shows good agreement  with experiments when all five non-dimensional parameters Re, Ri, Pr, 𝜃, 𝐴 are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This  database provides the complete set of flow variables all along the duct, which is a requirement  to properly study hydraulic processes in SID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  We consider flows at a fixed Reynolds number Re = 650 and fixed Prandtl number Pr = 7,  corresponding to temperature-stratified water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Four cases are examined, corresponding to  tilt angles 𝜃 = 2◦, 5◦, 6◦, 8◦, denoted by B2, B5, B6, B8, respectively in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Each of these four cases covers specific flow regimes: B2 is laminar (L), B5 has stationary  waves (SW), B6 has travelling waves (TW), and B8 has intermittent turbulence (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the  following, we refer to these datasets as L, SW, TW, and I respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We only briefly touch  upon more turbulent cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' B10 having Re = 1000 and 𝜃 = 10◦, referred to here as T),  as they deviate significantly from the assumptions of the model in this paper, that the flow  is primarily composed of two layers with a hydrostatic pressure field (discussed in §A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' For  more details about the flow regimes, statistics and spatio-temporal dynamics, see Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Two-layer model as a diagnostic tool  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Layer averaging procedure  We seek to reduce the dimensionality of our DNS datasets to a set of two layers in order to  interpret their dynamics using simple two-layer hydraulics, as sketched in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' To do  this, we will define the interface that separates the layers as the height 𝑧 = 𝜂(𝑥, 𝑡) where  𝜌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The streamwise velocities, heights and densities of each layer are 𝑢𝑖, ℎ𝑖, and 𝜌𝑖 (where  𝑖 = 1, 2 correspond to the upper and lower layer, respectively), are then obtained by averaging  the DNS data over the 𝑦-direction and in 𝑧 over the height of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Specifically, the flow  properties of the layers are obtained by  ℎ1(𝑥, 𝑡) = 1 − 𝜂(𝑥, 𝑡),  𝑢1(𝑥, 𝑡) = ⟨⟨𝑢⟩𝑦⟩𝑧1,  𝜌1(𝑥, 𝑡) = ⟨⟨𝜌⟩𝑦⟩𝑧1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1)  ℎ2(𝑥, 𝑡) = 1 + 𝜂(𝑥, 𝑡),  𝑢2(𝑥, 𝑡) = ⟨⟨𝑢⟩𝑦⟩𝑧2,  𝜌2(𝑥, 𝑡) = ⟨⟨𝜌⟩𝑦⟩𝑧2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2)  where the top-layer average is ⟨·⟩𝑧1 = (1/ℎ1)  ∫ 1  𝜂 · 𝑑𝑧, the bottom-layer average is ⟨·⟩𝑧2 =  (1/ℎ2)  ∫ 𝜂  −1 · 𝑑𝑧 and and the spanwise average is ⟨·⟩𝑦 = (1/2)  ∫ 1  −1 · 𝑑𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Recall that 𝑧 = 1 and  𝑧 = −1 are the non-dimensional height of the top and bottom walls, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Figure 3  shows a single snapshot at time 𝑡 = 110 of 𝑢 (colours) and 𝜌 (contours) at the 𝑦 = 0 midplane  Focus on Fluids articles must not exceed this page length  SID: two-layer hydraulics and instabilities  5  Figure 3: Snapshot at 𝑡 = 110 of streamwise velocity 𝑢 (blue to red shading) and density 𝜌  (colour contours) in the midplane (𝑦 = 0) for the (𝑎) laminar (L) flow and (𝑏) travelling  wave (TW) flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' the latter showing evidence of an jump (see the interface 𝜌 = 0  emphasised by the thick green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  for the two datasets L and TW, highlighting the 𝜌 = 0 density interface with a thick green  contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Layer-averaged DNS data and evidence of jumps  Figure 4 illustrates the results obtained after applying this layer averaging to the DNS  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We show 𝑥 − 𝑡 diagrams of the lower-layer height ℎ2 (top row) and lower-layer  velocity 𝑢2 > 0 (bottom row) for the laminar (L), stationary wave (SW), travelling wave  (TW), and intermittently turbulent (I) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The layer averaging (in 𝑦, 𝑧) was performed in  the range |𝑥| ⩽ 32, which includes the duct |𝑥| ⩽ 30 and extends slightly into the reservoirs  where the layers flow in and out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The L flow regime exhibits sudden changes in depth and speed only at the ends of the duct  𝑥 = ±30, as is expected from the flow exiting into the deep reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Figure 4(a) and an  instantaneous snapshot of the flow in figure 3(a) confirms that the interface 𝜂(𝑥) is steady  and gently sloping down (𝜂′ < 0) throughout the duct, and is symmetric in the duct centre,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 𝜂(𝑥, 𝑡) = 𝜂(𝑥) = −𝜂(−𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In contrast, for the SW and TW flow regimes, the upper layer thickness ℎ2 suddenly  increases along the direction of the flow (purple for 𝑥 < 0 to orange for 𝑥 > 0 in figure 4),  which is accompanied by a sudden drop in 𝑢2 (dark to light red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This discontinuity indicates  the presence of what is commonly called an ‘internal hydraulic jump’ (Baines 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Thorpe  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2018a), as can be seen in figure 3(b), where both layers experience a sudden expansion  and deceleration in their respective flow direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the TW flow (figure 3(b)) the interface  is sloping up (𝜂′ > 0) in the vicinity of the jump, and is negative (𝜂 < 0) throughout most of  the left-hand side of the duct, and vice versa, which is the opposite of what is found in the L  flow (figure 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figure 5 shows the temporal evolution of the interface in all four flows, with 𝜂(𝑥, 𝑡)  plotted at intervals separated by one ATU and vertically stacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the SW case, the jump  remains in the narrow interval 𝑥 = ±1 whereas in the TW case, it oscillates over a much larger  interval and sends off waves in either direction, hence the distinction between the ‘stationary’  and ‘travelling’ wave regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the I case, moving jumps are observed in the quiet phase  (150 ≲ 𝑡 ≲ 200), being initiated near both ends of the duct at 𝑡 ≈ 150 and progressively  moving toward the centre of the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These jumps merge at around 𝑡 = 200 and then stay  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  Figure 4: Spatio-temporal diagrams of (a-d) the lower-layer height ℎ2 and (e-h)  lower-layer velocity 𝑢2 from the DNS data for the four flow regimes (a,e) L, (b,f) SW,  (c,g) TW, (d,h) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' All data are for 𝑡 ∈ [80, 260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figure 5: Temporal evolution of the density interface 𝜂(𝑥, 𝑡) in (a) L, (b) SW, (c) TW, (d)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The interface curves are stacked in time at intervals of one ATU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Jumps are revealed in  (b)-(d) by the discontinuity in 𝜂(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (a) L  (b) SW  (c) TW  (d) I  250  h2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  200  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  150  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  (e) L  () SW  (g) TW  (h) I  250  U2  1  200  0  150  100  1  30-150  1530-30-15 0  1530-30-150  1530-30-15 0  15  30  x  x  x  x(a) L  (b) Sw  (C)  IW  (d)  260  80  30  0  0  0  30 -30  30 -30  30 -30  0  30  x  x  x  xSID: two-layer hydraulics and instabilities  7  Figure 6: The mass flux 𝑄𝑚 from the 15 DNS of Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2022) increasing with 𝑅𝑒 and  𝜃 until the upper bound of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The symbols, coded by the respective regime, show the  time-averaged value while the error bars depict the extreme values  at the middle of the duct 𝑥 = 0 for ≈ 20 ATU before the transition to turbulence occurs at  𝑡 ≈ 220 (the active stage of the intermittent cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Flow rate and evidence of maximal exchange  To a good approximation, the flow in SID has zero net (barotropic) volume flow rate  ⟨𝑢⟩𝑦,𝑧 ≡ 1  4  ∫ 1  −1  ∫ 1  −1  𝑢 𝑑𝑦 𝑑𝑧 = 1  2 (𝑢1ℎ1 + 𝑢2ℎ2) ≈ 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3)  but a non-zero exchange volume flow rate (or volume flux) 𝑄(𝑥, 𝑡) and mass flow rate (or  mass flux) 𝑄𝑚(𝑥, 𝑡) defined as  𝑄 ≡ ⟨|𝑢|⟩𝑦,𝑧 = 1  2 (𝑢2ℎ2 − 𝑢1ℎ1) ≈ −𝑢1ℎ1 ≈ 𝑢2ℎ2  using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4)  𝑄𝑚 ≡ ⟨𝜌𝑢⟩𝑦,𝑧 ≈ 𝜌1𝑢1ℎ1 ≈ 𝜌2𝑢2ℎ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5)  The approximation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5) comes from the fact that the layer averages of the product 𝜌𝑢 is not  exactly equal to the product 𝜌𝑖𝑢𝑖 of the layer averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Also, recall that the non-dimensional  density is defined such that the mean density is 0 and the minimum and maximum are −1  and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Hydraulic jumps in two-layer flows are often connected to the notion of maximal exchange,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' of an upper bound in the exchange volume flux 𝑄 and mass flux 𝑄𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This means that  flows lacking such jumps have a lower 𝑄, and that no realisable flow may have a higher  𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' While hydraulic jumps have not yet been investigated in detail in the experimental SID  literature, the mass flux 𝑄𝑚 has, due to the simplicity with which it can be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In figure 6 we show the dependence of the mass flux 𝑄𝑚 with RiRe sin 𝜃 ≈ (1/4)𝜃Re  using its temporal mean (symbols) and extreme values (error bars) in 15 different DNS  from Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2022), containing the four main datasets L, SW, TW and I, as well as 11  others including one turbulent dataset (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We find that 𝑄𝑚 increases approximately linearly  with 𝜃Re in the L regime (where little mixing ensures that 𝑄 ≈ 𝑄𝑚), in agreement with  the laminar analytical solution of Duran Matute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' For higher values of 𝜃Re it  reaches an upper bound 𝑄𝑚 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 in the W regime, before dropping below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 in the I and  T regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These observations are consistent with the corresponding experimental data of  Lefauve & Linden (2020) (their figures 5-6), if we exclude their data at small angles 𝜃 < 2◦  (which behave slightly differently and we did not simulate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This apparent upper bound is an  evidence of maximal exchange, since further increase in 𝜃 does not lead to an increase in the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6  T  西1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4  中中中  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2  LWIT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0  20  40  RiResin 08  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  velocity difference between the upper and lower layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' It is a significant departure from the  laminar solution based on the balance of gravitational forcing and viscous drag, suggesting  𝑄𝑚 ∝ 𝜃Re (Lefauve & Linden 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' As it will be shown later (section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3) this 𝑄𝑚 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  is the threshold of the long-wave instabilities and onset of hydraulic transitions and a further  increase in 𝑄𝑚 beyond this limit is taxed by turbulent dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  This large body of experimental and numerical evidence in favour of maximal exchange  in SID and 𝑄𝑚 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 at the transition between the L and W regime, combined with our  observations in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 of the existence of an internal jump, is strongly suggestive that hydraulic  effects dominate the flow in the W regime onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In the next section, we develop the two-layer hydraulics framework to study these jumps  and maximal exchange and their relation to the transition from laminar flow to waves and to  turbulence in SID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Two-layer equations: characteristics and instabilities  In this section, we aim to define and relate the characteristic velocity of the two-layer flows  in the context of SID to the hydraulic regime of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We also aim to provide a physical  implication of when this characteristic velocity is not purely real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We start by simplifying  the inviscid Navier-Stokes equations with shallow water (long wave) theory in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3  before comparing it to the Taylor-Goldstein (linear wave) theory in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4, and interpreting our  findings in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Shallow water equations: nonlinear long waves  The validity of this hydrostatic assumption is verified in our DNS data in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The  upper layer then obeys  𝜕ℎ1  𝜕𝑡 + 𝜕(ℎ1𝑢1)  𝜕𝑥  = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1)  𝜕(ℎ1𝑢1)  𝜕𝑡  + 𝜕  𝜕𝑥  �  ℎ1𝑢2  1 + Ri cos 𝜃𝜌1  ℎ2  1  2  �  = Ri sin 𝜃𝜌1ℎ1 − Ri cos 𝜃ℎ1  𝜕  𝜕𝑥 (𝑝𝑤 + 𝜌1ℎ2 + 𝜌1𝑏) ,  and the lower layer obeys  𝜕ℎ2  𝜕𝑡 + 𝜕(ℎ2𝑢2)  𝜕𝑥  = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2)  𝜕(ℎ2𝑢2)  𝜕𝑡  + 𝜕  𝜕𝑥  �  ℎ2𝑢2  2 + Ri cos 𝜃𝜌2  ℎ2  2  2  �  = Ri sin 𝜃𝜌2ℎ2 − Ri cos 𝜃ℎ2  𝜕  𝜕𝑥 (𝑝𝑤 + 𝜌1ℎ1 + 𝜌2𝑏) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Here, 𝑝𝑤 (𝑥, 𝑡) is the pressure at the upper wall and 𝑏(𝑥) is the elevation of the bottom wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Conveniently, the bottom wall in SID is fixed at 𝑏(𝑥) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We subtract the momentum  equations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2) to remove 𝑝𝑤 and reduce the number of unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The variation  of density of the upper and lower layers in 𝑥 is also neglected compared to variations in  height of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The shallow water equations become  𝜕ℎ2  𝜕𝑡 − 𝜕ℎ1  𝜕𝑡 + ℎ2  𝜕𝑢2  𝜕𝑥 − ℎ1  𝜕𝑢1  𝜕𝑥 + 𝑢2  𝜕ℎ2  𝜕𝑥 − 𝑢1  𝜕ℎ1  𝜕𝑥 = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3)  𝜕𝑢2  𝜕𝑡 − 𝜕𝑢1  𝜕𝑡 + 𝑢2  𝜕𝑢2  𝜕𝑥 − 𝑢1  𝜕𝑢1  𝜕𝑥 + Ri cos 𝜃(𝜌2 − 𝜌1) 𝜕ℎ2  𝜕𝑥  = Ri sin 𝜃(𝜌2 − 𝜌1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4)  −Ri cos 𝜃(𝜌2 − 𝜌1) 𝜕𝑏  𝜕𝑥 ,  SID: two-layer hydraulics and instabilities  9  with two auxiliary equations to satisfy the no-net (barotropic) flow condition and geometric  constraint inside the duct (𝜕𝑏/𝜕𝑥 = 0)  𝑢1ℎ1 + 𝑢2ℎ2 = 0,  ℎ1 + ℎ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5)  By taking the derivative of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5) with respect to 𝑥, these four equations can be written  compactly as  C 𝜕𝒒  𝜕𝑡 + A(𝒒, Δ𝜌) 𝜕𝒒  𝜕𝑥 = 𝒇,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6)  where the state vector 𝒒 and coefficient matrices A, C are  𝒒 =  �������  𝑢1  𝑢2  ℎ1  ℎ2  �������  , A =  ����  �  −𝑢1  𝑢2  0  Ri cos 𝜃 Δ𝜌  −ℎ1  ℎ2  −𝑢1  𝑢2  0  0  1  1  ℎ1  ℎ2  𝑢1  𝑢2  ����  �  , C =  ����  �  −1  1  0  0  0  0  −1  1  0  0  0  0  0  0  0  0  ����  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='7)  The shallow water equation (SWE) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6) is quasilinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' linear in the derivatives of 𝒒 but  with the coefficient matrix A dependent on 𝒒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The quasi-constant local density difference  between layers is defined as Δ𝜌(𝑥, 𝑡) ≡ 𝜌2−𝜌1 ∈ (0, 2) (specified by our layer-averaged DNS  data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This equation does not assume that interfacial waves have infinitesimal amplitudes,  but it does assume, through the hydrostatic assumption, that waves are long with respect to  the layer height (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' that their non-dimensional wavenumber 𝑘 ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the following, we  neglect the forcing  𝒇 = [Ri sin 𝜃 Δ𝜌, 0, 0, 0]𝑇 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='8)  recalling that sin 𝜃 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We will study (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6) in the homogeneous limit ( 𝒇 = 0), with a focus  on the eigenvalues of A, in which 𝒇 plays no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The role of forcing in shallow water theory  is an interesting question left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, the forcing proportional to sin 𝜃 does  influence the DNS, and is thus implicitly taken into account in the state vector 𝒒 obtained  after layer averaging the DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Characteristic curves and propagation of information  Consider a left eigenvector 𝒗 and eigenvalue 𝜆 associated with the matrix pair (A,C) such  that 𝒗𝐻A = 𝜆𝒗𝐻C, where 𝐻 denotes Hermitian transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6) by 𝒗𝐻 yields  𝒗𝐻C  � 𝜕𝒒  𝜕𝑡 + 𝜆(𝑥, 𝑡) 𝜕𝒒  𝜕𝑥  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='9)  The eigenvalues 𝜆 are called characteristic velocities, since they define characteristics curves  𝑠 in the (𝑥, 𝑡) plane along which the partial differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6) reduces to an ordinary  differential equation (Whitham 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' For this homogeneous equation, the combinations  of flow variables 𝒗𝐻C 𝑑𝒒/𝑑𝑠 = 0 is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These characteristic velocities, henceforth  simply referred to as ‘characteristics’, can be complex 𝜆 = 𝜆𝑅 + 𝑖𝜆𝐼 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Their real part  𝜆𝑅 ≡ ℜ(𝜆) represents the phase speed of shallow water waves, describing the trajectories 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Their imaginary part 𝜆𝐼 ≡ ℑ(𝜆) represents any potential growth (𝜆𝐼 > 0) or decay (𝜆𝐼 < 0)  of these waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The direction of information propagation is set by the sign of 𝜆𝑅: when  𝜆𝑅 > 0, information propagates rightward (towards increasing 𝑥), and vice versa, whereas  𝜆𝑅 = 0, indicates stationary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  Figure 7: Real and imaginary parts of the eigenvalues of the two-layer SWE in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='11) at (a)  positive (b) zero and (c) negative interface elevations 𝜂 as functions of the volumetric flow  rate 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The eigenvalues are always complex for 𝑄 ⩾ 𝑄𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5, and become complex for  slightly lower critical values 𝑄𝑐 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4 in the asymmetric cases (a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The characteristics 𝜆 are given by the two distinct solutions of det(𝑨 − 𝜆𝑪) = 0:  𝜆1,2(𝑥, 𝑡) =  ℎ1𝑢2 + ℎ2𝑢1  ℎ1 + ℎ2  ����������������������  convective velocity ≡ ¯𝜆  ±  √︂  Δ𝜌Ri cos 𝜃 ℎ1ℎ2  ℎ1 + ℎ2  �1 − 𝐹2  Δ  � ,  ��������������������������������������������������������������������  phase speed ≡ 𝛿𝜆  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10)  ≈  ������  −2𝜂𝑄  1 − 𝜂2  ±  ������������������������������������������������������������������  √︂  Δ𝜌  2 cos 𝜃(1 − 𝜂2)(1 − 𝐹2  Δ)  using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='11)  where  𝐹2  Δ(𝑥, 𝑡) =  (𝑢2 − 𝑢1)2  Δ𝜌 Ri cos 𝜃 (ℎ1 + ℎ2) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='12)  ≈  2  Δ𝜌 cos 𝜃  � 2𝑄  1 − 𝜂2  �2  using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='13)  Here, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='13) use the volume flux 𝑄 > 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4) (which is an approximation  relying on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3)) and the interface position instead of the layer heights and velocities, as well  as the fact that Ri = 1/4 and ℎ1 + ℎ2 = 2 in SID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We see that the characteristics consist  of a convective velocity ¯𝜆 and a phase speed 𝛿𝜆, which can be imaginary, depending on  the value of the ‘stability Froude number’ 𝐹2  Δ (Long 1956;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lawrence 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Dalziel 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='12) are slightly modified versions of those given in previous  studies (Long 1956;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Armi 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lawrence 1993) adapted to SID flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  If 𝐹2  Δ < 1 the two characteristics 𝜆1,2 = ¯𝜆 ± 𝛿𝜆 are real and information propagates in  both directions relative to the convective velocity ¯𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The absolute direction of propagation is  given by the sign of 𝜆1,2, which we shall return to in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  If 𝐹2  Δ > 1 the characteristics become complex conjugates 𝜆1,2 = 𝜆𝑅 ± 𝑖𝜆𝐼 = ¯𝜆 ± 𝑖|𝛿𝜆|,  indicating that the system is no longer hyperbolic and that waves grow temporally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The condition 𝐹2  Δ > 1 is sometimes known as Long’s instability criterion (Long 1956),  although it is quoted in Lamb (1932) and likely dates back to Helmholtz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The real part is the  convective velocity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' information only propagates in one direction, and the growth rate is  𝜆𝐼 = |𝛿𝜆| =  √︃  (Δ𝜌/2) cos 𝜃(1 − 𝜂2)(𝐹2  Δ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figure 7(a-c) shows how𝜆1,2 vary with the volume flux𝑄 and interface position 𝜂 following  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='11), assuming no mixing (Δ𝜌 = 2) and a horizontal duct (cos 𝜃 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We compare the case  where the interface is locally symmetric (𝜂 = 0, panel b) to the cases where it is asymmetric,  Rapids articles must not exceed this page length  (h) n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  (b) n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  (b) n=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  R  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  Q  Q  QSID: two-layer hydraulics and instabilities  11  Figure 8: Summary of the scenarios revealed by the stability Froude number 𝐹2  Δ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='12),  characteristics 𝜆1,2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10), and composite Froude number 𝐺2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' above or below the mid-level (𝜂 = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 in panels a and c, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Panel b shows that  with a symmetric interface 𝜆1,2 = ±  √︁  1 − 4𝑄2 are real (red curves) for 𝑄 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 and become  purely imaginary (blue curves) for 𝑄 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, when the interface is either below  or above the midplane (𝜂 = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5, panels a,c), this transition to instability occurs at a lower  critical volume flux 𝑄 > 𝑄𝑐 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In summary, instability is caused, for a given volume flux  𝑄, by an increasingly asymmetric interface |𝜂|, and vice versa, for a given interface position,  by an increasing volume flux 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This offers a possible explanation for the transition from L  to W flow observed in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Composite Froude number and hydraulic control  To further stress the importance of the characteristics 𝜆1,2, we return to the original SWE  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6), and note that a non-trivial steady solution 𝒒(𝑥) requires det A ≠ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  det A = 2Δ𝜌ℎ1ℎ2Ri cos 𝜃(𝐺2 − 1) ≠ 0  ⇔  𝐺2 ≠ 1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='14)  where 𝐺2 is the squared composite Froude number defined with the Froude numbers of the  upper and lower layers, 𝐹1, and 𝐹2, respectively,  𝐺2 = 𝐹2  1 + 𝐹2  2 ,  𝐹𝑖 =  𝑢𝑖  √︁  Δ𝜌 Ri cos 𝜃 ℎ𝑖  𝑖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='15)  Points at which 𝐺2 = 1 are called control points (Armi 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lawrence 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Dalziel 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  At control points, A is non-invertible and a regularity condition must exist to recover a steady  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The link between characteristics and the composite Froude number can be highlighted  with the identity  𝐺2 = 1 +  ℎ1 + ℎ2  (𝜌2 − 𝜌1)Ri cos 𝜃 ℎ1ℎ2  𝜆1𝜆2 = 1 +  2𝜆1𝜆2  Δ𝜌 cos 𝜃(1 − 𝜂2)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='16)  ≈ 1 +  √︄  2  Δ𝜌 cos 𝜃  𝜆1𝜆2𝐹Δ  2𝑄  using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='13) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='17)  From this expression, we deduce the following, illustrated in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  If the waves are stable (𝐹2  Δ < 1), the characteristics 𝜆1,2 are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The flow is called  subcritical and information propagates in both directions (along positive and negative 𝑥) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  𝜆1𝜆2 < 0 ⇔ 𝐺2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In other words, the absolute phase speed |𝛿𝜆| is larger than the absolute  G2<1  0>  subcriticalα「>「i  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 E R  =  stable  >→G>1  hyperbolic  supercritical 」  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2ECIR  >>1  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content="=  supercritical  unstable  non-hyperbolic  waves propagate at velocity aR - ^  grow at rate a' = I8a12  A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  convective velocity ¯𝜆 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Vice versa, the flow is called supercritical when information  propagates in only one direction i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 𝜆1𝜆2 > 0 ⇔ 𝐺2 > 1, and the absolute phase speed  |𝛿𝜆| is smaller than the absolute convective velocity ¯𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Note that for supercritical flow, the  direction of propagation associated with 𝜆1 and 𝜆2 is the same as that given by the convective  velocity ¯𝜆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' the waves are swept downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  If, on the other hand, the waves are unstable (𝐹2  Δ > 1), the characteristics are complex  conjugates 𝜆1,2 = 𝜆𝑅 ± 𝑖𝜆𝐼, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' information always propagates in a single direction given by  the sign of 𝜆𝑅 = ¯𝜆 and the flow is always supercritical (𝜆1𝜆2 = (𝜆𝑅)2+(𝜆𝐼)2 > 0 ⇔ 𝐺2 > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Under unstable conditions, control points where 𝐺2 = 1 cannot exist, but points can exist  where stationary (𝜆𝑅 = 0) waves grow (𝜆𝐼 > 0) and control the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In the next sections, we clarify the interpretation of unstable shallow water waves using  linear stability analysis around a locally parallel base flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4 we take the long-wave  limit of solutions of the Taylor-Goldstein equations, and in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 we linearise the shallow  water equations assuming the waves are sufficiently (but not excessively) long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Taylor-Goldstein equations: linear short and long waves  We relax the previous restriction that waves must be long (𝑘 ≪ 1) by studying waves of  possibly larger 𝑘 but of infinitesimal amplitude developing on a parallel base flow described  by a velocity profile U(𝑧) and a density profile R(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The perturbation streamfunction  ˆ𝜓(𝑧) exp𝑖𝑘(𝑥 − 𝑐𝑡) describing the evolution of these two-dimensional linear waves is given  by the inviscid Taylor-Goldstein equation (TGE)  (U − 𝑐)  � 𝑑2  𝑑𝑧2 − 𝑘2  �  ˆ𝜓 − U′′ ˆ𝜓 − Ri cos 𝜃 R′  U − 𝑐  ˆ𝜓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='18)  This can be analysed following standard methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Drazin 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Smyth & Carpenter 2019)  with details given in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Note that we assumed small tilt angles 0 < sin 𝜃 ≪ cos 𝜃,  although the more general TGE in appendix B shows that sin 𝜃 has a destabilising effect  (ignored here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the ordinary differential equation above, ‘′’ denotes differentiation with  respect to 𝑧, and 𝑐 ∈ C is the phase speed of the plane waves, akin to the characteristics 𝜆  of the SWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, we use a different notation for the characteristics of shallow water  nonlinear waves and the phase speed of TG linear waves to emphasise that while the former  implicitly assumes 𝑘 ≪ 1, the latter implicitly assumes 𝑘 ≫ 𝐴−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In other words, the TG  linear waves we investigate should be shorter than the duct length 𝐴 for the local analysis on  a parallel base flow to be sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In other words, 𝑘 ≫ 𝐴−1 ensures that the waves do not  ‘feel’ the streamwise variations of the base flow along the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  To make analytical progress and obtain solutions for 𝑐, we assume a two-layer flow bounded  by solid walls, with fixed layer heights ℎ1, ℎ2, velocities 𝑢1, 𝑢2, and an interface at 𝑧 = 0,  consistent with the two-layer model adopted throughout this paper  U(𝑧) =  �  𝑢1  0 < 𝑧 ⩽ ℎ1  𝑢2  −ℎ2 ⩽ 𝑧 < 0 ,  R(𝑧) =  �  𝜌1  0 < 𝑧 ⩽ ℎ1  𝜌2  −ℎ2 ⩽ 𝑧 < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='19)  Note that ℎ1 + ℎ2 = 2 and, by a simple vertical shift, the above model is equivalent to having  a domain restricted 𝑧 = ±1 and an interface at an arbitrary 𝑧 = 𝜂 (giving ℎ1,2 = 1 ± 𝜂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  By enforcing the matching conditions for the stream function and pressure at the interface  SID: two-layer hydraulics and instabilities  13  (see appendix B), we derive the dispersion relation for the complex phase speeds  𝑐1,2 = 𝑢1 + 𝑢2  2  + (𝜎5 − 𝜎6)(𝑢1 − 𝑢2) ± 𝜎2  2 (𝜎1 − 1)  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20)  𝜎1 = cosh (4 𝑘 ) ,  𝜎2 = sinh (2𝑘 ) Λ,  𝜎3 = sinh (2 𝑘 ℎ2) ,  𝜎4 = sinh (2 𝑘 ℎ1) ,  𝜎5 = cosh (2 𝑘 ℎ2) ,  𝜎6 = cosh (2 𝑘 ℎ1) ,  Λ =  √︂  −4  𝑘  �𝑘𝜎4𝜎3[𝑢1 − 𝑢2]2 + Ri cos 𝜃[𝜎4(1 − 𝜎5) + 𝜎3(1 − 𝜎6)]Δ𝜌�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  These waves are the well-known Kelvin-Helmholtz (KH) waves supported by a single vortex  sheet (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Smyth & Carpenter (2019) § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' They are modulated by stratification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  increasing Ri cos 𝜃Δ𝜌 always stabilises them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Importantly, the dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20)  describes waves in a domain bounded by the top and bottom walls, which as we will see,  strongly affect waves that have a wavelength comparable to or longer than the domain height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  For ‘short’ waves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' having a wavelength of the order of the duct height 𝑘 = 𝑂(1) or  shorter 𝑘 > 1, the dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) cannot be simplified further, and these waves  are dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These will be referred to simply as KH waves, as in the short wave limit they  are identical to those found in vertically unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  For long waves (which, for clarity, we do not call KH waves), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' having a wavelength  much longer than the duct height, but still shorter than the duct length 𝐴−1 ≪ 𝑘 ≪ 1, the  dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) simplifies as sinh 𝑘ℎ −→ 𝑘ℎ and cosh 𝑘ℎ −→ 1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) reduces  to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  𝑐1,2(𝑘) −→ 𝜆1,2  when 𝑘 ≪ 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='21)  and these waves are non-dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In other words, the characteristics 𝜆1,2 of nonlinear  shallow water waves can be identified to the phase speeds 𝑐1,2 of linear (infinitesimal) long  waves calculated assuming the local two-layer flow (𝑢1, 𝑢2, ℎ1, ℎ2) to be parallel and steady  as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) shows that there exists a smooth transition between short  KH waves and long shallow water waves, as those waves differ by their wavelength but not the  underlying physical mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We return to this smooth transition and plot the dispersion  relation in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Although the limit (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='21) can be inferred from the PhD thesis of Gu (2001) (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3) and is  briefly alluded to in Boonkasame & Milewski (2012) (§ 3), it does not appear to be widely  disseminated in the hydraulics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Link between complex characteristics and instability  This natural link between 𝜆1,2 and 𝑐1,2 in the long-wave limit can be further understood  by considering another limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' It is possible to directly linearise the SWE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6) to study the  evolution of infinitesimal perturbations 𝜖 ˜𝒒(𝑥, 𝑡) (0 < 𝜖 ≪ 1) on a parallel, steady base  flow 𝒒0 = (𝑢1, 𝑢2, ℎ1, ℎ2) akin to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We perform a first-order Taylor expansion of the  coefficient matrix from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='7) A(𝒒) = A(𝒒0 + 𝜖 ˜𝒒) and obtain  C 𝜕(𝒒0 + 𝜖 ˜𝒒)  𝜕𝑡  +  �  A(𝒒0) + 𝜖 ˜𝒒 𝜕A  𝜕𝒒0  � � 𝜕𝒒0  𝜕𝑥 + 𝜖 𝜕 ˜𝒒  𝜕𝑥  �  = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='22)  At order 𝜖 we have the linear, local SWE  C 𝜕 ˜𝒒  𝜕𝑡 + A0  𝜕 ˜𝒒  𝜕𝑥 = −˜𝒒 𝜕A  𝜕𝒒0  𝜕𝒒0  𝜕𝑥  ����  = 0  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='23)  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  Figure 9: (a) Summary of the interpretation of characteristics in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 using linear theory,  either by first linearising the NSE, and then taking the long-wave limit (TGE, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4) or vice  versa (SWE, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Both approaches yield the same result for waves much shorter than the  duct length but much longer than the duct height (range sketched in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  where A0 ≡ A(𝒒0) is the local constant coefficient matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Importantly, the right-hand side  coming from the Taylor expansion, and acting as a forcing term, vanishes when we assume  the base flow to vary slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Substituting the plane wave ansatz ˜𝒒 = ˆ𝒒 exp𝑖𝑘(𝑥 − 𝜍𝑡) gives  the phase speed 𝜍 as the eigenvalue of the matrix pair (A0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The two distinct solutions  𝜍1,2 of det(A0 − 𝜍C) = 0 then become identical to the local characteristics 𝜆1,2 (at a fixed  𝑥, 𝑡) derived in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In other words, the nonlinear shallow water wave (𝑘 ≪ 1) characteristics 𝜆1,2 can be  interpreted as the linear shallow water waves phase speed 𝜍1,2 propagating on a locally  parallel base flow (𝑘 ≫ 𝐴−1), which are themselves the long wave, non-dispersive limit case  of the Taylor-Goldstein two-layer phase speeds 𝑐1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This result ultimately stems from the  fact that, simply put, the linearisation and the long wave limit commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In particular, the  potential positive imaginary component of characteristics 𝜆𝐼 > 0 can be interpreted as the  exponential growth rate of such unstable waves satisfying 𝐴−1 ≪ 𝑘 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  This interpretation is summarised in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The long aspect ratio 𝐴 of SID (in this  paper 𝐴 = 30), ensures that the range of waves 𝐴−1 ≪ 𝑘 ≪ 1 exists, and therefore that this  interpretation is useful, unlike in shorter geometries having 𝐴 ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We also note in passing  that this interpretation can be generalised to any number of layers greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Two-layer hydraulics applied to DNS  In this section, we use the modelling results of §4 to understand the observations of hydraulic  jumps and maximal exchange of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1 we study the stability Froude number flagging  (a)  Shallow water  q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' t) = (ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' h2)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' t)  Inviscid Navier-Stokes  equations (SWE)  equations (NSE)  characteristics 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' t)  two layers  q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' t) = (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='p)  non-dispersive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' global  long waves k < 1  o = (u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' U2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' P2)  q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' t) = qo + eqexp ik(x - st)  q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' t) = qo + eq exp ik(x - ct)  parallel flow k > A-1  two-layer parallel flow k > A-1  linearise e < 1  linearise e < 1  plane waves with speed s  plane waves with speed c  21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 → C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 non-dispersive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' local  Taylor-Goldstein  long waves k < 1  2nQ  equations (TGE)  n2  phase speeds C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2(k)  Unstable F2 > 1 → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 = R ±ia!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  dispersive, local  wave propagation  wave growth  sign AR= - sign n α (1 -n2)(F2 - 1)  (b)  duct length  duct height  waves of interest  wavenumber  1  A-1  long waves k < 1  A-1<k1  parallel flow k > A-1SID: two-layer hydraulics and instabilities  15  Figure 10: Spatio-temporal diagram (𝑥 − 𝑡) of the stability Froude number 𝐹2  Δ for the (𝑎)  L, (𝑏) SW, (𝑐) TW, and (𝑑) I cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Long-wave instability is predicted for 𝐹2  Δ > 1 (red  colour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  long-wave instability, in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 we focus on the characteristics and composite Froude number  to diagnose internal jumps and hydraulic control, and in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3 we explain the observed flow  rate with notions of maximal exchange, viscous friction and mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Stability Froude number and instability  In figure 10 we plot the spatio-temporal diagram of the stability Froude number 𝐹2  Δ(𝑥, 𝑡)  given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='12) in our four datasets L, SW, TW, and I (all at 𝑅𝑒 = 650) to diagnose any long  wave instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The laminar flow (L) has 𝐹2  Δ < 1 everywhere (in blue in figure 10a), hence long waves  are stable at 𝜃 = 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In all other cases, 𝐹2  Δ > 1 (in red in figure 10) in most of the duct,  hence long waves are unstable at 𝑅𝑒 = 650 for 𝜃 > 𝜃𝑐 where 𝜃𝑐 ∈ [2◦, 5◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the wave  flows (SW and TW), the waves are most unstable (maximum 𝐹2  Δ, deep red) near the centre  of the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the I flow, long waves are very unstable (𝐹2  Δ ≫ 1) throughout most of the  duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These 𝐹2  Δ(𝑥, 𝑡) diagrams correspond to the absence of a jump in the duct in the L flow  (figure 4a) and the existence of a jump in the SW, TW, and I flows (figure 4b-d), although the  reasons for the jump remain to be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These diagrams also show that as the tilt angle  𝜃 is modestly increased between 5◦ and 8◦, further changes take place as the flow becomes  increasingly unstable to long waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Next, we delve deeper into the hydraulics analysis to  understand jumps by focusing on the characteristics in each flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Characteristics, composite Froude number and control  In figure 11 we plot the characteristic velocities 𝜆1,2(𝑥) given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='10) (real parts in panel  a and imaginary parts in panel b) and the composite Froude number 𝐺2(𝑥) given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='16)  (panel c) at time 𝑡 = 110, to diagnose the propagation of information and criticality of the  flow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In figure 12 we plot a set of discrete trajectories (white curves) by solving  𝑑𝑋  𝑑𝑡 = 𝜆𝑅(𝑋, 𝑡) for 𝑡 > 80,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1)  and initialising 𝑋(𝑡 = 80) as 61 equidistant points along the duct 𝑥 ∈ [−30, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Additionally,  we plot the local growth rate 𝜆𝐼 (𝑥, 𝑡) in background colours (dark blue to yellow) for  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figures 11(a) and 12(a) show that in the stable L flow, 𝜆 has two distinct real roots of  0  1  2  250  200  150  100  (b)  (n)  (c)  30  0  0  0  0  30  30-30  30-30  30-30  x  x  x  x16  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  Figure 11: Characteristics (a) real part 𝜆𝑅, (b) imaginary part 𝜆𝐼 (only the positive values  are shown), and (c) composite Froude number 𝐺2 of the L, SW, TW, I flows at 𝑡 = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Note that we also show data immediately outside the duct, up to 𝑥 = ±32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figure 12: Characteristic curves 𝑋(𝑡) (in white) obtained by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1) for (a) L, (b) SW, (c)  TW, (d) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The curves (in white) originate from 61 equally spaced positions between  𝑥 = −30 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The colour contours show the growth rate 𝜆𝐼 (𝑥, 𝑡) (which is zero in a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Note that we only show data inside the duct up to 𝑥 = ±30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  opposite signs throughout most of the duct (𝜆𝑅  1 𝜆𝑅  2 < 0 ⇔ 𝐺2 < 1), allowing characteristics to  cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Information propagates in both directions and the flow is subcritical inside the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The  speed of propagation is of order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' significantly lower than the advective velocity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, at the ends of the duct, the characteristics vanish locally (𝜆𝑅  1 𝜆𝑅  2 = 0 ⇔ 𝐺2 = 1 at  |𝑥| ≈ 30) signalling that long waves become stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Figure 11(a) shows that immediately  outside the duct, information propagates only in one direction (𝐺2 > 1 at |𝑥| ≳ 30), in fact,  leftward on the left-hand side (𝜆𝑅  1 , 𝜆𝑅  2 < 0) and rightward on the right-hand side (𝜆𝑅  1 , 𝜆𝑅  2 > 0),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' always away from the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The ends of the duct, therefore, act as control points, in the  sense that no information from the reservoirs can propagate into the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The existence  of two such hydraulic control points, with their respective characteristics pointed outwards,  means that the interior of the duct is ‘fully controlled’ in the hydraulic sense, isolating the  1, L --22,L ---SW  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='TW  R  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  (q)  ～ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  4  2  0  30  0  30  x0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6  (b) SW  (c) TW  (d) I  (a) L  入  250  200  150  100  30  0  30-30  0  30-30  0  30-30  0  30  x  x  x  xSID: two-layer hydraulics and instabilities  17  flow from hydrostatic disturbances within the reservoirs to either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This is the first direct  evidence that SID flows in the L regime are hydraulically controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In contrast, in the unstable SW, TW, and I cases, the roots are complex conjugates  throughout most of the duct (figure 11(a,b)), a consequence of instability (𝐹2  Δ > 1), causing  supercriticality (𝐺2 > 1, figure 11(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These unstable waves always move at the local  convective velocity of the flow 𝜆𝑅 = ¯𝜆(𝑥, 𝑡), which we recall from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='11) is non-zero if the  interface is not at mid-depth 𝜂(𝑥, 𝑡) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In the SW and TW cases, the unstable waves are initially carried rightward ( ¯𝜆 > 0)  throughout most of the left-hand side of the duct, and leftward ( ¯𝜆 < 0) throughout most of  the right-hand side of the duct, as seen in figure 12(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Thus, all unstable waves are carried  towards the centre (𝑥 = 0) where their characteristics converge, creating the hydraulic jump  observed in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The largest values of 𝜆𝐼 = |𝛿𝜆| are found in the region where the 𝜆𝑅 = ¯𝜆  convective components converge (see figure 11(b) and green-yellow shades in figure 12(b,c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Their growth rate is fast (𝜆𝐼 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5) and slightly higher in TW than in SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Such a jump,  bounded by supercritical regions on either side, is distinguished from the standard hydraulic  jumps which make the flow transition from a supercritical to a subcritical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' It can be  viewed as the limit of the length of the subcritical region tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The jumps in SW  and TW can be called ‘undular jumps’ because of their moderate ‘upstream’ Froude numbers  𝐹2  𝑖 ≈ 𝐺2/2 of each layer (between 1 and 2) and the small energy they dissipate, compared to  direct, breaking hydraulic jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Some jumps in I are stronger, as evidenced by their locally  higher 𝐺2 values and their visibly higher dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  This pattern of unstable characteristics converging toward the centre to form a jump can be  summarised by sign 𝜆𝑅 = −sign 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This can be understood first by sign 𝜆𝑅 = sign ¯𝜆 = −sign 𝜂  from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='11), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' the waves are carried at the local convective velocity, and second, by  sign 𝜂 = sign 𝑥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' the interface does not slope down as in L but is instead lowered on the  left-hand side of the duct, and lifted on the right-hand side (central jump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The main difference between SW (stationary wave regime) and TW (travelling wave  regime) lies in the behaviour of 𝜆𝑅 near the jump around 𝑥 = 0 (figure 11(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' While 𝜆𝑅(𝑥)  goes smoothly through zero in SW, it oscillates more in TW, suggesting that the location of  the jump is prone to oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This is confirmed by comparing the 𝑥 − 𝑡 trajectory of the  locus of the maximum 𝐹2  Δ in figure 10(b,c) or the maximum 𝜆𝐼 in figure 12(b,c) as a proxy  to the location of the jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Finally, we turn to the I flow, which exhibits more vigorous interfacial turbulence and  wave instability (especially between 𝑡 = 150 − 250) and greater variability in 𝑥 and 𝑡 than  SW and TW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The characteristics in figure 12(d) converge quickly to form a large number of  local jumps around 𝑡 ≈ 100, which then organise into three main clusters: a central stationary  cluster flanked by a left and a right cluster which themselves converge to form discrete jumps  around 𝑡 ≈ 160 while being carried to the centre, eventually converging into a single jump  𝑡 ≳ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The convergence of characteristics tends to coincide with the maximum instability  (𝜆𝐼 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' During the more stable (transitional) period at 𝑡 = 110 the multiple sign reversals  of 𝜆𝑅(𝑥) (figure 11(a)) hinder a straightforward interpretation of wave propagation along 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  This pattern by which unstable waves are carried in SW, TW and I flows differs so greatly  from the classical picture of hydraulic control in L flow that it prompts the question: since  information travels toward the duct centre, does it travel from the reservoirs into the duct,  and is the flow still hydraulically controlled?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Figure 11 shows that close to the duct exits  (|𝑥| ≈ 28), the composite Froude number 𝐺2 (panel c) of all the cases becomes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Meanwhile,  immediately outside the duct 30 < |𝑥| < 32 the waves are stable (𝜆𝐼 = 0, panel b), and both  the curves of the real characteristics (panel a) and of the composite Froude number 𝐺2 (panel  c) closely match those in the L flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In other words, the SW and TW flows also have a  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  control point (𝐺2 = 1), flanked by narrow regions of subcriticality (𝐺2 < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We conclude  that the flow within the duct remains isolated from the reservoirs , and hence that it is also  hydraulically controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This represents the first direct evidence that SID flows in the W and  I regimes, in addition to having a (supercritical-to-supercritical) jump in the centre of the  duct, are also hydraulically controlled at the ends of the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Maximal exchange and critical flow rate  In the previous section, we showed that all four flows cases (L, SW, TW and I) were  hydraulically controlled in the sense that control points at the ends of the duct isolated the  flow within the duct from processes in the reservoirs and prevented the flow inside the duct  from reaching velocities exceeding a maximal volume flux 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In this section, we seek to  explain the differences in the value of this critical volume flux (and by extension mass flux)  observed between the L, W, and I regimes in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In the stable L flow, we find a time-averaged 𝑄 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='31 well below the absolute upper bound  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 for instability given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This is understood by the frictional hydraulic theory  of Gu & Lawrence (2005), subsequently adapted to SID in Lefauve & Linden (2020) (their  § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In short, the relatively low values of the Reynolds number in these low-tilt L flows  mean that viscous friction at the duct walls and at the interface must be parameterised in  the shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This parameterisation allows a correct prediction of the sloping  interface 𝜂(𝑥) (a consequence of viscous friction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' loss of momentum along the flow of  each layer), which in turn allows prediction of 𝑄 by imposing the criticality condition at  the ends of the duct 𝐺2(𝑥 = ±𝐴) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Simply speaking, the lower 𝑅𝑒 and the longer the  duct aspect ratio 𝐴, the more friction occurs along the duct, the more offset the interface  |𝜂| becomes at the ends of the duct, and the lower the volume flux 𝑄 becomes to satisfy  𝐺2(|𝜂|, 𝑄) = 1 (since 𝐺2 is an increasing function of both |𝜂| and 𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In the unstable SW, TW and I flows, despite the existence of viscous friction, we find a  remarkably consistent time-averaged 𝑄 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='51 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='53, slightly above the critical 𝑄𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  upper bound for frictionless two-layer hydraulics and a flat interface 𝜂 = 0 (figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These  values require a different explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Although the Reynolds number of SW, TW, and I are  identical to L, their larger tilt angle 𝜃 pushes these flows beyond the instability threshold 𝐹2  Δ =  1, corresponding to the transition between ‘lazy’ and ‘forced’ flows (Lefauve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  However, 𝑄 does not continue to increase with 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Rather, beyond the instability threshold,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='13) suggests that, in the centre of the duct (where 𝜂 ≈ 0), 𝑄 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  √︁  (Δ𝜌)/2 cos 𝜃𝐹Δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  a linear increase with the stability Froude number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We deduce that, since 𝑄 never greatly  exceeds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 (the value reached the instability threshold), the subsequent increase in 𝐹Δ > 1  (indirectly caused by the forcing ∝ 𝑠𝑖𝑛𝜃 in the DNS) must be compensated by a decrease  in Δ𝜌/2 ∝ 1/𝐹2  Δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' by increased mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The data show that the average Δ𝜌/2 indeed  decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='79 in SW, to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='76 in TW, to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='68 in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This mixing in turn explains why 𝑄𝑚  (roughly ≈ (Δ𝜌/2)𝑄) stays robustly below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 in SID, and indeed decreases from the W to  the I regime (figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Applicability of unstable hydraulics  In this section, we study the applicability of the previous results to problems not usually  considered in two-layer hydraulics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1 we study the transition between long waves (the  propagation of which is identical to the local characteristics of the SWE) and shorter Kelvin-  Helmholtz waves (only predicted by the TGE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2 we study the waves diagnosed from  DNS run at different values of the Prandtl number to investigate their indirect dependence on  scalar diffusion and the thickness of the density interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3 we study how the growth  SID: two-layer hydraulics and instabilities  19  Figure 13: Dispersion relation of all (long and short) inviscid two-layer waves: growth rate  (colours) and phase speed (contours) solutions of the TGE varying with wavenumber 𝑘  and volume flux 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (a) Symmetric interface 𝜂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (b) Asymmetric interface 𝜂 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (c)  Symmetric interface but without solid top and bottom boundaries at 𝑧 = ±1, in which case  the long waves of SWE disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figure 14: Dispersion relation (growth rate only) predicted by the TGE two-layer model  applied to the DNS (a) L and (b) TW flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The dashed line at 𝑘 = 1 represents the  boundary between long and short waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Short waves are predicted to be most unstable by  this inviscid model but appear relatively stable in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  of long waves is impacted by smooth, diffuse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' not strictly two-layer) density and velocity  profiles, which are expected in all real-world flows (having a finite Re and Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Long vs short waves  Figure 13 shows the dispersion relation from the TGE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) with the phase speed ℜ(𝑐1) = 𝑐𝑅  1  (blue to red contours) and growth rate ℑ(𝑐1) = 𝑐𝐼  1 (colour map with deep blue being stable)  as functions of the wavenumber 𝑘 (vertical axis) and the volume flux 𝑄 (horizontal axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We  compare a symmetric interface (𝜂 = 0, panel a), an asymmetric interface (𝜂 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5, panel  b) and a case with a symmetric interface but without solid top and bottom boundaries (panel  c), whose dispersion relation (B 13) is derived analytically in §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  We recall that for 𝑘 ≪ 1 the phase speed 𝑐𝑅 and growth rate 𝑐𝐼 of TGE become identical  to 𝜆𝑅 and 𝜆𝐼 of SWE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In this case the 𝑘 = 10−2 data of figure 13(a,b) become  indistinguishable from those plotted in figure 7(b,c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  From figure 13(a-b) we recover the results from previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' First, long waves are  non-dispersive (the phase speed contours do not depend on 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Second, they become unstable  (lighter shades of blue and green) above a critical volume flux 𝑄 > 𝑄𝑐, equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 for  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2  一1  0  102  K  ¥100  (b)  ()  (a)  10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  Q  Q  Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (b)  100  10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  30  0  30-30  0  30  x  x20  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  a symmetric interface (panel a) and lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 for an asymmetric interface (panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Third, for a symmetric interface, all unstable waves (long and short) are stationary (absence  of contours), but all stable waves are travelling (presence of contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' For an asymmetric  interface, even unstable waves are travelling in the reference frame of the duct, as we explained  in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Fourth, the transition between short and long waves is smooth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' there is a continuity  between the long shallow water waves controlling the hydraulics of two-layer flows and the  short KH waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Panels (a,b) of figure 13 also give new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' First, short waves (𝑘 � 1) become unstable  at smaller values of 𝑄 compared to the long wave threshold 𝑄𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This transition to short waves  becomes noticeable from 𝑘 ≳ 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3 and is clear for 𝑘 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The shortest waves shown  here (𝑘 = 102) are predicted to become unstable above a very small volume flux 𝑄 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  However, we note that this threshold would be closer to the long-wave 𝑄𝑐 if we included  viscosity in the TGE, as viscosity would significantly damp the growth of short waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These  panels show that for a given value of 𝑄, the growth rate increases monotonically with 𝑘 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  the shortest waves are the most unstable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' As is often the case, this ‘ultraviolet catastrophe’  would be regularised by viscosity, with the possible existence of a maximum growth rate at  an intermediate 𝑘 for intermediate values of Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In figure 13(c), the absence of solid walls does not affect short waves (the colours and  contours at 𝑘 ≳ 3 are identical to panel a), because they do not ‘feel’ the presence of the  walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, the absence of solid walls precludes the existence of long waves 𝑘 ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3,  because this setup (despite being bounded at 𝑧 = ±1) approximates layers of infinite depth,  compared to which all waves are ‘short’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In other words, this analysis explicitly shows that the  presence of solid walls in SID are crucial to explain the leading order dynamics in the DNS  by allowing long-wave instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Adding walls (panel a) creates the long waves on which  hydraulic effects rely, an long waves transition smoothly into short waves as 𝑘 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Figure 14 shows the linear growth rate 𝑐𝐼 obtained by substituting into the TGE dispersion  relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) (function of 𝑘) the two-layer properties (as functions of 𝑥) extracted from  the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Diagnostics are shown for the L flow (panel a) and the TW flow (panel b) at time  𝑡 = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The TW flow is, unlike the L flow, unstable to long waves 𝑘 ≪ 1, with the maximal  growth rate found near the centre of the duct, as previously seen in figure 11(b) as we know  from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='21) that 𝑐𝐼 (𝑘 ≪ 1) −→ 𝜆𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, we also find that the L flow appears mildly  unstable to short waves (especially very short waves 𝑘 ≳ 10), and the TW flow appears even  more strongly unstable to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, we know that in the DNS the L flow is visibly stable  and does not have observable interfacial waves, while the TW flow is primarily unstable to  long waves, whereas short waves play a more minor role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This suggests that, at least for the  present values of Re = 650 and Pr = 7, the growth of short waves is sufficiently damped by  viscosity, mass diffusion and/or other effects not taken into account in this inviscid two-layer  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Low vs high Prandtl numbers  Although Pr does not appear explicitly in the SWE, the DNS dynamics from which the two-  layer properties are extracted certainly depend on Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In applications, three typical values  are of particular interest: Pr ≈ 1 (representative of temperature stratification in air), Pr ≈ 7  (representative of temperature stratification in water, as studied in this paper) and Pr ≈ 700  (representative of salt stratification in water).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  To study the impacts of diffusion, we carried out two additional DNS with parameters  identical to the TW flow (with Pr = 7) at Pr = 1 and Pr = 28 (the latter requiring a much  higher spatial resolution, hindering the study of higher Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In figure 15 we compare the  characteristics curves 𝑋(𝑡) and growth rates 𝜆𝐼 at these three different values of Pr using the  same visualisation as figure 12 (where the Pr = 7 data was already shown as TW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We find  SID: two-layer hydraulics and instabilities  21  Figure 15: Characteristic curves 𝑋(𝑡) and growth rate 𝜆𝐼 (colours) from DNS data at  Re = 650 𝜃 = 6◦ and (a) Pr = 1, (b) Pr = 7 (TW data), (c) Pr = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Note that for easier  comparison, (b) is identical to figure 12(c), though we only show data for 𝑡 ∈ [80, 220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Also note the reduced colour map maximum 𝜆𝐼 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 in this figure, compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6 in  figure 12)  Figure 16: Characteristics (a) 𝜆𝑅, (b) 𝜆𝐼 > 0, and (c) composite Froude number 𝐺2 of TW  flows at 𝑡 = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We compare three different Prandtl numbers Pr = 1, 7, 28 as in figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  that curves from the Pr = 1 flow initially converge into a central jump and a small number  of peripheral jumps, which eventually merge with the central jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This pattern resembles  that of the more stable SW flow from figure 12(b), except that it has a higher growth rate  than TW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The curves from the Pr = 28 flow converge into a larger number of intermediate,  travelling jumps before merging into a single jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This pattern resembles that of the more  unstable I flow from figure 12(d), except that it has a smaller growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' than TW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In figure 16 we compare the characteristics 𝜆𝑅(𝑥) (panel a) and 𝜆𝐼 (𝑥) (panel b) as well  as the composite Froude number 𝐺2(𝑥) at 𝑡 = 110 (the Pr = 7 data was already shown in  figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' All curves (solid red, dashed blue, and dotted green) have essentially the same  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  (a) Pr = 1  (b) Pr = 7 (TW)  (c) Pr = 28  200  ← 150  100  0  30 -30  0  30  30  0  30 -30  x  x  rPr= 1 --·Pr=7 ---Pr=28  R  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  (q)  ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  4  2  0  30  0  30  x22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  qualitative features described earlier in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, as noted in figure 15, the growth  rate appears to decrease slightly with Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The synoptic features of the flow governed by long waves, therefore, appear relatively  unaffected by Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This can be rationalised by the fact that Pr will primarily influence the  thickness of the density interface separating the two layers, rather than its location 𝜂 (the  locus of 𝜌 = 0) or the speed of the flow (𝑄), which are the two key model variables in the  SWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  We expect short waves to be more strongly influenced by a decreasing thickness of the  density interface with increasing Pr, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, the short waves observed in  our DNS at Pr = 28 and in experiments at 𝑃𝑟 = 700 are Holmboe waves, not the KH waves  supported by our two-layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Unlike the KH instability caused by a single vortex sheet,  the Holmboe instability is caused by the resonance of vorticity waves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' on the edges of a  diffuse velocity interface) with a non-collocated gravity wave (on a sharper density interface)  (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Lefauve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2018) performed a linear stability analysis on the  experimentally measured mean flow (at Re = 440, Pr = 700), including viscosity and scalar  diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' They found that intermediate 1 ≲ 𝑘 ≲ 2 Holmboe waves were most unstable (see  their figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, tackling Holmboe waves – and their presumed coexistence with  the long waves governing hydraulic processes at high Pr SID – would require a three-layer  model (for velocity) mixed with a two-layer model (for density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Sharp vs smooth two-layer flow profiles  Finally, we study the influence of smooth density and velocity profiles U(𝑧) and R(𝑧) on  the growth rate of long waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' To do so, we solve the eigenvalue problem from the Taylor-  Goldstein equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='18) before the two-layer base flow ansatz (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Numerical solutions  for the growth rate 𝑐𝐼 are shown in figure 17 as functions of 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In panel a, we show the  results for hyperbolic-tangent U(𝑧)/𝑄 = R(𝑧) = tanh 𝑧/𝛿 where the interface thickness is  progressively decreased from 1/80 (almost exactly two layers, solid lines) to 1/8 (dashed  lines) to 1/4 (thicker interface, dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We compare these ‘smooth tanh’ growth rates (in  black) to the equivalent ‘sharp two-layer’ growth rates (in red) obtained from the analytical  dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) by layer-averaging the tanh profiles (in which case 𝑐𝐼 = 𝜆𝐼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In  panel b, we keep the same density profiles but use U(𝑧)/𝑄 = − sin 𝜋𝑧, which is a good  approximation of the mean velocity at these relatively low values of Re = 𝑂(102 − 103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Both panels a and b show that the 𝑄𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 threshold for long-wave instability is virtually  unchanged by smooth profiles, with only a slight increase of a few percent for the thickest  interface 𝛿 = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This result supports the relevance of two-layer hydraulics even in ‘real-  world’ flows which depart significantly from the two-layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  However, the ‘smooth tanh’ growth rates are always lower than the corresponding ‘sharp  two-layer’ growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The thicker the interface, the lower the ‘smooth tanh’ growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Comparing the vertical scale in panels a and b, we conclude that the sinusoidal velocity profile  is more stable (by approximately a factor of 10) than the tanh profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The combination of  a sinusoidal velocity and a diffuse velocity interface (dotted blue line in panel b) yields the  slowest growth as 𝑄 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' As such profiles are a better approximation of the mean flow  of the DNS at low Pr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Pr = 1) than sharp two-layer profiles, these results warn us not to  interpret the large growth rates 𝜆𝐼 = 𝑂(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1) found in this paper too literally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In other words, although the qualitative predictions of two-layer hydraulics are robust (in  particular the long-wave instability threshold 𝑄𝑐) when the underlying data is not exactly  two-layer, the quantitative growth rates predictions are over-estimated when the interface is  diffuse, as in low-Pr flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  SID: two-layer hydraulics and instabilities  23  Figure 17: Growth rate 𝑐𝐼 of long waves on smooth velocity and density profiles, obtained  by a numerical solution of the TGE (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='18) as the volume flux 𝑄 is increased (black  curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (a) Hyperbolic-tangent profiles for velocity U(𝑧) and density R(𝑧) profiles of  increasing interface thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (b) Sinusoidal profiles for velocity U(𝑧) (and same R(𝑧) as  in (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The growth is always slower than it would be using the sharp two-layer analytical  solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20) (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Conclusions  In this paper, we employed a two-layer averaging procedure to extract a reduced-order  representation of four direct numerical simulations (DNS) datasets in the stratified inclined  duct (SID) at Re = 650 and Pr = 7 (with two supplementary datasets at Pr = 1 and Pr = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  This two-layer representation revealed in §3 that the flow is stable in the laminar regime  (L, tilt angle 𝜃 = 2◦), but develops an internal hydraulic jump (discontinuity in the layer  properties) in the centre of the duct in the stationary wave regime (SW, 𝜃 = 5◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This jump  moves around in the travelling wave (TW, 𝜃 = 6◦) regime, and causes further disorganised  wave breaking in the intermittently turbulent (I, 𝜃 = 8◦) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Modelling results  To understand these findings, in §4 we adapted to SID DNS the well-known inviscid  Boussinesq shallow water equations (SWE) governing the nonlinear evolution of long waves  (𝑘 ≪ 1) at a sharp density interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The SWE predict that information propagates along  a pair of trajectories, 𝜆1,2, that arise from the solution of a generalised eigenvalue problem  and depend on the local (𝑥) and instantaneous (𝑡) state of the two-layer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The  solutions can be written in the form 𝜆1,2 = ¯𝜆 ± 𝛿𝜆, where the convective velocity ¯𝜆 is  always real but the phase speed 𝛿𝜆 may be either real or imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' When 𝜆1,2 are real  (𝛿𝜆 ∈ R), they represent the propagation of two (neutrally stable) kinematic waves where  ¯𝜆 can be interpreted as a convective velocity and 𝛿𝜆 as the phase speed of waves relative  to ¯𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The respective signs of 𝜆1,2 determine the direction of information propagation and  whether the flow is subcritical (composite Froude number 𝐺2 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' product 𝜆1𝜆2 < 0)  with information propagating in both directions, or supercritical (𝐺2 > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 𝜆1𝜆2 > 0) with  information propagating only in the direction given by the sign of ¯𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' When 𝜆1,2 are complex  𝜆1,2 = 𝜆𝑅 ± 𝑖𝜆𝐼 = ¯𝜆 ± 𝑖|𝛿𝜆|, the real part still represents a convective velocity that carries  information while the positive imaginary part indicates that the flow is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Although in  )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='14  a  R= -tanh(80 z)  U  R = -tanh(80 z)  QU-  -- R = -tanh(8 z)  = R= -tanh(8 z)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' R = - tanh(4 z)  QU  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='8  = R= -tanh(4 z)  Q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='08  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='02  0  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  Q  Q24  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  this unstable case the SWE are no longer hyperbolic, the flow may be viewed as supercritical  (𝐺2 > 1) in the sense that information is propagated only in the direction given by ¯𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  To interpret the unstable SWE, we compared the characteristics with the dispersion  relation from the inviscid Taylor-Goldstein equation (TGE) governing the linear normal-  mode stability of a two-layer base flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We showed that the global nonlinear characteristics  𝜆(𝑥, 𝑡) defined on the non-parallel base flow and sloping interface of the SWE could be  interpreted locally as the phase speed and growth rate of linear waves in the long-wave limit  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 𝑘 ≪ 1), propagating on a base flow that is locally assumed parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Importantly, this  interpretation is only valid for waves that are much shorter than the duct length 2𝐴 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  𝑘 ≫ 𝐴−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' It provides a local, linear stability interpretation for unstable two-layer wave  characteristics satisfying 𝐴−1 ≪ 𝑘 ≪ 1, which is a relevant range in long ducts (𝐴−1 ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The dispersion relation for the dispersive TGE waves 𝑐(𝑘) tend to the non-dispersive SWE  characteristics 𝜆 as 𝑘 ≪ 1, but they also allow us to explore the smooth transition to shorter  (𝑘 � 1), non-hydrostatic Kelvin-Helmholtz (KH) waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Physical results  Applying this two-layer hydraulics and instability framework to the two-layer-averaged DNS  datasets yielded the main physical results of this paper in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We provided the first direct  evidence that SID flows are, in all regimes (L, SW, TW and I), hydraulically controlled at the  ends of the duct and thus in a state of maximal exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' At these control points, the flow is  locally supercritical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' thus information from the reservoirs cannot enter the duct and influence  the flow within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the SW, TW and I regime, the flow in the duct is always unstable to  long waves (𝐹2  Δ > 1) and thus supercritical (𝐺2 > 1), explaining the existence of an undular  jump within the duct, as a consequence of characteristic trajectories converging to a single  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' In the I regime, multiple local jumps gradually merge into clusters and eventually into  a single, stronger jump, resulting in greater instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The emergence of unstable, supercritical flow in SID beyond a certain tilt angle is  rationalised by the fact that gravitational forcing continuously provides a surplus of kinetic  energy which must be dissipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' From a hydraulics perspective, the required dissipation in a  supercritical flow (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' having a surplus of kinetic energy compared to potential energy) must  be associated with an decrease in kinetic energy and an increase in potential energy, hence a  thickening of both layers downstream of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The physical insight of energy surplus and  dissipation dates back to Meyer & Linden (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' It was later formalised by Lefauve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (2019) and Lefauve & Linden (2020), who showed using frictional two-layer hydraulics with  a tilt 𝜃 that the mid-duct interfacial slope obeyed 𝜂′(0) ∝ 𝜃 − 𝐹, where 𝐹 represents viscous  friction along the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The transition from subcritical to supercritical flow that we identified  corresponds to the transition from ‘lazy’ to ‘forced’ flows, which they identified based on  the relative importance of the tilt 𝜃 and the duct geometric angle 𝛼 = tan−1 𝐴−1 ≈ 2◦ in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' ‘Lazy’ flows are characterised by 𝜃 < 𝛼, and an interface gently sloping down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  ‘Forced’ flows are characterised by 𝜃 > 𝛼, and a relatively flat interface all along the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  In forced flows, the tendency of 𝜃 to tilt up the density interface exceeds the tendency of  frictional losses 𝐹 to tilt it down, thus 𝜂′(0) > 0, which causes central jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Next, we rationalised the values of the volume flux in all regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The value 𝑄 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3 in  the L regime (lazy flow) is explained by the offset of the interface at the ends of the duct  where control (𝐺2 = 1) takes place, recalling that the sloping interface is caused by viscous  friction along the duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The robust values 𝑄 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 in the unstable SW, TW, and I regimes  (forced flows) are all surprisingly close to the instability threshold 𝑄𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5 for a symmetric  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Increasing instability from SW to TW to I as the tilt angle 𝜃 is increased (which  theory predicts should increase 𝑄 above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5) appears balanced by increasing mixing in the  layers (which decreases 𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This explains why the maximal exchange threshold 𝑄 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  SID: two-layer hydraulics and instabilities  25  predicted by inviscid long wave theory remains a remarkably robust feature of SID flows,  even under turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Using the TGE analysis provided further physical insight into the applicability of unstable  hydraulics in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We showed that short inviscid KH waves are always predicted to be more  linearly unstable than long waves, despite the fact that long waves cause the internal hydraulic  jumps observed in SW, TW and I and appear to dominate the dynamics of these SID flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  We explained this paradox by the neglect of viscosity in TGE, which would damp the shortest  waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We also showed that DNS at lower Pr = 1 or higher Pr = 28 showed qualitatively (but  not quantitatively) similar two-layer long wave hydraulics to Pr = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We also highlighted that  experimental observations at Pr = 700 of the simultaneous existence of unstable long waves  (causing an internal jump and supercritical flow) with short finite-amplitude Holmboe waves  could not be explained by the two-layer model, because it does not support Holmboe waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Finally, we showed that the predictions of long wave instability (especially the threshold  𝑄𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5) were robust even in diffuse two-layer flows having a thick interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Outlook  These key ‘hydraulic’ features of SID flows, explaining the emergence of waves, increasingly  supercritical jumps, and ultimately turbulence, result from long wave instability which  (tautologically) relies on the existence of top and bottom solid boundaries confining the  flow in a long, tilted duct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This ‘long-wave’ pathway to turbulence in SID appears a priori  to differ from the classical ‘short-wave’ KH pathway in an unbounded stratified shear layer  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Caulfield & Peltier (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Mashayek & Peltier (2012)), often used as a paradigm  for ocean mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Further work is needed to clarify the relative importance of long and  short waves, and within short waves, of Kelvin-Helmholtz (two-layer) waves and Holmboe  (three-layer) waves, and how they contribute to the transition to turbulence under varying  𝜃, Re, Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The current formulation of the two-layer SWE does not account for the mixing layer that  develops and appears to be important beyond the wave regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The omission of a mixed  layer may reduce the accuracy when investigating detailed spatial features of the flow, such  as the localization of unstable wave regions in the centre of the duct in TW and SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' This  model is able to predict the formation of shocks and provide clues to the long-length-scale  dynamics and how they govern the synoptic features of the flow, but further work is needed  to accurately represent the non-hydrostatic processes of turbulent mixing itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Acknowledgments  We acknowledge support from the European Research Council (ERC) under the European  Union’s Horizon 2020 research and innovation Grant No 742480 ‘Stratified Turbulence And  Mixing Processes’ (STAMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Parts of the simulations were carried out with resources from  Compute/Calcul Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' is supported by a Leverhulme Trust Early Career Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  For the purpose of open access, the authors have applied a Creative Commons Attribution  (CC BY) licence to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Declaration of interests  The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Hydrostatic and non-hydrostatic pressure gradients  The assumptions behind the SWE require flows to be dominantly hydrostatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' To validate  this assumption, we explicitly decompose the non-dimensional pressure into a hydrostatic  26  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  Figure 18: Cumulative Distribution Function (CDF) of the ratio between non-hydrostatic  and hydrostatic streamwise pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  component defined by  𝑝ℎ(𝑥, 𝑧, 𝑡) = −𝑅𝑖 cos 𝜃  ∫  𝑧  −1  ⟨𝜌⟩𝑦(𝑥, 𝜉, 𝑡) d𝜉,  (A 1)  and the remaining non-hydrostatic (but still spanwise averaged) component  𝑝𝑛ℎ(𝑥, 𝑧, 𝑡) = ⟨𝑝⟩𝑦 − 𝑝ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (A 2)  Figure 18 shows the cumulative density function of the magnitude ratio between 𝜕𝑝ℎ/𝜕𝑥  and 𝜕𝑝𝑛ℎ/𝜕𝑥 in all five datasets L, SW, TW, I and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We find that the hydrostatic pressure  gradient dominates over the non-hydrostatic gradient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' |𝜕𝑝𝑛ℎ/𝜕𝑥|/|𝜕𝑝ℎ/𝜕𝑥| < 1 in  ⩾ 80% of the data in the L, SW, TW, and I cases, and ⩾ 60% of the data even in the  T case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Non-hydrostatic effects, therefore, play a secondary role in all but the T case, and the  two-layer SWE are expected to model adequately the primary two-layer dynamics of SID  flows forced by relatively small tilt angles 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' However, as the flow becomes turbulent (T  case), non-hydrostatic effects become important and the applicability of the shallow water  model breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Turbulence also creates a third layer of intermediate density (see Zhu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' (2022), figure 5(e,j)) and the two-layer model also breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' For these reasons, we  exclude this T case from the analyses of this paper and focus on the L, SW, TW, and I cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Derivation of the Taylor-Goldstein equation and solution  In this section, we provide additional details regarding the derivation of the inviscid  Boussinesq Taylor-Goldstein equation (TGE) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='18) for a two-layer SID flow and the  dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Governing equation  The linearised inviscid equations for two-dimensional wave-like perturbations [ ˜𝜓, ˜𝜌, ˜𝑝] =  [ ˆ𝜓, ˆ𝜌, ˆ𝑝](𝑧) exp𝑖𝑘(𝑥 − 𝑐𝑡) around a parallel base flow [U, R](𝑧) are  (U − 𝑐) ˆ𝜓′ − ˆ𝜓U′ = − ˆ𝑝 − 𝑖  𝑘 Ri sin 𝜃 ˆ𝜌,  (B 1)  𝑘2 (U − 𝑐) ˆ𝜓 = − ˆ𝑝′ − Ri cos 𝜃 ˆ𝜌,  (B 2)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='6  CDF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='4  TW  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='0  0pnh/0x//l0ph/axlSID: two-layer hydraulics and instabilities  27  (U − 𝑐) ˆ𝜌 − ˆ𝜓R′ = 0,  (B 3)  By taking the 𝑧 derivative of (B 1) and using (B 2) and (B 3) we derive the general TGE as  (U − 𝑐)  � 𝑑2  𝑑𝑧2 − 𝑘2  �  ˆ𝜓 − U′′ ˆ𝜓 − Ri cos 𝜃 R′  U − 𝑐  ˆ𝜓 = 𝐹,  (B 4)  with forcing 𝐹 = − 𝑖  𝑘 Ri sin 𝜃  �  R′  U − 𝑐  ˆ𝜓′ +  R′′  U − 𝑐  ˆ𝜓 −  U′R′  (U − 𝑐)2 ˆ𝜓  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  The streamwise component of the gravitational force Ri sin 𝜃 appears multiplied by 𝑖 such  that even if 𝑐 is real (the waves are stable) increasing the tilt angle will eventually lead to  instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' We neglect this effect here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  For small tilt angles, we assume for simplicity 𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' These unforced TG equations  under small tilt angles will be the focus of the following stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The unforced TG  equation then (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='e 𝐹 = 0) is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Taking the base flow as the two-layer piecewise constant profiles in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='19) leads to U′ =  (𝑢1−𝑢2) 𝛿(𝑧) and R′ = (𝜌1−𝜌2) 𝛿(𝑧), where 𝛿 is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Since U′′ = R′ = 0  everywhere except at the interface, the TGE becomes trivial  ˆ𝜓′′ − 𝑘2 ˆ𝜓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (B 5)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Solution with solid top and bottom walls  In this bounded duct configuration, we take a solution of the form  ˆ𝜓 =  �  𝐶1 sinh 𝑘(ℎ1 − 𝑧)  0 < 𝑧 ⩽ ℎ1,  𝐶2 sinh 𝑘(ℎ2 + 𝑧)  −ℎ2 ⩽ 𝑧 < 0,  (B 6)  which satisfies the no-penetration condition at 𝑧 = −ℎ2 and 𝑧 = ℎ1 ( ˆ𝑤 = 𝑖𝑘 ˆ𝜓 = 0) modelling  the presence of solid walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Following Drazin & Reid (2004), the matching conditions are derived by integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='18)  over the neighbouring region of the interface, and by using the integral property of the Dirac  delta function, leading to  ⟦  ˆ𝜓  U − 𝜆⟧0 = 0,  (B 7)  ⟦U ˆ𝜓′ − 𝑐 ˆ𝜓′⟧0 + Ri Δ𝜌 cos 𝜃  �  ˆ𝜓  U − 𝑐  �  𝑧=0  = 0,  (B 8)  where we recall that Δ𝜌 ≡ 𝜌2 − 𝜌1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The first condition (B 7) guarantees continuity of the  streamfunction across the interface (and thus the wall-normal velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Together, the first  and second term in the second condition (B 8) guarantees continuity of the pressure modes  ˆ𝑝 based on (B 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Note that ⟦U′ ˆ𝜓⟧0 vanishes as U′ = 0 on either side of the interface and  U′ → ∞ at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' The last term in (B 8) guarantees continuity of the density modes  ˆ𝜌 based on (B 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Using these two conditions we can solve for 𝐶1 and 𝐶2 leading to the  following algebraic system of equations:  𝐶1 sinh 𝑘ℎ1(𝑢2 − 𝑐) − 𝐶2 sinh 𝑘ℎ2(𝑢1 − 𝑐) = 0,  (B 9)  �  − (𝑢1 − 𝑐) 𝑘 cosh 𝑘ℎ1 + Ri Δ𝜌  2  cos 𝜃 sinh 𝑘ℎ1  𝑢1 − 𝑐  �  𝐶1 +  (B 10)  �  − (𝑢2 − 𝑐) 𝑘 cosh 𝑘ℎ2 + Ri Δ𝜌  2  cos 𝜃 sinh 𝑘ℎ2  𝑢2 − 𝑐  �  𝐶2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  28  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Atoufi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Zhu, and others  To have a non-trivial solution, the determinant of the above 2 × 2 system must be zero,  resulting in the dispersion relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Solution without solid top and bottom walls  In the unbounded configuration, we instead take a solution of the form  ˆ𝜓 = 𝐶 exp (−𝑘|𝑧|),  (B 11)  satisfying continuous and finite ˆ𝜓 for all 𝑧 (Smyth & Carpenter 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Substituting (B 11) into (B 8), we obtain  − (𝑢1 − 𝑐)2𝑘 − (𝑢2 − 𝑐)2𝑘 − Ri Δ𝜌 cos 𝜃 = 0,  (B 12)  and thus the dispersion relation for KH waves plotted in figure 13(c)  𝑐 = 𝑢1 + 𝑢2  2  ±  √︄  Ri Δ𝜌  2 cos 𝜃  𝑘  − (𝑢1 − 𝑢2)2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (B 13)  The KH instability (𝑐𝐼 ≠ 0) is thus found for  𝑘 > 2 Ri Δ𝜌 cos 𝜃  (𝑢1 − 𝑢2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  (B 14)  REFERENCES  Armi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content='  Baines, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Drazin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' 2001 Frictional exchange flow through a wide channel with application to the burlington ship canal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' Journal of Fluid  Mechanics 254, 605–633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' part 1 shear aligned convection, pairing, and braid instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' & Carpenter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Thorpe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 1968 A method of producing a shear flow in a stratified fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Journal of Fluid Mechanics  32 (04), 693–704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Thorpe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Malarkey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Voet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Alford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Girton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' & Carter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2018a Application  of a model of internal hydraulic jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Journal of Fluid Mechanics 834, 125–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Thorpe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Malarkey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Voet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Alford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Girton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' & Carter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2018b Application  of a model of internal hydraulic jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Journal of Fluid Mechanics 834, 125–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Whitham, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2011 Linear and nonlinear waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content='  Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Atoufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Lefauve, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Lawrence, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Dalziel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=', Kerswell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' &  Linden, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' 2022 Stratified inclined duct: direct numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
+page_content=' Submitted to Journal of  Fluid Mechanics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9FPT4oBgHgl3EQfOzQ4/content/2301.13035v1.pdf'}
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+IOP Publishing 
+Journal Title 
+Journal XX (XXXX) XXXXXX  
+https://doi.org/XXXX/XXXX 
+xxxx-xxxx/xx/xxxxxx 
+1 
+© xxxx IOP Publishing Ltd 
+ 
+Brain Model State Space Reconstruction Using an 
+LSTM Neural Network 
+Yueyang Liu1, Artemio Soto-Breceda2, Yun Zhao1, Phillipa Karoly2, Mark J. Cook3, David B. Grayden2, Daniel 
+Schmidt1, Levin Kuhlmann1 
+Abstract 
+Objective 
+Kalman filtering has previously been applied to track neural model states and parameters, particularly at the scale relevant 
+to electroencephalography (EEG). However, this approach lacks a reliable method to determine the initial filter conditions and 
+assumes that the distribution of states remains Gaussian. This study presents an alternative, data-driven method to track the 
+states and parameters of neural mass models (NMMs) from EEG recordings using deep learning techniques, specifically a Long 
+Short-Term Memory (LSTM) neural network. 
+Approach 
+An LSTM filter was trained on simulated EEG data generated by a neural mass model using a wide range of parameters. 
+With an appopriately customised loss function, the LSTM filter can learn the behaviour of NMMs. As a result, it can output 
+the state vector and parameters of NMMs given observation data as the input. 
+Main Results 
+Test results using simulated data yielded correlations with R squared of around 0.99 and verified that the method is robust 
+to noise and can be more accurate than a nonlinear Kalman filter when the initial conditions of the Kalman filter are not accurate. 
+As an example of real-world application, the LSTM filter was also applied to real EEG data that included epileptic seizures, 
+and revealed changes in connectivity strength parameters at the beginnings of seizures.  
+Significance 
+Tracking the state vector and parameters of mathematical brain models is of great importance in the area of brain modelling, 
+monitoring, imaging and control. This approach has no need to specify the initial state vector and parameters, which is very 
+difficult to do in practice because many of the variables being estimated cannot be measured directly in physiological 
+experiments. This method may be applied using any neural mass model and, therefore, provides a general, novel, efficient 
+approach to estimate brain model variables that are often difficult to measure.  
+Keywords: LSTM Neural Network, Neural Mass Model, Neurophysiological Process Imaging, EEG, Epilepsy 
+ 
+1 Department of Data Science and AI, Faculty of Information Technology, 
+Monash University, Clayton, Australia 
+2 Department of Biomedical Engineering, The University of Melbourne, 
+Parkville, Australia 
+3 Department of Medicine, St Vincent’s Hospital, The University of 
+Melbourne, Fitzroy, Australia 
+ 
+ 
+ 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+2 
+ 
+ 
+1. Introduction 
+Understanding the brain is one of the most challenging 
+problems of science, engineering and medicine. Currently, 
+due 
+to 
+limitations 
+of 
+existing 
+brain 
+imaging 
+and 
+neurophysiological techniques, many neurophysiological 
+variables cannot be measured directly (1,2). Moreover, if they 
+can be measured directly, we cannot do so for all neurons in 
+the brain let alone a small brain region. For instance, current 
+techniques relevant to human applications, such as scalp or 
+intracranial electroencephalography (EEG), record the 
+activities of all the neurons of different population types with 
+a heavy weighting towards cortical pyramidal neurons (3). 
+However, EEG does not have the resolution to track single cell 
+activity or connections between neurons or distinct 
+populations. Brain modelling provides an alternative way to 
+understand brain parameters and neural activity indirectly, and 
+can be used to study healthy brain function as well as diseases 
+like epilepsy, Alzheimer’s disease or Parkinson’s disease (4–
+8). One of the aims of modelling is to capture essential 
+temporal and/or spatial characteristics of the brain or brain 
+regions of interest, and to understand how they can emerge 
+from underlying neurophysiological variables such as neural 
+membrane potentials, synaptic strengths and time constants. 
+ 
+At the level of EEG modelling, several approaches have 
+been applied using NMMs to infer these crucial variables at 
+the neural population level (9–15). Different methods analyse 
+the EEG in either the time domain or the frequency domain. 
+One of the most common inference frameworks is Dynamic 
+Causal Modelling (DCM), which is usually applied in the 
+frequency 
+domain 
+(14). 
+Frequency-domain 
+inference 
+potentially offers greater stability in the estimates because 
+they are determined over windows of data.  
+On 
+the 
+other 
+hand, 
+time-domain 
+inference 
+of 
+neurophysiological variables affords the possibility of 
+capturing instantaneous transitions in these variables. This 
+could be used to understand temporal variations in neural 
+processing in the brain or enable timely control interventions 
+(16) such as electrical stimulation to steer the brain away from 
+an epileptic seizure state. These time-resolved variable 
+estimates, in particular model parameters such as population 
+averaged synaptic strengths, can also potentially provide 
+information about the current brain state (e.g., seizure, non-
+seizure, pre-seizure, short seizure, long seizure). This is 
+because inferred model parameters can be used to determine 
+where the brain dynamics are currently positioned with respect 
+to the model’s bifurcation space. Moreover, such information 
+could be used to make predictions, or to detect, different brain 
+states. 
+Inferring population averaged membrane potentials, 
+synaptic strengths and time constants, either locally or across 
+the brain from a limited number of EEG measurements, is a 
+challenging task with observability issues at play (17). 
+Depending on the spatial resolution of the population 
+averaging, there are typically fewer EEG measurements than 
+the number of variables being inferred. Although neural 
+population models can generate data resembling EEG, there 
+are no clear mathematical solutions for the model parameters 
+given the observation data. Indeed, there may be multiple 
+solutions within the state space given the observation data. 
+That is to say, there are multiple mathematical solutions to the 
+same observation and not all are correct. Even though some of 
+the solutions are mathematically achievable, it does not mean 
+they are biologically explainable. Developing a technique that 
+can track neurophysiological processes that is reasonable both 
+mathematically and biologically will open new vistas for 
+tracking and imaging brain states, as well as controlling them. 
+Regarding methods that have been used for time domain-
+based inference of neurophysiological processes given EEG 
+data, the Kalman filter and its nonlinear variants have been 
+applied (10,12,13,18). They address the problem of estimating 
+the state of a discrete-time controlled process through 
+stochastic difference equations (19). With the covariance 
+matrix of the process and measurement noise, a Kalman filter 
+balances the model and measurement noise to track the state 
+vector of the model. However, it is a local search method that 
+requires an initial guess of the expected state vector and its 
+covariance. As a result, state vector estimates can deviate far 
+from the true state vector if the initial guess is not close to the 
+correct state vector. Furthermore, a solution to the Kalman 
+gain and the covariance is needed. Since an arbitrary NMM 
+can potentially be very high dimensional, the solution is not 
+always easy to compute, and the initial guess is hard to 
+determine because the neurophysiological variables cannot be 
+measured in most cases. A solution that does not require an 
+initial guess and is easy to apply given an arbitrarily complex 
+model is needed.  
+Long Short-Term Memory (LSTM) is a recurrent neural 
+network that can “remember” older information in a time-
+series sequence (20). Therefore, it is a potential approach to 
+solve the problem of inferring underlying neurophysiological 
+processes (i.e., the state vector and parameters of NMMs) 
+from EEG data (21,22). In this paper, we compare the LSTM 
+approach with a nonlinear variant of Kalman Filtering. 
+ 
+2. Methodology 
+2.1 Brain Modelling 
+Scalp and intracranial EEG record electrical activity signals 
+captured on the scalp and on the surface of the brain, 
+respectively. 
+These 
+signals 
+primarily 
+represent 
+the 
+macroscopic activity of cerebral cortex. A single EEG signal 
+is recorded by a single electrode, and is a spatially and 
+temporally smoothed version of neuronally generated 
+electrical potentials (23). Intracranial EEG is invasive and 
+difficult to place enough electrophysiological recording 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+3 
+ 
+ 
+electrodes in a given cortical location to understand how the 
+underlying neurophysiological processes give rise to the EEG. 
+As a result, modelling is a way to bridge the EEG data and the 
+underlying neurophysiological processes. 
+There are predominantly two approaches to build a 
+mathematical model of the brain: the “bottom-up approach” 
+and the “top-down approach” (24). The bottom-up approach 
+forms a macroscopic model of the brain or a brain region by 
+coupling a network of microscopic models of individual 
+neurons (e.g., Hodgkin-Huxley neurons). To obtain the spatial 
+scale relevant to modelling EEG signals, the large number of 
+neurons involved in this kind of model, as well as the 
+heterogeneity and non-local connectivity of neurons, makes 
+this largely impractical for understanding which key 
+population-level neurophysiological variables influence 
+macroscopic dynamics.  
+On the other hand, the top-down approach defines 
+dynamical models based on phenomenological observations 
+and emergent behaviour. This means that the mathematical 
+model is built to model large-scale signals, such as EEG, 
+through the interactions of neural populations, where the 
+neural populations only coarsely model the average properties 
+of the underlying detailed neuronal networks. Although it 
+would be useful to model and infer the detailed microscopic 
+neurophysiological processes, this is difficult to do as a result 
+of the aforementioned experimental and observability issues 
+as well as computational tractability. As a result, we focus here 
+on inferring population-level neurophysiological processes 
+using top-down models. 
+We focus on one of the most common NMMs used to 
+model cerebral cortex, the Jansen-Rit model (9,25). The 
+Jansen-Rit model was developed based on the model proposed 
+by Lopes Da Silva et al. (26), who proposed a two-population 
+neural field model to study spontaneous EEG generation and 
+alpha frequency rhythms, and reduced the complexity of the 
+model by aggregating the activities within a single cortical 
+column to perform analyses. Jansen et al. (25) constructed a 
+similar model of pyramidal neurons to show that the 
+mechanisms that guide the evolution of evoked potentials and 
+spontaneous EEG signal generation are the same. Jansen and 
+Rit (9) extended the model with a third population of local 
+excitatory interneurons that provide feedback to the pyramidal 
+neuron population. The latter model has three populations: 
+pyramidal neurons, inhibitory interneurons and excitatory 
+interneurons. 
+Figure 1 illustrates the model. The spatially averaged post-
+synaptic potential of population 𝑚 arising as a result of input 
+from pre-synaptic population 𝑛 is expressed as 
+ 
+𝑣𝑚𝑛(𝑡) = α𝑚𝑛 ∫
+ℎ𝑚𝑛(𝑡 − 𝑡′)ϕ(𝑣𝑛(𝑡′))𝑑𝑡′
+𝑡
+−∞
+,   
+[1] 
+ 
+where α𝑚𝑛 is the population averaged synaptic connectivity 
+strength. The post-synaptic response kernel is given by 
+ 
+ℎ𝑚𝑛(𝑡) = η(𝑡)
+𝑡
+τ𝑚𝑛 exp (−
+𝑡
+τ𝑚𝑛),   
+[2] 
+ 
+Figure 1. Jansen & Rit neural mass model and time series recordings and parameter values. Structure of the model. The model consists 
+of three populations: pyramidal neurons, inhibitory interneurons and excitatory interneurons. The output of each population is a firing 
+rate, which is transformed to changes in the mean membrane potentials of connected neural populations.  
+
+external input
+Vpi
+supragranularlayers
+inhibitory
+inhibitory interneurons
+interneurons
+excitatory
+αpi
+granular layer IV
+interneurons
+excitatoryinterneurons
+Vpe
+αpi
+u
+αip
+ape
+infragranularlayers
+pyramidal cells
+Vep
+Vip
+pyramidal
+population
+EEGJournal XX (XXXX) XXXXXX 
+Author et al  
+ 
+4 
+ 
+ 
+where η(𝑡) is the Heaviside step function and τ𝑚𝑛  is the 
+population averaged synaptic time constant. Moreover, ϕ(𝑣) 
+is a sigmoid function that transforms the membrane potential 
+to a firing rate given by 
+ 
+ϕ(𝑣) =
+1
+2 (erf (
+𝑣−𝑣0
+ς ) + 1), 
+[3] 
+ 
+where 𝑣0  is a threshold and ς is the slope of the sigmoid 
+function, which is also the variance of firing thresholds of the 
+population assuming the firing thresholds follow a Gaussian 
+distribution. The mean membrane potential of population 𝑚 is 
+the sum of the synaptic contributions, 
+ 
+𝑣𝑚(𝑡) = ∑
+𝑣𝑚𝑛(𝑡)
+𝑁
+𝑛=1
+. 
+[4] 
+ 
+The convolution in equation [1] can be also written as two 
+coupled, first-order, ordinary differential equations (27), 
+ 
+𝑑𝑣𝑚𝑛
+𝑑𝑡
+= 𝑧𝑚𝑛 
+[5] 
+ 
+𝑑𝑧𝑚𝑛
+𝑑𝑡
+=
+α𝑚𝑛
+τ𝑚𝑛 ϕ𝑚𝑛 −
+2
+τ𝑚𝑛 𝑧𝑚𝑛 −
+1
+τ𝑚𝑛
+2
+𝑣𝑚𝑛, 
+[6] 
+ 
+where τ𝑚𝑛 is a lumped time constant, and 𝑚 and 𝑛 indicate 
+the pre-synaptic and post-synaptic neural population, 
+respectively. 
+ 
+Furthermore, we can express the neural mass model in 
+matrix-vector notation so the operations can be expressed 
+more compactly. The neural mass model can be expressed in 
+matrix notation as 
+ 
+𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝜙⃗ (𝐶𝑥(𝑡)) 
+[7] 
+𝑦(𝑡) = 𝐻𝑥(𝑡) + 𝑣(𝑡), 
+[8] 
+ 
+where 𝐻  is the observation matrix, 𝑣(𝑡)~𝑁(0, 𝑅)  is the 
+observation noise (10) and 𝑦(𝑡) is the membrane potential of 
+the pyramidal population, which is considered to be the 
+contributor to the generation of the EEG signal in our model. 
+The matrix 𝐴 is defined by the membrane time constants, 
+𝐴 =  diag(Ψ1 … Ψ𝑛), 
+[9] 
+ 
+where  
+ 
+Ψ𝑛 = [
+0
+1
+−
+1
+τ2𝑚𝑛
+−
+2
+τ𝑚𝑛
+]. 
+[10] 
+ 
+The matrix 𝐵 is the synaptic gains from internal inputs, 
+which has the form  
+ 
+𝐵 = diag(0 a1 … 0 a𝑛). 
+[11] 
+ 
+The matrix 𝐶  is the adjacency matrix that defines the 
+connectivity structure of the model, 
+ 
+𝐶 =
+[
+ 
+ 
+ 
+ 0
+0
+…
+0
+0
+𝑐2,1
+0
+ 
+𝑐2,𝑁𝑥−1
+0
+⋮
+ 
+⋱
+ 
+⋮
+0
+0
+ 
+0
+0
+𝐶𝑁𝑥,1
+0
+⋯
+𝐶𝑁𝑥,𝑁𝑥−1
+0]
+ 
+ 
+ 
+ 
+, 
+[12] 
+ 
+which is a matrix of zeros and ones, where one indicates there 
+is a connection between two populations and zero indicates 
+there is no connection 
+In order to apply a nonlinear Kalman filter to this model to 
+infer both the state vector and the parameters, we define a 
+vector of parameters as 𝜃 = [𝑢 𝛼𝑝𝑒 𝛼𝑝𝑖 𝛼𝑖𝑝 𝛼𝑒𝑝]
+𝑇. This set of 
+parameters corresponds to the input 𝑢 to the model and the 
+population averaged connection strengths between the three 
+populations. Moreover, we assume the time constants of the 
+model to be constant to simplify the estimation problem. The 
+above parameters were combined with the state vector 𝑥 to 
+form the augmented state vector, 
+ 
+ξ = [XTθT]𝑇. 
+[13] 
+ 
+The augmented state-space model is then 
+ 
+ξ𝑡 = 𝐴θξ𝑡−1 + 𝐵θϕ(𝐶θξ𝑡−1) + 𝑊𝑡−1, 
+[14] 
+ 
+where 𝑊𝑡 is Gaussian noise. 
+ 
+2.2 Analytic Kalman Filter 
+Before defining the novel LSTM-based filter for state 
+vector and parameter estimation, this section outlines the 
+Kalman filter applied as a benchmark.  In particular, a filter 
+referred to as the Analytical Kalman filter (AKF) is applied. 
+This filter is highly stable and accurate and was developed in 
+prior work (2,10,27) that evolved from deriving the Kalman 
+filter for general nonlinear NMMs using the specific 
+sigmoidal non-linearity in equation (15). For the sake of 
+brevity, we refer the reader to these prior works for greater 
+mathematical insight. Moreover, links to code provided at the 
+end of this paper give the exact specification of the 
+implementation of the AKF. Here, a brief description of the 
+main computations of the AKF are provided.  
+The aim of the AKF is to find the most likely posterior 
+distribution of the augmented state given the previous 
+measurements under Gaussian assumptions. Such a posterior 
+distribution is characterised by its a posteriori state vector 
+estimate and its covariance, 
+ 
+𝜉̂𝑡
++ = 𝐸[𝜉𝑡 | 𝑦1, 𝑦2, ⋯ , 𝑦𝑡]  
+[16] 
+ 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+5 
+ 
+ 
+𝑃̂𝑡
++ = 𝐸 [(𝜉𝑡 − 𝜉̂𝑡
++)(𝜉𝑡 − 𝜉̂𝑡
++)
+⊤]. 
+[17] 
+ 
+The AKF proceeds in two steps: prediction and update. 
+During prediction, the prior distribution (obtained from the 
+previous estimate) is propagated through the model equations. 
+This step provides the so called a priori estimate, which is also 
+a Gaussian distribution with mean and covariance, 
+ 
+𝜉̂𝑡
+− = 𝐸[𝜉𝑡 | 𝑦1, 𝑦2, ⋯ , 𝑦𝑡−1] 
+[18] 
+ 
+𝑃̂𝑡
+− = 𝐸 [(𝜉𝑡 − 𝜉̂𝑡
+−)(𝜉𝑡 − 𝜉̂𝑡
+−)
+⊤]. 
+[19] 
+ 
+During update, the a posteriori state vector estimate is 
+determined by correcting the a priori state vector estimate 
+with recorded (EEG) data by 
+ 
+𝜉̂𝑡
++ = 𝜉̂𝑡
+− + 𝛫𝑡(𝑦𝑡 − 𝐻𝜉̂𝑡
+−), 
+[20] 
+ 
+where 𝐾𝑡 is the Kalman gain (18), 
+ 
+𝛫𝑡 =
+𝑃̂𝑡
+−𝐻⊤
+𝐻𝑃̂𝑡
+−𝐻⊤+𝑅. 
+[21] 
+ 
+The a posteriori state vector estimate covariance is then 
+updated by using the Kalman gain, 
+ 
+𝑃̂𝑡
++ = (𝛪 − 𝛫𝑡𝐻)𝑃̂𝑡
+−. 
+[22] 
+ 
+After each time step, the a posteriori estimate becomes the 
+prior distribution for the next time step, and the process 
+repeats for each new recorded data point. The AKF requires 
+the a posteriori state vector estimate and its covariance to be 
+initialised at time 𝑡 = 0. The special features of the AKF are 
+that the state vector estimate and its covariance, along with the 
+Kalman gain, were derived using the fully non-linear Jansen-
+Rit model.  
+While the AKF framework can be applied to multichannel 
+EEG data in a multivariate fashion, for simplicity this paper 
+focuses on estimating the state vector and parameters of the 
+Jansen-Rit model using a single channel of EEG data 
+(analogous to the output observation/measurement of the 
+Jansen-Rit model) as input to the filter. 
+2.3 A novel LSTM filter for state and parameter estimation 
+The Recurrent Neural Network (RNN) was introduced 
+mainly to deal with time series data such as speech recognition 
+and natural language processing (28). An ordinary feed 
+forward neural network is only processes independent data 
+points. However, when data points depend on previous data 
+point, in the case of time series data, an RNN incorporates the 
+dependency by providing feedback to the next timestep. In this 
+way, the final output depends not only by the input but also on 
+the outputs of previous neurons. 
+Long Short-Term Memory (LSTM) is a variation of the 
+RNN that has improved its performance (20). Since gradient 
+descent is applied to train neural networks, the gradient might 
+explode or vanish after applying the sigmoid function over and 
+over again, limiting to a certain number of discrete time steps 
+to avoid this problem (29,30). However, the LSTM provides 
+stability to bridge many more discrete time steps by enforcing 
+constant error flow within special memory cells. 
+LSTM neural networks incorporate memory cells and gate 
+units to convey useful information about previous states of the 
+neural network to the current state. An LSTM layer consists 
+of multiple recurrently connected memory blocks and three 
+multiplicative gates, known as input, output and forget gates, 
+that learn to open and close access to the error flow within the 
+memory cell (20,31). For example, an input gate could use 
+inputs from other memory cells to decide whether it should 
+store information in its own memory cell.  
+An extension of the LSTM model is the bidirectional 
+LSTM model that provide more accurate and improved results 
+(32–34). One of the shortcomings of RNNs is that they can 
+only learn from the context of the previous time steps and not 
+anything after the current time step. Bidirectional LSTM can 
+process data in both directions with separated hidden layers, 
+which are then fed to the same output layer (35). Also, since 
+the LSTM is free to access as much or as little of the data 
+within the given time window as needed, predefining a 
+specific time window for the model is not required (31).  
+ 
+Here, a bidirectional LSTM filter is constructed that takes 
+as input simulated or real EEG and predicts the state vector 
+and parameters of the Jansen-Rit model as well as simulated 
+or real EEG. Similar to the AKF, while the framework that 
+follows could potentially be applied to multichannel EEG data 
+in a multivariate fashion, this paper, for simplicity, focuses on 
+using the LSTM filter to estimate the state vector and 
+parameters of the Jansen-Rit model for a single channel of 
+EEG data. 
+The structure of the LSTM model is designed to achieve 
+high performance while maintaining good time efficiency. 
+Since the only non-linearity of the neural mass model is the 
+firing rate function in equation (3), and to achieve high 
+performance 
+while 
+predicting 
+parameters 
+given 
+the 
+observation, we would expect that at least two layers of 
+artificial neural network are required. A grid search 
+experiment was conducted to test the number of neurons in 
+each layer. The selected structure was 128 and 32 neurons in 
+layers 1 and 2, respectively. 
+2.4 Training Data 
+As real EEG data provides no ground truth about the 
+neurophysiological variables that correspond to the state 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+6 
+ 
+ 
+vector and parameters of the Jansen-Rit model, the LSTM 
+filter is trained on simulation data was required. This allows 
+the prediction of the state vector and parameters of the model 
+when inputting either simulated or real EEG into the LSTM 
+filter. It is important to generate a variety of training data with 
+different patterns and a wide range of parameters so the model 
+can learn the relationship between signal patterns and how the 
+state and parameters change. Furthermore, it is also crucial to 
+generate data containing oscillations, since there are likely to 
+be many solutions for a constant signal, or fixed point. This is 
+because when there is no oscillation, different combinations 
+of changes in either the external input, input from the 
+inhibitory or excitatory interneurons may result in the same 
+observation signal. However, changes of the parameters could 
+affect the ranges of external input that generate oscillations, 
+which makes it difficult to find the desired range of external 
+input when there can be different combinations of parameters. 
+This is because the time constants affect the transformation 
+from firing rate to post-synaptic membrane potential thus 
+affecting the ranges over which the population produces 
+oscillations. Thus, an effective method to generate oscillatory 
+data with a wide range of parameters is required. 
+Different combinations of inhibitory and excitatory time 
+constants can generate data with different waveforms, 
+frequencies and amplitudes. Normally, the ranges of both 
+excitatory and inhibitory time constants are considered 
+between 10-60 ms (36). In addition, external input also plays 
+an important role in data generation. External input to the 
+model is also affected by the time constant, such that the range 
+of the external input that can produce oscillations is affected 
+by different combinations of time constants, but the exact 
+range for each combination is unknown. To determine the 
+range of external input and whether the given combination of 
+time constants is able to generate oscillation, an automatic data 
+generation process is designed. 
+Statistical hypothesis testing is used to determine whether 
+an oscillation exists in a given time-series recording. Two tests 
+are used. If the simulated EEG signal is noise-like, then it will 
+have a Gaussian distribution so the Anderson-Darling test (37) 
+is used to test if the data is not Gaussian distributed, which 
+means there could be an oscillation. The other test is the 
+Ljung-Box test (38), which tests whether any autocorrelations 
+of the time series are different from zero. Either of the tests 
+can help determine whether the generated data contains an 
+oscillation and, since it is more important to ensure there are 
+no false positives (data is Gaussian distributed or data has non-
+zero correlation but we reject this hypothesis) to avoid data 
+being generated without including oscillations. We set the 
+significance level to 𝛼 = 10−4. 
+A flow chart used to generate the training data is shown in 
+Figure 2.  We generate a short recording first to check whether 
+this combination of time constants and external input is able 
+to generate data containing oscillations. If not, then the 
+external input is increased until it does. If it still does not 
+produce oscillations for 15 consecutive increases of the input, 
+the current combination of time constants is considered to not 
+be able to produce the desired data and so the next 
+combination is used. If the generated data includes oscillations 
+then the upper and lower bounds of the external input are 
+recorded, and training data is generated with the recorded 
+parameters.  
+The size of the dataset is as large as possible within the size 
+of physical memory, which means that the gap between each 
+combination is kept relatively small and as many examples as 
+possible are provided for training (39). The generated data was 
+shuffled randomly and divided into training, validation and 
+testing data with ratio of 80:10:10. 
+To improve the performance of the model with different 
+amplitudes, the dataset is standardised. As in many cases with 
+either real or simulated data, the raw amplitude of the EEG 
+Figure 2. Flow chart of the generation of training and testing data. 
+The same flow chart is used on different NMMs (the Jansen-Rit 
+model is used in this paper). The amount of the increased external 
+input is not fixed, as the upper bound could be high when the 
+excitatory time constant is high. Thus, the amount is low when the 
+input is low, and is higher when it is high. 
+
+New
+combinations of
+Yes
+time constants
+Simulate a
+Not
+short
+rejectedfor15
+recording
+times?
+No
+Has it ever
+Hypothesis
+No
+Test
+been rejected?
+Reject
+Yes
+Rejected
+No
+for the first
+Generate
+time?
+Data
+Yes
+No
+Record as the
+Record as the
+upper bound
+lowerbound
+Increase
+external inputJournal XX (XXXX) XXXXXX 
+Author et al  
+ 
+7 
+ 
+ 
+data may be on arbitrary scales. To avoid potential issues due 
+to variations in amplitude scale or values out of the training 
+range, The mean and standard error were computed for each 
+feature 
+and 
+target 
+variable, 
+and 
+standardisation 
+is 
+implemented for each of them using 
+𝑥−μ
+σ . The other reason for 
+doing this is to ensure the loss is reduced for all variables 
+without any bias, as different amplitudes might cause the loss 
+of some of the target variables to be weighted more than 
+others. For example, if one variable has a higher amplitude 
+than the other, its error is higher and thus will be reduced 
+more. 
+2.5 Loss Function 
+The training of an artificial neural network aims to 
+minimise the loss function over all training data. By default, 
+the loss function is the mean squared error of all target 
+variables, which are the state variables and parameters in our 
+case. However, it is not enough for the model error to be 
+minimised only on the training data, as we would like the 
+LSTM model to learn whether the predicted state variables 
+and parameters can represent the observations. More 
+importantly, it should also learn to behave like the NMM. 
+Given an observation signal, the trained LSTM model should 
+be able to predict the state vector and parameters that can 
+recreate the observation signal using the NMM. Since the 
+generation of the training data is based on setting the discrete 
+parameters, even though we can lower the step within each 
+parameter, it is still possible that the model cannot learn the 
+mathematical relationship between all state variables and 
+parameters. Thus, we link the LSTM model with the NMM. 
+To link the LSTM model with the NMM, we customise the 
+loss function to allow the LSTM model to know the 
+mathematical relationship between the observation and the 
+state. Since the LSTM model is able to produce the state for 
+the next timestep 𝑡 given the state of the current time step 𝑡 −
+1, it is possible to generate all 𝑡 states given the state at the 
+current timestep. By comparing the state at time 𝑡 with the 
+state predicted by the LSTM model, we can know the error 
+between the LSTM model prediction and the neural mass 
+model prediction. If we can link the neural mass model to the 
+loss function, the LSTM model can learn to minimise both the 
+error of the LSTM predicted values and the neural mass model 
+predicted values.  
+The squared error at a single timestep is defined as 
+ 
+Squared Error = [(ξt − ξ̂𝑡)
+2,  (𝑦𝑡 − 𝐻ξ̂𝑡)
+2], 
+[22] 
+ 
+where ξ is the augmented state, 𝑦 is the observation of the 
+training data and ξ̂  and 𝐻ξ̂𝑡  are the augmented state and 
+observation predicted by the LSTM model, respectively. The 
+Squared Error term compares the predictions with the true 
+values. By minimising the error, the predictions of both states 
+and measurements will be closer to the truth. 
+We then add to the loss function the model error, which is 
+the error between the state of the LSTM model and the state 
+generated by the neural mass model (40,41), 
+ 
+Model Error = [(ξ̂𝑡 − (𝐴θξ̂𝑡−1 + 𝐵θϕ(𝐶θξ̂𝑡−1)))
+2
+, 0] [23] 
+ 
+With the augmented state-space model, the estimated 
+augmented state vector at the current time step ξ̂𝑡−1 can be 
+converted to the next time step. Minimising the squared error 
+between ξ̂𝑡  and the converted state vector 𝐴θξ̂𝑡−1 +
+𝐵θϕ(𝐶θξ̂𝑡−1) links the estimation to the neural mass model, 
+since the training has to minimise the error between its own 
+prediction and the state generated by the NMM. 
+Note that, although 𝐵θ , 𝐶θ  and ϕ are considered to be 
+constant, the matrix 𝐴  can vary depending on the time 
+constants; i.e., the 𝐴  has to change over time. In the 
+customised loss function, 𝐴 is calculated at each timestep so 
+the change in state and parameters can be reflected over time.  
+The last thing aspect of the loss function is the rate of 
+change of the connectivity strength parameters, as they are 
+considered to be slowly changing parameters compared to the 
+membrane potentials and simulated EEG. It is possible to add 
+the standard deviation of the parameters to the loss function to 
+limit the amounts that they change. However, the parameters 
+need to be adjusted rapidly when they differ substantially from 
+the neural mass model. Thus, the standard deviation is 
+combined with the model error as 
+ 
+Std = 𝑠(𝛼) [(𝜉̂𝑡 − (𝐴𝜃𝜉̂𝑡−1 + 𝐵𝜃𝜙(𝐶𝜃𝜉̂𝑡−1)))
+2
+, 0] ∗ 𝑘
+ 
+[24] 
+ 
+𝑠(𝛼) = [0, … 0, std(𝛼),0]𝑇, 
+[25] 
+ 
+where 𝛼  is the vector of the four connectivity strengths 
+parameters and 𝑘 is an adjustable weight that is set to 0.1 as 
+default. 
+The final loss is the summation of equations [22], [23] and 
+[24]. 
+2.6 Testing on Simulated EEG Data 
+On top of the testing data split from the training dataset, 
+another testing dataset with the same range of parameters was 
+generated, but the steps of τ𝑒 and τ𝑖 are set to be smaller. This 
+dataset avoids using the exact parameters of the original 
+training dataset.  
+We tested both the AKF and the LSTM model on the same 
+dataset. To evaluate the impact of incorrect initialisation of the 
+AKF, two settings were used. The first setting was correctly 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+8 
+ 
+ 
+initialised Kalman filters that were initialised to be exactly the 
+same as the parameter initial values. The second setting was 
+default parameters, where the excitatory and inhibitory time 
+constants were 0.02 s and 0.01 s, respectively, which 
+corresponds to the parameter values that generate a strong 
+alpha-like (8-12 Hz) rhythm (9). 
+To ensure all state vectors and parameters are on the same 
+amplitude scale numerically, the testing results are 
+standardised. We compute the Root Mean Squared Error 
+(RMSE) between the truth and the prediction for each state 
+variable and parameter, and find the difference between the 
+AKF and the LSTM model result. The RMSE for each 
+parameter is 
+ 
+RMSE =√∑
+(
+𝑥𝑖−𝜇
+𝜎
+−
+𝑥̂𝑖−𝜇
+𝜎 )
+2
+𝑛
+𝑖=1
+/𝑛, 
+[26] 
+ 
+where 𝜇 and 𝜎 are the mean and standard deviation of the 
+testing dataset for each parameter, respectively, and 𝑥𝑖 and 𝑥̂𝑖 
+are the truth and prediction, respectively. Equation (26) is used 
+for each estimation method to extract the raw RMSE and the 
+values are compared across methods. 
+Testing the robustness of the model is also of great 
+importance, since typically there will be strong noise in real 
+EEG data. As the standard deviation of the testing observation 
+data is around 40, we add Gaussian noise with a mean of 0 and 
+a standard deviation of 4, representing 10% Gaussian noise 
+added to the testing data. Both the LSTM model and the 
+perfectly initialised Kalman Filter are tested using the noisy 
+testing data. 
+ 
+We also test the LSTM model on randomly generated data 
+involving time-dependent parameter changes. The time 
+constants are randomly chosen from the range 0.01-0.06 s, and 
+the model is simulated with the selected time contents held 
+fixed for 5 s before the next combination of time constants. 
+Note that there is a 5 s buffer between each 5 s fixed time 
+constant simulation segment, where we connect the parameter 
+values between segments with a straight line to simulate 
+transitions between different brain states, where the 
+parameters are not constant. We assume the parameters are 
+slowly-changing, so a straight line between different stable 
+states would be enough for simulation purposes. Since the 
+training was done on constant parameters with noise only, 
+testing on randomly generated data is needed to know whether 
+the LSTM filter can work in a scenario where parameters are 
+time varying. 
+ 
+2.7 Testing on Real EEG Data 
+Apart from simulation data, we also use intracranial EEG 
+(iEEG) data collected from a patient with epilepsy chosen 
+from data recorded continuously from 15 patients for up to 
+three years (42). The patients were implanted with 16 
+intracranial electrodes around their seizure onset areas with 
+sampling rate 400 Hz. The electrode array was wirelessly 
+connected to a portable device. Seizures were automatically 
+detected by the advisory device and were reviewed by clinical 
+experts. We denote the iEEG data from this study as “real 
+data”. 
+To demonstrate the practical utility of the LSTM filter, it is 
+applied to a 1 h intracranial EEG recording containing an 
+epileptic seizure to show whether or not we can observe a 
+transition between normal and ictal brain states. Although the 
+actual values of the state variables and parameters are not 
+known, there is an obvious brain state change with seizure 
+onset, so we can see if the LSTM model may be used to detect 
+or even predict a seizure. Since normally iEEG data becomes 
+more rhythmic when it enters a seizure compared to normal 
+activity, we expect the parameter estimates to become stable 
+after entering the seizure state. 
+ 
+ 
+ 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+9 
+ 
+ 
+ 
+3. Results 
+We have chosen some typical time series to show the 
+different parameter estimation results between the perfectly 
+initialised AKF and the LSTM model. As shown in Figure 3, 
+the first column is the default condition where excitatory and 
+inhibitory time constants are 0.01 s and 0.02 s, respectively. 
+The second and the third columns are examples where the 
+LSTM performs better and where the AKF performs better, 
+respectively. The simulated EEG signals can be fit closely for 
+all scenarios. However, the fits of the parameters are very 
+different. The LSTM model tends to converge rapidly in about 
+0.1 s, while the AKF takes more time to find the optimal 
+solution. The LSTM model also appears to fluctuate more 
+compared to the AKF, normally because the LSTM model has 
+to compute using the provided input which is the observation 
+signal. When the input has oscillations, the LSTM model can 
+only minimise the changes, but it cannot make it constant like 
+the truth. However, the results of the LSTM model are 
+generally within a reasonable range with predictions close to 
+the truth. The perfectly initialised AKF tracks the parameters 
+very well, but the filter drifts far away from the correct 
+solution for some cases, as shown in the second column. 
+Figure 4 shows comparisons between the proposed LSTM 
+method against the AKF. The two time constants were 
+selected as the axes of the plots since the time constants have 
+a fixed range (0.01 s to 0.06 s) compared to the external input, 
+which has a more flexible range depending on the time 
+constants. The median of the RMSE is computed over 
+different external inputs, as sometimes the AKF results in 
+errors when it cannot find the nearest positive definite 
+covariance matrix which leads to extremely large RMSE. The 
+covariance matrix must remain positive definite to ensure 
+inversion is possible, so a nearest positive definite matrix has 
+to be found when it is not. Due to the standardisation of the 
+parameters in the RMSE calculation, most of the raw RMSE 
+results are within the range 0-1. Hence, the minimum and the 
+maximum of the colour bars are set to 0 and 1, respectively. 
+The minimum and the maximum of the colour bars for the 
+RMSE difference are set to -1 and 1, respectively. 
+As seen in Figure 4 (a), the raw RMSE are mostly low for 
+the LSTM model with a few exceptions. The performance is 
+better in the centre of the parameter space compared to the 
+edges. The centre of the space is close to the default parameter 
+point (𝜏𝑒 = 0.01, 𝜏𝑖 = 0.02) for the Jansen-Rit model. The 
+regions of low performance (red area) are because very little 
+or no oscillations are detected in the observation signal. The 
+perfectly initialised AKF has a nearly perfect performance for 
+lower values of the excitatory time constant (the upper parts 
+of the plots), but the result is worse when the excitatory time 
+constant increases. On the other hand, the fixed AKF 
+initialised to the default parameter point only performs well 
+around the region of the default setting. 
+ Figure 4 (b) show the differences between the LSTM filter 
+and the two initialisation cases of the AKF. The red zone 
+indicates that the LSTM model has a higher RMSE than the 
+Figure 3. Examples of time series recordings and predictions using Kalman filters and LSTM models. Different settings (columns) are given 
+to the NMM to generate artificial EEG recordings. Observations and parameters are displayed to show the correct parameters (truth) and 
+those estimated by the Kalman filters and the LSTM models. 
+
+Truth
+Kalman Filter
+LSTMJournal XX (XXXX) XXXXXX 
+Author et al  
+ 
+10 
+ 
+ 
+AKF. For the LSTM and the perfectly initialised AKF, despite 
+the lower part of the graph where the LSTM model has a much 
+lower RMSE compared to the AKF, the LSTM model usually 
+has a slightly higher RMSE. Compared with the fixed Kalman 
+Filter, the LSTM model usually shows much better 
+performance. Thus, we can conclude that the proposed LSTM 
+model can be considered as a generally better tool to estimate 
+the EEG data without knowledge of the initial state variables 
+and parameters. 
+Figure 5 shows the performance with Gaussian noise added 
+to all recordings with 0 mean and 10% of the standard 
+deviation of the original data. We can see that the performance 
+of the LSTM model did not change much, but the AKF did 
+have more areas where the performance was worse compared 
+Figure 4. Heatmaps of the LSTM and Kalman Filter errors. (a) Raw RMSE between the prediction and the truth. The test was run on the 
+LSTM model (top), the perfectly initialised Kalman Filter (middle) and the fixed Kalman Filter (bottom) whose initialisation was set as 
+default. The colours indicate the amount of error ranging from 0 (blue) to 1 (red). (b) The RMSE difference between the LSTM and the 
+Kalman Filters. The difference was compared between the LSTM and the perfectly initialised Kalman Filter (top), and between the LSTM 
+and the fixed Kalman Filter (bottom). The comparison was done by using the RMSE of the LSTM minus the RMSE of the Kalman Filter, 
+which means a positive number indicates the LSTM model having a higher error. The colours indicate the RMSE difference ranging from -1 
+(blue) to 1 (red), with no difference shown as grey. 
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+!1Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+11 
+ 
+ 
+to no noise. Furthermore, we can also observe from Figure 5 
+(b) that there are more blue areas compared to those in Figure 
+4 (b), showing that the LSTM model is more robust when the 
+data is noisier. 
+To show the performance of the LSTM filter in response to 
+time-varying parameters, we also test using randomly 
+generated recording with changing parameters as presented in 
+Figure 6. The AKF is initialised with the initial parameters of 
+the simulation data. Its parameter estimates are stable at the 
+beginning of the recording when the true parameters are 
+stable, but its parameter estimates diverge when the true 
+parameters change. Although AKF can follow the observation 
+at the beginning (before 20 s), it does not follow the parameter 
+space correctly after this period. The LSTM model shows a 
+much better fit to both the observation and the parameters. 
+There are some times when the parameters are not followed 
+precisely but, overall, the LSTM model tracks within a 
+reasonable range. As a result, the figure shows that the LSTM 
+model is able to track time-varying changes in the true 
+parameter values that might be observed when the brain 
+transitions between different dynamical states. 
+The final test involves using real intracranial EEG data (42) 
+containing an epileptic seizure to show whether the LSTM 
+filter can detect brain state changes. In Figure 7, the seizure 
+start and end times are labelled with vertical red bars, though 
+they are difficult to differentiate in the plot of the full hour of 
+data. In Figure 7 (a), which shows the estimation results for 
+one hour of data, we can see all parameters fluctuate 
+frequently and in a large range as well, which reflects the noisy 
+EEG recording with no stable state to track. However, upon 
+entering the seizure state, the changes in the parameters were 
+limited as seen from Figure 7 (b), where the parameters are 
+more stable during the seizure, and fluctuate more before and 
+after the seizure. We can also see that from Figure 7 (b), where 
+we zoom into the minute when the seizure was about to begin, 
+the parameters transition between different values. This shows 
+some promise that the LSTM model may be able to detect 
+epileptic seizure transitions. 
+ 
+4. Discussion 
+A novel LSTM filter has been presented to provide efficient 
+neurophysiological variable estimates derived from EEG data, 
+which does not require initialisation of the filter and can track 
+dynamic changes in brain states. Moreover, the approach is 
+sufficiently general that it can potentially be applied to other 
+neural mass models. 
+Although the overall performance of the LSTM model is 
+better or at least similar to the perfectly initialised AKF, there 
+are some noticeable drawbacks. In most cases, we can see that 
+the performance of the estimated external input is usually the 
+worst among all state variables and parameters, especially in 
+the heatmaps shown in Figures 4 and 5. This is because there 
+are few mathematical constraints regarding the external 
+Figure 5. Heatmaps of the LSTM and the Kalman Filter after adding Gaussian noise. The same method was applied and the perfectly 
+initialised Kalman Filter was tested. (a) Raw RMSE between the prediction and the truth. The test was run on the LSTM model (top) and 
+the perfectly initialised Kalman Filter (bottom). The colours indicate the amount of error ranging from 0 (blue) to 1 (red).  (b) The RMSE 
+Difference between the LSTM and the Kalman Filter. The comparison was done by using the RMSE of the LSTM minus the RMSE of the 
+Kalman Filter, which means a positive number indicates the LSTM model having a higher error. The colours indicate the RMSE difference 
+ranging from -1 (blue) to 1 (red), with no difference shown as grey. 
+
+(a)
+LSTM RMSE
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+!1Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+12 
+ 
+ 
+inputs, as it acts as a way to adjust DC shift that is sometimes 
+not achievable by the membrane potential linked to the 
+pyramidal population. Other parameters that are more 
+constrained mathematically by the relationship between the 
+neural mass model equations and the EEG output of the model 
+have better estimation performance. On the other hand, 
+converting the firing rate of the external input to the membrane 
+potential via the post-synaptic potential kernel depends on the 
+excitatory time constant, so the external input has a dynamic 
+range to vary instead of a fixed range like the time constant, 
+which makes it harder to track. However, the situation 
+improves when the data becomes noisier as the LSTM is still 
+robust compared to the AKF. 
+Time efficiency is also one of the important features of the 
+artificial neural network. The AKF only considers one 
+previous timestep, but the LSTM model considers 400 
+timesteps. This means the computation of the LSTM model is 
+more complex. We have tested the result when both methods 
+are computed via CPU, the AKF takes about 516 seconds to 
+run on a one-hour recording, while the LSTM model takes 
+about three times longer: about 1720 seconds. With the 
+utilisation of GPU, the LSTM model is significantly faster, 
+and is able to run on the same recording in 14 seconds. This 
+suggests the LSTM filter can be scaled up for larger more 
+complex neural mass models for more detailed inference 
+based imaging tasks (2), while still providing tractable run 
+times. 
+Another advantage of the LSTM filter is that it can 
+potentially be trained on a very specific region of the 
+parameter space, in particular only within biologically 
+realistic regions of the model parameter space. This means it 
+will be much more likely to produce estimates in such a region 
+also. The AKF on the other hand is dependent on numerical 
+stability of the neural mass model associated with the 
+sampling rate being used, and it also is not necessarily 
+constrained to producing estimates outside of biologically 
+plausible regions of the parameter space. Thus, in some cases 
+the AKF can produce unrealistic estimates. In this paper, the 
+training data generated by the neural mass model sometimes 
+did not comply with biological realism. For example, the 
+firing rate sometimes achieved rates as high as ten thousand 
+spikes per second, which is not biologically realistic. This is 
+because the purpose of the experiment was to show whether 
+the LSTM filter is able to track the state vector and parameters 
+regardless of the range of the parameters in general. 
+Nevertheless, it would be straightforward to train the LSTM 
+filter on completely biologically realistic regions of parameter 
+space, in order to only produce biologically realistic estimates. 
+Since there is only one input dimension, the observation in our 
+training data, we would like to consider whether adding more 
+features would benefit. We could extract the band power from 
+one second prior to the current timestep to represent the 
+change in the strength over different frequency bands. Since 
+the frequency changes might be minor, especially in the low 
+frequency area, we would like to compute a narrow bandwidth 
+within low frequencies and wide bandwidth for the high 
+frequencies (43). However, this feature extraction process 
+takes a long time leading to time efficiency issues. 
+The LSTM filter presented here is accessible through the 
+link that is provided in the Code Availability section. The 
+package has been prepared such that the LSTM filter can be 
+applied in a univariate fashion to multichannel EEG, 
+intracranial EEG or EEG/MEG source imaging data to derive 
+single channel neurophysiological variable estimates across 
+all channels. Future work will consider the multivariate 
+multichannel estimate problem. 
+ 
+ 
+Figure 6. Test on a randomly generated recording. (a) True 
+parameters and the prediction made by the LSTM and the AKF. The 
+AKF is perfectly initialised at the beginning, but starts to lose track 
+as the change of parameters happens, while the LSTM keeps mostly 
+on track. (b) Observation signals. The LSTM model follows the truth 
+much better than the AKF. 
+
+Kalman Filter
+LSTMJournal XX (XXXX) XXXXXX 
+Author et al  
+ 
+13 
+ 
+ 
+ 
+5. Conclusion 
+The proposed methodology has demonstrated competitive 
+accuracy, high time efficiency, and the potential to be applied 
+to real world scenarios such as epileptic seizure detection. The 
+result has shown the accuracy of the proposed approach is 
+comparable with a perfectly initialised AKF, and is much 
+better than the AKF that is initialised with default parameters. 
+In a parameter-changing environment, the LSTM filter is able 
+to track the changing parameters much better than the AKF 
+even if the AKF is perfectly initialised. With GPU 
+acceleration, the time efficiency is also greatly improved, with 
+the time cost reduced by 96%. Finally, after testing on real 
+data with an epileptic seizure, the LSTM filter can detect 
+instantaneous transitions in brain states and, therefore, holds 
+promise as a potential application for detecting or predicting 
+seizures. Further research will be focused on applying the 
+proposed method to different scenarios to infer, image and 
+understand the neurophysiological processes underlying 
+different kinds of electrophysiological and electromagnetic 
+brain recordings. 
+ 
+6. Code Availability 
+Figure 7. Test on real data with a seizure. (a) A one-hour recording containing an epileptic seizure. Seizure is indicated as a vertical red bar. 
+(b) Zoomed in for the minute where the seizure happened. The vertical red bars indicate the start and the end of the seizure. 
+
+Journal XX (XXXX) XXXXXX 
+Author et al  
+ 
+14 
+ 
+ 
+The data generation, model training and random recording 
+testing 
+codes 
+are 
+available 
+at 
+https://github.com/yueyang6/brain_state_reconstruction.  
+7. References 
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+
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+page_content='org/XXXX/XXXX  xxxx-xxxx/xx/xxxxxx  1  © xxxx IOP Publishing Ltd  Brain Model State Space Reconstruction Using an  LSTM Neural Network  Yueyang Liu1, Artemio Soto-Breceda2, Yun Zhao1, Phillipa Karoly2, Mark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Cook3, David B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Grayden2, Daniel  Schmidt1, Levin Kuhlmann1  Abstract  Objective  Kalman filtering has previously been applied to track neural model states and parameters, particularly at the scale relevant  to electroencephalography (EEG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, this approach lacks a reliable method to determine the initial filter conditions and  assumes that the distribution of states remains Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This study presents an alternative, data-driven method to track the  states and parameters of neural mass models (NMMs) from EEG recordings using deep learning techniques, specifically a Long  Short-Term Memory (LSTM) neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Approach  An LSTM filter was trained on simulated EEG data generated by a neural mass model using a wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  With an appopriately customised loss function, the LSTM filter can learn the behaviour of NMMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As a result, it can output  the state vector and parameters of NMMs given observation data as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Main Results  Test results using simulated data yielded correlations with R squared of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='99 and verified that the method is robust  to noise and can be more accurate than a nonlinear Kalman filter when the initial conditions of the Kalman filter are not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  As an example of real-world application, the LSTM filter was also applied to real EEG data that included epileptic seizures,  and revealed changes in connectivity strength parameters at the beginnings of seizures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Significance  Tracking the state vector and parameters of mathematical brain models is of great importance in the area of brain modelling,  monitoring, imaging and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This approach has no need to specify the initial state vector and parameters, which is very  difficult to do in practice because many of the variables being estimated cannot be measured directly in physiological  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This method may be applied using any neural mass model and, therefore, provides a general, novel, efficient  approach to estimate brain model variables that are often difficult to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Keywords: LSTM Neural Network, Neural Mass Model, Neurophysiological Process Imaging, EEG, Epilepsy  1 Department of Data Science and AI, Faculty of Information Technology,  Monash University, Clayton, Australia  2 Department of Biomedical Engineering, The University of Melbourne,  Parkville, Australia  3 Department of Medicine, St Vincent’s Hospital, The University of  Melbourne, Fitzroy, Australia  Journal XX (XXXX) XXXXXX  Author et al  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Introduction  Understanding the brain is one of the most challenging  problems of science, engineering and medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Currently,  due  to  limitations  of  existing  brain  imaging  and  neurophysiological techniques, many neurophysiological  variables cannot be measured directly (1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Moreover, if they  can be measured directly, we cannot do so for all neurons in  the brain let alone a small brain region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' For instance, current  techniques relevant to human applications, such as scalp or  intracranial electroencephalography (EEG), record the  activities of all the neurons of different population types with  a heavy weighting towards cortical pyramidal neurons (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  However, EEG does not have the resolution to track single cell  activity or connections between neurons or distinct  populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Brain modelling provides an alternative way to  understand brain parameters and neural activity indirectly, and  can be used to study healthy brain function as well as diseases  like epilepsy, Alzheimer’s disease or Parkinson’s disease (4–  8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' One of the aims of modelling is to capture essential  temporal and/or spatial characteristics of the brain or brain  regions of interest, and to understand how they can emerge  from underlying neurophysiological variables such as neural  membrane potentials, synaptic strengths and time constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  At the level of EEG modelling, several approaches have  been applied using NMMs to infer these crucial variables at  the neural population level (9–15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Different methods analyse  the EEG in either the time domain or the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  One of the most common inference frameworks is Dynamic  Causal Modelling (DCM), which is usually applied in the  frequency  domain  (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Frequency-domain  inference  potentially offers greater stability in the estimates because  they are determined over windows of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  On  the  other  hand,  time-domain  inference  of  neurophysiological variables affords the possibility of  capturing instantaneous transitions in these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This  could be used to understand temporal variations in neural  processing in the brain or enable timely control interventions  (16) such as electrical stimulation to steer the brain away from  an epileptic seizure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' These time-resolved variable  estimates, in particular model parameters such as population  averaged synaptic strengths, can also potentially provide  information about the current brain state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=', seizure, non-  seizure, pre-seizure, short seizure, long seizure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This is  because inferred model parameters can be used to determine  where the brain dynamics are currently positioned with respect  to the model’s bifurcation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Moreover, such information  could be used to make predictions, or to detect, different brain  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Inferring population averaged membrane potentials,  synaptic strengths and time constants, either locally or across  the brain from a limited number of EEG measurements, is a  challenging task with observability issues at play (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Depending on the spatial resolution of the population  averaging, there are typically fewer EEG measurements than  the number of variables being inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Although neural  population models can generate data resembling EEG, there  are no clear mathematical solutions for the model parameters  given the observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Indeed, there may be multiple  solutions within the state space given the observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  That is to say, there are multiple mathematical solutions to the  same observation and not all are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Even though some of  the solutions are mathematically achievable, it does not mean  they are biologically explainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Developing a technique that  can track neurophysiological processes that is reasonable both  mathematically and biologically will open new vistas for  tracking and imaging brain states, as well as controlling them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Regarding methods that have been used for time domain-  based inference of neurophysiological processes given EEG  data, the Kalman filter and its nonlinear variants have been  applied (10,12,13,18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' They address the problem of estimating  the state of a discrete-time controlled process through  stochastic difference equations (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' With the covariance  matrix of the process and measurement noise, a Kalman filter  balances the model and measurement noise to track the state  vector of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, it is a local search method that  requires an initial guess of the expected state vector and its  covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As a result, state vector estimates can deviate far  from the true state vector if the initial guess is not close to the  correct state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Furthermore, a solution to the Kalman  gain and the covariance is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since an arbitrary NMM  can potentially be very high dimensional, the solution is not  always easy to compute, and the initial guess is hard to  determine because the neurophysiological variables cannot be  measured in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' A solution that does not require an  initial guess and is easy to apply given an arbitrarily complex  model is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Long Short-Term Memory (LSTM) is a recurrent neural  network that can “remember” older information in a time-  series sequence (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Therefore, it is a potential approach to  solve the problem of inferring underlying neurophysiological  processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=', the state vector and parameters of NMMs)  from EEG data (21,22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In this paper, we compare the LSTM  approach with a nonlinear variant of Kalman Filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Methodology  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='1 Brain Modelling  Scalp and intracranial EEG record electrical activity signals  captured on the scalp and on the surface of the brain,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  These  signals  primarily  represent  the  macroscopic activity of cerebral cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' A single EEG signal  is recorded by a single electrode, and is a spatially and  temporally smoothed version of neuronally generated  electrical potentials (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Intracranial EEG is invasive and  difficult to place enough electrophysiological recording  Journal XX (XXXX) XXXXXX  Author et al  3  electrodes in a given cortical location to understand how the  underlying neurophysiological processes give rise to the EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  As a result, modelling is a way to bridge the EEG data and the  underlying neurophysiological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  There are predominantly two approaches to build a  mathematical model of the brain: the “bottom-up approach”  and the “top-down approach” (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The bottom-up approach  forms a macroscopic model of the brain or a brain region by  coupling a network of microscopic models of individual  neurons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=', Hodgkin-Huxley neurons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' To obtain the spatial  scale relevant to modelling EEG signals, the large number of  neurons involved in this kind of model, as well as the  heterogeneity and non-local connectivity of neurons, makes  this largely impractical for understanding which key  population-level neurophysiological variables influence  macroscopic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  On the other hand, the top-down approach defines  dynamical models based on phenomenological observations  and emergent behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This means that the mathematical  model is built to model large-scale signals, such as EEG,  through the interactions of neural populations, where the  neural populations only coarsely model the average properties  of the underlying detailed neuronal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Although it  would be useful to model and infer the detailed microscopic  neurophysiological processes, this is difficult to do as a result  of the aforementioned experimental and observability issues  as well as computational tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As a result, we focus here  on inferring population-level neurophysiological processes  using top-down models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  We focus on one of the most common NMMs used to  model cerebral cortex, the Jansen-Rit model (9,25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  Jansen-Rit model was developed based on the model proposed  by Lopes Da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (26), who proposed a two-population  neural field model to study spontaneous EEG generation and  alpha frequency rhythms, and reduced the complexity of the  model by aggregating the activities within a single cortical  column to perform analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Jansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (25) constructed a  similar model of pyramidal neurons to show that the  mechanisms that guide the evolution of evoked potentials and  spontaneous EEG signal generation are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Jansen and  Rit (9) extended the model with a third population of local  excitatory interneurons that provide feedback to the pyramidal  neuron population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The latter model has three populations:  pyramidal neurons, inhibitory interneurons and excitatory  interneurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Figure 1 illustrates the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The spatially averaged post-  synaptic potential of population 𝑚 arising as a result of input  from pre-synaptic population 𝑛 is expressed as  𝑣𝑚𝑛(𝑡) = α𝑚𝑛 ∫  ℎ𝑚𝑛(𝑡 − 𝑡′)ϕ(𝑣𝑛(𝑡′))𝑑𝑡′  𝑡  −∞  ,  [1]  where α𝑚𝑛 is the population averaged synaptic connectivity  strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The post-synaptic response kernel is given by  ℎ𝑚𝑛(𝑡) = η(𝑡)  𝑡  τ𝑚𝑛 exp (−  𝑡  τ𝑚𝑛),  [2]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Jansen & Rit neural mass model and time series recordings and parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The model consists  of three populations: pyramidal neurons, inhibitory interneurons and excitatory interneurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The output of each population is a firing  rate, which is transformed to changes in the mean membrane potentials of connected neural populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  external input  Vpi  supragranularlayers  inhibitory  inhibitory interneurons  interneurons  excitatory  αpi  granular layer IV  interneurons  excitatoryinterneurons  Vpe  αpi  u  αip  ape  infragranularlayers  pyramidal cells  Vep  Vip  pyramidal  population  EEGJournal XX (XXXX) XXXXXX  Author et al  4  where η(𝑡) is the Heaviside step function and τ𝑚𝑛  is the  population averaged synaptic time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Moreover, ϕ(𝑣)  is a sigmoid function that transforms the membrane potential  to a firing rate given by  ϕ(𝑣) =  1  2 (erf (  𝑣−𝑣0  ς ) + 1),  [3]  where 𝑣0  is a threshold and ς is the slope of the sigmoid  function, which is also the variance of firing thresholds of the  population assuming the firing thresholds follow a Gaussian  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The mean membrane potential of population 𝑚 is  the sum of the synaptic contributions,  𝑣𝑚(𝑡) = ∑  𝑣𝑚𝑛(𝑡)  𝑁  𝑛=1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [4]  The convolution in equation [1] can be also written as two  coupled, first-order, ordinary differential equations (27),  𝑑𝑣𝑚𝑛  𝑑𝑡  = 𝑧𝑚𝑛  [5]  𝑑𝑧𝑚𝑛  𝑑𝑡  =  α𝑚𝑛  τ𝑚𝑛 ϕ𝑚𝑛 −  2  τ𝑚𝑛 𝑧𝑚𝑛 −  1  τ𝑚𝑛  2  𝑣𝑚𝑛,  [6]  where τ𝑚𝑛 is a lumped time constant, and 𝑚 and 𝑛 indicate  the pre-synaptic and post-synaptic neural population,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Furthermore, we can express the neural mass model in  matrix-vector notation so the operations can be expressed  more compactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The neural mass model can be expressed in  matrix notation as  𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝜙⃗ (𝐶𝑥(𝑡))  [7]  𝑦(𝑡) = 𝐻𝑥(𝑡) + 𝑣(𝑡),  [8]  where 𝐻  is the observation matrix, 𝑣(𝑡)~𝑁(0, 𝑅)  is the  observation noise (10) and 𝑦(𝑡) is the membrane potential of  the pyramidal population, which is considered to be the  contributor to the generation of the EEG signal in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The matrix 𝐴 is defined by the membrane time constants,  𝐴 =  diag(Ψ1 … Ψ𝑛),  [9]  where  Ψ𝑛 = [  0  1  −  1  τ2𝑚𝑛  −  2  τ𝑚𝑛  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [10]  The matrix 𝐵 is the synaptic gains from internal inputs,  which has the form  𝐵 = diag(0 a1 … 0 a𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [11]  The matrix 𝐶  is the adjacency matrix that defines the  connectivity structure of the model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  𝐶 =  [  0  0  …  0  0  𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='1  0  𝑐2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='𝑁𝑥−1  0  ⋮  ⋱  ⋮  0  0  0  0  𝐶𝑁𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='1  0  ⋯  𝐶𝑁𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='𝑁𝑥−1  0]  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [12]  which is a matrix of zeros and ones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' where one indicates there  is a connection between two populations and zero indicates  there is no connection  In order to apply a nonlinear Kalman filter to this model to  infer both the state vector and the parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' we define a  vector of parameters as 𝜃 = [𝑢 𝛼𝑝𝑒 𝛼𝑝𝑖 𝛼𝑖𝑝 𝛼𝑒𝑝]  𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This set of  parameters corresponds to the input 𝑢 to the model and the  population averaged connection strengths between the three  populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Moreover, we assume the time constants of the  model to be constant to simplify the estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  above parameters were combined with the state vector 𝑥 to  form the augmented state vector,  ξ = [XTθT]𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [13]  The augmented state-space model is then  ξ𝑡 = 𝐴θξ𝑡−1 + 𝐵θϕ(𝐶θξ𝑡−1) + 𝑊𝑡−1,  [14]  where 𝑊𝑡 is Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='2 Analytic Kalman Filter  Before defining the novel LSTM-based filter for state  vector and parameter estimation, this section outlines the  Kalman filter applied as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In particular, a filter  referred to as the Analytical Kalman filter (AKF) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  This filter is highly stable and accurate and was developed in  prior work (2,10,27) that evolved from deriving the Kalman  filter for general nonlinear NMMs using the specific  sigmoidal non-linearity in equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' For the sake of  brevity, we refer the reader to these prior works for greater  mathematical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Moreover, links to code provided at the  end of this paper give the exact specification of the  implementation of the AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Here, a brief description of the  main computations of the AKF are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The aim of the AKF is to find the most likely posterior  distribution of the augmented state given the previous  measurements under Gaussian assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Such a posterior  distribution is characterised by its a posteriori state vector  estimate and its covariance,  𝜉̂𝑡  + = 𝐸[𝜉𝑡 | 𝑦1, 𝑦2, ⋯ , 𝑦𝑡]  [16]  Journal XX (XXXX) XXXXXX  Author et al  5  𝑃̂𝑡  + = 𝐸 [(𝜉𝑡 − 𝜉̂𝑡  +)(𝜉𝑡 − 𝜉̂𝑡  +)  ⊤].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [17]  The AKF proceeds in two steps: prediction and update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  During prediction, the prior distribution (obtained from the  previous estimate) is propagated through the model equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  This step provides the so called a priori estimate, which is also  a Gaussian distribution with mean and covariance,  𝜉̂𝑡  − = 𝐸[𝜉𝑡 | 𝑦1, 𝑦2, ⋯ , 𝑦𝑡−1]  [18]  𝑃̂𝑡  − = 𝐸 [(𝜉𝑡 − 𝜉̂𝑡  −)(𝜉𝑡 − 𝜉̂𝑡  −)  ⊤].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [19]  During update, the a posteriori state vector estimate is  determined by correcting the a priori state vector estimate  with recorded (EEG) data by  𝜉̂𝑡  + = 𝜉̂𝑡  − + 𝛫𝑡(𝑦𝑡 − 𝐻𝜉̂𝑡  −),  [20]  where 𝐾𝑡 is the Kalman gain (18),  𝛫𝑡 =  𝑃̂𝑡  −𝐻⊤  𝐻𝑃̂𝑡  −𝐻⊤+𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [21]  The a posteriori state vector estimate covariance is then  updated by using the Kalman gain,  𝑃̂𝑡  + = (𝛪 − 𝛫𝑡𝐻)𝑃̂𝑡  −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  [22]  After each time step, the a posteriori estimate becomes the  prior distribution for the next time step, and the process  repeats for each new recorded data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The AKF requires  the a posteriori state vector estimate and its covariance to be  initialised at time 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The special features of the AKF are  that the state vector estimate and its covariance, along with the  Kalman gain, were derived using the fully non-linear Jansen-  Rit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  While the AKF framework can be applied to multichannel  EEG data in a multivariate fashion, for simplicity this paper  focuses on estimating the state vector and parameters of the  Jansen-Rit model using a single channel of EEG data  (analogous to the output observation/measurement of the  Jansen-Rit model) as input to the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='3 A novel LSTM filter for state and parameter estimation  The Recurrent Neural Network (RNN) was introduced  mainly to deal with time series data such as speech recognition  and natural language processing (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' An ordinary feed  forward neural network is only processes independent data  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, when data points depend on previous data  point, in the case of time series data, an RNN incorporates the  dependency by providing feedback to the next timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In this  way, the final output depends not only by the input but also on  the outputs of previous neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Long Short-Term Memory (LSTM) is a variation of the  RNN that has improved its performance (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since gradient  descent is applied to train neural networks, the gradient might  explode or vanish after applying the sigmoid function over and  over again, limiting to a certain number of discrete time steps  to avoid this problem (29,30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, the LSTM provides  stability to bridge many more discrete time steps by enforcing  constant error flow within special memory cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  LSTM neural networks incorporate memory cells and gate  units to convey useful information about previous states of the  neural network to the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' An LSTM layer consists  of multiple recurrently connected memory blocks and three  multiplicative gates, known as input, output and forget gates,  that learn to open and close access to the error flow within the  memory cell (20,31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' For example, an input gate could use  inputs from other memory cells to decide whether it should  store information in its own memory cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  An extension of the LSTM model is the bidirectional  LSTM model that provide more accurate and improved results  (32–34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' One of the shortcomings of RNNs is that they can  only learn from the context of the previous time steps and not  anything after the current time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Bidirectional LSTM can  process data in both directions with separated hidden layers,  which are then fed to the same output layer (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Also, since  the LSTM is free to access as much or as little of the data  within the given time window as needed, predefining a  specific time window for the model is not required (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Here, a bidirectional LSTM filter is constructed that takes  as input simulated or real EEG and predicts the state vector  and parameters of the Jansen-Rit model as well as simulated  or real EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Similar to the AKF, while the framework that  follows could potentially be applied to multichannel EEG data  in a multivariate fashion, this paper, for simplicity, focuses on  using the LSTM filter to estimate the state vector and  parameters of the Jansen-Rit model for a single channel of  EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The structure of the LSTM model is designed to achieve  high performance while maintaining good time efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Since the only non-linearity of the neural mass model is the  firing rate function in equation (3), and to achieve high  performance  while  predicting  parameters  given  the  observation, we would expect that at least two layers of  artificial neural network are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' A grid search  experiment was conducted to test the number of neurons in  each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The selected structure was 128 and 32 neurons in  layers 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='4 Training Data  As real EEG data provides no ground truth about the  neurophysiological variables that correspond to the state  Journal XX (XXXX) XXXXXX  Author et al  6  vector and parameters of the Jansen-Rit model, the LSTM  filter is trained on simulation data was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This allows  the prediction of the state vector and parameters of the model  when inputting either simulated or real EEG into the LSTM  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' It is important to generate a variety of training data with  different patterns and a wide range of parameters so the model  can learn the relationship between signal patterns and how the  state and parameters change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Furthermore, it is also crucial to  generate data containing oscillations, since there are likely to  be many solutions for a constant signal, or fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This is  because when there is no oscillation, different combinations  of changes in either the external input, input from the  inhibitory or excitatory interneurons may result in the same  observation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, changes of the parameters could  affect the ranges of external input that generate oscillations,  which makes it difficult to find the desired range of external  input when there can be different combinations of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  This is because the time constants affect the transformation  from firing rate to post-synaptic membrane potential thus  affecting the ranges over which the population produces  oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Thus, an effective method to generate oscillatory  data with a wide range of parameters is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Different combinations of inhibitory and excitatory time  constants can generate data with different waveforms,  frequencies and amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Normally, the ranges of both  excitatory and inhibitory time constants are considered  between 10-60 ms (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In addition, external input also plays  an important role in data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' External input to the  model is also affected by the time constant, such that the range  of the external input that can produce oscillations is affected  by different combinations of time constants, but the exact  range for each combination is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' To determine the  range of external input and whether the given combination of  time constants is able to generate oscillation, an automatic data  generation process is designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Statistical hypothesis testing is used to determine whether  an oscillation exists in a given time-series recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Two tests  are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' If the simulated EEG signal is noise-like, then it will  have a Gaussian distribution so the Anderson-Darling test (37)  is used to test if the data is not Gaussian distributed, which  means there could be an oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The other test is the  Ljung-Box test (38), which tests whether any autocorrelations  of the time series are different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Either of the tests  can help determine whether the generated data contains an  oscillation and, since it is more important to ensure there are  no false positives (data is Gaussian distributed or data has non-  zero correlation but we reject this hypothesis) to avoid data  being generated without including oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We set the  significance level to 𝛼 = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  A flow chart used to generate the training data is shown in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We generate a short recording first to check whether  this combination of time constants and external input is able  to generate data containing oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' If not, then the  external input is increased until it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' If it still does not  produce oscillations for 15 consecutive increases of the input,  the current combination of time constants is considered to not  be able to produce the desired data and so the next  combination is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' If the generated data includes oscillations  then the upper and lower bounds of the external input are  recorded, and training data is generated with the recorded  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The size of the dataset is as large as possible within the size  of physical memory, which means that the gap between each  combination is kept relatively small and as many examples as  possible are provided for training (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The generated data was  shuffled randomly and divided into training, validation and  testing data with ratio of 80:10:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  To improve the performance of the model with different  amplitudes, the dataset is standardised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As in many cases with  either real or simulated data, the raw amplitude of the EEG  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Flow chart of the generation of training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The same flow chart is used on different NMMs (the Jansen-Rit  model is used in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The amount of the increased external  input is not fixed, as the upper bound could be high when the  excitatory time constant is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Thus, the amount is low when the  input is low, and is higher when it is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  New  combinations of  Yes  time constants  Simulate a  Not  short  rejectedfor15  recording  times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  No  Has it ever  Hypothesis  No  Test  been rejected?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Reject  Yes  Rejected  No  for the first  Generate  time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Data  Yes  No  Record as the  Record as the  upper bound  lowerbound  Increase  external inputJournal XX (XXXX) XXXXXX  Author et al  7  data may be on arbitrary scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' To avoid potential issues due  to variations in amplitude scale or values out of the training  range, The mean and standard error were computed for each  feature  and  target  variable,  and  standardisation  is  implemented for each of them using  𝑥−μ  σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The other reason for  doing this is to ensure the loss is reduced for all variables  without any bias, as different amplitudes might cause the loss  of some of the target variables to be weighted more than  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' For example, if one variable has a higher amplitude  than the other, its error is higher and thus will be reduced  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='5 Loss Function  The training of an artificial neural network aims to  minimise the loss function over all training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' By default,  the loss function is the mean squared error of all target  variables, which are the state variables and parameters in our  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, it is not enough for the model error to be  minimised only on the training data, as we would like the  LSTM model to learn whether the predicted state variables  and parameters can represent the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' More  importantly, it should also learn to behave like the NMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Given an observation signal, the trained LSTM model should  be able to predict the state vector and parameters that can  recreate the observation signal using the NMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since the  generation of the training data is based on setting the discrete  parameters, even though we can lower the step within each  parameter, it is still possible that the model cannot learn the  mathematical relationship between all state variables and  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Thus, we link the LSTM model with the NMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  To link the LSTM model with the NMM, we customise the  loss function to allow the LSTM model to know the  mathematical relationship between the observation and the  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since the LSTM model is able to produce the state for  the next timestep 𝑡 given the state of the current time step 𝑡 −  1, it is possible to generate all 𝑡 states given the state at the  current timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' By comparing the state at time 𝑡 with the  state predicted by the LSTM model, we can know the error  between the LSTM model prediction and the neural mass  model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' If we can link the neural mass model to the  loss function, the LSTM model can learn to minimise both the  error of the LSTM predicted values and the neural mass model  predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The squared error at a single timestep is defined as  Squared Error = [(ξt − ξ̂𝑡)  2,  (𝑦𝑡 − 𝐻ξ̂𝑡)  2],  [22]  where ξ is the augmented state, 𝑦 is the observation of the  training data and ξ̂  and 𝐻ξ̂𝑡  are the augmented state and  observation predicted by the LSTM model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  Squared Error term compares the predictions with the true  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' By minimising the error, the predictions of both states  and measurements will be closer to the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  We then add to the loss function the model error, which is  the error between the state of the LSTM model and the state  generated by the neural mass model (40,41),  Model Error = [(ξ̂𝑡 − (𝐴θξ̂𝑡−1 + 𝐵θϕ(𝐶θξ̂𝑡−1)))  2  , 0] [23]  With the augmented state-space model, the estimated  augmented state vector at the current time step ξ̂𝑡−1 can be  converted to the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Minimising the squared error  between ξ̂𝑡  and the converted state vector 𝐴θξ̂𝑡−1 +  𝐵θϕ(𝐶θξ̂𝑡−1) links the estimation to the neural mass model,  since the training has to minimise the error between its own  prediction and the state generated by the NMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Note that, although 𝐵θ , 𝐶θ  and ϕ are considered to be  constant, the matrix 𝐴  can vary depending on the time  constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=', the 𝐴  has to change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In the  customised loss function, 𝐴 is calculated at each timestep so  the change in state and parameters can be reflected over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The last thing aspect of the loss function is the rate of  change of the connectivity strength parameters, as they are  considered to be slowly changing parameters compared to the  membrane potentials and simulated EEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' It is possible to add  the standard deviation of the parameters to the loss function to  limit the amounts that they change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, the parameters  need to be adjusted rapidly when they differ substantially from  the neural mass model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Thus, the standard deviation is  combined with the model error as  Std = 𝑠(𝛼) [(𝜉̂𝑡 − (𝐴𝜃𝜉̂𝑡−1 + 𝐵𝜃𝜙(𝐶𝜃𝜉̂𝑡−1)))  2  , 0] ∗ 𝑘  [24]  𝑠(𝛼) = [0, … 0, std(𝛼),0]𝑇,  [25]  where 𝛼  is the vector of the four connectivity strengths  parameters and 𝑘 is an adjustable weight that is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='1 as  default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The final loss is the summation of equations [22], [23] and  [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='6 Testing on Simulated EEG Data  On top of the testing data split from the training dataset,  another testing dataset with the same range of parameters was  generated, but the steps of τ𝑒 and τ𝑖 are set to be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This  dataset avoids using the exact parameters of the original  training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  We tested both the AKF and the LSTM model on the same  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' To evaluate the impact of incorrect initialisation of the  AKF, two settings were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The first setting was correctly  Journal XX (XXXX) XXXXXX  Author et al  8  initialised Kalman filters that were initialised to be exactly the  same as the parameter initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The second setting was  default parameters, where the excitatory and inhibitory time  constants were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='02 s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='01 s, respectively, which  corresponds to the parameter values that generate a strong  alpha-like (8-12 Hz) rhythm (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  To ensure all state vectors and parameters are on the same  amplitude scale numerically, the testing results are  standardised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We compute the Root Mean Squared Error  (RMSE) between the truth and the prediction for each state  variable and parameter, and find the difference between the  AKF and the LSTM model result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The RMSE for each  parameter is  RMSE =√∑  (  𝑥𝑖−𝜇  𝜎  −  𝑥̂𝑖−𝜇  𝜎 )  2  𝑛  𝑖=1  /𝑛,  [26]  where 𝜇 and 𝜎 are the mean and standard deviation of the  testing dataset for each parameter, respectively, and 𝑥𝑖 and 𝑥̂𝑖  are the truth and prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Equation (26) is used  for each estimation method to extract the raw RMSE and the  values are compared across methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Testing the robustness of the model is also of great  importance, since typically there will be strong noise in real  EEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As the standard deviation of the testing observation  data is around 40, we add Gaussian noise with a mean of 0 and  a standard deviation of 4, representing 10% Gaussian noise  added to the testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Both the LSTM model and the  perfectly initialised Kalman Filter are tested using the noisy  testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  We also test the LSTM model on randomly generated data  involving time-dependent parameter changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The time  constants are randomly chosen from the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='06 s, and  the model is simulated with the selected time contents held  fixed for 5 s before the next combination of time constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Note that there is a 5 s buffer between each 5 s fixed time  constant simulation segment, where we connect the parameter  values between segments with a straight line to simulate  transitions between different brain states, where the  parameters are not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We assume the parameters are  slowly-changing, so a straight line between different stable  states would be enough for simulation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since the  training was done on constant parameters with noise only,  testing on randomly generated data is needed to know whether  the LSTM filter can work in a scenario where parameters are  time varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='7 Testing on Real EEG Data  Apart from simulation data, we also use intracranial EEG  (iEEG) data collected from a patient with epilepsy chosen  from data recorded continuously from 15 patients for up to  three years (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The patients were implanted with 16  intracranial electrodes around their seizure onset areas with  sampling rate 400 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The electrode array was wirelessly  connected to a portable device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Seizures were automatically  detected by the advisory device and were reviewed by clinical  experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We denote the iEEG data from this study as “real  data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  To demonstrate the practical utility of the LSTM filter, it is  applied to a 1 h intracranial EEG recording containing an  epileptic seizure to show whether or not we can observe a  transition between normal and ictal brain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Although the  actual values of the state variables and parameters are not  known, there is an obvious brain state change with seizure  onset, so we can see if the LSTM model may be used to detect  or even predict a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since normally iEEG data becomes  more rhythmic when it enters a seizure compared to normal  activity, we expect the parameter estimates to become stable  after entering the seizure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Journal XX (XXXX) XXXXXX  Author et al  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Results  We have chosen some typical time series to show the  different parameter estimation results between the perfectly  initialised AKF and the LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As shown in Figure 3,  the first column is the default condition where excitatory and  inhibitory time constants are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='01 s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='02 s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The second and the third columns are examples where the  LSTM performs better and where the AKF performs better,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The simulated EEG signals can be fit closely for  all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, the fits of the parameters are very  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The LSTM model tends to converge rapidly in about  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='1 s, while the AKF takes more time to find the optimal  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The LSTM model also appears to fluctuate more  compared to the AKF, normally because the LSTM model has  to compute using the provided input which is the observation  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' When the input has oscillations, the LSTM model can  only minimise the changes, but it cannot make it constant like  the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, the results of the LSTM model are  generally within a reasonable range with predictions close to  the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The perfectly initialised AKF tracks the parameters  very well, but the filter drifts far away from the correct  solution for some cases, as shown in the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Figure 4 shows comparisons between the proposed LSTM  method against the AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The two time constants were  selected as the axes of the plots since the time constants have  a fixed range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='01 s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='06 s) compared to the external input,  which has a more flexible range depending on the time  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The median of the RMSE is computed over  different external inputs, as sometimes the AKF results in  errors when it cannot find the nearest positive definite  covariance matrix which leads to extremely large RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  covariance matrix must remain positive definite to ensure  inversion is possible, so a nearest positive definite matrix has  to be found when it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Due to the standardisation of the  parameters in the RMSE calculation, most of the raw RMSE  results are within the range 0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Hence, the minimum and the  maximum of the colour bars are set to 0 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The minimum and the maximum of the colour bars for the  RMSE difference are set to -1 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  As seen in Figure 4 (a), the raw RMSE are mostly low for  the LSTM model with a few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The performance is  better in the centre of the parameter space compared to the  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The centre of the space is close to the default parameter  point (𝜏𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='01, 𝜏𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='02) for the Jansen-Rit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  regions of low performance (red area) are because very little  or no oscillations are detected in the observation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  perfectly initialised AKF has a nearly perfect performance for  lower values of the excitatory time constant (the upper parts  of the plots), but the result is worse when the excitatory time  constant increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' On the other hand, the fixed AKF  initialised to the default parameter point only performs well  around the region of the default setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Figure 4 (b) show the differences between the LSTM filter  and the two initialisation cases of the AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The red zone  indicates that the LSTM model has a higher RMSE than the  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Examples of time series recordings and predictions using Kalman filters and LSTM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Different settings (columns) are given  to the NMM to generate artificial EEG recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Observations and parameters are displayed to show the correct parameters (truth) and  those estimated by the Kalman filters and the LSTM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Truth  Kalman Filter  LSTMJournal XX (XXXX) XXXXXX  Author et al  10  AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' For the LSTM and the perfectly initialised AKF, despite  the lower part of the graph where the LSTM model has a much  lower RMSE compared to the AKF, the LSTM model usually  has a slightly higher RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Compared with the fixed Kalman  Filter, the LSTM model usually shows much better  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Thus, we can conclude that the proposed LSTM  model can be considered as a generally better tool to estimate  the EEG data without knowledge of the initial state variables  and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Figure 5 shows the performance with Gaussian noise added  to all recordings with 0 mean and 10% of the standard  deviation of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We can see that the performance  of the LSTM model did not change much, but the AKF did  have more areas where the performance was worse compared  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Heatmaps of the LSTM and Kalman Filter errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (a) Raw RMSE between the prediction and the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The test was run on the  LSTM model (top), the perfectly initialised Kalman Filter (middle) and the fixed Kalman Filter (bottom) whose initialisation was set as  default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The colours indicate the amount of error ranging from 0 (blue) to 1 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (b) The RMSE difference between the LSTM and the  Kalman Filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The difference was compared between the LSTM and the perfectly initialised Kalman Filter (top), and between the LSTM  and the fixed Kalman Filter (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The comparison was done by using the RMSE of the LSTM minus the RMSE of the Kalman Filter,  which means a positive number indicates the LSTM model having a higher error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The colours indicate the RMSE difference ranging from -1  (blue) to 1 (red), with no difference shown as grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  (a)  LSTM RMSE  αp  αpl  EP  αpE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
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+page_content='8  Perfectly Initialised Kalman Filter RMSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
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+page_content='05  0  加加加加  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='1Journal XX (XXXX) XXXXXX  Author et al  11  to no noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Furthermore, we can also observe from Figure 5  (b) that there are more blue areas compared to those in Figure  4 (b), showing that the LSTM model is more robust when the  data is noisier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  To show the performance of the LSTM filter in response to  time-varying parameters, we also test using randomly  generated recording with changing parameters as presented in  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The AKF is initialised with the initial parameters of  the simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Its parameter estimates are stable at the  beginning of the recording when the true parameters are  stable, but its parameter estimates diverge when the true  parameters change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Although AKF can follow the observation  at the beginning (before 20 s), it does not follow the parameter  space correctly after this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The LSTM model shows a  much better fit to both the observation and the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  There are some times when the parameters are not followed  precisely but, overall, the LSTM model tracks within a  reasonable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' As a result, the figure shows that the LSTM  model is able to track time-varying changes in the true  parameter values that might be observed when the brain  transitions between different dynamical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The final test involves using real intracranial EEG data (42)  containing an epileptic seizure to show whether the LSTM  filter can detect brain state changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In Figure 7, the seizure  start and end times are labelled with vertical red bars, though  they are difficult to differentiate in the plot of the full hour of  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In Figure 7 (a), which shows the estimation results for  one hour of data, we can see all parameters fluctuate  frequently and in a large range as well, which reflects the noisy  EEG recording with no stable state to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, upon  entering the seizure state, the changes in the parameters were  limited as seen from Figure 7 (b), where the parameters are  more stable during the seizure, and fluctuate more before and  after the seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We can also see that from Figure 7 (b), where  we zoom into the minute when the seizure was about to begin,  the parameters transition between different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This shows  some promise that the LSTM model may be able to detect  epileptic seizure transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Discussion  A novel LSTM filter has been presented to provide efficient  neurophysiological variable estimates derived from EEG data,  which does not require initialisation of the filter and can track  dynamic changes in brain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Moreover, the approach is  sufficiently general that it can potentially be applied to other  neural mass models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Although the overall performance of the LSTM model is  better or at least similar to the perfectly initialised AKF, there  are some noticeable drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In most cases, we can see that  the performance of the estimated external input is usually the  worst among all state variables and parameters, especially in  the heatmaps shown in Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This is because there  are few mathematical constraints regarding the external  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Heatmaps of the LSTM and the Kalman Filter after adding Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The same method was applied and the perfectly  initialised Kalman Filter was tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (a) Raw RMSE between the prediction and the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The test was run on the LSTM model (top) and  the perfectly initialised Kalman Filter (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The colours indicate the amount of error ranging from 0 (blue) to 1 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (b) The RMSE  Difference between the LSTM and the Kalman Filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The comparison was done by using the RMSE of the LSTM minus the RMSE of the  Kalman Filter, which means a positive number indicates the LSTM model having a higher error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The colours indicate the RMSE difference  ranging from -1 (blue) to 1 (red), with no difference shown as grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  (a)  LSTM RMSE  αpl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='01  CEF  αpE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
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+page_content='01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
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+page_content='1Journal XX (XXXX) XXXXXX  Author et al  12  inputs, as it acts as a way to adjust DC shift that is sometimes  not achievable by the membrane potential linked to the  pyramidal population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Other parameters that are more  constrained mathematically by the relationship between the  neural mass model equations and the EEG output of the model  have better estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' On the other hand,  converting the firing rate of the external input to the membrane  potential via the post-synaptic potential kernel depends on the  excitatory time constant, so the external input has a dynamic  range to vary instead of a fixed range like the time constant,  which makes it harder to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, the situation  improves when the data becomes noisier as the LSTM is still  robust compared to the AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Time efficiency is also one of the important features of the  artificial neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The AKF only considers one  previous timestep, but the LSTM model considers 400  timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This means the computation of the LSTM model is  more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We have tested the result when both methods  are computed via CPU, the AKF takes about 516 seconds to  run on a one-hour recording, while the LSTM model takes  about three times longer: about 1720 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' With the  utilisation of GPU, the LSTM model is significantly faster,  and is able to run on the same recording in 14 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This  suggests the LSTM filter can be scaled up for larger more  complex neural mass models for more detailed inference  based imaging tasks (2), while still providing tractable run  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Another advantage of the LSTM filter is that it can  potentially be trained on a very specific region of the  parameter space, in particular only within biologically  realistic regions of the model parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This means it  will be much more likely to produce estimates in such a region  also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The AKF on the other hand is dependent on numerical  stability of the neural mass model associated with the  sampling rate being used, and it also is not necessarily  constrained to producing estimates outside of biologically  plausible regions of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Thus, in some cases  the AKF can produce unrealistic estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' In this paper, the  training data generated by the neural mass model sometimes  did not comply with biological realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' For example, the  firing rate sometimes achieved rates as high as ten thousand  spikes per second, which is not biologically realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' This is  because the purpose of the experiment was to show whether  the LSTM filter is able to track the state vector and parameters  regardless of the range of the parameters in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Nevertheless, it would be straightforward to train the LSTM  filter on completely biologically realistic regions of parameter  space, in order to only produce biologically realistic estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Since there is only one input dimension, the observation in our  training data, we would like to consider whether adding more  features would benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' We could extract the band power from  one second prior to the current timestep to represent the  change in the strength over different frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Since  the frequency changes might be minor, especially in the low  frequency area, we would like to compute a narrow bandwidth  within low frequencies and wide bandwidth for the high  frequencies (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' However, this feature extraction process  takes a long time leading to time efficiency issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  The LSTM filter presented here is accessible through the  link that is provided in the Code Availability section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  package has been prepared such that the LSTM filter can be  applied in a univariate fashion to multichannel EEG,  intracranial EEG or EEG/MEG source imaging data to derive  single channel neurophysiological variable estimates across  all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Future work will consider the multivariate  multichannel estimate problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Test on a randomly generated recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (a) True  parameters and the prediction made by the LSTM and the AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  AKF is perfectly initialised at the beginning, but starts to lose track  as the change of parameters happens, while the LSTM keeps mostly  on track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (b) Observation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The LSTM model follows the truth  much better than the AKF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Kalman Filter  LSTMJournal XX (XXXX) XXXXXX  Author et al  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Conclusion  The proposed methodology has demonstrated competitive  accuracy, high time efficiency, and the potential to be applied  to real world scenarios such as epileptic seizure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The  result has shown the accuracy of the proposed approach is  comparable with a perfectly initialised AKF, and is much  better than the AKF that is initialised with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  In a parameter-changing environment, the LSTM filter is able  to track the changing parameters much better than the AKF  even if the AKF is perfectly initialised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' With GPU  acceleration, the time efficiency is also greatly improved, with  the time cost reduced by 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Finally, after testing on real  data with an epileptic seizure, the LSTM filter can detect  instantaneous transitions in brain states and, therefore, holds  promise as a potential application for detecting or predicting  seizures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Further research will be focused on applying the  proposed method to different scenarios to infer, image and  understand the neurophysiological processes underlying  different kinds of electrophysiological and electromagnetic  brain recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Code Availability  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Test on real data with a seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' (a) A one-hour recording containing an epileptic seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' Seizure is indicated as a vertical red bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  (b) Zoomed in for the minute where the seizure happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content=' The vertical red bars indicate the start and the end of the seizure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  Journal XX (XXXX) XXXXXX  Author et al  14  The data generation, model training and random recording  testing  codes  are  available  at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='com/yueyang6/brain_state_reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE_T4oBgHgl3EQf_Rw9/content/2301.08391v1.pdf'}
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diff --git a/wtE_T4oBgHgl3EQf_hye/content/tmp_files/2301.08393v1.pdf.txt b/wtE_T4oBgHgl3EQf_hye/content/tmp_files/2301.08393v1.pdf.txt
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+arXiv:2301.08393v1  [eess.SP]  20 Jan 2023
+1
+Robust Symbol Level Precoding for Overlay
+Cognitive Radio Networks
+Lu Liu, Student Member, IEEE, Christos Masouros, Senior Member, IEEE,
+and A. Lee Swindlehurst, Fellow, IEEE
+Abstract
+This paper focuses on designing robust symbol-level precoding (SLP) in the downlink of an overlay
+cognitive radio (CR) network, where a primary base station (PBS) serving primary users (PUs) and a
+cognitive base station (CBS) serving cognitive users (CUs) share the same frequency band. When the
+PBS shares data and perfect channel state information (CSI) with the CBS, an SLP approach which
+minimizes the CR transmission power and satisfies symbol-wise Safety Margin (SM) constraints of both
+PUs and CUs, is obtained in a low-complexity quadratic formulation. Then for the case of imperfect CSI
+from the PBS to CBS, we propose robust SLP schemes. First, with a norm-bounded CSI error model
+to approximate uncertain channels at the PBS, we adopt the max-min philosophy to conservatively
+achieve robust SLP constraints. Second, we use the additive quantization noise model (AQNM) to
+describe the statistics of the quantized PBS CSI, and we employ a stochastic constraint to formulate
+the problem, where the SM constraints are converted to be deterministic. Simulation results show that
+the proposed robust SLP schemes help enable PUs to mitigate negative effect of the quantization noise
+and simultaneously offer CR transmission with significant improvements in energy efficiency compared
+to non-robust methods.
+Index Terms
+Cognitive radio, symbol-level precoding, constructive interference, robust precoding, quantization.
+L. Liu and A. Swindlehurst are with Center for Pervasive Communications and Computing, University of California, Irvine,
+USA (e-mail: {liul22,swindle}@uci.edu).
+C. Masouros is with the Department of Electronic & Electrical Engineering, University College London, London WC1E7JE,
+U.K. (e-mail: c.masouros@ucl.ac.uk).
+This work was supported by the National Science Foundation under grants CCF-2008714 and CCF-2225575.
+
+2
+I. INTRODUCTION
+As the number of wireless devices and their applications grow exponentially, the availability of
+unoccupied radio spectrum is becoming increasingly scarce and occupied bands are increasingly
+congested. Over the past two decades, cognitive radio (CR) technology has been extensively
+studied as a means to alleviate this problem through more efficient, flexible and comprehensive
+use of the spectrum [1]–[5].
+The fundamental challenge lies in balancing the interference generated by the CR at the
+primary users (PUs) with the quality of service (QoS) of the cognitive users (CUs). To address
+this issue, both the inter-system and inter-user interference need to be successfully managed. In
+the standard (non-cognitive) multiuser downlink scenario, beamforming or precoding at the multi-
+antenna transmitter can be employed to mitigate the multiuser interference (MUI) and compensate
+for its adverse affect on the received signals [6], [7]. Existing precoding schemes can be classified
+as either block-level precoding (BLP) or symbol-level precoding (SLP). In recent decades, many
+approaches have been proposed to implement block-level precoders that only depend on the
+current channel state information (CSI), such as maximum ratio transmission (MRT), zero-forcing
+(ZF), regularized ZF and optimum interference-constrained or power-constrained precoding [8]–
+[13]. These approaches all treat the MUI as a detrimental effect that is to be suppressed as much
+as possible.
+Unlike BLP, SLP techniques exploit information about the symbols to be transmitted in
+addition to the CSI, which can significantly improve performance at the expense of increased
+complexity at the transmitter [14], [15]. The additional degrees of freedom (DoF) provided
+by the symbol-level information make it possible to exploit the constructive component of the
+MUI, converting it into constructive interference (CI) that can move the received signals further
+from the decision thresholds of the constellation points [16]–[18]. CI-based SLP recasts the
+traditional viewpoint of interference as a source of degradation to one where interference is a
+potential resource that can be exploited.
+Constructive interference regions (CIRs), which define the degree to which the received
+symbols will be robust to noise and unmodeled perturbations, are fundamental to SLP designs.
+While early CI-based SLP approaches are intended to increase the distance of the CIRs from
+the symbol decision boundaries, they did not directly optimize the CIR. More recent techniques
+have focused on designing the precoder to directly optimize this distance [19]–[28], which has
+
+3
+been referred to as the safety margin (SM). Optimal Maximum Safety Margin (MSM) precoders
+generally result in a non-linear mapping between the symbols and the transmitted waveforms, and
+can be shown to minimize an upper bound on the symbol error rate (SER) [24]. This is contrasted
+with algorithms that minimize the mean squared-error (MMSE) between the transmitted and
+received symbols, which do not offer the same guarantee [18], [19]. MSM precoders are in
+general able to achieve a better QoS for the same level of transmit power, or the equivalently
+the same QoS with less power consumption. There has been limited work that studies CI-based
+SLP in CR systems. Although the linear precoder proposed in [29] for an overlay CR network
+is based on the use of CI with the MMSE criterion, the PU performance is impaired compared
+to the primary-only case, which violates the principle of CR design that we follow.
+The performance of both BLP and SLP are sensitive to channel uncertainties due for example
+to channel estimation errors, quantization noise or latency-related effects [30]–[32]. To mitigate
+the impact of such errors, robust designs are needed that properly model the errors and account
+for their effect in the optimization of the precoders. Two general approaches for doing so include
+exploiting known statistical properties of the CSI error or assuming worst-case bounded error
+models. In the former case, a particular distribution (e.g., Gaussian) is assumed for the error, and
+Bayesian or other probabilistic approaches [33] are employed to optimize the quality of service
+(QoS) or transmit power under certain stochastic signal-to-interference-plus-noise-ratio (SINR) or
+rate-outage probability constraints [34], [35]. In this case, the probability-constraint formulation
+is typically not deterministic and various technique must be used to obtain a tractable problem
+[36], [37]. The second approach involves the use of deterministic CSI error bounds that assume
+the error is confined to a convex uncertainty region (typically an ellipsoid) surrounding the true
+CSI [33]. In these approaches, robustness is achieved by constraining the users’ QoS or other
+design objectives to be satisfied for all channel realizations in the convex uncertainty region,
+effectively minimizing the impact of the worst-case channel [38]. This max-min philosophy can
+lead to a relatively conservative design depending on the tightness of the priori error bound.
+In either of the two cases described above, the penalty paid for increasing the robustness to
+imperfect CSI is increased transmit power.
+In overlay or cooperative CR systems, CSI errors beyond those due to channel estimation
+are anticipated due to the limited cooperation between the PBS and CBS. While robust BLP
+designs for traditional MIMO or CR scenarios have been widely investigated [39]–[42], robust
+SLP algorithms for general CR scenarios have not been considered. Prior work on robust designs
+
+4
+for SLP includes [19], which derived a robust SLP algorithm suitable for imperfect CSI with
+bounded CSI errors, but it is based on a multicast formulation without fully taking advantage
+of CI. The work described in [43] and [44] considered a linear channel distortion model with
+bounded additive noise and Gaussian-distributed channel uncertainties. They designed robust
+SLP schemes to minimize transmission power subject to CI constraints as well as QoS or SINR
+requirements. While not focused on CR applications, this prior work demonstrates that robust
+SLP designs can be formulated to improve and achieve a better balance between QoS and power
+consumption.
+In this paper, we propose a robust CR SLP algorithm for each of two different CSI error
+models that account for the quantization error in the CSI shared by the PBS with the CBS. In
+particular, we focus on overlay CR downlink channels [3] where the PBS shares with the CBS its
+CSI to the PUs and CUs, as well as its data intended for the PUs. The shared CSI is assumed to
+be quantized, which is known to often make achieving the desired user QoS constraints infeasible
+without introducing robustness into the problem formulation [45], [46]. The use of SLP for this
+scenario has not been considered previously in the literature. Additionally, for CR-based SLP
+designs, the imperfect CSI shared by the PBS will often cause the noise-free received symbols
+at both PUs and CUs to fall outside the desired CIR. To derive a robust SLP formulation for CR
+systems, we formulate the problem as one of minimizing the transmit power at the CBS while
+simultaneously satisfying the SM constraints at both the PUs and CUs to guarantee the worst-
+case user’s QoS. The SM at each user is calculated for two different channel error models. We
+first consider the case where the quantization error is norm-bounded as in [33], and then we study
+a stochastic approach based on the additive quantization noise model (AQNM) [47], [48] that
+is sufficiently accurate to approximate the quantization error at low and medium signal-to-noise
+ratios and has been widely used in the analysis of quantized MIMO systems [49]–[51].
+Below we summarize the main contributions of the paper:
+1) We apply the SLP idea to the overlay CR problem where the PBS shares (possibly imperfect)
+information with the CBS to facilitate the co-existence of the two networks. This problem
+has not been addressed previously in the literature. The work in [29] is the most related
+prior effort, but it requires that the CBS directly transmit the PBS data together with its own
+data, which is not as energy efficient as our proposed approach. In addition, [29] does not
+consider the impact of the PBS interference at the cognitive users, it uses a less effective
+SLP approach, and it does not take into account the fact that the PBS CSI exploited at the
+
+5
+CBS may be imperfect due to quantization or other effects. Our proposed SLP approach
+enables the PUs to exploit constructive interference as well as the CUs, and thus we can
+demonstrate that the presence of the cognitive network can actually improve the PU network
+performance rather than degrade it. This result is unique to the literature on CR.
+2) We first derive a power-minimizing SLP approach for overlay CR with SM constraints at
+both the PUs and CUs assuming perfect CSI, leading to a quadratic optimization problem
+with linear constraints that can be efficienty solved.
+3) Assuming a norm-bounded quantization error in the shared CSI from the PBS to the CBS,
+we derive a robust SLP algorithm based on maximizing the worst case SM. This leads to
+a conservative design that trades transmit power for increased protection of the PUs from
+PBS interference due to the quantized CSI.
+4) As an alternative approach, we employ AQNM to approximate the quantized CSI from the
+PBS. In this case, the SM of the PUs and CUs are constrained to meet a preset threshold with
+a certain probability. We then apply the Safe Approximation I method in [44] to reformulate
+the intractable probabilistic constraints as deterministic constraints and finally construct an
+optimization problem to obtain the robust SLP solution [52].
+5) To validate the effectiveness of our proposed robust precoders, we conduct simulations
+assuming the PBS channel is quantized using the scalar Lloyd Max algorithm which
+minimizes the average quantization noise power [53], [54]. Our simulations demonstrate the
+flexibility of the proposed robust SLP algorithms in trading transmit power for improved
+performance when quantized CSI is present. Furthermore, inspired by the results of [55],
+we study the problem of allocating bits to the CSI of the PBS to the PUs and CUs, and
+demonstrate that the bit allocation strategy in our robust SLP algorithm is not as important
+as that in the non-robust methods.
+Notation: Bold lower case and upper case letters indicate vectors and matrices, and non-bold
+letters express scalars. The N × N identity (zero) matrix is denoted by IN (0N×N). The N
+dimensional vector of ones (zeroes) is denoted by 1N (0N). Amn denotes the (m, n)-th element
+in the matrix A and am denotes the m-th element in the vector a. The operators (·)∗, (·)−1,
+(·)T and (·)H stand for the conjugation, the inverse, the transpose and the Hermitian transpose
+operations, respectively. Cm×n represents the space of complex matrices of dimension m×n. E(·),
+|(·)| and ∥ · ∥ respectively represent the expectation operator, absolute value and the Euclidean
+
+6
+PBS
+CBS
+PU1
+PU2
+PUNp
+CU1
+CU2
+CUNc
+Mp
+Mc
+Hpp
+Hcc
+Hcp
+Hpc
+Figure 1: Cognitive Radio System Model
+norm. CN (µ, σ2) denotes the complex normal distribution with mean µ and variance σ2. The
+functions tr{·} and diag{·} respectively indicate the trace of a matrix and a vector composed of
+the diagonal elements of a square matrix, while diag{a} denotes a square diagonal matrix with
+the elements of vector a on the main diagonal. R{·} and I{·} denote the real and imaginary
+parts of a complex number, respectively. For matrices and vectors, ≥ and ≤ denote element-wise
+inequalities.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+We consider a downlink CR network with an Mc-antenna CBS serving Nc single-antenna
+CUs. The CR network is granted access to share the primary system spectrum in which an
+Mp-antenna PBS is communicating with Np single-antenna PUs. The system model is depicted
+in Fig. 1. The direct primary and cognitive channels are assumed to be respectively denoted by
+the following flat-fading model:
+Hpp =
+�
+hT
+pp,1
+· · ·
+hT
+pp,Np
+�T
+∈ CNp×Mp
+(1)
+Hcc =
+�
+hT
+cc,1
+· · ·
+hT
+cc,Np
+�T
+∈ CNc×Mc
+(2)
+The corresponding interference channels are defined as
+Hpc =
+�
+hT
+pc,1
+· · ·
+hT
+pc,Np
+�T
+∈ CNc×Mp
+(3)
+Hcp =
+�
+hT
+cp,1
+· · ·
+hT
+cp,Np
+�T
+∈ CNp×Mc
+(4)
+
+7
+from the PBS to CUs and the CBS to PUs, respectively. We will leave further specification of
+the channel models until later. The vectors sp(t) = [sp,1(t), sp,2(t), · · · , sp,Np(t)]T and sc(t) =
+[sc,1(t), sc,2(t), · · · , sc,Nc(t)]T contain the modulated signals to be transmitted to the individual
+PUs and CUs.
+In this work we assume for simplicity that all transmitted symbols are uncorrelated and drawn
+from a D-PSK constellation with unit magnitude, i.e., sl,m(t) ∈ {s|s = exp(jπ(2d+1))/D, d ∈
+{0, · · · , D − 1}} where l ∈ {p, c} denotes the primary or cognitive system, and m denotes
+the user index in the corresponding system. The sets K = {1, · · · , Np} and J = {1, · · · , Nc}
+enumerate the PUs and CUs, respectively. The idea of CI precoding can in principle be applied
+to any constellation design [27], e.g., QAM or otherwise, but is most easily formulated for the
+case of PSK signals.
+At time slot t, the received signals at the PUs and CUs can be respectively written as
+yp(t) = Hppxp(t) + Hcpxc(t) + np(t)
+(5)
+yc(t) = Hccxc(t) + Hpcxp(t) + nc(t)
+(6)
+where xp(t) ∈ CMp×1 and xc(t) ∈ CMc×1 are the transmitted signals at the PBS and CBS after
+precoding and power loading, and np(t) ∼ CN (0, σ2
+p) and nc(t) ∼ CN (0, σ2
+c) are additive white
+Gaussian noise (AWGN) vectors. In order to simplify the notation, in what follows we drop the
+time index t.
+A. CR-PALP
+Conventional precoding methods such as MMSE, ZF and maximum-SINR beamforming are
+designed with the objective of minimizing the inter-user interference so that the received symbols
+lie as close as possible to the nominal constellation points (or scaled versions thereof in the case
+of PSK). This is effectively equivalent to ensuring that for each user m, the noise-free received
+signal rm = hmx lies within a circle centered at its corresponding constellation point sm [19],
+as depicted in Fig. 2. The shaded area inside the circle is referred to as the symbol region (SR),
+a down-scaled version of the decision region for sm.
+The method described in [29] is based on the MMSE criterion and the early PALP technique
+for SLP [17]; it is the prior approach most closely related to the algorithms we present in
+this paper for cognitive radio scenarios. However, a modification to the PALP approach in [29]
+is necessary for a fair comparison, and to allow the algorithm to protect the PUs from the CR
+
+8
+Im
+Re
+rm
+θ
+sm
+constellation point
+noise-free symbol
+symbol region boundary
+decision boundary
+symbol region
+Figure 2: Symbol region for conventional precoding
+interference. In particular, we tailor [29] (hereafter referred to as CR-PALP) by allowing different
+instantaneous power scaling factors at the PBS and CBS:
+fp =
+�
+Pp
+trace{WpspsH
+p WH
+p }, fc =
+�
+Pc
+trace{WcscsH
+c WH
+c }
+(7)
+where fp and fc are respectively the instantaneous transmitted power scaling factors for the
+primary and cogntive systems, and s =
+�
+sT
+p
+sT
+c
+�T
+denotes the symbol vector transmitted by the
+CBS, which in [29] includes both the CU and PU data.
+The generalized MSE criterion for CR-PALP thus becomes
+ǫ = E{∥Vps + HWcs − (A + BQφ)s∥2}
+(8)
+where A = diag{[1, · · · , 1, 0, · · · , 0]} is a diagonal matrix whose first Np diagonal elements
+equal 1, B = diag{[0, · · · , 0, 1, · · · , 1]} is a diagonal matrix whose last Nc elements equal 1,
+and Qφ = diag(s) · |HHH| · diag(s)H contains the phase-corrected correlation elements. The
+precoding matrix at the CBS derived from the MMSE criterion is given by
+Wc = HH(HHH)−1(A + BQφ − Vp) ,
+(9)
+and the received signals at the PUs and CUs are
+yp = fpsp + np
+(10)
+yc = (fp − fc)HpcWpsp + fcQφ
+c sc + nc .
+(11)
+
+m9
+Im
+Re
+θ
+Figure 3: Symbol region for CI-Based SLP
+Im
+Re
+θ
+zmI
+δ
+δm
+Figure 4: Symbol region and safety margin in a modified coordinate ststem
+The performance of the PUs in our CR scenario will not be impaired by CR using this modified
+CR-PALP approach, which is different from using the method of [29] directly.
+B. SM-constrained SLP
+For PSK constellations, it is not necessary that rm be close to sm in order to be decoded
+correctly, as long as it lies in the correct decision region with a given level of certainty. Thus,
+it is not necessary that all of the inter-user interference be eliminated, since some interference
+components could add constructively and push the received symbol further into the decision
+region, making it more robust to noise and interference external to the system. We can thus
+redefine the SR as, for example, in Fig. 3, where the SR becomes a displaced version of the
+circular sector of angular extent 2π/D centered at the origin and corresponding to sm. This
+
+10
+displaced sector has an infinite radius, and all points within it are at least a certain distance δm
+from the decision boundaries for sm. This region is referred to as a constructive interference
+region (CIR) with safety margin (SM) δm [56]. The larger δm, the more robust the received
+signal will be to noise, interference, modeling errors, or other impairments.
+In order to mathematically interpret the CIR and SM in a unified way, we rotate the original
+coordinate system by the phase of the desired constellation symbol, i.e., ∡sm, to obtain the
+modified coordinate system in Fig. 4. After rotation, sm is placed at 1 on the real axis, and rm
+is relocated to
+zm = rms∗
+m .
+(12)
+Then we can easily calculate the SM of the noise-free symbol at user m as
+δm = R{zm} sin θ − |I{zm}| cos θ.
+(13)
+Ideally, the SM should be large enough to sufficiently reduce the probability that noise or other
+impairments will push the noise-free signal outside the desired detection region; the larger the
+SM, the smaller the SER. To design the precoder, one can constrain the SM to be above
+a certain threshold to ensure a given target SER. The fact that the CIR in Fig. 4 is much
+larger than the SR in Fig. II-B means that increased flexibility is available to achieve the given
+performance objective. In this paper, we will consider the following type of SLP optimization,
+which minimizes the transmit power to achieve a certain desired SM:
+min
+x
+∥x∥2
+(14)
+subject to
+δm ≥ δm,0
+∀m ∈ M
+(15)
+where δm,0 is the desired minimum SM for user m and M = {1, · · · , M} indexes the users. For
+simplicity, we will assume in the following that the same threshold is employed for all users,
+i.e., δm,0 = δ0
+p for the PUs and δm,0 = δ0
+c for the CUs, although this assumption can be easily
+relaxed.
+III. POWER MINIMIZATION SLP IN CR
+Before considering the robust SLP design, we first examine the simple case where the PBS
+shares its data and perfect CSI with the CBS. The SM for the PUs and CUs is constrained to be
+at least δ0
+p and δ0
+c, corresponding to target SERs for these users. Here we focus on SLP designs
+that minimize the transmit power at the CBS and achieve the SM QoS constraints at the both
+the PUs and CUs.
+
+11
+A. Primary System
+The rotated symbols received at the PUs can be expressed as
+zp,k = s∗
+p,krp,k = s∗
+p,k(hpp,kxp + hcp,kxc)
+(16)
+for k ∈ K. Defining
+˜hpp,k ≜ s∗
+p,khpp,k,
+˜hcp,k ≜ s∗
+p,khcp,k ,
+(17)
+we have
+zp,k = ˜hpp,kxp + ˜hcp,kxc.
+(18)
+The constraint ensuring the QoS of the PUs can be expressed as
+δp,k = R{zp,k} sin θ − |I{zp,k}| cos θ ≥ δ0
+p , ∀k ∈ K,
+(19)
+which is equivalent to
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+R{˜hpp,kxp} sin θ − I{˜hpp,kxp} cos θ
++R{˜hcp,kxc} sin θ − I{˜hcp,kxc} cos θ ≥ δ0
+p
+(20a)
+R{˜hpp,kxp} sin θ + I{˜hpp,kxp} cos θ
++R{˜hcp,kxc} sin θ + I{˜hcp,kxc} cos θ ≥ δ0
+p
+.
+(20b)
+For any given complex vector x, we define the operator
+℧(x) ≜
+
+R{x} sin θ − I{x} cos θ
+−R{x} cos θ − I{x} sin θ
+R{x} sin θ + I{x} cos θ
+R{x} cos θ − I{x} sin θ
+
+
+(21)
+and denote
+�H℧
+pp,k = ℧(˜hpp,k),
+�H℧
+cp,k = ℧(˜hcp,k).
+(22)
+Using the following real-valued notation,
+ˇxp =
+
+R{xp}
+I{xp}
+
+ ,
+ˇxc =
+
+R{xc}
+I{xc}
+
+ ,
+(23)
+the constraint in Eq. (19) can be simplified as
+�H℧
+pp,kˇxp + �H℧
+cp,kˇxc ≥ δ0
+p12
+∀k ∈ K .
+(24)
+
+12
+B. Cognitive System
+Similarly, the rotated symbols at the CUs can be written as
+zc,j = s∗
+c,jrc,j = s∗
+c,j(hcc,jxc + hpc,jxp)
+(25)
+= ˜hcc,jxc + ˜hpc,jxp ,
+(26)
+where
+˜hcc,j ≜ s∗
+c,jhcc,j
+˜hpc,j ≜ s∗
+c,jhpc,j .
+(27)
+The SM constraints at the CUs can be expressed as
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+R{˜hcc,jxc} sin θ − I{˜hcc,jxc} cos θ
++R{˜hpc,jxp} sin θ − I{˜hpc,jxp} cos θ ≥ δ0
+c
+(28a)
+R{˜hcc,jxc} sin θ + I{˜hcc,jxc} cos θ
++R{˜hpc,jxp} sin θ + I{˜hpc,jxp} cos θ ≥ δ0
+c
+(28b)
+for j ∈ J , which can again be written more compactly using the operator in Eq. (21):
+�H℧
+cc,jˇxc + �H℧
+pc,jˇxp ≥ δ0
+c12
+∀j ∈ J .
+(29)
+Combining all of the above notation together, we can express the general power minimization
+SLP problem with perfect CSI as follows:
+minimize
+ˇxc
+∥ˇxc∥2
+(P1)
+subject to
+
+− �H℧
+cp
+− �H℧
+cc
+
+ ˇxc ≤
+
+
+�H℧
+pp
+�H℧
+pc
+
+ ˇxp −
+
+δ0
+p12Np
+δ0
+c12Nc
+
+
+(C1)
+where �H℧
+cp ≜ ℧(diag(s∗
+p)Hcp), �H℧
+cc ≜ ℧(diag(s∗
+c)Hcc), �H℧
+pp ≜ ℧(diag(s∗
+p)Hpp), and �H℧
+pc ≜
+℧(diag(s∗
+c)Hpc). This is a quadratic programming problem with linear inequality constraints and
+can be efficiently solved using a variety of numerical methods.
+IV. ROBUST SLP FOR NORM-BOUNDED CSI ERRORS
+In practice, the CSI shared by the PBS with the CBS will be imperfect due for example to
+quantization, or somewhat outdated due to delays required for processing and transmission. As
+a result, robust precoding designs are critical for overlay systems. We address such a design in
+
+13
+this section for the case where the imperfect CSI can be described in terms of a norm-bounded
+error. We model the CSI shared by the PBS with the CBS as follows:
+hpp,k = ˆhpp,k + ep,k
+(30)
+hpc,j = ˆhpc,j + ec,j,
+(31)
+where the ˆ· indicates the shared CSI and ep,k, ec,j are norm-bounded CSI error vectors, i.e.,
+∥ep,k∥2 ≤ ǫp,k and ∥ec,j∥2 ≤ ǫc,j. No other assumption regarding the channels is required. Using
+Eq. (21), it is easy to show that
+�H℧
+pp,k = ℧(˜hpp,k) = ℧(s∗
+p,k(ˆhpp,k + ep,k))
+(32)
+= ¯H℧
+pp,k + ˜E℧
+p,k
+(33)
+�H℧
+pc,j = ℧(˜hpc,j) = ℧(s∗
+c,j(ˆhpc,j + ec,j))
+(34)
+= ¯H℧
+pc,j + ˜E℧
+c,j,
+(35)
+where ¯H℧
+pp,k ≜ ℧(s∗
+p,kˆhpp,k), ˜E℧
+p,k ≜ ℧(s∗
+p,kep,k), ¯H℧
+pc,j ≜ ℧(s∗
+c,jˆhpc,j), and ˜E℧
+c,j ≜ ℧(s∗
+c,jec,j).
+With this notation, the constraints in P1 can be reformulated as
+( ¯H℧
+pp,k + ˜E℧
+p,k)ˇxp + �H℧
+cp,kˇxc ≥ δ0
+p12, ∀k ∈ K
+(36)
+�H℧
+cc,jˇxc + ( ¯H℧
+pc,j + ˜E℧
+c,j)ˇxp ≥ δ0
+c12, ∀j ∈ J .
+(37)
+For a robust bounded-CSI-error design, we desire that the above constraints hold for every
+possible error realization and every user:
+−˜E℧
+p,kˇxp ≤ ¯H℧
+pp,kˇxp + �H℧
+cp,kˇxc − δ0
+p12 ,
+∀∥ep,k∥2 ≤ ǫp,k, ∀k ∈ K ,
+(38)
+−˜E℧
+c,jˇxp ≤ �H℧
+cc,jˇxc + ¯H℧
+pc,jˇxp − δ0
+c12 ,
+∀∥ec,j∥2 ≤ ǫc,j, ∀j ∈ J .
+(39)
+We separate the operator ℧(x) into two parts, as follows:
+℧(x) ≜
+
+℧1(x)
+℧2(x)
+
+ ≜
+
+x℧1
+x℧2
+
+ ,
+(40)
+where
+x℧1 =
+�
+R{x} sin θ − I{x} cos θ
+−R{x} cos θ − I{x} sin θ
+�
+= ℧1(x),
+(41)
+
+14
+x℧2 =
+�
+R{x} sin θ + I{x} cos θ
+R{x} cos θ − I{x} sin θ
+�
+= ℧2(x),
+(42)
+so that constraint (38) can be rewritten in two parts as
+−˜e℧1
+p,kˇxp ≤ ¯h℧1
+pp,kˇxp + �h℧1
+cp,kˇxc − δ0
+p ,
+∀∥ep,k∥2 ≤ ǫp,k, ∀k ∈ K ,
+(43)
+−˜e℧2
+p,kˇxp ≤ ¯h℧2
+pp,kˇxp + �h℧2
+cp,kˇxc − δ0
+p ,
+∀∥ec,j∥2 ≤ ǫc,j, ∀j ∈ J ,
+(44)
+where
+−˜e℧1
+p,kˇxp ≤ ∥ − ˜e℧1
+p,k∥2∥ˇxp∥2
+(45)
+=
+������
+�
+R{eT
+p,k}
+I{eT
+p,k}
+�
+
+ sin θ
+− cosθ
+− cos θ
+− sin θ
+
+
+������
+F
+∥ˇxp∥2
+(46)
+≤
+���
+�
+R{eT
+p,k}
+I{eT
+p,k}
+����
+F
+������
+
+ sin θ
+− cosθ
+− cosθ
+− sin θ
+
+
+������
+F
+∥ˇxp∥2
+(47)
+≤
+√
+2ǫp,k∥ˇxp∥2 .
+(48)
+Similarly, (39) becomes
+− ˜e℧2
+p,kˇxp ≤
+√
+2ǫp,k∥ˇxp∥2 .
+(49)
+Thus, the constraints in (38) and (39) can be expressed as
+¯H℧
+pp,kˇxp + �H℧
+cp,kˇxc ≥ (
+√
+2ǫp,k∥ˇxp∥2 + δ0
+p)12, ∀k ∈ K
+(50)
+¯H℧
+pc,jˇxp + �H℧
+cc,jˇxc ≥ (
+√
+2ǫc,j∥ˇxp∥2 + δ0
+c)12, ∀j ∈ J .
+(51)
+Finally, the robust precoder can be obtained by solving the following optimization problem:
+minimize
+ˇxc
+∥ˇxc∥2
+(52)
+subject to ¯H℧
+pp,kˇxp + �H℧
+cp,kˇxc ≥ (
+√
+2ǫp,k∥ˇxp∥2 + δ0
+p)12
+∀k ∈ K
+(53)
+¯H℧
+pc,jˇxp + �H℧
+cc,jˇxc ≥ (
+√
+2ǫc,j∥ˇxp∥2 + δ0
+c)12
+∀j ∈ J .
+(54)
+As in the case with perfect CSI, the robust SLP design can be found via a quadratic program
+with linear inequality constraints.
+
+15
+Table I: Distortion Factors for Different Quantization Bit Resolutions [53]
+b
+1
+2
+3
+4
+5
+ρ
+0.3634
+0.1175
+0.03454
+0.009497
+0.002499
+V. ROBUST SLP FOR STOCHASTIC CSI ERRORS
+The bounded error model above is a very conservative approach, given its goal of ensuring that
+the SM constraints are met for all possible CSI error realizations. A less conservative approach
+that allows constraint violations with some acceptably small probability is to assume a statistical
+CSI error model. As an example, in this section we consider the case where such a model for
+the PBS CSI error is available due to knowledge of how the channel is quantized. In particular,
+we assume that the channels hQ
+pp,k and hQ
+pc,j shared by the PBS are element-wise quantized,
+and we use the approximate additive quantization noise model (AQNM) [47], [48] to describe
+their resulting statistics. Other models are possible based on the specific quantization method
+employed.
+We assume that the channels are Gaussian with zero mean and covariances given by
+Rhpp,k = E{hH
+pp,khpp,k} = βpIMp
+(55)
+Rhpc,j = E{hH
+pc,jhpc,j} = βcIMp .
+(56)
+Using AQNM, the quantized CSI from the PBS after rotation is expressed as
+�hQ
+pp,k = Q(�hpp,k) ≈ αp�hpp,k + �nQ
+pp,k
+(57)
+�hQ
+pc,j = Q(�hpc,j) ≈ αc�hpc,j + �nQ
+pc,j,
+(58)
+where Q(·) is a scalar quantization function applied element-wise and separately to the real and
+imaginary parts of the input. The vectors �nQ
+pp,k ≜ s∗
+p,knQ
+pp,k ∈ C1×Mp and �nQ
+pc,j ≜ s∗
+c,jnQ
+pc,j ∈
+C1×Mp denote the zero-mean Gaussian-distributed quantization noise vectors, and both are as-
+sumed to be uncorrelated with �hpp,k and �hpc,j. The gains αk = 1−ρk for k ∈ {p, c} are functions
+of the following distortion factors [49]:
+ρp =
+E{∥hpp,k − hQ
+pp,k∥2}
+E{∥hpp,k∥2}
+, ρc = E{∥hpc,j − hQ
+pc,j∥2}
+E{∥hpc,j∥2}
+.
+(59)
+
+16
+The value of ρ is given in Table I for different bit resolutions b assuming an optimal non-
+uniform Lloyd-Max quantizer [53]. The phase rotation does not alter the covariance matrices of
+the quantization noise, which are given by [47]
+R{�nQ
+pp,k} = αpρpdiag{Rhpp,k} = αpρpβpIMp,
+(60)
+R{�nQ
+pc,j} = αcρcdiag{Rhpc,j} = αcρcβcIMp.
+(61)
+Based on Eq. (57) and Eq. (58), we can derive
+�hpp,k =
+�hQ
+pp,k − �nQ
+pp,k
+αp
+= ¯αp�hQ
+pp,k − ¯αp�nQ
+pp,k
+(62)
+�hpc,j =
+�hQ
+pc,j − �nQ
+pc,j
+αc
+= ¯αc�hQ
+pc,j − ¯αc�nQ
+pc,j ,
+(63)
+where ¯αp =
+1
+αp and ¯αc =
+1
+αc. Therefore,
+�H℧
+pp,k = ℧( ¯αp�hQ
+pp,k − ¯αp�nQ
+pp,k) = ¯αp( �HQ,℧
+pp,k − �NQ,℧
+pp,k)
+(64)
+�H℧
+pc,j = ℧( ¯αc�hQ
+pc,j − ¯αc�nQ
+pc,j) = ¯αc( �HQ,℧
+pc,j − �NQ,℧
+pc,j),
+(65)
+where �HQ,℧
+pp,k ≜ ℧(�hQ
+pp,k), �NQ,℧
+pp,k ≜ ℧(�nQ
+pp,k), �HQ,℧
+pc,j ≜ ℧(�hQ
+pc,j), and �NQ,℧
+pc,j ≜ ℧(�nQ
+pc,j). Substituting
+Eq. (64) and Eq. (65) in (C1), we have
+¯αp( �HQ,℧
+pp,k − �NQ,℧
+pp,k)ˇxp + �H℧
+cp,kˇxc ≥ δ0
+p12, ∀k ∈ K
+(C2-1)
+�H℧
+cc,jˇxc + ¯αc( �HQ,℧
+pc,j − �NQ,℧
+pc,j)ˇxp ≥ δ0
+c12, ∀j ∈ J .
+(C2-2)
+The constraints above are expressed in terms of the unknown random quantization noise, and
+thus cannot be directly enforced. Instead, we choose to pose the problem such that the constraints
+are achieved with a certain probability. In particular, we rewrite (C2-1) and (C2-2) as follows:
+P{αp �H℧
+cp,kˇxc + �HQ,℧
+pp,kˇxp − αpδ0
+p12 ≥ �NQ,℧
+pp,kˇxp} ≥ v1,
+(C3-1)
+P{αc �H℧
+cc,jˇxc + �HQ,℧
+pc,j ˇxp − αcδ0
+c12 ≥ �NQ,℧
+pc,j ˇxp} ≥ v2,
+(C3-2)
+where P{A} denotes the probability of event A, and {v1, v2} ∈ (0.5, 1] represent the probability
+thresholds. In the following two subsections, we find expressions for the probabilities in (C3-1)
+and (C3-2).
+
+17
+A. Primary System
+It is easy to show that
+E{ �NQ,℧
+pp,k} = 02×2Mp
+(66)
+R �NQ,℧
+pp,k = αpρpI2Mp
+(67)
+E{ �NQ,℧
+pp,k( �NQ,℧
+pp,k)H} = Mpαpρpβp
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+ .
+(68)
+The vector
+qp,k ≜ �NQ,℧
+pp,kˇxp ≜
+
+q1
+p,k
+q2
+p,k
+
+
+(69)
+is a bivariate correlated Gaussian random vector with mean
+E{qp,k} = E{ �NQ,℧
+pp,kˇxp} = E{ �NQ,℧
+pp,k}ˇxp = 02×1
+(70)
+and covariance
+Rqp,k = E{qp,kqT
+p,k}
+(71)
+= E{ �NQ,℧
+pp,kˇxp( �NQ,℧
+pp,kˇxp)T}
+(72)
+= Pp
+2Mp
+E{ �NQ,℧
+pp,kI2Mp( �NQ,℧
+pp,k)T}
+(73)
+= Pp
+2Mp
+× Mpαpρpβp
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+
+(74)
+= Pp
+2 αpρpβp
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+ .
+(75)
+To simplify the notation, we define
+wp,k(ˇxc) ≜ αp �H℧
+cp,kˇxc + �HQ,℧
+pp,kˇxp − αpδ0
+p12 ≜
+
+w1
+p,k
+w2
+p,k
+
+
+(76)
+which is affine in ˇxc. Using the new notation, the chance constraint (C3-1) can be rewritten as
+P{wp,k(ˇxc) ≥ qp,k} ≥ v1.
+(77)
+No explicit closed-form expression for this probability seems possible, so we resort to finding a
+tractable approximation using the following lemma.
+
+18
+Lemma 1. P{wp,k(ˇxc) ≥ qp,k} ≥ v1 can be approximated by the inequality
+αp �H℧
+cp,kˇxc + �HQ,℧
+pp,kˇxp ≥ ηpR
+1
+2qp,k12 + αpδ0
+p12
+(78)
+where η ≜
+√
+2 erf−1 �
+2√v1 − 1
+�
+.
+Proof. See Appendix A.
+B. Cognitive System
+The constraint in (C3-2) for the cognitive system can be found in an identical fashion. It is
+easy to show that
+E{ �NQ,℧
+pc,j} = 02×2Mp
+(79)
+R �NQ,℧
+pc,j = αcρcβcI2Mp
+(80)
+E{ �NQ,℧
+pc,j( �NQ,℧
+pc,j)H} = Mpαcρcβc
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+ ,
+(81)
+and we define
+qc,j ≜ �NQ,℧
+pc,j ˇxp ≜
+
+q1
+c,j
+q2
+c,j
+
+ .
+(82)
+Similar to the above, we can show that qc,j is a bivariate correlated Gaussian random variable
+with mean
+E{qc,j} = 02×1
+(83)
+and covariance
+Rqc,j = Pp
+2 αcρcβc
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+ .
+(84)
+Furthermore, we define
+wc,j(ˇxc) ≜ αc �H℧
+cc,jˇxc + �HQ,℧
+pc,j ˇxp − αcδ0
+c12 ≜
+
+w1
+c,j
+w2
+c,j
+
+
+(85)
+which is affine in ˇxc. Using the new notation, the chance constraint (C3-2) can be rewritten as
+P{wc,j(ˇxc) ≥ qc,j} ≥ v2 .
+(86)
+As with the primary system, we define ¯wc,j(ˇxc) ≜ R
+− 1
+2
+qc,jwc,j(ˇxc) and ¯qc,j ≜ R
+− 1
+2
+qc,jqc,j, and we
+obtain the following lemma.
+
+19
+Lemma 2. P{ ¯wc,j(ˇxc) ≥ ¯qc,j} ≥ v2 can be approximated by the inequality
+αc �H℧
+cc,jˇxc + �HQ,℧
+pc,j ˇxp ≥ ηcR
+1
+2qc,j12 + αcδ0
+c12,
+(87)
+where ηc =
+√
+2 erf−1 �
+2√v2 − 1
+�
+is a preset constant.
+Proof. The proof follows directly from that of Lemma 1 exactly.
+C. Optimization Problem for Stochastic CSI Error Model
+We can now formulate the robust SLP design with the probabilistic constraints by replacing
+(C3-1) and (C3-2) with (78) and (87), as follows:
+minimize
+ˇxc
+∥ˇxc∥2
+(PQ)
+subject to
+αp �H℧
+cp,kˇxc + �HQ,℧
+pp,kˇxp ≥ ηpR
+1
+2qp,k12 + αpδ0
+p12
+∀k ∈ K
+(CQ-1)
+αc �H℧
+cc,jˇxc + �HQ,℧
+pc,j ˇxp ≥ ηcR
+1
+2qc,j12 + αcδ0
+c12
+∀j ∈ J .
+(CQ-2)
+As with the other problems studied above, the result is a quadratic program with linear inequality
+constraints.
+VI. NUMERICAL RESULTS
+In this section, we assess the performance of our proposed power-minimizing SLP (PMSLP)
+approaches. Monte-Carlo simulations are conducted over 1000 independent channel realizations,
+each employing a block of T = 100 symbols. The channels Hpp, Hcp, Hpc and Hcc are
+composed of i.i.d. Gaussian random variables with zero mean and unit variance. The complex
+Gaussian noise is assumed to have the same power (σp = σc = 1) for all PUs and CUs. The
+PBS transmission power is set at Pp = 10 dBW. Both the PBS and CBS are assumed to have
+Mp = Mc = 8 antennas and the number of PUs and CUs are both set at Np = Nc = 4.
+Since for SLP we work with finite alphabet constellations, we will analyze the block trans-
+mission performance of the system using the throughput τ as calculated in [57]:
+τ = (1 − PB) × c × T × N,
+(88)
+
+20
+0
+5
+10
+15
+20
+25
+30
+CBS Transmission Power (dBW)
+10-3
+10-2
+10-1
+100
+CU BER
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+PU BER
+10-3
+CR-PALP
+PMSLP
+CR-PALP
+PMSLP
+Primay Only
+Figure 5: BER of CU (left) or PU (right) vs. CBS transmit power where Pp = 10 dBW, QPSK
+modulation.
+where PB is the block error rate (BLER), c = log2 D is the number of bits per modulation
+symbol, T is the block length and N is the number of receivers. In each block for each user,
+there are C = c × T data message bits transmitted from the BS. For PSK modulation, assuming
+a binomial distribution of errors in each block, the probability of more than q errors occurring
+in one block of C bits is expressed as
+Pe(q, C) = 1 −
+q
+�
+i=0
+�C
+i
+�
+P i
+b(1 − Pb)C−i
+(89)
+where Pb is the BER. If the receiver detects errors without correction, a block is received correctly
+only if all C bits in the block are received correctly, and thus the BLER is PB = Pe(0, C). On
+the other hand, if the receiver is capable of correcting up to Q errors in each block, then the
+BLER is given by PB = Pe(Q, C) [58].
+We begin in Fig. 5 assuming perfect CSI, plotting the average BER of the users versus the
+CBS transmission power, and comparing PMSLP assuming minimum safety margin δ0
+p = 1.9
+with the performance of the CR-PALP algorithm described in Section II-A. With these settings,
+even when the CBS increases its transmit power to better serve the CUs, it can still avoid any
+negative impact on the PUs such that the BER of PUs is not greater than that in the primary-only
+case as shown in Fig. 5. Moreover, the BER of the PUs remains nearly unchanged for both types
+of precoders, although the PUs actually enjoy some benefit with PMSLP since it exploits CI from
+the CBS signals which can further improve the SM for the PUs. Meanwhile, PMSLP provides
+a much lower BER for the CUs; the CBS can save more than 15dBW of power to achieve an
+
+21
+-1.9
+-1.7
+-1.5
+-1.3
+-1.1
+-0.9
+-0.7
+-0.5
+10-5
+10-4
+10-3
+10-2
+PU BER
+Non-robust PMSLP, 
+p=0.3, c=
+Robust PMSLP, p=0.3, c=
+Non-robust PMSLP, 
+p= , c=0.3
+Robust PMSLP, p= , c=0.3
+Figure 6: BER of PU vs. error norm bound where Pp = 10 dBW, δ0
+p = δ0
+c = 1.5, QPSK
+modulation.
+-1.9
+-1.7
+-1.5
+-1.3
+-1.1
+-0.9
+-0.7
+-0.5
+10-5
+10-4
+10-3
+10-2
+10-1
+CU BER
+Non-robust PMSLP, 
+p=0.3, c=
+Robust PMSLP, p=0.3, c=
+Non-robust PMSLP, 
+p= , c=0.3
+Robust PMSLP, p= , c=0.3
+Figure 7: BER of CU vs. error norm bound where Pp = 10 dBW, δ0
+p = δ0
+c = 1.5, QPSK
+modulation.
+uncoded BER of 10−2, compared to the CR-PALP method which allows only the CUs to benefit
+from the inter-user CI in the cognitive system, but in general will not entirely eliminate the
+destructive interference from the PBS. On the contrary, our proposed PMSLP precoder can take
+advantage of CI not only from the CU MUI but also what is available (although not optimized)
+from the primary system. Moreover, the PUs can also benefit from the CI coming from the
+cognitive system. CR-PALP requires that the CBS act as a relay transmitting not only the CUs’
+but also the PUs’ signals, which is unnecessary in our PMSLP design.
+Next we consider the norm-bounded CSI error model discussed in Section IV. With the SM
+threshold for the PUs and CUs set to 1.5, we plot the BER of the PUs and CUs as the norm of
+
+22
+-1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1
+-1
+-0.9 -0.8 -0.7 -0.6 -0.5
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+CBS Transmission Power (W)
+Non-robust PMSLP, 
+p=0.3, c=
+Robust PMSLP, p=0.3, c=
+Non-robust PMSLP, 
+p= , c=0.3
+Robust PMSLP, p= , c=0.3
+Figure 8: Transmit power vs. error norm bound where Pp = 10 dBW, δ0
+p = δ0
+c = 1.5, QPSK
+modulation.
+one error (ǫp,k or ǫc,j) changes as log10 ǫ while the norm of the other is fixed to 0.3. For simplicity,
+we assume that the CSI error bounds are the same for all users: ǫp,k = ǫp and ǫc,j = ǫc. Fig. 6
+and Fig. 7 respectively show the BER for the PUs and CUs as a function of the error bound,
+and demonstrate that the proposed robust precoder can mitigate the CSI uncertainty and provide
+a much lower BER compared to the non-robust precoder. With the robustness introduced, the
+BER of the PUs and CUs actually decrease as the norm of the corresponding error increases.
+This can be explained by examining Eq. (54), where we see that a larger error bound creates
+a larger effective SM in the constraint, which provides the robustness necessary to account for
+the imperfect channel. There is however a price to be paid for this robustness, as clearly seen in
+Fig. 8, which shows that the robust schemes require the CBS to operate with significantly more
+power, especially as the error bound increases. It is clear from these results that the worst-case
+approach based on the norm-bounded CSI error leads to a conservative design.
+In order to quantify the power-performance trade-off between the robust and non-robust
+designs, in Fig. 9 we plot the energy efficiency (EE) of the approaches, defined as the ratio
+between the throughput calculated from Eq. (88) and the transmit power per channel:
+EE =
+τ
+T × ∥ˇxc∥2.
+(90)
+We see that despite the increase in transmit power, the proposed robust SLP algorithm achieves
+a significantly higher energy efficiency. When the uncertainty ǫc in Hpc is fixed, the energy
+efficiency at the CBS decreases with greater uncertainty in Hpp since the CBS needs to consume
+
+23
+-1.9
+-1.7
+-1.5
+-1.3
+-1.1
+-0.9
+-0.7
+-0.5
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+CBS Energy Efficency (bit/Hz/J/channel use)
+Non-robust PMSLP, p=0.3, c=
+Robust PMSLP, p=0.3, c=
+Non-robust PMSLP, p= , c=0.3
+Robust PMSLP, p= , c=0.3
+Figure 9: Energy Efficiency vs. Norm of the Error where Pp = 10 dBW, δ0
+p = δ0
+c = 1.5, QPSK.
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+Preset 
+c
+0
+10-4
+10-3
+10-2
+10-1
+CU BER
+Non-robust b p=bc=2
+Robust b p=bc=2, v1=v2=0.8
+Non-robust b p=bc=3
+Robust b p=bc=3, v1=v2=0.95
+Figure 10: BER of CUs vs. preset SM at the CUs where Pp = 10 dBW, δ0
+p = 1.5, QPSK
+modulation.
+more power to meet the SM constraint at the PUs. Interestingly, in our setting with δ0
+p = δ0
+c = 1.5,
+if the uncertainty ǫp in Hpp is fixed, the energy efficiency at the CBS first increases as the error
+in Hpc increases and then decreases when log10(ǫc) ≥ −0.7. Considering that the effective SM
+(right side of Eq. (54)) in the robust PMSLP algorithm is decided by both the norm bound of the
+CSI error and the preset SM, when log10(ǫc) ≤ −1.1, the preset δ0
+c = 1.5 is not large enough for
+the CIR to account for the CSI error in Hpc, which results in a high BER for the CUs and lower
+EE. This illustrates the importance of selecting a suitable SM based on the CSI error bound.
+The remaining examples use the probabilistic SM constraints discussed in Section V based on
+the AQNM approximation, although the actual quantized CSI is generated using a non-uniform
+Lloyd Max quantizer [53], [54]. Fig. 10 shows the BER of the CUs as a function of the preset SM
+
+24
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+Preset 
+c
+0
+0
+10
+20
+30
+40
+50
+60
+70
+80
+CBS Transmission Power (W/symbol)
+Non-robust b p=bc=2
+Robust b p=bc=2, v1=v2=0.8
+Non-robust b p=bc=3
+Robust b p=bc=3, v1=v2=0.95
+Figure 11: Transmit Power at CBS vs. preset SM at the CUs where Pp = 10 dBW, δ0
+p = 1.5,
+QPSK modulation.
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+Preset 
+c
+0
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+Cognitive Energy Efficency (bit/Hz/J/channel use)
+Non-robust b p=bc=2
+Robust b p=bc=2, v1=v2=0.8
+Non-robust b p=bc=3
+Robust b p=bc=3, v1=v2=0.95
+Figure 12: EE at CBS vs. preset SM at the CUs where Pp = 10 dBW, δ0
+p = 1.5, QPSK
+modulation.
+threshold at the CUs, assuming either b = 2 or b = 3 quantization bits per channel coefficient
+and different probability constraints. We see that the CUs reap benefits from the robust SLP
+design, achieving significantly lower BER. Again illustrating the trade-off of robustness with
+increase power, we see that in Fig. 11 as the preset δ0
+c increases, the CBS in the robust SLP
+approach requires more power to meet the SM constraint the non-robust SLP. In order to fairly
+compare different SLP methods, we plot the EE at the CBS in Fig. 12. It is clear that the greater
+the preset SM, or the higher the quantization resolution, the higher the EE. The robust SLP
+approach performs particuarly well for b = 3, δ0
+c = 1.7, yielding an increase in EE by a factor
+of about 10 compared with the non-robust SLP.
+
+25
+2
+3
+4
+5
+Quantization Bits (b)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+Cognitive Energy Efficency (bit/Hz/J/channel use)
+Non-robust PMSLP, b p=b, b c=3
+Robust PMSLP, bp=b, b c=3
+Non-obust PMSLP, b p=3, bc=b
+Robust PMSLP, bp=3, bc=b
+bp=b, b c=3
+bp=b, b c=3
+bp=3, bc=b
+bp=3, bc=b
+Figure 13: Energy Efficiency at the CBS vs. quantization resolution where Pp = 10 dBW,
+δ0
+p = δ0
+c = 1.5, v1 = v2 = 0.9, QPSK modulation.
+In the final example, we study the allocation of the quantization bits on the system performance
+[55]. In particular, in Fig. 13 we plot the energy efficiency at the CBS when the direct hpp,k
+and interference channels hpc,j are quantized with different resolutions. For the non-robust SLP
+schemes (blue curves), the CBS achieves higher EE in cases where bc > bp, indicating that for a
+fixed number of quantization bits, it is more energy efficient for the cognitive system to receive
+a more accurate representation of the interference channel than the direct channel. However,
+the robust-SLP schemes are less sensitive to the allocation of the quantization bits, and show
+roughly the same performance regardless of which channel is more accurately represented.
+VII. CONCLUSION
+In this paper, we have designed non-robust and robust SLP schemes for CR system with the
+goal of minimizing the transmission power and simultaneously ensuring the QoS of all users.
+Unlike traditional CR precoding techniques, we set the SM threshold in the interference con-
+straints instead of using SINR or BER metrics in order to fully exploit constructive interference
+as much as possible. First, under the assumption of perfect CSI, we propose an SLP algorithm
+that performs significantly better than a prior CR-based SLP approach modified to address our
+overlay problem. In the proposed algorithm, not only the CUs but also the PUs benefit from the
+constructive interference.. Then, using two different CSI error models, we derive two robust SLP
+methods, one based on a max-min optimization of the worst-case CSI error and the other on a
+probability-constraint problem using AQNM to approximate the impact of the CSI quantization.
+
+26
+All of the proposed optimization problems result in a quadratic objective function with linear
+inequality constraints that can be efficiently solved.. Our numerical results demonstrate that our
+robust SLP schemes can deal with various types of CSI error and still maintain a high energy
+efficiency. For the stochastic CSI model, our robust SLP algorithm can maintain a lower BER
+at both the PUs and CUs. A key observation from our results is that, by enabling the PUs to
+exploit constructive interference as well as the CUs, the presence of the cognitive network can
+actually improve the PU network performance rather than degrade it.
+APPENDIX
+A. Proof of Lemma 1
+From [44] we note that eliminating the (possible) correlation between the entries of qp,k by
+applying a whitening transform can ease the difficulty of finding the desired approximation. In
+particular, we will apply the optimal whitening matrix from [59] in terms of mean-squared error,
+i.e.,
+R
+− 1
+2
+qp,k =
+√
+2
+�
+αpρpβpPp
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+
+− 1
+2
+(91)
+=
+1
+sin 2θ
+�
+αpρpβpPp
+
+sin θ + cos θ
+cos θ − sin θ
+cos θ − sin θ
+sin θ + cos θ
+
+
+(92)
+The determinant of
+
+
+1
+− cos 2θ
+− cos 2θ
+1
+
+ is 1 − cos2 2θ = sin2 2θ is non-negative, and is
+non-zero for θ ̸= 90◦, and thus Rqp,k is non-singular, positive definite and invertible. As a result,
+the probability expression in (77) can be equivalently written as
+P{wp,k(ˇxc) ≥ qp,k} = P{wp,k(ˇxc) ≥ R
+1
+2qp,kR
+− 1
+2
+qp,kqp,k}
+(93)
+= P{R
+− 1
+2
+qp,kwp,k(ˇxc) ≥ R
+− 1
+2
+qp,kqp,k}
+(94)
+= P{ ¯wp,k(ˇxc) ≥ ¯qp,k}
+(95)
+where ¯wp,k(ˇxc) ≜ R
+− 1
+2
+qp,kwp,k(ˇxc) and ¯qp,k ≜ R
+− 1
+2
+qp,kqp,k. Consequently, the chance constraint (77)
+is equivalent to
+P{ ¯wp,k(ˇxc) ≥ ¯qp,k} ≥ v1
+(96)
+with ¯qp,k ∼ N (0, I).
+
+27
+To obtain an efficiently computable constraint, we apply the Safe Approximation I method in
+[44]. The two entries of ¯qp,k are uncorrelated and independent. Defining
+¯qp,k ≜
+
+¯q1
+p,k
+¯q2
+p,k
+
+ ,
+¯wp,k(ˇxc) ≜
+
+ ¯w1
+p,k
+¯w2
+p,k
+
+
+(97)
+the Gaussian cumulative distribution function can be used to calculate the joint probability in (95)
+as follows:
+P{ ¯wp,k(ˇxc) ≥ ¯qp,k} = P{ ¯w1
+p,k ≥ ¯q1
+p,k}P{ ¯w2
+p,k ≥ ¯q2
+p,k}
+(98)
+=
+1 + erf(
+¯w1
+p,k
+√
+2 )
+2
+×
+1 + erf(
+¯w2
+p,k
+√
+2 )
+2
+(99)
+where the error function is given by erf(x) =
+2
+√π
+� x
+0 exp(−t2) dt. Due to the monotonicity of
+erf(x), the desired probability is bounded below by
+P{ ¯wp,k(ˇxc) ≥ ¯qp,k} ≥
+
+1 + erf(
+min{ ¯w1
+p,k, ¯w2
+p,k}
+√
+2
+)
+2
+
+
+2
+.
+(100)
+In order to satisfy the chance constraint (96), it is sufficient to consider the deterministic constraint
+
+
+1 + erf
+� min{ ¯w1
+p,k, ¯w2
+p,k}
+√
+2
+�
+2
+
+
+2
+≥ v1.
+(101)
+Due to ¯wp,k(ˇxc) ≜ R
+− 1
+2
+qp,kwp,k(ˇxc), the constraint can be rewritten as
+min
+�
+R
+− 1
+2
+qp,kwp,k(ˇxc)
+�
+≥
+√
+2 erf−1 (2√v1 − 1)
+(C4-1)
+where erf−1(·) denotes the inverse error function and min[·] denotes the element-wise minimum.
+We thus finally arrive at the following linear inequality constraint:
+αp �H℧
+cp,kˇxc + �HQ,℧
+pp,kˇxp ≥ ηpR
+1
+2qp,k12 + αpδ0
+p12
+(102)
+where η ≜
+√
+2 erf−1 �
+2√v1 − 1
+�
+.
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+
diff --git a/wtE_T4oBgHgl3EQf_hye/content/tmp_files/load_file.txt b/wtE_T4oBgHgl3EQf_hye/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c3f8c059af6368274bb68a322a661dd600581518
--- /dev/null
+++ b/wtE_T4oBgHgl3EQf_hye/content/tmp_files/load_file.txt
@@ -0,0 +1,1102 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf,len=1101
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='08393v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='SP]  20 Jan 2023  1  Robust Symbol Level Precoding for Overlay  Cognitive Radio Networks  Lu Liu, Student Member, IEEE, Christos Masouros, Senior Member, IEEE,  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Lee Swindlehurst, Fellow, IEEE  Abstract  This paper focuses on designing robust symbol-level precoding (SLP) in the downlink of an overlay  cognitive radio (CR) network, where a primary base station (PBS) serving primary users (PUs) and a  cognitive base station (CBS) serving cognitive users (CUs) share the same frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' When the  PBS shares data and perfect channel state information (CSI) with the CBS, an SLP approach which  minimizes the CR transmission power and satisfies symbol-wise Safety Margin (SM) constraints of both  PUs and CUs, is obtained in a low-complexity quadratic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Then for the case of imperfect CSI  from the PBS to CBS, we propose robust SLP schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' First, with a norm-bounded CSI error model  to approximate uncertain channels at the PBS, we adopt the max-min philosophy to conservatively  achieve robust SLP constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Second, we use the additive quantization noise model (AQNM) to  describe the statistics of the quantized PBS CSI, and we employ a stochastic constraint to formulate  the problem, where the SM constraints are converted to be deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Simulation results show that  the proposed robust SLP schemes help enable PUs to mitigate negative effect of the quantization noise  and simultaneously offer CR transmission with significant improvements in energy efficiency compared  to non-robust methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Index Terms  Cognitive radio, symbol-level precoding, constructive interference, robust precoding, quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Liu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Swindlehurst are with Center for Pervasive Communications and Computing, University of California, Irvine,  USA (e-mail: {liul22,swindle}@uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Masouros is with the Department of Electronic & Electrical Engineering, University College London, London WC1E7JE,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (e-mail: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='masouros@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  This work was supported by the National Science Foundation under grants CCF-2008714 and CCF-2225575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' INTRODUCTION  As the number of wireless devices and their applications grow exponentially, the availability of  unoccupied radio spectrum is becoming increasingly scarce and occupied bands are increasingly  congested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Over the past two decades, cognitive radio (CR) technology has been extensively  studied as a means to alleviate this problem through more efficient, flexible and comprehensive  use of the spectrum [1]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  The fundamental challenge lies in balancing the interference generated by the CR at the  primary users (PUs) with the quality of service (QoS) of the cognitive users (CUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' To address  this issue, both the inter-system and inter-user interference need to be successfully managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In  the standard (non-cognitive) multiuser downlink scenario, beamforming or precoding at the multi-  antenna transmitter can be employed to mitigate the multiuser interference (MUI) and compensate  for its adverse affect on the received signals [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Existing precoding schemes can be classified  as either block-level precoding (BLP) or symbol-level precoding (SLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In recent decades, many  approaches have been proposed to implement block-level precoders that only depend on the  current channel state information (CSI), such as maximum ratio transmission (MRT), zero-forcing  (ZF), regularized ZF and optimum interference-constrained or power-constrained precoding [8]–  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' These approaches all treat the MUI as a detrimental effect that is to be suppressed as much  as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Unlike BLP, SLP techniques exploit information about the symbols to be transmitted in  addition to the CSI, which can significantly improve performance at the expense of increased  complexity at the transmitter [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The additional degrees of freedom (DoF) provided  by the symbol-level information make it possible to exploit the constructive component of the  MUI, converting it into constructive interference (CI) that can move the received signals further  from the decision thresholds of the constellation points [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' CI-based SLP recasts the  traditional viewpoint of interference as a source of degradation to one where interference is a  potential resource that can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Constructive interference regions (CIRs), which define the degree to which the received  symbols will be robust to noise and unmodeled perturbations, are fundamental to SLP designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  While early CI-based SLP approaches are intended to increase the distance of the CIRs from  the symbol decision boundaries, they did not directly optimize the CIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' More recent techniques  have focused on designing the precoder to directly optimize this distance [19]–[28], which has  3  been referred to as the safety margin (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Optimal Maximum Safety Margin (MSM) precoders  generally result in a non-linear mapping between the symbols and the transmitted waveforms, and  can be shown to minimize an upper bound on the symbol error rate (SER) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This is contrasted  with algorithms that minimize the mean squared-error (MMSE) between the transmitted and  received symbols, which do not offer the same guarantee [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' MSM precoders are in  general able to achieve a better QoS for the same level of transmit power, or the equivalently  the same QoS with less power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' There has been limited work that studies CI-based  SLP in CR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Although the linear precoder proposed in [29] for an overlay CR network  is based on the use of CI with the MMSE criterion, the PU performance is impaired compared  to the primary-only case, which violates the principle of CR design that we follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  The performance of both BLP and SLP are sensitive to channel uncertainties due for example  to channel estimation errors, quantization noise or latency-related effects [30]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' To mitigate  the impact of such errors, robust designs are needed that properly model the errors and account  for their effect in the optimization of the precoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Two general approaches for doing so include  exploiting known statistical properties of the CSI error or assuming worst-case bounded error  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In the former case, a particular distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=', Gaussian) is assumed for the error, and  Bayesian or other probabilistic approaches [33] are employed to optimize the quality of service  (QoS) or transmit power under certain stochastic signal-to-interference-plus-noise-ratio (SINR) or  rate-outage probability constraints [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In this case, the probability-constraint formulation  is typically not deterministic and various technique must be used to obtain a tractable problem  [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The second approach involves the use of deterministic CSI error bounds that assume  the error is confined to a convex uncertainty region (typically an ellipsoid) surrounding the true  CSI [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In these approaches, robustness is achieved by constraining the users’ QoS or other  design objectives to be satisfied for all channel realizations in the convex uncertainty region,  effectively minimizing the impact of the worst-case channel [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This max-min philosophy can  lead to a relatively conservative design depending on the tightness of the priori error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In either of the two cases described above, the penalty paid for increasing the robustness to  imperfect CSI is increased transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In overlay or cooperative CR systems, CSI errors beyond those due to channel estimation  are anticipated due to the limited cooperation between the PBS and CBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' While robust BLP  designs for traditional MIMO or CR scenarios have been widely investigated [39]–[42], robust  SLP algorithms for general CR scenarios have not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Prior work on robust designs  4  for SLP includes [19], which derived a robust SLP algorithm suitable for imperfect CSI with  bounded CSI errors, but it is based on a multicast formulation without fully taking advantage  of CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The work described in [43] and [44] considered a linear channel distortion model with  bounded additive noise and Gaussian-distributed channel uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' They designed robust  SLP schemes to minimize transmission power subject to CI constraints as well as QoS or SINR  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' While not focused on CR applications, this prior work demonstrates that robust  SLP designs can be formulated to improve and achieve a better balance between QoS and power  consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In this paper, we propose a robust CR SLP algorithm for each of two different CSI error  models that account for the quantization error in the CSI shared by the PBS with the CBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In  particular, we focus on overlay CR downlink channels [3] where the PBS shares with the CBS its  CSI to the PUs and CUs, as well as its data intended for the PUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The shared CSI is assumed to  be quantized, which is known to often make achieving the desired user QoS constraints infeasible  without introducing robustness into the problem formulation [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The use of SLP for this  scenario has not been considered previously in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Additionally, for CR-based SLP  designs, the imperfect CSI shared by the PBS will often cause the noise-free received symbols  at both PUs and CUs to fall outside the desired CIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' To derive a robust SLP formulation for CR  systems, we formulate the problem as one of minimizing the transmit power at the CBS while  simultaneously satisfying the SM constraints at both the PUs and CUs to guarantee the worst-  case user’s QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The SM at each user is calculated for two different channel error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We  first consider the case where the quantization error is norm-bounded as in [33], and then we study  a stochastic approach based on the additive quantization noise model (AQNM) [47], [48] that  is sufficiently accurate to approximate the quantization error at low and medium signal-to-noise  ratios and has been widely used in the analysis of quantized MIMO systems [49]–[51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Below we summarize the main contributions of the paper:  1) We apply the SLP idea to the overlay CR problem where the PBS shares (possibly imperfect)  information with the CBS to facilitate the co-existence of the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This problem  has not been addressed previously in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The work in [29] is the most related  prior effort, but it requires that the CBS directly transmit the PBS data together with its own  data, which is not as energy efficient as our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In addition, [29] does not  consider the impact of the PBS interference at the cognitive users, it uses a less effective  SLP approach, and it does not take into account the fact that the PBS CSI exploited at the  5  CBS may be imperfect due to quantization or other effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Our proposed SLP approach  enables the PUs to exploit constructive interference as well as the CUs, and thus we can  demonstrate that the presence of the cognitive network can actually improve the PU network  performance rather than degrade it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This result is unique to the literature on CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  2) We first derive a power-minimizing SLP approach for overlay CR with SM constraints at  both the PUs and CUs assuming perfect CSI, leading to a quadratic optimization problem  with linear constraints that can be efficienty solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  3) Assuming a norm-bounded quantization error in the shared CSI from the PBS to the CBS,  we derive a robust SLP algorithm based on maximizing the worst case SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This leads to  a conservative design that trades transmit power for increased protection of the PUs from  PBS interference due to the quantized CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  4) As an alternative approach, we employ AQNM to approximate the quantized CSI from the  PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In this case, the SM of the PUs and CUs are constrained to meet a preset threshold with  a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We then apply the Safe Approximation I method in [44] to reformulate  the intractable probabilistic constraints as deterministic constraints and finally construct an  optimization problem to obtain the robust SLP solution [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  5) To validate the effectiveness of our proposed robust precoders, we conduct simulations  assuming the PBS channel is quantized using the scalar Lloyd Max algorithm which  minimizes the average quantization noise power [53], [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Our simulations demonstrate the  flexibility of the proposed robust SLP algorithms in trading transmit power for improved  performance when quantized CSI is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Furthermore, inspired by the results of [55],  we study the problem of allocating bits to the CSI of the PBS to the PUs and CUs, and  demonstrate that the bit allocation strategy in our robust SLP algorithm is not as important  as that in the non-robust methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Notation: Bold lower case and upper case letters indicate vectors and matrices, and non-bold  letters express scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The N × N identity (zero) matrix is denoted by IN (0N×N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The N  dimensional vector of ones (zeroes) is denoted by 1N (0N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Amn denotes the (m, n)-th element  in the matrix A and am denotes the m-th element in the vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The operators (·)∗, (·)−1,  (·)T and (·)H stand for the conjugation, the inverse, the transpose and the Hermitian transpose  operations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Cm×n represents the space of complex matrices of dimension m×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' E(·),  |(·)| and ∥ · ∥ respectively represent the expectation operator, absolute value and the Euclidean  6  PBS  CBS  PU1  PU2  PUNp  CU1  CU2  CUNc  Mp  Mc  Hpp  Hcc  Hcp  Hpc  Figure 1: Cognitive Radio System Model  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' CN (µ, σ2) denotes the complex normal distribution with mean µ and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The  functions tr{·} and diag{·} respectively indicate the trace of a matrix and a vector composed of  the diagonal elements of a square matrix, while diag{a} denotes a square diagonal matrix with  the elements of vector a on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' R{·} and I{·} denote the real and imaginary  parts of a complex number, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' For matrices and vectors, ≥ and ≤ denote element-wise  inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  We consider a downlink CR network with an Mc-antenna CBS serving Nc single-antenna  CUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The CR network is granted access to share the primary system spectrum in which an  Mp-antenna PBS is communicating with Np single-antenna PUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The system model is depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The direct primary and cognitive channels are assumed to be respectively denoted by  the following flat-fading model:  Hpp =  �  hT  pp,1  · ·  hT  pp,Np  �T  ∈ CNp×Mp  (1)  Hcc =  �  hT  cc,1  · ·  hT  cc,Np  �T  ∈ CNc×Mc  (2)  The corresponding interference channels are defined as  Hpc =  �  hT  pc,1  · ·  hT  pc,Np  �T  ∈ CNc×Mp  (3)  Hcp =  �  hT  cp,1  · ·  hT  cp,Np  �T  ∈ CNp×Mc  (4)  7  from the PBS to CUs and the CBS to PUs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We will leave further specification of  the channel models until later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The vectors sp(t) = [sp,1(t), sp,2(t), · · · , sp,Np(t)]T and sc(t) =  [sc,1(t), sc,2(t), · · · , sc,Nc(t)]T contain the modulated signals to be transmitted to the individual  PUs and CUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In this work we assume for simplicity that all transmitted symbols are uncorrelated and drawn  from a D-PSK constellation with unit magnitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=', sl,m(t) ∈ {s|s = exp(jπ(2d+1))/D, d ∈  {0, · · · , D − 1}} where l ∈ {p, c} denotes the primary or cognitive system, and m denotes  the user index in the corresponding system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The sets K = {1, · · · , Np} and J = {1, · · · , Nc}  enumerate the PUs and CUs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The idea of CI precoding can in principle be applied  to any constellation design [27], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=', QAM or otherwise, but is most easily formulated for the  case of PSK signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  At time slot t, the received signals at the PUs and CUs can be respectively written as  yp(t) = Hppxp(t) + Hcpxc(t) + np(t)  (5)  yc(t) = Hccxc(t) + Hpcxp(t) + nc(t)  (6)  where xp(t) ∈ CMp×1 and xc(t) ∈ CMc×1 are the transmitted signals at the PBS and CBS after  precoding and power loading, and np(t) ∼ CN (0, σ2  p) and nc(t) ∼ CN (0, σ2  c) are additive white  Gaussian noise (AWGN) vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In order to simplify the notation, in what follows we drop the  time index t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' CR-PALP  Conventional precoding methods such as MMSE, ZF and maximum-SINR beamforming are  designed with the objective of minimizing the inter-user interference so that the received symbols  lie as close as possible to the nominal constellation points (or scaled versions thereof in the case  of PSK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This is effectively equivalent to ensuring that for each user m, the noise-free received  signal rm = hmx lies within a circle centered at its corresponding constellation point sm [19],  as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The shaded area inside the circle is referred to as the symbol region (SR),  a down-scaled version of the decision region for sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  The method described in [29] is based on the MMSE criterion and the early PALP technique  for SLP [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' it is the prior approach most closely related to the algorithms we present in  this paper for cognitive radio scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' However, a modification to the PALP approach in [29]  is necessary for a fair comparison, and to allow the algorithm to protect the PUs from the CR  8  Im  Re  rm  θ  sm  constellation point  noise-free symbol  symbol region boundary  decision boundary  symbol region  Figure 2: Symbol region for conventional precoding  interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In particular, we tailor [29] (hereafter referred to as CR-PALP) by allowing different  instantaneous power scaling factors at the PBS and CBS:  fp =  �  Pp  trace{WpspsH  p WH  p }, fc =  �  Pc  trace{WcscsH  c WH  c }  (7)  where fp and fc are respectively the instantaneous transmitted power scaling factors for the  primary and cogntive systems, and s =  �  sT  p  sT  c  �T  denotes the symbol vector transmitted by the  CBS, which in [29] includes both the CU and PU data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  The generalized MSE criterion for CR-PALP thus becomes  ǫ = E{∥Vps + HWcs − (A + BQφ)s∥2}  (8)  where A = diag{[1, · · · , 1, 0, · · · , 0]} is a diagonal matrix whose first Np diagonal elements  equal 1, B = diag{[0, · · · , 0, 1, · · · , 1]} is a diagonal matrix whose last Nc elements equal 1,  and Qφ = diag(s) · |HHH| · diag(s)H contains the phase-corrected correlation elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The  precoding matrix at the CBS derived from the MMSE criterion is given by  Wc = HH(HHH)−1(A + BQφ − Vp) ,  (9)  and the received signals at the PUs and CUs are  yp = fpsp + np  (10)  yc = (fp − fc)HpcWpsp + fcQφ  c sc + nc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (11)  m9  Im  Re  θ  Figure 3: Symbol region for CI-Based SLP  Im  Re  θ  zmI  δ  δm  Figure 4: Symbol region and safety margin in a modified coordinate ststem  The performance of the PUs in our CR scenario will not be impaired by CR using this modified  CR-PALP approach, which is different from using the method of [29] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' SM-constrained SLP  For PSK constellations, it is not necessary that rm be close to sm in order to be decoded  correctly, as long as it lies in the correct decision region with a given level of certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Thus,  it is not necessary that all of the inter-user interference be eliminated, since some interference  components could add constructively and push the received symbol further into the decision  region, making it more robust to noise and interference external to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We can thus  redefine the SR as, for example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 3, where the SR becomes a displaced version of the  circular sector of angular extent 2π/D centered at the origin and corresponding to sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This  10  displaced sector has an infinite radius, and all points within it are at least a certain distance δm  from the decision boundaries for sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This region is referred to as a constructive interference  region (CIR) with safety margin (SM) δm [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The larger δm, the more robust the received  signal will be to noise, interference, modeling errors, or other impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In order to mathematically interpret the CIR and SM in a unified way, we rotate the original  coordinate system by the phase of the desired constellation symbol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=', ∡sm, to obtain the  modified coordinate system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' After rotation, sm is placed at 1 on the real axis, and rm  is relocated to  zm = rms∗  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (12)  Then we can easily calculate the SM of the noise-free symbol at user m as  δm = R{zm} sin θ − |I{zm}| cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (13)  Ideally, the SM should be large enough to sufficiently reduce the probability that noise or other  impairments will push the noise-free signal outside the desired detection region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' the larger the  SM, the smaller the SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' To design the precoder, one can constrain the SM to be above  a certain threshold to ensure a given target SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The fact that the CIR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 4 is much  larger than the SR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' II-B means that increased flexibility is available to achieve the given  performance objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In this paper, we will consider the following type of SLP optimization,  which minimizes the transmit power to achieve a certain desired SM:  min  x  ∥x∥2  (14)  subject to  δm ≥ δm,0  ∀m ∈ M  (15)  where δm,0 is the desired minimum SM for user m and M = {1, · · · , M} indexes the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' For  simplicity, we will assume in the following that the same threshold is employed for all users,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=', δm,0 = δ0  p for the PUs and δm,0 = δ0  c for the CUs, although this assumption can be easily  relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' POWER MINIMIZATION SLP IN CR  Before considering the robust SLP design, we first examine the simple case where the PBS  shares its data and perfect CSI with the CBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The SM for the PUs and CUs is constrained to be  at least δ0  p and δ0  c, corresponding to target SERs for these users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Here we focus on SLP designs  that minimize the transmit power at the CBS and achieve the SM QoS constraints at the both  the PUs and CUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  11  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Primary System  The rotated symbols received at the PUs can be expressed as  zp,k = s∗  p,krp,k = s∗  p,k(hpp,kxp + hcp,kxc)  (16)  for k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Defining  ˜hpp,k ≜ s∗  p,khpp,k,  ˜hcp,k ≜ s∗  p,khcp,k ,  (17)  we have  zp,k = ˜hpp,kxp + ˜hcp,kxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (18)  The constraint ensuring the QoS of the PUs can be expressed as  δp,k = R{zp,k} sin θ − |I{zp,k}| cos θ ≥ δ0  p , ∀k ∈ K,  (19)  which is equivalent to  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  R{˜hpp,kxp} sin θ − I{˜hpp,kxp} cos θ  +R{˜hcp,kxc} sin θ − I{˜hcp,kxc} cos θ ≥ δ0  p  (20a)  R{˜hpp,kxp} sin θ + I{˜hpp,kxp} cos θ  +R{˜hcp,kxc} sin θ + I{˜hcp,kxc} cos θ ≥ δ0  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (20b)  For any given complex vector x, we define the operator  ℧(x) ≜  \uf8ee  \uf8f0R{x} sin θ − I{x} cos θ  −R{x} cos θ − I{x} sin θ  R{x} sin θ + I{x} cos θ  R{x} cos θ − I{x} sin θ  \uf8f9  \uf8fb  (21)  and denote  �H℧  pp,k = ℧(˜hpp,k),  �H℧  cp,k = ℧(˜hcp,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (22)  Using the following real-valued notation,  ˇxp =  \uf8ee  \uf8f0R{xp}  I{xp}  \uf8f9  \uf8fb ,  ˇxc =  \uf8ee  \uf8f0R{xc}  I{xc}  \uf8f9  \uf8fb ,  (23)  the constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (19) can be simplified as  �H℧  pp,kˇxp + �H℧  cp,kˇxc ≥ δ0  p12  ∀k ∈ K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (24)  12  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Cognitive System  Similarly, the rotated symbols at the CUs can be written as  zc,j = s∗  c,jrc,j = s∗  c,j(hcc,jxc + hpc,jxp)  (25)  = ˜hcc,jxc + ˜hpc,jxp ,  (26)  where  ˜hcc,j ≜ s∗  c,jhcc,j  ˜hpc,j ≜ s∗  c,jhpc,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (27)  The SM constraints at the CUs can be expressed as  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  R{˜hcc,jxc} sin θ − I{˜hcc,jxc} cos θ  +R{˜hpc,jxp} sin θ − I{˜hpc,jxp} cos θ ≥ δ0  c  (28a)  R{˜hcc,jxc} sin θ + I{˜hcc,jxc} cos θ  +R{˜hpc,jxp} sin θ + I{˜hpc,jxp} cos θ ≥ δ0  c  (28b)  for j ∈ J , which can again be written more compactly using the operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (21):  �H℧  cc,jˇxc + �H℧  pc,jˇxp ≥ δ0  c12  ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (29)  Combining all of the above notation together, we can express the general power minimization  SLP problem with perfect CSI as follows:  minimize  ˇxc  ∥ˇxc∥2  (P1)  subject to  \uf8ee  \uf8f0− �H℧  cp  − �H℧  cc  \uf8f9  \uf8fb ˇxc ≤  \uf8ee  \uf8f0  �H℧  pp  �H℧  pc  \uf8f9  \uf8fb ˇxp −  \uf8ee  \uf8f0δ0  p12Np  δ0  c12Nc  \uf8f9  \uf8fb  (C1)  where �H℧  cp ≜ ℧(diag(s∗  p)Hcp), �H℧  cc ≜ ℧(diag(s∗  c)Hcc), �H℧  pp ≜ ℧(diag(s∗  p)Hpp), and �H℧  pc ≜  ℧(diag(s∗  c)Hpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This is a quadratic programming problem with linear inequality constraints and  can be efficiently solved using a variety of numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' ROBUST SLP FOR NORM-BOUNDED CSI ERRORS  In practice, the CSI shared by the PBS with the CBS will be imperfect due for example to  quantization, or somewhat outdated due to delays required for processing and transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' As  a result, robust precoding designs are critical for overlay systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We address such a design in  13  this section for the case where the imperfect CSI can be described in terms of a norm-bounded  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We model the CSI shared by the PBS with the CBS as follows:  hpp,k = ˆhpp,k + ep,k  (30)  hpc,j = ˆhpc,j + ec,j,  (31)  where the ˆ· indicates the shared CSI and ep,k, ec,j are norm-bounded CSI error vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=',  ∥ep,k∥2 ≤ ǫp,k and ∥ec,j∥2 ≤ ǫc,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' No other assumption regarding the channels is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (21), it is easy to show that  �H℧  pp,k = ℧(˜hpp,k) = ℧(s∗  p,k(ˆhpp,k + ep,k))  (32)  = ¯H℧  pp,k + ˜E℧  p,k  (33)  �H℧  pc,j = ℧(˜hpc,j) = ℧(s∗  c,j(ˆhpc,j + ec,j))  (34)  = ¯H℧  pc,j + ˜E℧  c,j,  (35)  where ¯H℧  pp,k ≜ ℧(s∗  p,kˆhpp,k), ˜E℧  p,k ≜ ℧(s∗  p,kep,k), ¯H℧  pc,j ≜ ℧(s∗  c,jˆhpc,j), and ˜E℧  c,j ≜ ℧(s∗  c,jec,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  With this notation, the constraints in P1 can be reformulated as  ( ¯H℧  pp,k + ˜E℧  p,k)ˇxp + �H℧  cp,kˇxc ≥ δ0  p12, ∀k ∈ K  (36)  �H℧  cc,jˇxc + ( ¯H℧  pc,j + ˜E℧  c,j)ˇxp ≥ δ0  c12, ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (37)  For a robust bounded-CSI-error design, we desire that the above constraints hold for every  possible error realization and every user:  −˜E℧  p,kˇxp ≤ ¯H℧  pp,kˇxp + �H℧  cp,kˇxc − δ0  p12 ,  ∀∥ep,k∥2 ≤ ǫp,k, ∀k ∈ K ,  (38)  −˜E℧  c,jˇxp ≤ �H℧  cc,jˇxc + ¯H℧  pc,jˇxp − δ0  c12 ,  ∀∥ec,j∥2 ≤ ǫc,j, ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (39)  We separate the operator ℧(x) into two parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' as follows:  ℧(x) ≜  \uf8ee  \uf8f0℧1(x)  ℧2(x)  \uf8f9  \uf8fb ≜  \uf8ee  \uf8f0x℧1  x℧2  \uf8f9  \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (40)  where  x℧1 =  �  R{x} sin θ − I{x} cos θ  −R{x} cos θ − I{x} sin θ  �  = ℧1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (41)  14  x℧2 =  �  R{x} sin θ + I{x} cos θ  R{x} cos θ − I{x} sin θ  �  = ℧2(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (42)  so that constraint (38) can be rewritten in two parts as  −˜e℧1  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxp ≤ ¯h℧1  pp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxp + �h℧1  cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxc − δ0  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  ∀∥ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k∥2 ≤ ǫp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' ∀k ∈ K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (43)  −˜e℧2  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxp ≤ ¯h℧2  pp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxp + �h℧2  cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxc − δ0  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  ∀∥ec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='j∥2 ≤ ǫc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' ∀j ∈ J ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (44)  where  −˜e℧1  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='kˇxp ≤ ∥ − ˜e℧1  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k∥2∥ˇxp∥2  (45)  =  ������  �  R{eT  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k}  I{eT  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k}  �  \uf8ee  \uf8f0 sin θ  − cosθ  − cos θ  − sin θ  \uf8f9  \uf8fb  ������  F  ∥ˇxp∥2  (46)  ≤  ���  �  R{eT  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k}  I{eT  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k}  ����  F  ������  \uf8ee  \uf8f0 sin θ  − cosθ  − cosθ  − sin θ  \uf8f9  \uf8fb  ������  F  ∥ˇxp∥2  (47)  ≤  √  2ǫp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='k∥ˇxp∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (48)  Similarly, (39) becomes  − ˜e℧2  p,kˇxp ≤  √  2ǫp,k∥ˇxp∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (49)  Thus, the constraints in (38) and (39) can be expressed as  ¯H℧  pp,kˇxp + �H℧  cp,kˇxc ≥ (  √  2ǫp,k∥ˇxp∥2 + δ0  p)12, ∀k ∈ K  (50)  ¯H℧  pc,jˇxp + �H℧  cc,jˇxc ≥ (  √  2ǫc,j∥ˇxp∥2 + δ0  c)12, ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (51)  Finally, the robust precoder can be obtained by solving the following optimization problem:  minimize  ˇxc  ∥ˇxc∥2  (52)  subject to ¯H℧  pp,kˇxp + �H℧  cp,kˇxc ≥ (  √  2ǫp,k∥ˇxp∥2 + δ0  p)12  ∀k ∈ K  (53)  ¯H℧  pc,jˇxp + �H℧  cc,jˇxc ≥ (  √  2ǫc,j∥ˇxp∥2 + δ0  c)12  ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (54)  As in the case with perfect CSI, the robust SLP design can be found via a quadratic program  with linear inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  15  Table I: Distortion Factors for Different Quantization Bit Resolutions [53]  b  1  2  3  4  5  ρ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3634  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='03454  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='009497  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='002499  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' ROBUST SLP FOR STOCHASTIC CSI ERRORS  The bounded error model above is a very conservative approach, given its goal of ensuring that  the SM constraints are met for all possible CSI error realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' A less conservative approach  that allows constraint violations with some acceptably small probability is to assume a statistical  CSI error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' As an example, in this section we consider the case where such a model for  the PBS CSI error is available due to knowledge of how the channel is quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In particular,  we assume that the channels hQ  pp,k and hQ  pc,j shared by the PBS are element-wise quantized,  and we use the approximate additive quantization noise model (AQNM) [47], [48] to describe  their resulting statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Other models are possible based on the specific quantization method  employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  We assume that the channels are Gaussian with zero mean and covariances given by  Rhpp,k = E{hH  pp,khpp,k} = βpIMp  (55)  Rhpc,j = E{hH  pc,jhpc,j} = βcIMp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (56)  Using AQNM, the quantized CSI from the PBS after rotation is expressed as  �hQ  pp,k = Q(�hpp,k) ≈ αp�hpp,k + �nQ  pp,k  (57)  �hQ  pc,j = Q(�hpc,j) ≈ αc�hpc,j + �nQ  pc,j,  (58)  where Q(·) is a scalar quantization function applied element-wise and separately to the real and  imaginary parts of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The vectors �nQ  pp,k ≜ s∗  p,knQ  pp,k ∈ C1×Mp and �nQ  pc,j ≜ s∗  c,jnQ  pc,j ∈  C1×Mp denote the zero-mean Gaussian-distributed quantization noise vectors, and both are as-  sumed to be uncorrelated with �hpp,k and �hpc,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The gains αk = 1−ρk for k ∈ {p, c} are functions  of the following distortion factors [49]:  ρp =  E{∥hpp,k − hQ  pp,k∥2}  E{∥hpp,k∥2}  , ρc = E{∥hpc,j − hQ  pc,j∥2}  E{∥hpc,j∥2}  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (59)  16  The value of ρ is given in Table I for different bit resolutions b assuming an optimal non-  uniform Lloyd-Max quantizer [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The phase rotation does not alter the covariance matrices of  the quantization noise, which are given by [47]  R{�nQ  pp,k} = αpρpdiag{Rhpp,k} = αpρpβpIMp,  (60)  R{�nQ  pc,j} = αcρcdiag{Rhpc,j} = αcρcβcIMp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (61)  Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (57) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (58), we can derive  �hpp,k =  �hQ  pp,k − �nQ  pp,k  αp  = ¯αp�hQ  pp,k − ¯αp�nQ  pp,k  (62)  �hpc,j =  �hQ  pc,j − �nQ  pc,j  αc  = ¯αc�hQ  pc,j − ¯αc�nQ  pc,j ,  (63)  where ¯αp =  1  αp and ¯αc =  1  αc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Therefore,  �H℧  pp,k = ℧( ¯αp�hQ  pp,k − ¯αp�nQ  pp,k) = ¯αp( �HQ,℧  pp,k − �NQ,℧  pp,k)  (64)  �H℧  pc,j = ℧( ¯αc�hQ  pc,j − ¯αc�nQ  pc,j) = ¯αc( �HQ,℧  pc,j − �NQ,℧  pc,j),  (65)  where �HQ,℧  pp,k ≜ ℧(�hQ  pp,k), �NQ,℧  pp,k ≜ ℧(�nQ  pp,k), �HQ,℧  pc,j ≜ ℧(�hQ  pc,j), and �NQ,℧  pc,j ≜ ℧(�nQ  pc,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Substituting  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (64) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (65) in (C1), we have  ¯αp( �HQ,℧  pp,k − �NQ,℧  pp,k)ˇxp + �H℧  cp,kˇxc ≥ δ0  p12, ∀k ∈ K  (C2-1)  �H℧  cc,jˇxc + ¯αc( �HQ,℧  pc,j − �NQ,℧  pc,j)ˇxp ≥ δ0  c12, ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (C2-2)  The constraints above are expressed in terms of the unknown random quantization noise, and  thus cannot be directly enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Instead, we choose to pose the problem such that the constraints  are achieved with a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In particular, we rewrite (C2-1) and (C2-2) as follows:  P{αp �H℧  cp,kˇxc + �HQ,℧  pp,kˇxp − αpδ0  p12 ≥ �NQ,℧  pp,kˇxp} ≥ v1,  (C3-1)  P{αc �H℧  cc,jˇxc + �HQ,℧  pc,j ˇxp − αcδ0  c12 ≥ �NQ,℧  pc,j ˇxp} ≥ v2,  (C3-2)  where P{A} denotes the probability of event A, and {v1, v2} ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, 1] represent the probability  thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In the following two subsections, we find expressions for the probabilities in (C3-1)  and (C3-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  17  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Primary System  It is easy to show that  E{ �NQ,℧  pp,k} = 02×2Mp  (66)  R �NQ,℧  pp,k = αpρpI2Mp  (67)  E{ �NQ,℧  pp,k( �NQ,℧  pp,k)H} = Mpαpρpβp  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (68)  The vector  qp,k ≜ �NQ,℧  pp,kˇxp ≜  \uf8ee  \uf8f0q1  p,k  q2  p,k  \uf8f9  \uf8fb  (69)  is a bivariate correlated Gaussian random vector with mean  E{qp,k} = E{ �NQ,℧  pp,kˇxp} = E{ �NQ,℧  pp,k}ˇxp = 02×1  (70)  and covariance  Rqp,k = E{qp,kqT  p,k}  (71)  = E{ �NQ,℧  pp,kˇxp( �NQ,℧  pp,kˇxp)T}  (72)  = Pp  2Mp  E{ �NQ,℧  pp,kI2Mp( �NQ,℧  pp,k)T}  (73)  = Pp  2Mp  × Mpαpρpβp  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb  (74)  = Pp  2 αpρpβp  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (75)  To simplify the notation, we define  wp,k(ˇxc) ≜ αp �H℧  cp,kˇxc + �HQ,℧  pp,kˇxp − αpδ0  p12 ≜  \uf8ee  \uf8f0w1  p,k  w2  p,k  \uf8f9  \uf8fb  (76)  which is affine in ˇxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Using the new notation, the chance constraint (C3-1) can be rewritten as  P{wp,k(ˇxc) ≥ qp,k} ≥ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (77)  No explicit closed-form expression for this probability seems possible, so we resort to finding a  tractable approximation using the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  18  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' P{wp,k(ˇxc) ≥ qp,k} ≥ v1 can be approximated by the inequality  αp �H℧  cp,kˇxc + �HQ,℧  pp,kˇxp ≥ ηpR  1  2qp,k12 + αpδ0  p12  (78)  where η ≜  √  2 erf−1 �  2√v1 − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Cognitive System  The constraint in (C3-2) for the cognitive system can be found in an identical fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' It is  easy to show that  E{ �NQ,℧  pc,j} = 02×2Mp  (79)  R �NQ,℧  pc,j = αcρcβcI2Mp  (80)  E{ �NQ,℧  pc,j( �NQ,℧  pc,j)H} = Mpαcρcβc  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb ,  (81)  and we define  qc,j ≜ �NQ,℧  pc,j ˇxp ≜  \uf8ee  \uf8f0q1  c,j  q2  c,j  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (82)  Similar to the above, we can show that qc,j is a bivariate correlated Gaussian random variable  with mean  E{qc,j} = 02×1  (83)  and covariance  Rqc,j = Pp  2 αcρcβc  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (84)  Furthermore, we define  wc,j(ˇxc) ≜ αc �H℧  cc,jˇxc + �HQ,℧  pc,j ˇxp − αcδ0  c12 ≜  \uf8ee  \uf8f0w1  c,j  w2  c,j  \uf8f9  \uf8fb  (85)  which is affine in ˇxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Using the new notation, the chance constraint (C3-2) can be rewritten as  P{wc,j(ˇxc) ≥ qc,j} ≥ v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (86)  As with the primary system, we define ¯wc,j(ˇxc) ≜ R  − 1  2  qc,jwc,j(ˇxc) and ¯qc,j ≜ R  − 1  2  qc,jqc,j, and we  obtain the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  19  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' P{ ¯wc,j(ˇxc) ≥ ¯qc,j} ≥ v2 can be approximated by the inequality  αc �H℧  cc,jˇxc + �HQ,℧  pc,j ˇxp ≥ ηcR  1  2qc,j12 + αcδ0  c12,  (87)  where ηc =  √  2 erf−1 �  2√v2 − 1  �  is a preset constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The proof follows directly from that of Lemma 1 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Optimization Problem for Stochastic CSI Error Model  We can now formulate the robust SLP design with the probabilistic constraints by replacing  (C3-1) and (C3-2) with (78) and (87), as follows:  minimize  ˇxc  ∥ˇxc∥2  (PQ)  subject to  αp �H℧  cp,kˇxc + �HQ,℧  pp,kˇxp ≥ ηpR  1  2qp,k12 + αpδ0  p12  ∀k ∈ K  (CQ-1)  αc �H℧  cc,jˇxc + �HQ,℧  pc,j ˇxp ≥ ηcR  1  2qc,j12 + αcδ0  c12  ∀j ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (CQ-2)  As with the other problems studied above, the result is a quadratic program with linear inequality  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, we assess the performance of our proposed power-minimizing SLP (PMSLP)  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Monte-Carlo simulations are conducted over 1000 independent channel realizations,  each employing a block of T = 100 symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The channels Hpp, Hcp, Hpc and Hcc are  composed of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Gaussian random variables with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The complex  Gaussian noise is assumed to have the same power (σp = σc = 1) for all PUs and CUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The  PBS transmission power is set at Pp = 10 dBW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Both the PBS and CBS are assumed to have  Mp = Mc = 8 antennas and the number of PUs and CUs are both set at Np = Nc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Since for SLP we work with finite alphabet constellations, we will analyze the block trans-  mission performance of the system using the throughput τ as calculated in [57]:  τ = (1 − PB) × c × T × N,  (88)  20  0  5  10  15  20  25  30  CBS Transmission Power (dBW)  10-3  10-2  10-1  100  CU BER  1  2  3  4  5  6  7  8  9  10  PU BER  10-3  CR-PALP  PMSLP  CR-PALP  PMSLP  Primay Only  Figure 5: BER of CU (left) or PU (right) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' CBS transmit power where Pp = 10 dBW, QPSK  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  where PB is the block error rate (BLER), c = log2 D is the number of bits per modulation  symbol, T is the block length and N is the number of receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In each block for each user,  there are C = c × T data message bits transmitted from the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' For PSK modulation, assuming  a binomial distribution of errors in each block, the probability of more than q errors occurring  in one block of C bits is expressed as  Pe(q, C) = 1 −  q  �  i=0  �C  i  �  P i  b(1 − Pb)C−i  (89)  where Pb is the BER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' If the receiver detects errors without correction, a block is received correctly  only if all C bits in the block are received correctly, and thus the BLER is PB = Pe(0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' On  the other hand, if the receiver is capable of correcting up to Q errors in each block, then the  BLER is given by PB = Pe(Q, C) [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  We begin in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 5 assuming perfect CSI, plotting the average BER of the users versus the  CBS transmission power, and comparing PMSLP assuming minimum safety margin δ0  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  with the performance of the CR-PALP algorithm described in Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' With these settings,  even when the CBS increases its transmit power to better serve the CUs, it can still avoid any  negative impact on the PUs such that the BER of PUs is not greater than that in the primary-only  case as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Moreover, the BER of the PUs remains nearly unchanged for both types  of precoders, although the PUs actually enjoy some benefit with PMSLP since it exploits CI from  the CBS signals which can further improve the SM for the PUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Meanwhile, PMSLP provides  a much lower BER for the CUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' the CBS can save more than 15dBW of power to achieve an  21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  10-5  10-4  10-3  10-2  PU BER  Non-robust PMSLP,  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Robust PMSLP, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Non-robust PMSLP,  p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Robust PMSLP, p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Figure 6: BER of PU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' error norm bound where Pp = 10 dBW, δ0  p = δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, QPSK  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  10-5  10-4  10-3  10-2  10-1  CU BER  Non-robust PMSLP,  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Robust PMSLP, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Non-robust PMSLP,  p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Robust PMSLP, p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Figure 7: BER of CU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' error norm bound where Pp = 10 dBW, δ0  p = δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, QPSK  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  uncoded BER of 10−2, compared to the CR-PALP method which allows only the CUs to benefit  from the inter-user CI in the cognitive system, but in general will not entirely eliminate the  destructive interference from the PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' On the contrary, our proposed PMSLP precoder can take  advantage of CI not only from the CU MUI but also what is available (although not optimized)  from the primary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Moreover, the PUs can also benefit from the CI coming from the  cognitive system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' CR-PALP requires that the CBS act as a relay transmitting not only the CUs’  but also the PUs’ signals, which is unnecessary in our PMSLP design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Next we consider the norm-bounded CSI error model discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' With the SM  threshold for the PUs and CUs set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, we plot the BER of the PUs and CUs as the norm of  22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='6 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='4 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  15  20  25  30  35  40  45  50  55  60  CBS Transmission Power (W)  Non-robust PMSLP,  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Robust PMSLP, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Non-robust PMSLP,  p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Robust PMSLP, p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Figure 8: Transmit power vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' error norm bound where Pp = 10 dBW, δ0  p = δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, QPSK  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  one error (ǫp,k or ǫc,j) changes as log10 ǫ while the norm of the other is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' For simplicity,  we assume that the CSI error bounds are the same for all users: ǫp,k = ǫp and ǫc,j = ǫc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 6  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 7 respectively show the BER for the PUs and CUs as a function of the error bound,  and demonstrate that the proposed robust precoder can mitigate the CSI uncertainty and provide  a much lower BER compared to the non-robust precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' With the robustness introduced, the  BER of the PUs and CUs actually decrease as the norm of the corresponding error increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  This can be explained by examining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (54), where we see that a larger error bound creates  a larger effective SM in the constraint, which provides the robustness necessary to account for  the imperfect channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' There is however a price to be paid for this robustness, as clearly seen in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 8, which shows that the robust schemes require the CBS to operate with significantly more  power, especially as the error bound increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' It is clear from these results that the worst-case  approach based on the norm-bounded CSI error leads to a conservative design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In order to quantify the power-performance trade-off between the robust and non-robust  designs, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 9 we plot the energy efficiency (EE) of the approaches, defined as the ratio  between the throughput calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (88) and the transmit power per channel:  EE =  τ  T × ∥ˇxc∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (90)  We see that despite the increase in transmit power, the proposed robust SLP algorithm achieves  a significantly higher energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' When the uncertainty ǫc in Hpc is fixed, the energy  efficiency at the CBS decreases with greater uncertainty in Hpp since the CBS needs to consume  23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='25  CBS Energy Efficency (bit/Hz/J/channel use)  Non-robust PMSLP, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Robust PMSLP, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3, c=  Non-robust PMSLP, p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Robust PMSLP, p= , c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Figure 9: Energy Efficiency vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Norm of the Error where Pp = 10 dBW, δ0  p = δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, QPSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  2  Preset  c  0  10-4  10-3  10-2  10-1  CU BER  Non-robust b p=bc=2  Robust b p=bc=2, v1=v2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8  Non-robust b p=bc=3  Robust b p=bc=3, v1=v2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='95  Figure 10: BER of CUs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' preset SM at the CUs where Pp = 10 dBW, δ0  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, QPSK  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  more power to meet the SM constraint at the PUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Interestingly, in our setting with δ0  p = δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5,  if the uncertainty ǫp in Hpp is fixed, the energy efficiency at the CBS first increases as the error  in Hpc increases and then decreases when log10(ǫc) ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Considering that the effective SM  (right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' (54)) in the robust PMSLP algorithm is decided by both the norm bound of the  CSI error and the preset SM, when log10(ǫc) ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1, the preset δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5 is not large enough for  the CIR to account for the CSI error in Hpc, which results in a high BER for the CUs and lower  EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' This illustrates the importance of selecting a suitable SM based on the CSI error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  The remaining examples use the probabilistic SM constraints discussed in Section V based on  the AQNM approximation, although the actual quantized CSI is generated using a non-uniform  Lloyd Max quantizer [53], [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 10 shows the BER of the CUs as a function of the preset SM  24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  2  Preset  c  0  0  10  20  30  40  50  60  70  80  CBS Transmission Power (W/symbol)  Non-robust b p=bc=2  Robust b p=bc=2, v1=v2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8  Non-robust b p=bc=3  Robust b p=bc=3, v1=v2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='95  Figure 11: Transmit Power at CBS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' preset SM at the CUs where Pp = 10 dBW, δ0  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5,  QPSK modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9  2  Preset  c  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='16  Cognitive Energy Efficency (bit/Hz/J/channel use)  Non-robust b p=bc=2  Robust b p=bc=2, v1=v2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='8  Non-robust b p=bc=3  Robust b p=bc=3, v1=v2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='95  Figure 12: EE at CBS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' preset SM at the CUs where Pp = 10 dBW, δ0  p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, QPSK  modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  threshold at the CUs, assuming either b = 2 or b = 3 quantization bits per channel coefficient  and different probability constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' We see that the CUs reap benefits from the robust SLP  design, achieving significantly lower BER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Again illustrating the trade-off of robustness with  increase power, we see that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 11 as the preset δ0  c increases, the CBS in the robust SLP  approach requires more power to meet the SM constraint the non-robust SLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In order to fairly  compare different SLP methods, we plot the EE at the CBS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' It is clear that the greater  the preset SM, or the higher the quantization resolution, the higher the EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The robust SLP  approach performs particuarly well for b = 3, δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='7, yielding an increase in EE by a factor  of about 10 compared with the non-robust SLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  25  2  3  4  5  Quantization Bits (b)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='3  Cognitive Energy Efficency (bit/Hz/J/channel use)  Non-robust PMSLP, b p=b, b c=3  Robust PMSLP, bp=b, b c=3  Non-obust PMSLP, b p=3, bc=b  Robust PMSLP, bp=3, bc=b  bp=b, b c=3  bp=b, b c=3  bp=3, bc=b  bp=3, bc=b  Figure 13: Energy Efficiency at the CBS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' quantization resolution where Pp = 10 dBW,  δ0  p = δ0  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='5, v1 = v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='9, QPSK modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  In the final example, we study the allocation of the quantization bits on the system performance  [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In particular, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' 13 we plot the energy efficiency at the CBS when the direct hpp,k  and interference channels hpc,j are quantized with different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' For the non-robust SLP  schemes (blue curves), the CBS achieves higher EE in cases where bc > bp, indicating that for a  fixed number of quantization bits, it is more energy efficient for the cognitive system to receive  a more accurate representation of the interference channel than the direct channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' However,  the robust-SLP schemes are less sensitive to the allocation of the quantization bits, and show  roughly the same performance regardless of which channel is more accurately represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' CONCLUSION  In this paper, we have designed non-robust and robust SLP schemes for CR system with the  goal of minimizing the transmission power and simultaneously ensuring the QoS of all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  Unlike traditional CR precoding techniques, we set the SM threshold in the interference con-  straints instead of using SINR or BER metrics in order to fully exploit constructive interference  as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' First, under the assumption of perfect CSI, we propose an SLP algorithm  that performs significantly better than a prior CR-based SLP approach modified to address our  overlay problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In the proposed algorithm, not only the CUs but also the PUs benefit from the  constructive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='. Then, using two different CSI error models, we derive two robust SLP  methods, one based on a max-min optimization of the worst-case CSI error and the other on a  probability-constraint problem using AQNM to approximate the impact of the CSI quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  26  All of the proposed optimization problems result in a quadratic objective function with linear  inequality constraints that can be efficiently solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='. Our numerical results demonstrate that our  robust SLP schemes can deal with various types of CSI error and still maintain a high energy  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' For the stochastic CSI model, our robust SLP algorithm can maintain a lower BER  at both the PUs and CUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' A key observation from our results is that, by enabling the PUs to  exploit constructive interference as well as the CUs, the presence of the cognitive network can  actually improve the PU network performance rather than degrade it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Proof of Lemma 1  From [44] we note that eliminating the (possible) correlation between the entries of qp,k by  applying a whitening transform can ease the difficulty of finding the desired approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' In  particular, we will apply the optimal whitening matrix from [59] in terms of mean-squared error,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=',  R  − 1  2  qp,k =  √  2  �  αpρpβpPp  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb  − 1  2  (91)  =  1  sin 2θ  �  αpρpβpPp  \uf8ee  \uf8f0sin θ + cos θ  cos θ − sin θ  cos θ − sin θ  sin θ + cos θ  \uf8f9  \uf8fb  (92)  The determinant of  \uf8ee  \uf8f0  1  − cos 2θ  − cos 2θ  1  \uf8f9  \uf8fb is 1 − cos2 2θ = sin2 2θ is non-negative, and is  non-zero for θ ̸= 90◦, and thus Rqp,k is non-singular, positive definite and invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' As a result,  the probability expression in (77) can be equivalently written as  P{wp,k(ˇxc) ≥ qp,k} = P{wp,k(ˇxc) ≥ R  1  2qp,kR  − 1  2  qp,kqp,k}  (93)  = P{R  − 1  2  qp,kwp,k(ˇxc) ≥ R  − 1  2  qp,kqp,k}  (94)  = P{ ¯wp,k(ˇxc) ≥ ¯qp,k}  (95)  where ¯wp,k(ˇxc) ≜ R  − 1  2  qp,kwp,k(ˇxc) and ¯qp,k ≜ R  − 1  2  qp,kqp,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Consequently, the chance constraint (77)  is equivalent to  P{ ¯wp,k(ˇxc) ≥ ¯qp,k} ≥ v1  (96)  with ¯qp,k ∼ N (0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  27  To obtain an efficiently computable constraint, we apply the Safe Approximation I method in  [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' The two entries of ¯qp,k are uncorrelated and independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Defining  ¯qp,k ≜  \uf8ee  \uf8f0¯q1  p,k  ¯q2  p,k  \uf8f9  \uf8fb ,  ¯wp,k(ˇxc) ≜  \uf8ee  \uf8f0 ¯w1  p,k  ¯w2  p,k  \uf8f9  \uf8fb  (97)  the Gaussian cumulative distribution function can be used to calculate the joint probability in (95)  as follows:  P{ ¯wp,k(ˇxc) ≥ ¯qp,k} = P{ ¯w1  p,k ≥ ¯q1  p,k}P{ ¯w2  p,k ≥ ¯q2  p,k}  (98)  =  1 + erf(  ¯w1  p,k  √  2 )  2  ×  1 + erf(  ¯w2  p,k  √  2 )  2  (99)  where the error function is given by erf(x) =  2  √π  � x  0 exp(−t2) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content=' Due to the monotonicity of  erf(x), the desired probability is bounded below by  P{ ¯wp,k(ˇxc) ≥ ¯qp,k} ≥  \uf8ee  \uf8f01 + erf(  min{ ¯w1  p,k, ¯w2  p,k}  √  2  )  2  \uf8f9  \uf8fb  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (100)  In order to satisfy the chance constraint (96), it is sufficient to consider the deterministic constraint  \uf8ee  \uf8f0  1 + erf  � min{ ¯w1  p,k, ¯w2  p,k}  √  2  �  2  \uf8f9  \uf8fb  2  ≥ v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  (101)  Due to ¯wp,k(ˇxc) ≜ R  − 1  2  qp,kwp,k(ˇxc), the constraint can be rewritten as  min  �  R  − 1  2  qp,kwp,k(ˇxc)  �  ≥  √  2 erf−1 (2√v1 − 1)  (C4-1)  where erf−1(·) denotes the inverse error function and min[·] denotes the element-wise minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
+page_content='  We thus finally arrive at the following linear inequality constraint:  αp �H℧  cp,kˇxc + �HQ,℧  pp,kˇxp ≥ ηpR  1  2qp,k12 + αpδ0  p12  (102)  where η ≜  √  2 erf−1 �  2√v1 − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
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+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE_T4oBgHgl3EQf_hye/content/2301.08393v1.pdf'}
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diff --git a/xdAzT4oBgHgl3EQfQfvA/content/tmp_files/2301.01201v1.pdf.txt b/xdAzT4oBgHgl3EQfQfvA/content/tmp_files/2301.01201v1.pdf.txt
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+UNCERTAINTY IN REAL-TIME SEMANTIC SEGMENTATION ON EMBEDDED SYSTEMS
+Ethan Goan, Clinton Fookes
+ej.goan@qut.edu.au
+ABSTRACT
+Application for semantic segmentation models in areas such
+as autonomous vehicles and human computer interaction re-
+quire real-time predictive capabilities. The challenges of ad-
+dressing real-time application is amplified by the need to op-
+erate on resource constrained hardware. Whilst development
+of real-time methods for these platforms has increased, these
+models are unable to sufficiently reason about uncertainty
+present. This paper addresses this by combining deep feature
+extraction from pre-trained models with Bayesian regression
+and moment propagation for uncertainty aware predictions.
+We demonstrate how the proposed method can yield mean-
+ingful uncertainty on embedded hardware in real-time whilst
+maintaining predictive performance.
+Index Terms— semantic segmentation, uncertainty, real-
+time, Bayesian
+1. INTRODUCTION
+Development and capabilities of semantic segmentation mod-
+els has increased dramatically, with models based on deep
+learning applied to domains such as autonomous vehicles
+[1, 2, 3] and human computer interaction [4, 2]. Practical ap-
+plication in these areas requires systems to provide real-time
+performance, and in safety critical domains, meaningful un-
+certainty information is essential. The challenge of providing
+this uncertainty information becomes increasingly challeng-
+ing for real-time operation, as obtaining this uncertainty in-
+formation can considerably increase compute demands. A
+natural way to represent uncertainty is through a Bayesian
+framework, where uncertainty in the model is propagated to
+predictions.
+Complete Bayesian inference is intractable for deep learn-
+ing models, meaning expensive Monte Carlo integration is of-
+ten used for approximate inference [5, 6, 7]. These sampling
+based techniques considerably increase the time required for
+prediction, making them unsuitable for real-time operation.
+Deterministic methods have been proposed to alleviate the ex-
+pensive Monte Carlo estimates [8], though requires approxi-
+mating epistemic uncertainty through the use of an additional
+Gaussian-Discriminant Analysis model. Other research have
+proposed utilising uncertainty information during training
+[9, 10], though are unable to sufficiently reason about epis-
+input image
+segmentation
+epistemic uncertainty
+aleatoric uncertainty
+Fig. 1: Example of semantic segmentation results obtainable
+from proposed real-time semantic segmentation method and
+the forms of uncertainty it permits in real-time applications.
+temic uncertainty present during predictions. Whilst address-
+ing the issues of avoiding expensive sampling approaches, all
+these methods have yet to demonstrate real-time predictive
+performance for semantic segmentation on traditional com-
+puting platforms, let alone resource constrained embedded
+hardware. Given the primary practical application of many
+semantic segmentation models is performed on edge comput-
+ing devices, it is crucial that we are able to deliver this uncer-
+tainty information on these platforms in real time.
+This work aims to address this issue by building from
+the work of the Gaussian Process treatments of deep ker-
+nel learning and Bayesian optimization [11, 12], where the
+tasks of feature extraction is computed deterministically with
+fixed model parameters, which are then used as inputs to a
+probabilistic classification module. Complexity for Gaussian
+processes is cubic in the number of training samples, thus
+making them infeasible for the intended applications.
+In-
+stead we opt for parameterised probabilistic regression mod-
+ule combined with a moment propegating non-linearity for
+classification. This reduces the inference overhead for high-
+dimensional data whilst providing analytic results when us-
+ing conjugate models, avoiding the need for expensive Monte
+Carlo approximations. We demonstrate how this approach
+can be applied to existing real-time semantic segmentation
+models on embedded hardware whilst providing meaningful
+uncertainty measures with minimal compute overhead. An
+arXiv:2301.01201v1  [cs.CV]  20 Dec 2022
+
+f(x; θ)
+µα
+σα
+ADFSoftmax
+µt
+σt
+Fig. 2: Summary of the proposed model. The pink box repre-
+sent the deterministic feature extractor f(x; θ) that can be any
+pretrained semantic segmentation network. The blue convo-
+lutional layer represents our probabilistic layer, which allows
+for analytic predictive inference, and is used to generate a
+mean and variance representation of the predictive logits α.
+These are then passed through the ADFSoftmax layer to rep-
+resent final predictive categories t and epistemic uncertainty
+information.
+example of the predictions and uncertainty measures for the
+presented models is shown in Figure 1.
+The contributions of this paper are as follows,
+• Propose a light-weight method for uncertainty in se-
+mantic segmentation by combining Bayesian methods
+within moment propagation,
+• Develop meaningful real-time aleatoric and epistemic
+uncertainty metrics and investigation into how they can
+inform end users and decision protocols
+• Demonstrate how these methods can be easily adapted
+to pre-trained models, enabling them to provide real
+time uncertainty quantification on embedded devices.
+2. REAL-TIME UNCERTAINTY QUANTIFICATION
+We pose the uncertainty quantification within the Bayesian
+framework, where we propagate uncertainty in the model pa-
+rameters to predictions. For this we require a posterior of our
+model latent variables ω such that,
+p(ω|X, Y) ∝ p(ω)p(X, Y|ω)
+(1)
+where X is the set of our training inputs and Y is a matrix of
+our predictive labels. We can then form a predictive distribu-
+tion for new test data (ˆx, ˆy) as,
+p(ˆy|ˆx, X, Y) =
+�
+p(ω|X, Y)p(ˆx, ˆy|ω)dω
+(2)
+For the case of a neural network representing our likelihood,
+the integrals required for Eqns. (1) (2) when treating all net-
+work parameters as random variables is intractable. For the
+case of (1), the posterior may be approximated by a tractable
+form qθ(ω), though the predictive posterior in (2) for deep
+neural networks remains intractable. As previously stated, the
+commonly used sampling methods to approximate these inte-
+grals are computationally prohibitive for real-time models.
+The goal of this work is to find a simpler representation of
+model predictions for semantic segmentation that permits for
+analytic computation, thus avoiding the need for expensive
+sampling approximations. To achieve this, we propose sim-
+plifying the model to that of linear model with respect to la-
+tent variables, where the basis function applied to the input
+is a neural network f(·; θ) parameterised by all but the last
+layer of the neural network θ. These parameters are treated
+as fixed and known values. We then replace the final linear
+layer within a given network with a probabilistic layer param-
+eterised by our latent variables ω for which we will perform
+inference. This can be seen as separating the model into a
+feature extraction stage with fixed parameters θ, and the clas-
+sification stage with parameters treated as random variables
+ω. The latent parameters in ω will represent the weights and
+bias of a final convolutional layer applied to the model. Fig-
+ure 2 illustrates the proposed model, and can be summarised
+as,
+Φ(x) = f(x; θ)
+(3)
+α = Φ(x)ω
+(4)
+t = Softmax(α),
+(5)
+where Φ(x) represents the design matrix for input x gener-
+ated from a neural network with fixed parameters θ, α and t
+represents the corresponding logits and final predictive cate-
+gorical probability respectively, and ω ∼ p(ω|X, Y, θ) is the
+latent variables we wish to perform inference for. We rep-
+resent the multiplication between the design matrix and the
+probabilistic parameters as an inner product, but note that this
+inner product can be computed with an equivalent convolu-
+tion operation. Since inference is only required for the final
+latent variables, we are able to leverage pretrained networks
+f(x; θ) for the generation of design matricies.
+For the work presented within, we frame inference in the
+logit space as apposed to the final categorical distribution.
+This allows us to frame the inference problem as a simple
+linear regression model, such that our posterior of latent vari-
+ables ω with N data points is,
+p(ω|X, Y, θ) ∝
+� N−1
+�
+i=0
+N
+�
+ω|f(xi; θ)ω, σ2��
+p(ω).
+(6)
+With a conjugate Gaussian prior placed on latent variables ω,
+our posterior will also be Gaussian [13],
+p(ω|X, Y, θ) = N(µπ, Σπ).
+(7)
+The advantage of representing our model in this way is
+that it allows us to represent predictive probabilities over the
+
+logits for a single test image ˆX analytically as,
+p(ˆα|ˆx, X, Y, θ) =
+�
+N
+�
+ω|Φ(ˆx)ω, σ
+�
+p(ω|X, Y, θ)dω
+(8)
+= N(ˆα|Φ(ˆx)µπ, σN(ˆx))
+(9)
+σ2
+N(ˆx) = σ2 + ˆxT Σπˆx
+(10)
+2.1. Inference of Latent Variables
+A challenge with this modelling scheme is that since we are
+performing inference in the logit space, we cannot directly de-
+fine a likelihood based on the data, as the labels for our data is
+represented as final categorical probabilites and the softmax
+function applied to logits to obtain this predictive probabil-
+ity is not bijective. To address this, we instead propose to
+approximate samples from the conditional posterior of our la-
+tent variables ω using the diagonal SWAG method [14].
+Given that with a conjugate prior, we know that the pos-
+terior of our latent variables ω will be a Gaussian, meaning
+it can be fully described using only the first two moments.
+To summarise these moments, we apply the diagonal SWAG
+method to approximately sample from our model parameters
+using SGD iterates to compute ωSWA = �T
+i=1 ωi. From this,
+we can compute an empirical variance estimate for our proba-
+bilistic parameters as Σdiag = diag(¯ω2 −Σ2
+SWG). In this work,
+we differ from the original SWAG method, where instead of
+computing the empirical mean from the SWAG iterates, we
+use the pretrained weights for the last convolutional layer in
+our network ¯ω = ωpretrained. This reduces the need to record
+many parameter values during training, and encourages mean
+predictions from the probabilistic model to be similar to the
+point estimate network. With this empirical variance param-
+eters, we can represent our approximate posterior as a fac-
+torised Gaussian distribution,
+p(ω|X, Y, θ) ≈ N(ωi|¯ω, Σdiag).
+(11)
+This allows us to perform approximate inference in the logit
+space by utilising the categorical labels available for our data
+sets. Furthermore, the use of a factorised approximation ac-
+celerates computation of predictive posterior, as the covari-
+ance matrix Σπ in (10) becomes a diagonal matrix, such that,
+σ2
+N(ˆx) ≈ σ2 + ˆxT Σdiagˆx.
+(12)
+Whilst eliminating covariance within our posterior ap-
+proximation, the use of a diagonal covariance matrix consid-
+erably reduces the number of model parameters, allowing for
+models using this our to be more easily adapted to low mem-
+ory applications and accelerate prediction on resource con-
+strained.
+2.2. Predictive Distributions and Measures of Uncer-
+tainty
+Whilst providing an analytic solution to the predictive prob-
+abilities of our logits, we ultimately wish to represent a pre-
+dictive distribution for final categorical probabilities. An ex-
+act analytical solution when using the softmax activation is
+not known. We address this by building on the methods of
+moment propegation used in the works of Assumed Density
+Filtering, where instead of computing exact distributions, we
+approximate the moments of random variables after applying
+certain functions, and use these moments to approximate the
+transformed random variables as Gaussians. Similar to [15],
+we compute intermediate moments of the softmax using prop-
+erties of the Log-Normal distribution.
+For a Gaussian random variable γ ∼ N(µγ, σ2
+γ), β =
+exp(γ) ∼ Lognormal(µβ, σ2
+β), where the mean and vari-
+ance of β is µβ = exp
+�
+µγ + σ2
+γ/2
+�
+and σ2
+β =
+�
+exp σ2
+γ −
+1
+�
+exp(2µγ + σ2
+γ) respectively. We can use these two mo-
+ments to approximate the transformed random variables as a
+Gaussian β ˙∼N(µβ, σ2
+β). From this we can approximate the
+distribution of the output for each class as,
+tj =
+exp(αj)
+�N−1
+i=0 exp(αi)
+=
+yj
+�N−1
+i=0 yi
+(13)
+where αj is a single logit from our probabilistic layer and
+yi ∼ N(µy,i, σ2
+y,i) using the properties of the Log-Normal
+distribution. To find the mean and variance for our output ti
+using moment propagation, we first find the mean and vari-
+ance of the denominator. Assuming independence in our out-
+puts, the denominator distribution in Equation (13) can be
+represented as,
+N−1
+�
+i=0
+yi ∼ N(
+�
+µy,i,
+�
+σ2
+y,i) = N(µd, σ2
+d).
+(14)
+The application of moment propagation used in [15] does
+not aim to approximate the moments from the ratio distribu-
+tion encountered within a probabilistic treatment of the soft-
+max, and instead only aim to ensure the resulting mean can be
+used as parameter to a categorical distribution. We rectify this
+by following ratio distribution approximation of [16], where a
+Gaussian approximation is derived from a Taylor expansion.
+With this we can approximate the mean and variance of each
+softmax output indexed by j as,
+tj ˙∼N(µj
+µd
+, µj
+µd
+�
+σ2
+j + σ2
+d),
+(15)
+where µj and σj are found from the Log-Normal properties.
+We label the use of this moment propegation as the Assumed
+Density Filtered Softmax (ADFSoftmax).
+Whilst having the ability to representing model uncer-
+tainty in our output, we also wish to form a suitable cate-
+gorical distribution over outputs for classification using the
+conventional argmax operator.
+
+Proposition 1. With the probabilistic output in eq. (15), we
+can create a valid k−dimensional categorical distribution
+such that,
+C ∼ Cat(k, E[t]).
+(16)
+Proof. For Cat(k, E[t]) to be a valid categorical distribution,
+we require that the parameters generated by E[t] satisfy the
+condition that E[t]j ≤ 1 for 0 ≤ j < k and �k−1
+i=0 E[t]i = 1.
+We first observe that expectation for the jth component is,
+E[t]j = µj
+µd
+=
+exp
+�
+µj + σ2
+j /2
+�
+�k−1
+i=0 exp
+�
+µi + σ2
+i /2
+� = exp
+�
+µj + σ2
+j /2
+�
+µd
+,
+(17)
+where µd is that of Equation (14). Similar to the traditional
+softmax function, the denominator is shared amoungst all
+components, and the exponential function ensures all com-
+ponents are strictly positive. Following the same arguments
+of the softmax, this implies that exp
+�
+µj + σ2
+j /2
+�
+for valid
+index j, and that �k−1
+i
+E[t]i = 1 as required.
+With this categorical representation, we can easily per-
+form final classification similar to point estimate networks,
+whilst also delivering important uncertainty information in
+these predictions. In the next section, we discuss the uncer-
+tainty metrics available.
+2.3. Computing Predictive Uncertainty
+With our representation of predictive probabilities, we want
+to be able to reason about the different types of uncertainty
+present. It is common to refer to the epistemic and aleatoric
+uncertainty [17], where epistemic uncertainty is a represen-
+tation of reducible uncertainty identified by the model, and
+aleatoric uncertainty is irreducible uncertainty caused by the
+data present. With the probabilistic model presented within,
+we are able to reason about both forms of uncertainty.
+We propose to capture epistemic uncertainty using the
+variance information available in our model predictions. An
+advantage of our approach is that we can measure this uncer-
+tainty in both the logit and prediction space. Given we are
+modelling these spaces as Gaussians, we can succinctly sum-
+marise this uncertainty using the entropy of these distribu-
+tions. The entropy for a D-dimensional multivariate Gaussian
+is,
+H[x] = 1
+2 log |Σ| + D
+2 (1 + log 2π).
+(18)
+We see from eq. (18) that the entropy depends only on the co-
+variance information in the distribution. This is advantageous
+as it provides a succinct and natural form to summarise the
+uncertainty induced by our latent variables in our predictions.
+Furthermore, given that we are representing our distributions
+over our logits and final predictions as independent Gaussian
+variables, the trace operator can be reduced to a product of
+the variance components for each component in the relevant
+distribution.
+Table 1: Summary of predictive performance of real-time
+semantic segmentation methods on Jetson Xavier embedded
+hardware.
+Model
+Dataset
+Input Size
+mIOU
+fps
+BiSeNetV1
+Cityscapes
+512x1024
+74.5457
+65.8129
+CocoStuff
+512x1024
+30.6171
+72.415
+ADE20k
+512x1024
+35.197
+62.48
+Bayes
+BiSeNetV1
+Cityscapes
+512x1024
+74.0748
+57.359
+CocoStuff
+512x512
+30.786
+43.334
+ADE20k
+512x512
+35.1449
+54.245
+BiSeNetV2
+Cityscapes
+512x1024
+74.6672
+49.5015
+CocoStuff
+512x1024
+26.7878
+58.6345
+ADE20k
+512x1024
+32.3957
+59.5024
+Bayes
+BiSeNetV2
+Cityscapes
+512x1024
+74.3651
+42.6472
+CocoStuff
+512x512
+28.5445
+41.2032
+ADE20k
+512x512
+32.1453
+41.8055
+Entropy provides a succinct summary for our uncertainty
+over all categories of interest, though there are instances
+where we may wish to view uncertainty information pertain-
+ing to a single class of interest.
+For example, for an au-
+tonomous vehicle scenario we may be interested in observing
+the uncertainty for safety critical classes such as other vehi-
+cles or pedestrians. We can summarise the class conditional
+uncertainty utilising the corresponding variance in our predic-
+tive probability in eq. (15) for classes of interest.
+For aleatoric uncertainty, a common method used in point
+estimate models is to compute the entropy in the final cate-
+gorical distribution obtained. We follow this approach in this
+work, were we use the categorical distribution represented
+in (16).
+3. EXPERIMENTS
+With our probabilistic model, inference scheme and uncer-
+tainty measures defined, we now demonstrate how they can all
+be combined to deliver uncertainty in real-time compute con-
+strained devices. The NVIDIA Jetson AGX Xavier embedded
+platform is the used as the computing platform. Given we are
+targeting lightweight real-time methods, we build upon the
+BiSeNet model variants [18, 19] as our pre-trained backbone
+for the deep feature extractors f(x; θ), and with pre-trained
+weights in the final layer serving as the mean for our prob-
+abilistic classificlation layer ¯ω.
+Pre-trained backbones for
+these variants are available for the CityScapes[3], ADE20k
+[4] and CocoStuff [2] datasets, which we build upon for the
+following experimentations.
+For our inference procedure, we perform training on the
+existing models to obtain our diagonal SWAG samples with
+the with Ohem cross-entropy loss [20]. We perform a total
+of 10,000 SGD iterations, with a warmup of 1,000 iterations
+
+input
+predicted segmentation
+epistemic uncertainty
+aleatoric uncertainty
+class uncertainty
+Fig. 3: Examples of predictions from the proposed model using the BiSeNetv1 backbone. The rows represent samples from
+the Cityscapes, ADE20k and CocoStuff datasets respectively. From left to right, the columns represent the input, segmentation,
+epistemic uncertainty, aleatoric uncertainty and class conditional epistemic uncertainty. For class conditional uncertainty, we
+show the uncertainty for the classes “car”, “tree” and “dog” across the datasets.
+with a linear increase of the learning rate. After this warmup
+period, the learning rate remains constant. The parameters are
+observed every 200 iterations for the remainder of training for
+inference utilising diagonal SWAG. These recorded parame-
+ters are then used to estimate the empirical covariance ma-
+trix for the final probabilistic convolutional layer. Our prior
+over our latent variables is a Gaussian distribution N(0, 1−4),
+which is implemented using weight decay.
+After training is completed, the models are then compiled
+to the TensorRT format to be used on the Jetson board for
+calculation of inference speed metrics. During compilation,
+the models are converted to half-precision floating point val-
+ues to accelerate computation. To measure predictive speed,
+we perform time the computation required for 1,000 forward
+passes. Our primary measures for evaluating model perfor-
+mance is the mean intersection over union (mIOU), and pre-
+dictive frames-per-second (fps). We summarise the predictive
+performance for these probabilistic models in Table 1, and
+compare them against the point estimate networks.
+From the results in Table 1, we see comparable perfor-
+mance amongst the probabilistic models and their point esti-
+mate equivalents. Whilst the predictive speed does decrease
+across the Bayesian models, we see that predictive perfor-
+mance remains suitability for real time application.
+Fur-
+thermore, we note that the predictive performance of the
+BiSeNetv1 model frequently outperforms in terms of pre-
+dictive accuracy and inference speed, despite the BiSeNetv2
+models having considerably fewer parameters. We attribute
+this decrease in predictive performance on the Jetson hard-
+ware to the depth-wise convolutional operations used in the
+BiSeNetv2 model not being as thoroughly optimised within
+the TensorRT framework. We highlight this as an important
+engineering research topic for future work.
+With our predictive performance measured quantitatively,
+we now investigate the uncertainty measures qualitatively.
+In Section 2.3, we state how the proposed modelling method
+is capable of producing measures for aleatoric, epistemic and
+class conditional epistemic uncertainty. Figure 3 illustrates
+the predictive performance over the datasets examined within
+this work and the relevant uncertainty measures. We can see
+from these figures that the measured epistemic uncertainty is
+concentrated within the objects of interest, whilst the aleatoric
+uncertainty is larger around the edges of these objects. We
+further see from the class conditional uncertainty measures
+that when targeting individual classes, the uncertainty is tar-
+geted towards the edge of the objects for the class of interest.
+4. CONCLUSION
+Within this research, we have targeted the need for mean-
+ingful uncertainty information for challenging semantic seg-
+mentation tasks on resource constrained hardware. We pro-
+pose a combination of a deterministic feature extractor with
+a probabilistic regression module, that can be combined with
+a moment propagation softmax module to generate final pre-
+dictive outputs and uncertainties. We evaluate these models
+for multiple datasets on embedded hardware, and demonstrate
+how the proposed probabilistic model can mitigate the expen-
+sive compute required for a complete probabilistic treatment
+of deep segmentation models to provide real-time perfor-
+
+mance. Qualitative evaluation of obtained uncertainty mea-
+sures demonstrate how they can be computed and used for
+demanding and high risk real-time scenarios.
+5. REFERENCES
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+
diff --git a/xdAzT4oBgHgl3EQfQfvA/content/tmp_files/load_file.txt b/xdAzT4oBgHgl3EQfQfvA/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf,len=233
+page_content='UNCERTAINTY IN REAL-TIME SEMANTIC SEGMENTATION ON EMBEDDED SYSTEMS  Ethan Goan, Clinton Fookes  ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='goan@qut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='au  ABSTRACT  Application for semantic segmentation models in areas such  as autonomous vehicles and human computer interaction re-  quire real-time predictive capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The challenges of ad-  dressing real-time application is amplified by the need to op-  erate on resource constrained hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Whilst development  of real-time methods for these platforms has increased, these  models are unable to sufficiently reason about uncertainty  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' This paper addresses this by combining deep feature  extraction from pre-trained models with Bayesian regression  and moment propagation for uncertainty aware predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  We demonstrate how the proposed method can yield mean-  ingful uncertainty on embedded hardware in real-time whilst  maintaining predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Index Terms— semantic segmentation, uncertainty, real-  time, Bayesian  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' INTRODUCTION  Development and capabilities of semantic segmentation mod-  els has increased dramatically, with models based on deep  learning applied to domains such as autonomous vehicles  [1, 2, 3] and human computer interaction [4, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Practical ap-  plication in these areas requires systems to provide real-time  performance, and in safety critical domains, meaningful un-  certainty information is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The challenge of providing  this uncertainty information becomes increasingly challeng-  ing for real-time operation, as obtaining this uncertainty in-  formation can considerably increase compute demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' A  natural way to represent uncertainty is through a Bayesian  framework, where uncertainty in the model is propagated to  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Complete Bayesian inference is intractable for deep learn-  ing models, meaning expensive Monte Carlo integration is of-  ten used for approximate inference [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' These sampling  based techniques considerably increase the time required for  prediction, making them unsuitable for real-time operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Deterministic methods have been proposed to alleviate the ex-  pensive Monte Carlo estimates [8], though requires approxi-  mating epistemic uncertainty through the use of an additional  Gaussian-Discriminant Analysis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Other research have  proposed utilising uncertainty information during training  [9, 10], though are unable to sufficiently reason about epis-  input image  segmentation  epistemic uncertainty  aleatoric uncertainty  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' 1: Example of semantic segmentation results obtainable  from proposed real-time semantic segmentation method and  the forms of uncertainty it permits in real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  temic uncertainty present during predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Whilst address-  ing the issues of avoiding expensive sampling approaches, all  these methods have yet to demonstrate real-time predictive  performance for semantic segmentation on traditional com-  puting platforms, let alone resource constrained embedded  hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Given the primary practical application of many  semantic segmentation models is performed on edge comput-  ing devices, it is crucial that we are able to deliver this uncer-  tainty information on these platforms in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  This work aims to address this issue by building from  the work of the Gaussian Process treatments of deep ker-  nel learning and Bayesian optimization [11, 12], where the  tasks of feature extraction is computed deterministically with  fixed model parameters, which are then used as inputs to a  probabilistic classification module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Complexity for Gaussian  processes is cubic in the number of training samples, thus  making them infeasible for the intended applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  In-  stead we opt for parameterised probabilistic regression mod-  ule combined with a moment propegating non-linearity for  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' This reduces the inference overhead for high-  dimensional data whilst providing analytic results when us-  ing conjugate models, avoiding the need for expensive Monte  Carlo approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We demonstrate how this approach  can be applied to existing real-time semantic segmentation  models on embedded hardware whilst providing meaningful  uncertainty measures with minimal compute overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' An  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='01201v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='CV]  20 Dec 2022  f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ)  µα  σα  ADFSoftmax  µt  σt  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' 2: Summary of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The pink box repre-  sent the deterministic feature extractor f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ) that can be any  pretrained semantic segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The blue convo-  lutional layer represents our probabilistic layer, which allows  for analytic predictive inference, and is used to generate a  mean and variance representation of the predictive logits α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  These are then passed through the ADFSoftmax layer to rep-  resent final predictive categories t and epistemic uncertainty  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  example of the predictions and uncertainty measures for the  presented models is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  The contributions of this paper are as follows,  Propose a light-weight method for uncertainty in se-  mantic segmentation by combining Bayesian methods  within moment propagation,  Develop meaningful real-time aleatoric and epistemic  uncertainty metrics and investigation into how they can  inform end users and decision protocols  Demonstrate how these methods can be easily adapted  to pre-trained models, enabling them to provide real  time uncertainty quantification on embedded devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' REAL-TIME UNCERTAINTY QUANTIFICATION  We pose the uncertainty quantification within the Bayesian  framework, where we propagate uncertainty in the model pa-  rameters to predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' For this we require a posterior of our  model latent variables ω such that,  p(ω|X, Y) ∝ p(ω)p(X, Y|ω)  (1)  where X is the set of our training inputs and Y is a matrix of  our predictive labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We can then form a predictive distribu-  tion for new test data (ˆx, ˆy) as,  p(ˆy|ˆx, X, Y) =  �  p(ω|X, Y)p(ˆx, ˆy|ω)dω  (2)  For the case of a neural network representing our likelihood,  the integrals required for Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' (1) (2) when treating all net-  work parameters as random variables is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' For the  case of (1), the posterior may be approximated by a tractable  form qθ(ω), though the predictive posterior in (2) for deep  neural networks remains intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' As previously stated, the  commonly used sampling methods to approximate these inte-  grals are computationally prohibitive for real-time models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  The goal of this work is to find a simpler representation of  model predictions for semantic segmentation that permits for  analytic computation, thus avoiding the need for expensive  sampling approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' To achieve this, we propose sim-  plifying the model to that of linear model with respect to la-  tent variables, where the basis function applied to the input  is a neural network f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ) parameterised by all but the last  layer of the neural network θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' These parameters are treated  as fixed and known values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We then replace the final linear  layer within a given network with a probabilistic layer param-  eterised by our latent variables ω for which we will perform  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' This can be seen as separating the model into a  feature extraction stage with fixed parameters θ, and the clas-  sification stage with parameters treated as random variables  ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The latent parameters in ω will represent the weights and  bias of a final convolutional layer applied to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Fig-  ure 2 illustrates the proposed model, and can be summarised  as,  Φ(x) = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ)  (3)  α = Φ(x)ω  (4)  t = Softmax(α),  (5)  where Φ(x) represents the design matrix for input x gener-  ated from a neural network with fixed parameters θ, α and t  represents the corresponding logits and final predictive cate-  gorical probability respectively, and ω ∼ p(ω|X, Y, θ) is the  latent variables we wish to perform inference for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We rep-  resent the multiplication between the design matrix and the  probabilistic parameters as an inner product, but note that this  inner product can be computed with an equivalent convolu-  tion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Since inference is only required for the final  latent variables, we are able to leverage pretrained networks  f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ) for the generation of design matricies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  For the work presented within, we frame inference in the  logit space as apposed to the final categorical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  This allows us to frame the inference problem as a simple  linear regression model, such that our posterior of latent vari-  ables ω with N data points is,  p(ω|X, Y, θ) ∝  � N−1  �  i=0  N  �  ω|f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ)ω, σ2��  p(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (6)  With a conjugate Gaussian prior placed on latent variables ω,  our posterior will also be Gaussian [13],  p(ω|X, Y, θ) = N(µπ, Σπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (7)  The advantage of representing our model in this way is  that it allows us to represent predictive probabilities over the  logits for a single test image ˆX analytically as,  p(ˆα|ˆx, X, Y, θ) =  �  N  �  ω|Φ(ˆx)ω, σ  �  p(ω|X, Y, θ)dω  (8)  = N(ˆα|Φ(ˆx)µπ, σN(ˆx))  (9)  σ2  N(ˆx) = σ2 + ˆxT Σπˆx  (10)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Inference of Latent Variables  A challenge with this modelling scheme is that since we are  performing inference in the logit space, we cannot directly de-  fine a likelihood based on the data, as the labels for our data is  represented as final categorical probabilites and the softmax  function applied to logits to obtain this predictive probabil-  ity is not bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' To address this, we instead propose to  approximate samples from the conditional posterior of our la-  tent variables ω using the diagonal SWAG method [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Given that with a conjugate prior, we know that the pos-  terior of our latent variables ω will be a Gaussian, meaning  it can be fully described using only the first two moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  To summarise these moments, we apply the diagonal SWAG  method to approximately sample from our model parameters  using SGD iterates to compute ωSWA = �T  i=1 ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' From this,  we can compute an empirical variance estimate for our proba-  bilistic parameters as Σdiag = diag(¯ω2 −Σ2  SWG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' In this work,  we differ from the original SWAG method, where instead of  computing the empirical mean from the SWAG iterates, we  use the pretrained weights for the last convolutional layer in  our network ¯ω = ωpretrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' This reduces the need to record  many parameter values during training, and encourages mean  predictions from the probabilistic model to be similar to the  point estimate network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' With this empirical variance param-  eters, we can represent our approximate posterior as a fac-  torised Gaussian distribution,  p(ω|X, Y, θ) ≈ N(ωi|¯ω, Σdiag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (11)  This allows us to perform approximate inference in the logit  space by utilising the categorical labels available for our data  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Furthermore, the use of a factorised approximation ac-  celerates computation of predictive posterior, as the covari-  ance matrix Σπ in (10) becomes a diagonal matrix, such that,  σ2  N(ˆx) ≈ σ2 + ˆxT Σdiagˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (12)  Whilst eliminating covariance within our posterior ap-  proximation, the use of a diagonal covariance matrix consid-  erably reduces the number of model parameters, allowing for  models using this our to be more easily adapted to low mem-  ory applications and accelerate prediction on resource con-  strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Predictive Distributions and Measures of Uncer-  tainty  Whilst providing an analytic solution to the predictive prob-  abilities of our logits, we ultimately wish to represent a pre-  dictive distribution for final categorical probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' An ex-  act analytical solution when using the softmax activation is  not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We address this by building on the methods of  moment propegation used in the works of Assumed Density  Filtering, where instead of computing exact distributions, we  approximate the moments of random variables after applying  certain functions, and use these moments to approximate the  transformed random variables as Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Similar to [15],  we compute intermediate moments of the softmax using prop-  erties of the Log-Normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  For a Gaussian random variable γ ∼ N(µγ, σ2  γ), β =  exp(γ) ∼ Lognormal(µβ, σ2  β), where the mean and vari-  ance of β is µβ = exp  �  µγ + σ2  γ/2  �  and σ2  β =  �  exp σ2  γ −  1  �  exp(2µγ + σ2  γ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We can use these two mo-  ments to approximate the transformed random variables as a  Gaussian β ˙∼N(µβ, σ2  β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' From this we can approximate the  distribution of the output for each class as,  tj =  exp(αj)  �N−1  i=0 exp(αi)  =  yj  �N−1  i=0 yi  (13)  where αj is a single logit from our probabilistic layer and  yi ∼ N(µy,i, σ2  y,i) using the properties of the Log-Normal  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' To find the mean and variance for our output ti  using moment propagation, we first find the mean and vari-  ance of the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Assuming independence in our out-  puts, the denominator distribution in Equation (13) can be  represented as,  N−1  �  i=0  yi ∼ N(  �  µy,i,  �  σ2  y,i) = N(µd, σ2  d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (14)  The application of moment propagation used in [15] does  not aim to approximate the moments from the ratio distribu-  tion encountered within a probabilistic treatment of the soft-  max, and instead only aim to ensure the resulting mean can be  used as parameter to a categorical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We rectify this  by following ratio distribution approximation of [16], where a  Gaussian approximation is derived from a Taylor expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  With this we can approximate the mean and variance of each  softmax output indexed by j as,  tj ˙∼N(µj  µd  , µj  µd  �  σ2  j + σ2  d),  (15)  where µj and σj are found from the Log-Normal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  We label the use of this moment propegation as the Assumed  Density Filtered Softmax (ADFSoftmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Whilst having the ability to representing model uncer-  tainty in our output, we also wish to form a suitable cate-  gorical distribution over outputs for classification using the  conventional argmax operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' With the probabilistic output in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' (15), we  can create a valid k−dimensional categorical distribution  such that,  C ∼ Cat(k, E[t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (16)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' For Cat(k, E[t]) to be a valid categorical distribution,  we require that the parameters generated by E[t] satisfy the  condition that E[t]j ≤ 1 for 0 ≤ j < k and �k−1  i=0 E[t]i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  We first observe that expectation for the jth component is,  E[t]j = µj  µd  =  exp  �  µj + σ2  j /2  �  �k−1  i=0 exp  �  µi + σ2  i /2  � = exp  �  µj + σ2  j /2  �  µd  ,  (17)  where µd is that of Equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Similar to the traditional  softmax function, the denominator is shared amoungst all  components, and the exponential function ensures all com-  ponents are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Following the same arguments  of the softmax, this implies that exp  �  µj + σ2  j /2  �  for valid  index j, and that �k−1  i  E[t]i = 1 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  With this categorical representation, we can easily per-  form final classification similar to point estimate networks,  whilst also delivering important uncertainty information in  these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' In the next section, we discuss the uncer-  tainty metrics available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Computing Predictive Uncertainty  With our representation of predictive probabilities, we want  to be able to reason about the different types of uncertainty  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' It is common to refer to the epistemic and aleatoric  uncertainty [17], where epistemic uncertainty is a represen-  tation of reducible uncertainty identified by the model, and  aleatoric uncertainty is irreducible uncertainty caused by the  data present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' With the probabilistic model presented within,  we are able to reason about both forms of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  We propose to capture epistemic uncertainty using the  variance information available in our model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' An  advantage of our approach is that we can measure this uncer-  tainty in both the logit and prediction space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Given we are  modelling these spaces as Gaussians, we can succinctly sum-  marise this uncertainty using the entropy of these distribu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The entropy for a D-dimensional multivariate Gaussian  is,  H[x] = 1  2 log |Σ| + D  2 (1 + log 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  (18)  We see from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' (18) that the entropy depends only on the co-  variance information in the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' This is advantageous  as it provides a succinct and natural form to summarise the  uncertainty induced by our latent variables in our predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Furthermore, given that we are representing our distributions  over our logits and final predictions as independent Gaussian  variables, the trace operator can be reduced to a product of  the variance components for each component in the relevant  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Table 1: Summary of predictive performance of real-time  semantic segmentation methods on Jetson Xavier embedded  hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Model  Dataset  Input Size  mIOU  fps  BiSeNetV1  Cityscapes  512x1024  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='5457  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='8129  CocoStuff  512x1024  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='6171  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='415  ADE20k  512x1024  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='197  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='48  Bayes  BiSeNetV1  Cityscapes  512x1024  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='0748  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='359  CocoStuff  512x512  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='786  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='334  ADE20k  512x512  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='1449  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='245  BiSeNetV2  Cityscapes  512x1024  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='6672  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='5015  CocoStuff  512x1024  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='7878  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='6345  ADE20k  512x1024  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='3957  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='5024  Bayes  BiSeNetV2  Cityscapes  512x1024  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='3651  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='6472  CocoStuff  512x512  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='5445  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='2032  ADE20k  512x512  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='1453  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='8055  Entropy provides a succinct summary for our uncertainty  over all categories of interest, though there are instances  where we may wish to view uncertainty information pertain-  ing to a single class of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  For example, for an au-  tonomous vehicle scenario we may be interested in observing  the uncertainty for safety critical classes such as other vehi-  cles or pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We can summarise the class conditional  uncertainty utilising the corresponding variance in our predic-  tive probability in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' (15) for classes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  For aleatoric uncertainty, a common method used in point  estimate models is to compute the entropy in the final cate-  gorical distribution obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We follow this approach in this  work, were we use the categorical distribution represented  in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' EXPERIMENTS  With our probabilistic model, inference scheme and uncer-  tainty measures defined, we now demonstrate how they can all  be combined to deliver uncertainty in real-time compute con-  strained devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The NVIDIA Jetson AGX Xavier embedded  platform is the used as the computing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Given we are  targeting lightweight real-time methods, we build upon the  BiSeNet model variants [18, 19] as our pre-trained backbone  for the deep feature extractors f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' θ), and with pre-trained  weights in the final layer serving as the mean for our prob-  abilistic classificlation layer ¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Pre-trained backbones for  these variants are available for the CityScapes[3], ADE20k  [4] and CocoStuff [2] datasets, which we build upon for the  following experimentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  For our inference procedure, we perform training on the  existing models to obtain our diagonal SWAG samples with  the with Ohem cross-entropy loss [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We perform a total  of 10,000 SGD iterations, with a warmup of 1,000 iterations  input  predicted segmentation  epistemic uncertainty  aleatoric uncertainty  class uncertainty  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' 3: Examples of predictions from the proposed model using the BiSeNetv1 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The rows represent samples from  the Cityscapes, ADE20k and CocoStuff datasets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' From left to right, the columns represent the input, segmentation,  epistemic uncertainty, aleatoric uncertainty and class conditional epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' For class conditional uncertainty, we  show the uncertainty for the classes “car”, “tree” and “dog” across the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  with a linear increase of the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' After this warmup  period, the learning rate remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' The parameters are  observed every 200 iterations for the remainder of training for  inference utilising diagonal SWAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' These recorded parame-  ters are then used to estimate the empirical covariance ma-  trix for the final probabilistic convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Our prior  over our latent variables is a Gaussian distribution N(0, 1−4),  which is implemented using weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  After training is completed, the models are then compiled  to the TensorRT format to be used on the Jetson board for  calculation of inference speed metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' During compilation,  the models are converted to half-precision floating point val-  ues to accelerate computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' To measure predictive speed,  we perform time the computation required for 1,000 forward  passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Our primary measures for evaluating model perfor-  mance is the mean intersection over union (mIOU), and pre-  dictive frames-per-second (fps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We summarise the predictive  performance for these probabilistic models in Table 1, and  compare them against the point estimate networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  From the results in Table 1, we see comparable perfor-  mance amongst the probabilistic models and their point esti-  mate equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Whilst the predictive speed does decrease  across the Bayesian models, we see that predictive perfor-  mance remains suitability for real time application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  Fur-  thermore, we note that the predictive performance of the  BiSeNetv1 model frequently outperforms in terms of pre-  dictive accuracy and inference speed, despite the BiSeNetv2  models having considerably fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We attribute  this decrease in predictive performance on the Jetson hard-  ware to the depth-wise convolutional operations used in the  BiSeNetv2 model not being as thoroughly optimised within  the TensorRT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We highlight this as an important  engineering research topic for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  With our predictive performance measured quantitatively,  we now investigate the uncertainty measures qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='3, we state how the proposed modelling method  is capable of producing measures for aleatoric, epistemic and  class conditional epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Figure 3 illustrates  the predictive performance over the datasets examined within  this work and the relevant uncertainty measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We can see  from these figures that the measured epistemic uncertainty is  concentrated within the objects of interest, whilst the aleatoric  uncertainty is larger around the edges of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We  further see from the class conditional uncertainty measures  that when targeting individual classes, the uncertainty is tar-  geted towards the edge of the objects for the class of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' CONCLUSION  Within this research, we have targeted the need for mean-  ingful uncertainty information for challenging semantic seg-  mentation tasks on resource constrained hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We pro-  pose a combination of a deterministic feature extractor with  a probabilistic regression module, that can be combined with  a moment propagation softmax module to generate final pre-  dictive outputs and uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' We evaluate these models  for multiple datasets on embedded hardware, and demonstrate  how the proposed probabilistic model can mitigate the expen-  sive compute required for a complete probabilistic treatment  of deep segmentation models to provide real-time perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content=' Qualitative evaluation of obtained uncertainty mea-  sures demonstrate how they can be computed and used for  demanding and high risk real-time scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdAzT4oBgHgl3EQfQfvA/content/2301.01201v1.pdf'}
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